Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds

1. Introduction

In the mathematical optimization theory, the concept of variational inequality problems was initially introduced by Hartman and Stampacchia [1]. Giannessi [2,3] introduced the vector-valued versions of Stampacchia [4] and Minty [5] variational inequality problems in the framework of Euclidean space. Due to several real-life applications of vector variational inequality problems, for instance, in traffic equilibrium problems [6], vector equilibria [7], and vector optimization problems [8], vector variational inequality problems have been extensively studied by several researchers in various frameworks (see, for instance, [9,10,11,12,13,14,15] and the references cited therein). Furthermore, the notion of hemivariational inequality problem was first introduced by Panagiotopoulos [16], which is based on the properties of Clarke generalized subdifferential for locally Lipschitz functions. These problems have various applications in the field of science and engineering, such as structured analysis, nonconvex optimization, mechanics, and contact problems (see, for instance, [17,18]). As a result, hemivariational inequality problems emerged as a significant area of research (see, for instance, [19,20,21] and the references cited therein).

Variational-hemivariational inequality problems are a fusion of variational and hemivariational inequality problems, which incorporate both convex and nonconvex functions. The initial conceptualization and formulation of variational-hemivariational inequality problems are attributed to the work of Motreanu and Rǎdulescu [22]. Furthermore, these problems have several applications in mathematical modeling and contact mechanics (see, for instance, [23,24]). In recent years, the theory of variational-hemivariational inequality problems has gained significant attention from several researchers (see, for instance, [17,19,25] and the references cited therein). Tang and Huang [26] have studied existence results for variational-hemivariational inequality problems by employing the KKM theorem in reflexive Banach spaces. Moreover, Migórski et al. [27] have investigated the existence and uniqueness of the solution for variational-hemivariational inequality problems in the setting of reflexive Banach spaces.

It is well known that gap functions play an important role in the study of existence of solutions to variational inequality problems, as one can transform variational inequality into an optimization problem by employing gap functions (see, [28]). Moreover, gap functions are used to derive the error bounds, which provide an upper estimate of the distance between an arbitrary point of the feasible set and the solution set of an optimization problem (see, [29,30]). The notions of gap functions and regularized gap functions for variational inequality problems have been introduced by Auslender [31] and Fukushima [32] in the setting of Euclidean space. Moreover, Yamashita and Fukushima [30] introduced Moreau-Yosida type regularized gap functions for variational inequality problems and further, derived several global error bounds for the solution of the considered problem. Li and He [33] have formulated gap functions and derived the existence results for generalized vector variational inequality problems under the monotonicity assumption on topological vector spaces. Various gap functions for vector variational inequality problems have been studied by Charitha et al. [34] in the Euclidean space setting. Furthermore, Hung et al. [35] have studied gap functions and global error bounds for variational-hemivariational inequality problems in the setting of reflexive Banach spaces. For a detailed exposition regarding gap functions and global error bounds for variational and vector variational inequality problems in various settings, we refer the readers to [29,36,37,38,39] and the references cited therein.

Over the last few decades, it has been observed that various real-life optimization problems arising in the field of engineering and science can be effectively formulated in the setting of Riemannian manifolds rather than in the Euclidean space setting, see, for instance, [40,41] and the references cited therein. The extension of optimization concepts from the Euclidean space setting to the framework of Riemannian manifolds is associated with several crucial advantages, such as one can transform nonconvex optimization problems in the setting of Euclidean space into convex optimization problems in the framework of Riemannian manifolds. Moreover, constrained optimization problems in the Euclidean space setting can be reformulated as unconstrained optimization problems in the Riemannian manifold framework (see, for instance, [42,43] and the references cited therein). Udrişte [41] introduced the notions of geodesic convex functions on Hadamard manifolds corresponding to the notions of convex functions in the setting of Euclidean space. For further detail related to the extension of optimization techniques from linear spaces to Riemannian manifolds, we refer the readers to [9,14,15,44,45] and the references cited therein.

In the setting of Hadamard manifolds, Németh [46] introduced the concept of variational inequality problems and discussed the existence of solutions for the considered problem. Since then, numerous scholars have investigated variational and vector variational inequality problems in the framework of Hadamard manifolds, see, for instance, [9,14,15,38,47,48] and the references cited therein. Chen and Huang [10] have studied existence results for vector variational inequality problems by employing the KKM lemma in the framework of Hadamard manifolds. Moreover, Jayswal et al. [44] have studied existence results and gap functions for nonsmooth mixed vector variational inequality problems via Clarke subdifferentials. Existence results for the solution of hemivariational inequality problems have been established by Tang et al. [21] in the Hadamard manifolds setting. Further, the notions of gap functions and global error bounds for generalized mixed variational inequality problems have been studied by Li et al. [49]. Ansari et al. [50] have formulated gap functions and derived global error bounds for nonsmooth variational inequality problems in terms of bifunctions. Moroever, Hung et al. [20] have derived global error bounds for the solution of considered mixed quasi-hemivariational inequality problems in the setting of Hadamard manifolds.

It is significant to note that several authors have studied existence results for the solutions of vector variational inequality problems and hemivariational inequality problems in various frameworks (see, for instance, [10,21,26,35,44]). However, the existence results for the solution of WVVHVIP, which belongs to a broader class of vector variational as well as hemivariational inequality problems, have not been investigated before in the setting of Hadamard manifolds. Moreover, gap functions and global error bounds for variational inequality problems, vector variational inequality problems, and hemivariational inequality problems in the framework of manifolds have been studied by numerous researchers (see, for instance, [20,33,34,35,44,49,50] and the references cited therein). However, the notions of gap functions and global error bounds for vector variational-hemivariational inequality problems involving nonsmooth locally Lipschitz functions, in particular, WVVHVIP, have not been studied yet in the framework of Hadamard manifolds. Furthermore, it is imperative to note that manifolds, in general, are not equipped with a linear structure. Consequently, suitable new advancements are essential to deal with the non-linear structure of manifolds. Therefore, in this article, we aim to address these research gaps and challenges by establishing existence results, gap functions, and global error bounds for WVVHVIP via Clarke subdifferentials in the Hadamard manifolds setting.

Inspired by the results established in [20,30,31,34,35], in this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (WVVHVIP) in the framework of Hadamard manifolds. We introduce the notion of the generalized geodesic strong monotonicity of a set-valued vector field in the Hadamard manifold setting. By employing an analogous to KKM lemma, we establish the existence of a solution for WVVHVIP without using any monotonicity assumptions. Moreover, we establish the uniqueness of the solution to WVVHVIP by using generalized geodesic strong monotonicity assumptions. We formulate several gap functions, in particular, Auslender type, regularized, and Moreau-Yosida type regularized gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of a solution to the considered problem, namely, WVVHVIP. Furthermore, we derive the global error bounds for the solution of WVVHVIP by employing regularized gap functions under the assumptions of generalized geodesic strong monotonicity.

The novelty and contributions of this paper are fourfold: Firstly, the existence results for the solution of WVVHVIP generalize the corresponding results derived in [10,44] from nonsmooth weak vector variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problems in the Hadamard manifolds setting. Secondly, the uniqueness results for the solution of WVVHVIP generalize the corresponding results derived by Hung et al. [35] from variational-hemivariational inequality problems to a broader category of vector variational-hemivariational inequality problems, namely, WVVHVIP, as well as generalize them from the setting of reflexive Banach space to the framework of Hadamard manifolds. Thirdly, several results related to gap functions and global error bounds for WVVHVIP generalize the corresponding results derived in [30,31,32] from the Euclidean space setting to the framework of Hadamard manifolds, as well as generalize them from variational inequality problems to WVVHVIP, which belongs to a broader class of vector variational and hemivariational inequality problems. Fourthly, the gap functions and error bounds results investigated in this article generalize various corresponding results derived by Charitha et al. [34] from vector variational inequality problems to WVVHVIP and from the setting of Euclidean space to the framework of Hadamard manifolds. To the best of our knowledge, existence results, uniqueness results, as well as gap functions and global error bounds for WVVHVIP, have not been investigated before in the setting of Hadamard manifolds. Consequently, the results established in this paper are applicable to study more general classes of vector variational inequality problems and hemivariational inequality problems, as compared to the results existing in the available literature (see, for instance, [21,29,31,35,44] and the references cited therein).

The rest of this article is structured as follows. In Section 2, we recall some basic definitions and mathematical preliminaries related to Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence results for the solutions of WVVHVIP in Section 3. Moreover, a uniqueness result for the solution of WVVHVIP is established under assumptions of generalized geodesic strong monotonicity. Furthermore, we formulate various gap functions and regularized gap functions, in particular, Auslender, regularized, and Moreau-Yosida type regularized gap functions for WVVHVIP in Section 4. Subsequently, in Section 5, we derive the global error bounds for the solution of WVVHVIP in terms of regularized gap functions under generalized geodesic strong monotonicity hypotheses. Finally, in Section 6, we draw conclusions and suggest all possible future research avenues.

2. Notations and Mathematical Preliminaries

The following definitions and fundamental concepts related to Riemannian and Hadamard manifolds are from [41,51,52].

Let ${\mathbb{R}}^n$ be the $n-$dimensional Euclidean space and ${\mathbb{R}}_{+}^n$ be the non-negative orthant of ${\mathbb{R}}^n$. The notations $\langle ·,·\rangle$ and $int{\mathbb{R}}_{+}^n$ are employed to denote the standard inner product in ${\mathbb{R}}^n$ and interior of $int{\mathbb{R}}_{+}^n$, respectively. The notation ∅ is used to denote an empty set.

Let ${\mathbb{H}}^n$ be the symbol used to represent an $n-$dimensional connected Riemannian manifold endowed with the Riemannian metric $\mathcal{G}.$ The tangent space at $u\in {\mathbb{H}}^n$ is denoted by $T_u{\mathbb{H}}^n,$ which is a vector space of dimension n over $\mathbb{R}.$ The dual space of $T_u{\mathbb{H}}^n$ is represented by $T_u^{\ast }{\mathbb{H}}^n.$ The tangent bundle $T{\mathbb{H}}^n$ is a disjoint union of all the tangent spaces at every point in the Riemannian manifold. That is, $T{\mathbb{H}}^n=\bigcup _{u\in {\mathbb{H}}^n}{T}_u{\mathbb{H}}^n.$ The symbols ${\langle ·,·\rangle }_u$ and $\left|\right|·{\left|\right|}_u$ are used to represent the inner product and its associated norm on the tangent space $T_u{\mathbb{H}}^n$ for every $u\in {\mathbb{H}}^n,$ respectively. Let $\mathsf{\Gamma }:[0,1]\to {\mathbb{H}}^n$ be a piecewise differentiable curve joining u and v in ${\mathbb{H}}^n$. That is, $\mathsf{\Gamma }(0)=u$ and $\mathsf{\Gamma }(1)=v$. The length of the curve $\mathsf{\Gamma }$ is denoted by $L(\mathsf{\Gamma })$, and is defined as:

$$L(\mathsf{\Gamma }):={\int }_0^1∥{\mathsf{\Gamma }}^{\prime }(t)∥dt,$$

where ${\mathsf{\Gamma }}^{\prime }(t)={\displaystyle \frac{d\mathsf{\Gamma }(t)}{dt}},\mathrm{for}\mathrm{all}t\in (0,1).$ Moreover, the Riemannian distance between u and v in ${\mathbb{H}}^n$ is defined as follows:

$$d(u,v):=inf\{L(\mathsf{\Gamma }):\mathsf{\Gamma }\mathrm{is}\mathrm{a}\mathrm{piecewise}\mathrm{smooth}\mathrm{curve}\mathrm{joining}u\mathrm{and}vin{\mathbb{H}}^n\}.$$

A differentiable curve $\mathsf{\Gamma }$ is said to be a geodesic if its vector field is parallel to itself, that is, ${\nabla }_{{\mathsf{\Gamma }}^{\prime }}{\mathsf{\Gamma }}^{\prime }=0,$ where ∇ is the unique Levi-Civita connection on ${\mathbb{H}}^n$. Furthermore, a geodesic curve joining $u,v\in {\mathbb{H}}^n$ is termed as a minimal geodesic if its length is equal to the Riemannian distance between them. The Riemannian manifold ${\mathbb{H}}^n$ is called geodesically complete at $u\in {\mathbb{H}}^n,$ if every geodesic $\mathsf{\Gamma }$ emanating from u is defined for every real number (see, [51]). Therefore, ${\mathbb{H}}^n$ is said to be geodesically complete if it is geodesically complete at every point in ${\mathbb{H}}^n.$ From Hopf-Rinow theorem (see, [51]), every geodesically complete Riemannian manifold is a complete metric space, and there always exists a minimal geodesic between any two points in ${\mathbb{H}}^n.$ A Riemannian manifold is called a Hadamard manifold if it is simply connected, complete, and has non-positive sectional curvature everywhere.

From now onwards, we assume that ${\mathbb{H}}^n$ is an n-dimensional Hadamard manifold.

For any $u\in {\mathbb{H}}^n$, the exponential map ${exp}_u:T_u{\mathbb{H}}^n\to {\mathbb{H}}^n$ is given by ${exp}_u(\nu )={\mathsf{\Gamma }}_{\nu }(1),\mathrm{for}\mathrm{all}\nu \in T_u{\mathbb{H}}^n,$ where ${\mathsf{\Gamma }}_{\nu }$ is the unique geodesic such that ${\mathsf{\Gamma }}_{\nu }(0)=u$ and ${\mathsf{\Gamma }}_{\nu }^{\prime }(0)=\nu .$ Moreover, the inverse of the exponential map ${exp}_u^{-1}:{\mathbb{H}}^n\to T_u{\mathbb{H}}^n$ satisfies ${exp}_u^{-1}u=0_u,$ where $0_u$ is the zero tangent vector in tangent space $T_u{\mathbb{H}}^n.$ In addition, for any $u,v\in {\mathbb{H}}^n$, the distance between u and v is $d(u,v)=\left|\right|{exp}_u^{-1}(v){\left|\right|}_u.$ The parallel transport from u to v along a geodesic $\mathsf{\Gamma }$ is a linear isometry $L_{\mathsf{\Gamma }}^{u,v}:T_u{\mathbb{H}}^n\to T_v{\mathbb{H}}^n$. The symbol $L_{u,v}$ is used to represent a parallel transport from u to v along a minimal geodesic joining u and $v.$

A function $\mathsf{\Psi }:{\mathbb{H}}^n\to \mathbb{R}$ is known as a locally Lipschitz function at $u\in {\mathbb{H}}^n$ with rank K$(K\in \mathbb{R},K>0)$, if there exists a neighborhood U of u such that

$$|\mathsf{\Psi }(v)-\mathsf{\Psi }(w)|\le Kd(v,w),\mathrm{for}\mathrm{all}v,w\in U.$$

A function $\mathsf{\Psi }$ is known as locally Lipschitz on ${\mathbb{H}}^n$ with rank K if it is locally Lipschitz at every $u\in {\mathbb{H}}^n$ with rank K.

In the following definition, we recall the notion of a geodesic convex set from Udrişte [41].

Definition 1. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n,$ and $u,w$ be any two points in $\mathcal{F}.$ Then, $\mathcal{F}$ is said to be a geodesic convex set, provided that the following inclusion holds:

$${exp}_u(t{exp}_u^{-1}(w))\in \mathcal{F},\mathrm{for}\mathrm{all}t\in [0,1].$$

The notion of the geodesic convex hull of a subset of ${\mathbb{H}}^n$ is recalled in the following definition (see, for instance, [52,53]).

Definition 2. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n$ be a nonempty set. The geodesic convex hull of a set $\mathcal{F}$ is the smallest geodesic convex subset of ${\mathbb{H}}^n$ containing $\mathcal{F}.$

Remark 1. 

The geodesic convex hull of a set $\mathcal{F}\subseteq {\mathbb{H}}^n$ can be defined as the intersection of all the geodesic convex sets containing $\mathcal{F}$ (see, for instance, [52,53]).

From now onwards, the geodesic convex hull of a set $\mathcal{F}\subseteq {\mathbb{H}}^n$ is represented by $conv(\mathcal{F}),$ unless specified otherwise.

The notion of the geodesic convex combination of a finite number of elements in ${\mathbb{H}}^n$ is recalled in the following definition (see, for instance, [50,53]).

Definition 3. 

Let $u_1,u_2,\cdots ,u_n(n\in \mathbb{N})$ be a finite number of elements chosen arbitrarily from ${\mathbb{H}}^n.$ Then, the geodesic convex combination of $u_1,u_2,\dots ,u_n$ is the geodesic joining $u_n$ to any geodesic convex combination of $u_1,u_2,\dots ,u_{n-1}$, defined as follows:

$${comb}_{u_1,u_2,\dots ,u_n}(t_2,t_3,\dots ,t_n):={exp}_{u_n}(t_n{exp}_{u_n}^{-1}({comb}_{u_1,u_2,\dots ,u_{n-1}}(t_2,t_3,\dots ,t_{n-1}))),$$

for every $t_i\in [0,1]$ with $i=2,3,\dots ,n.$

A relationship between geodesic convex hull, geodesic convex set, and geodesic convex combination is provided in the following lemmas (see, [53]).

Lemma 1. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n.$ Then $\mathcal{F}$ is said to be a geodesic convex set if and only if it contains all the geodesic convex combinations of its elements.

Lemma 2. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n$ be any set. Then, geodesic convex hull of $\mathcal{F}$ consists of all the geodesic convex combinations of elements of $\mathcal{F}.$

In the following definitions, we recall the notions of generalized directional derivative and generalized subdifferential of a locally Lipschitz function from Bento et al. [54].

Definition 4. 

Let $\mathsf{\Psi }:{\mathbb{H}}^n\to \mathbb{R}$ be a locally Lipschitz function on ${\mathbb{H}}^n$.

(i)

For any $u,w\in {\mathbb{H}}^n,$ the generalized directional derivative of Ψ at u in some direction $\nu \in T_u{\mathbb{H}}^n$ is denoted by ${\mathsf{\Psi }}^o(u;\nu )$, and is defined as follows :

$${\mathsf{\Psi }}^{\circ }(u;\nu ):=\underset{w\to u}{lim}\underset{t\downarrow 0}{sup}{\displaystyle \frac{\mathsf{\Psi }({exp}_{w}t{(D{exp}_u)}_{{exp}_u^{-1}(w)}\nu )-\mathsf{\Psi }(w)}{t}},$$

where ${(D{exp}_u)}_{{exp}_u^{-1}(w)}:T_{{exp}_u^{-1}(w)}(T_{u}M)\to T_{w}M$ is the differential of exponential map at ${exp}_u^{-1}(w).$

(ii)

The generalized subdifferential of function Ψ at $u\in {\mathbb{H}}^n$ is denoted by ${\partial }_c\mathsf{\Psi }(u)$, and is given by:

$${\partial }_c\mathsf{\Psi }(u):=\{\xi \in T_u{\mathbb{H}}^n|{\mathsf{\Psi }}^o(u;\nu )\ge \langle \xi ,\nu \rangle ,\mathrm{for}\mathrm{all}\nu \in T_u{\mathbb{H}}^n\}.$$

In the following lemmas, we discuss some basic properties of the generalized directional derivative and generalized subdifferential for a locally Lipschitz function from Hosseini and Pouryayevali [55].

Lemma 3. 

Let $\mathsf{\Psi }:{\mathbb{H}}^n\to \mathbb{R}$ be a locally Lipschitz function on ${\mathbb{H}}^n$ with rank K. Then, the following conditions hold:

(i)

The function ${\mathsf{\Psi }}^{\circ }(u;\nu )$ is finite, positively homogeneous and subadditive on tangent space $T_u{\mathbb{H}}^n,$ for every $u\in {\mathbb{H}}^n.$ Moreover, ${\mathsf{\Psi }}^{\circ }(u;\nu )$ satisfies the following condition:

$$\parallel {\mathsf{\Psi }}^{\circ }{(u;\nu )\parallel }_u\le K{\parallel \nu \parallel }_u,\mathrm{for}\mathrm{all}\nu \in T_u{\mathbb{H}}^n.$$

(ii)

${\mathsf{\Psi }}^{\circ }(u,\nu )$ is a locally Lipschitz function with rank K on $T_u{\mathbb{H}}^n$, for every $u\in {\mathbb{H}}^n.$ Further, ${\mathsf{\Psi }}^{\circ }(·,·)$ is upper semicontinuous on ${\mathbb{H}}^n\times T_u{\mathbb{H}}^n.$

(iii)

${\mathsf{\Psi }}^{\circ }(u;-\nu )={(-\mathsf{\Psi })}^{\circ }(u;\nu )$, for every $u\in {\mathbb{H}}^n$ and $\nu \in T_u{\mathbb{H}}^n.$

Lemma 4. 

Let $\mathsf{\Psi }:{\mathbb{H}}^n\to \mathbb{R}$ be a locally Lipschitz function on ${\mathbb{H}}^n$ with rank K and $u\in {\mathbb{H}}^n$ be an arbitrary element. Then,

(i)

The Clarke subdifferential ${\partial }_c\mathsf{\Psi }(u)$ is a nonempty, convex, compact subset of $T_u{\mathbb{H}}^n$ and ${\parallel \zeta \parallel }_u\le K$, for every $\zeta \in {\partial }_c\mathsf{\Psi }(u).$

(ii)

Let ${\{u_i\}}_{i=1}^{\infty }$ and ${\{{\zeta }_i\}}_{i=1}^{\infty }$ be arbitrary sequences in ${\mathbb{H}}^n$ and $T{\mathbb{H}}^n,$ respectively, such that ${\zeta }_i\in {\partial }_c\mathsf{\Psi }(u_i)$ for each i, and ${\{u_i\}}_{i=1}^{\infty }$ converges to u. Further, we assume that ζ is a weak*-cluster point of the sequence ${\{L_{u_i,u}{\zeta }_i\}}_{i=1}^{\infty }.$ Then, we have $\zeta \in {\partial }_c\mathsf{\Psi }(u).$

The following definition of a geodesic convex function is from Udrişte [41].

Definition 5. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n$ be a geodesic convex set and $\mathsf{\Psi }:\mathcal{F}\to \mathbb{R}$ be any function. Then, Ψ is said to be a geodesic convex function if for any $u,w\in \mathcal{F}$, the following condition holds:

$$\mathsf{\Psi }({exp}_u(\tau {exp}_u^{-1}(w)))\le (1-\tau )\mathsf{\Psi }(u)+\tau \mathsf{\Psi }(w),\mathrm{for}\mathrm{all}\tau \in [0,1].$$

Let us define a set-valued vector field $\mathcal{A}:{\mathbb{H}}^n\to 2^{T{\mathbb{H}}^n}$, such that $\mathcal{A}(u)\subseteq T_u{\mathbb{H}}^n,$ for every $u\in {\mathbb{H}}^n.$ The domain of $\mathcal{A}$, denoted by $\mathcal{D}(\mathcal{A})$ is defined as follows:

$$\mathcal{D}(\mathcal{A}):=\{u\in {\mathbb{H}}^n|\mathcal{A}(u)\ne \varnothing \}.$$

In the following definition, we introduce the generalized version of geodesic strong monotonicity of a set-valued vector field in the setting of Hadamard manifolds. For further details related to monotone vector fields, we refer the readers to [56,57].

Definition 6. 

Let $\mathcal{A}:{\mathbb{H}}^n\to 2^{T{\mathbb{H}}^n}$ be any set-valued vector field and $\sigma \in {\bigcap }_{u\in {\mathbb{H}}^n}{T}_u{\mathbb{H}}^n.$ Then, $\mathcal{A}$ is said to be geodesic strongly monotone with respect to σ, if there exists a positive constant $m_{\mathcal{A}}^{\sigma },$ such that for every $u,w\in \mathcal{D}(\mathcal{A}),$ and for every $\xi \in \mathcal{A}(u),\zeta \in \mathcal{A}(w),$ the following condition holds:

$$\begin{array}{c}\hfill {\langle \xi -\sigma ,{exp}_u^{-1}(w)\rangle }_u+{\langle \zeta -\sigma ,{exp}_w^{-1}(u)\rangle }_w\le -m_{\mathcal{A}}^{\sigma }\left|\right|{exp}_w^{-1}(u){\left|\right|}_w^2.\end{array}$$

Remark 2. 

1.

If $\sigma =0,$ then geodesic strong monotonicity of $\mathcal{A}$ with respect to σ reduces to the monotonicity of the vector field $\mathcal{A},$ presented by Li et al. [56].

2.

If $\sigma =0,$ then geodesic strong monotonicity of set-valued vector field $\mathcal{A}$ with respect to σ reduces to the corresponding definition of strong monotonicity presented by Barani [58].

3.

If ${\mathbb{H}}^n$ is a finite-dimensional real Hilbert space and $\sigma =0,$ then ${\bigcap }_{u\in {\mathbb{H}}^n}{T}_u{\mathbb{H}}^n={\mathbb{H}}^n$, $T{\mathbb{H}}^n={\mathbb{H}}^n$, ${exp}_u^{-1}(w)=w-u\mathrm{for}\mathrm{all}w,u\in {\mathbb{H}}^n$. In this case, geodesic strong monotonicity of $\mathcal{A}$ with respect to σ reduces to the corresponding definition of strong monotonicity presented by Tang and Huang [29].

In the following example, we demonstrate the validity of the Definition 6.

Example 1. 

Let ${\mathcal{P}}_{+}^2$ and ${\mathbb{S}}^2$ denote the sets of all symmetric positive definite matrices and symmetric matrices of order $2\times 2,$ respectively. For any matrix $U\in {\mathcal{P}}_{+}^2,$$\mathrm{trace}U$ and $\det U$ denote the trace and determinant of matrix U, respectively. Equivalently, ${\mathcal{P}}_{+}^2$ can be defined as follows:

$${\mathcal{P}}_{+}^2:=\left\{U=\left[\begin{array}{cc}{u}_1& u_2\\ u_3& u_4\end{array}\right]|u_i\in \mathbb{R},\mathrm{for}\mathrm{all}i=1,2,3,4,u_1,u_4>0,u_3=u_2,detU>0\right\}.$$

Moreover, ${\mathcal{P}}_{+}^2$ is a Riemannian manifold endowed with the following Riemannian metric (see, [42])

$${\langle Y,W\rangle }_U=\mathrm{trace}(Y{U}^{-1}W{U}^{-1}),\mathrm{for}\mathrm{all}W,Y\in T_U{\mathcal{P}}_{+}^2,U\in {\mathcal{P}}_{+}^2.$$

From [42], it follows that ${\mathcal{P}}_{+}^2$ is a Hadamard manifold with tangent space $T_U{\mathcal{P}}_{+}^2={\mathbb{S}}^2,\mathrm{for}\mathrm{all}U\in {\mathcal{P}}_{+}^2.$ Therefore, ${\bigcap }_{U\in {\mathcal{P}}_{+}^2}{T}_U{\mathcal{P}}_{+}^2={\mathbb{S}}^2,$ which is a nonempty set.The exponential map ${exp}_U:T_U{\mathcal{P}}_{+}^2\to {\mathcal{P}}_{+}^2$ for $U\in {\mathcal{P}}_{+}^2$ is defined as follows:

$${exp}_U(W):=U^{{\displaystyle \frac{1}{2}}}\mathrm{Exp}(U^{-{\displaystyle \frac{1}{2}}}W{U}^{-{\displaystyle \frac{1}{2}}})U^{{\displaystyle \frac{1}{2}}},\mathrm{for}\mathrm{all}W\in T_U{\mathcal{P}}_{+}^2,$$

where Exp denotes the usual exponential map. The inverse of the exponential map ${exp}_U^{-1}:{\mathcal{P}}_{+}^2\to T_U{\mathcal{P}}_{+}^2$ is defined as follows:

$${exp}_U^{-1}(\mathcal{X}):=U^{{\displaystyle \frac{1}{2}}}\mathrm{Log}(U^{-{\displaystyle \frac{1}{2}}}\mathcal{X}{U}^{-{\displaystyle \frac{1}{2}}})U^{{\displaystyle \frac{1}{2}}},\mathrm{for}\mathrm{all}\mathcal{X}\in {\mathcal{P}}_{+}^2,$$

where Log denotes the usual logarithmic function on ${\mathcal{P}}_{+}^2.$ Let $\mathsf{\Psi }:{\mathcal{P}}_{+}^2\to \mathbb{R}$ be a real-valued function. Then, the Riemannian gradient of function $\mathsf{\Psi }:{\mathcal{P}}_{+}^2\to \mathbb{R}$ is given as follows (see, for instance, [42]):

$$grad\mathsf{\Psi }(U)=U\mathsf{\nabla }\mathsf{\Psi }(U)U,$$

where $\mathsf{\nabla }\mathsf{\Psi }(U)$ denotes the Euclidean gradient of Ψ at $U.$ Let $\mathcal{F}:=\{U\in {\mathcal{P}}_{+}^2|U=\mathrm{Exp}(cI),c\in [-1,2]\}$ and let $\mathcal{A}:\mathcal{F}\to 2^{T{\mathcal{P}}_{+}^2}$ be defined as follows:

$$\mathcal{A}(U):=\left\{2(1-t)UlndetU+I|t\in \left[0,{\displaystyle \frac{1}{2}}\right]\right\}\subseteq T_U{\mathcal{P}}_{+}^2.$$

Let $\sigma =\left[\begin{array}{cc}-1& 0\\ 0& 5\end{array}\right]\in {\bigcap }_{U\in {\mathcal{P}}_{+}^2}{T}_U{\mathcal{P}}_{+}^2={\mathbb{S}}^2.$ Then, one can verify that for every $U,W\in \mathcal{F}$ and $\xi \in A(U),\zeta \in A(W),$ there exists a positive constant $m_{\mathcal{A}}^{\sigma }=1,$ such that the following inequality holds:

$$\begin{array}{cc}\hfill {〈\zeta -\sigma ,{exp}_W^{-1}(U)〉}_W+{〈\xi -\sigma ,{exp}_U^{-1}(W)〉}_U\ge & m_{\mathcal{A}}^{\sigma }{∥{exp}_W^{-1}(U)∥}_W^2\hfill \end{array}$$

Therefore, $\mathcal{A}$ is a geodesic strongly monotone vector field with respect to $\sigma =\left[\begin{array}{cc}-1& 0\\ 0& 5\end{array}\right].$

Remark 3. 

It is worth noting that $\mathcal{A}$ is not a monotone vector field for $\sigma \ne 0$, which was introduced by Li et al. [56] in the Hadamard manifold setting. Therefore, the results derived in the present article can be applied to a broader class of variational inequalities in the framework of a more general space, namely Hadamard manifolds. However, the results derived in the existing literature (see, for instance, [10,20,21,49]) are not applicable for the considered vector variational inequality problems, particularly for $\sigma \ne 0$.

The following lemma from Li et al. [56] will be used in the sequel.

Lemma 5. 

Let $u,v,w,$ and z be arbitrary elements of ${\mathbb{H}}^n$ such that $d(u,z)=d(v,z)={\displaystyle \frac{d(u,v)}{2}}.$ Then, one has

$$d^2(w,z)\le {\displaystyle \frac{1}{2}}{d}^2(w,u)+{\displaystyle \frac{1}{2}}{d}^2(w,v)-{\displaystyle \frac{1}{4}}{d}^2(u,v).$$

Remark 4. 

In view of Lemma 5, we have (see, for instance, [49])

$${\displaystyle \frac{1}{2}}{d}^2(u,v)\le d^2(v,z)+d^2(u,z),\mathrm{for}\mathrm{all}u,v,z\in {\mathbb{H}}^n.$$

(1)

Now, we recall the definition of the KKM map and an analogous to the KKM lemma from Zhou and Huang [59]. For further details, one may refer to [60].

Definition 7. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n$ be a closed geodesic convex set and $\{u_1,u_2,\cdots ,u_k\}\subseteq \mathcal{F}$ be any finite set. Further, assume that $\mathcal{G}:\mathcal{F}\to 2^{\mathcal{F}}$ is a set-valued map. Then, $\mathcal{G}$ is known as a KKM map if the following inclusion holds:

$$conv\left(\{u_1,u_2,\cdots ,u_k\}\right)\subseteq \bigcup _{i=1}^k\mathcal{G}(u_i).$$

Lemma 6. 

Let $\mathcal{F}\subseteq {\mathbb{H}}^n$ be a closed geodesic convex set and $\mathcal{G}:\mathcal{F}\to 2^{\mathcal{F}}$ be a KKM map. Further, we assume that the following assertions hold:

(i)

$\mathcal{G}(u)$ is a closed set for every $u\in \mathcal{F}$.

(ii)

There exists $u_0\in \mathcal{F}$, such that $\mathcal{G}(u_0)$ is a compact set.

Then, we have

$$\bigcap _{u\in \mathcal{F}}\mathcal{G}(u)\ne \varnothing .$$

3. Existence and Uniqueness Results for WVVHVIP

In this section, by employing an analogous to the KKM lemma, we derive the existence of a solution for WVVHVIP via Clarke subdifferentials. Moreover, the uniqueness of the solution of WVVHVIP is established under generalized geodesic strong monotonicity assumptions.

To formulate nonsmooth weak vector variational-hemivariational inequality problems in the framework of Hadamard manifold ${\mathbb{H}}^n,$ we assume that the following assumptions hold:

(B1)

Let $\mathcal{F}$ be a nonempty, compact, and geodesic convex subset of ${\mathbb{H}}^n$. Moreover, we assume that $\sigma \in {\bigcap }_{u\in {\mathbb{H}}^n}{T}_u{\mathbb{H}}^n$.

(B2)

Let $\mathcal{I}:=\{1,\cdots ,l\}$ be an index set and $h_i:\mathcal{F}\to \mathbb{R},$${\mathcal{P}}_i:{\mathbb{H}}^n\to \mathbb{R}$ be locally Lipschitz functions on $\mathcal{F}$ for every $i\in \mathcal{I}$.

(B3)

Let for every $i\in \mathcal{I},$${\mathsf{\Psi }}_i(·,·):\mathcal{F}\times \mathcal{F}\to \mathbb{R}$ be a continuous function.

Now, we consider a nonsmooth weak vector variational-hemivariational inequality problem in the framework of Hadamard manifolds as follows:

Nonsmooth Weak Vector Variational-Hemivariational Inequality Problem (WVVHVIP): Find $u^{\ast }\in \mathcal{F}$, such that for every $w\in \mathcal{F}$ there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u^{\ast })(i\in \mathcal{I})$, satisfying:

$$\begin{array}{cc}\hfill & \left({〈{\xi }_1^{\ast }-\sigma ,{exp}_{u^{\ast }}^{-1}(w)〉}_{u^{\ast }}+{\mathsf{\Psi }}_1(u^{\ast },w)-{\mathsf{\Psi }}_1(u^{\ast },u^{\ast })+{\mathcal{P}}_1^{\circ }(u^{\ast };{exp}_{u^{\ast }}^{-1}(w)),\dots ,\right.\hfill \\ & \left.{〈{\xi }_l^{\ast }-\sigma ,{exp}_{u^{\ast }}^{-1}(w)〉}_{u^{\ast }}+{\mathsf{\Psi }}_l(u^{\ast },w)-{\mathsf{\Psi }}_l(u^{\ast },u^{\ast })+{\mathcal{P}}_l^{\circ }(u^{\ast };{exp}_{u^{\ast }}^{-1}(w))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(2)

Remark 5. 

• It is worthwhile to note that if for every $i\in \mathcal{I},$${\mathcal{P}}_i^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in {\mathbb{H}}^n,$$\sigma =0,$ and ${\mathsf{\Psi }}_i(u,w)={\mathsf{\Psi }}_i(w),\mathrm{for}\mathrm{all}u,w\in {\mathbb{H}}^n,$ then WVVHVIP reduces to the mixed weak vector variational inequality of the form: Find $u^{\ast }\in \mathcal{F}$ and ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u^{\ast })$ such that

  $$\begin{array}{cc}& \left({\langle {\xi }_1^{\ast }.{exp}_{u^{\ast }}^{-1}(w)\rangle }_{u^{\ast }}+{\mathsf{\Psi }}_1(w)-{\mathsf{\Psi }}_1(u^{\ast }),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }.{exp}_{u^{\ast }}^{-1}(w)\rangle }_{u^{\ast }}+{\mathsf{\Psi }}_l(w)-{\mathsf{\Psi }}_l(u^{\ast })\right)\notin -int{\mathbb{R}}_{+}^l,\mathrm{for}\mathrm{all}w\in \mathcal{F},\hfill \end{array}$$
  as considered by Jayswal et al. [44].
• If $\sigma =0,$${\mathsf{\Psi }}_i(u,w)=0,\mathrm{for}\mathrm{all}u,w\in \mathcal{F},\mathrm{for}\mathrm{all}i\in \mathcal{I},$ and ${\mathcal{P}}_i^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in {\mathbb{H}}^n,\mathrm{for}\mathrm{all}i\in \mathcal{I}$, then WVVHVIP reduces to weak Stampacchia vector variational inequality problem as discussed in [9,10].
• If $\mathcal{I}=\{1\},$$\sigma =0,$${\mathsf{\Psi }}_1(u,w)=0$, for every $u,w\in \mathcal{F}$ and ${\partial }_c{h}_1(u)=\mathcal{A}(u),$ where $\mathcal{A}$ is a single-valued vector field, then WVVHVIP reduces to a hemivariational inequality problem (HVIP($\mathcal{A},{\mathcal{P}}_1,\mathcal{F}$)) of the form: Find $u^{\ast }\in \mathcal{F}$ such that

  $$\langle \mathcal{A}(u^{\ast }),{exp}_{u^{\ast }}^{-1}(w)\rangle +{\mathcal{P}}_1^{\circ }(u^{\ast };{exp}_{u^{\ast }}^{-1}(w))\ge 0,\mathrm{for}\mathrm{all}w\in \mathcal{F},$$
  as considered by Tang et al. [21].
• If $\mathcal{I}=\{1\},$$\sigma =0,$${\mathsf{\Psi }}_1(u,w)=0,$ for every $u,w\in \mathcal{F}$, ${\partial }_c{h}_1(u)=\mathcal{A}(u)$ is a single-valued vector field, and ${\mathcal{P}}_1$ is a constant function, then ${\mathcal{P}}_1^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in {\mathbb{H}}^n.$ In this case, WVVHVIP reduces to a variational inequality problem of the form: Find $u^{\ast }\in \mathcal{F}$ such that

  $${\langle \mathcal{A}(u^{\ast }),{exp}_{u^{\ast }}^{-1}(w)\rangle }_{u^{\ast }}\ge 0,\mathrm{for}\mathrm{all}w\in \mathcal{F},$$
  as introduced by Németh [46].
• If $\mathcal{I}=\{1\},$$\sigma =0,$${\partial }_c{h}_1(u)=\mathcal{A}(u)$, $\mathrm{for}\mathrm{all}u\in \mathcal{F}$, ${\mathcal{P}}_1$ is a constant function, and ${\mathsf{\Psi }}_1(u,w)={\mathsf{\Psi }}_1(w)$ for all $u,w\in \mathcal{F},$ then WVVHVIP reduces to the generalized mixed variational inequality of the form: Find $u^{\ast }\in \mathcal{F}$, such that there exists ${\xi }^{\ast }\in \mathcal{A}(u^{\ast })$, satisfying:

  $${\langle {\xi }^{\ast },{exp}_{u^{\ast }}^{-1}(w)\rangle }_{u^{\ast }}+{\mathsf{\Psi }}_1(w)-{\mathsf{\Psi }}_1(u^{\ast })\ge 0,\mathrm{for}\mathrm{all}w\in \mathcal{F},$$
  as studied by Li et al. [49].
• If ${\mathbb{H}}^n={\mathbb{R}}^n,\sigma =0,{\mathsf{\Psi }}_i(u,w)=0,\mathrm{for}\mathrm{all}u,w\in \mathcal{F},$${\partial }_c{h}_i(u)=F_i(u),\mathrm{for}\mathrm{all}u\in \mathcal{F},$ where $F_i:{\mathbb{R}}^n\to {\mathbb{R}}^l(i\in \mathcal{I})$ is a vector-valued map, and ${\mathcal{P}}_i^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in \mathcal{F},(i\in \mathcal{I})$, then WVVHVIP reduces to the vector variational inequality problem of the form: Find $u^{\ast }\in \mathcal{F}$ such that

  $$\left(\langle F_1(u^{\ast }),w-u^{\ast }\rangle ,\dots ,\langle F_l(u^{\ast }),w-u^{\ast }\rangle \right)\notin -int{\mathbb{R}}_{+}^l,\mathrm{for}\mathrm{all}w\in \mathcal{F},$$
  as considered by Charitha et al. [34].
• If ${\mathbb{H}}^n={\mathbb{R}}^n$, $\sigma =0,I=\{1\}$, ${\mathsf{\Psi }}_1(u,w)=0,\mathrm{for}\mathrm{all}u,w\in \mathcal{F}$ and ${\mathcal{P}}_1$ is a constant function, then ${\mathcal{P}}_1^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in {\mathbb{H}}^n,$${exp}_u^{-1}(w)=w-u,\mathrm{for}\mathrm{all}w,u\in {\mathbb{H}}^n.$ Moreover, ${\partial }_c{h}_1(u)=F(u)$, where $F:{\mathbb{H}}^n\to {\mathbb{R}}^n$ is a vector-valued function, then WVVHVIP reduces to the variational inequality problem of the form: Find $u^{\ast }\in \mathcal{F}$ such that

  $$\langle F(u^{\ast }),w-u^{\ast }\rangle \ge 0,\mathrm{for}\mathrm{all}w\in \mathcal{F},$$
  as considered by Yamashita and Fukushima [30].

In the following theorem, we establish the existence of a solution to WVVHVIP without relying on the monotonicity assumption of ${\partial }_c{h}_i(i\in \mathcal{I}).$

Theorem 1. 

Let for every $i\in \mathcal{I},$${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}$ be a geodesic convex function on $\mathcal{F}$ for every $u\in \mathcal{F}$. Moreover, we suppose that for every $u\in \mathcal{F}$ and for every $i\in \mathcal{I},$ the following set

$$\begin{array}{cc}\hfill B_i& :=\left\{w\in \mathcal{F}|\forall {\xi }_i\in {\partial }_c{h}_i(u):{\langle {\xi }_i-\sigma ,{exp}_u^{-1}(w)\rangle }_u+{\mathsf{\Psi }}_i(u,w)-{\mathsf{\Psi }}_i(u,u)\right.\hfill \\ & \left.+{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))<0\right\},\hfill \end{array}$$

is a geodesic convex set. Then, WVVHVIP has a solution in $\mathcal{F}.$

Proof. 

Let $u\in \mathcal{F}$ be an arbitrary element. A set-valued map $\mathcal{G}:\mathcal{F}\to 2^{\mathcal{F}}$ is defined as follows:

$$\begin{array}{cc}\hfill & \mathcal{G}(u):=\{w\in \mathcal{F}|\exists {\xi }_i^{\ast }\in {\partial }_c{h}_i(w)(i\in \mathcal{I}):\hfill \\ & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_1(w,u)-{\mathsf{\Psi }}_1(w,w)+{\mathcal{P}}_1^{\circ }(w;{exp}_w^{-1}(u)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_l(w,u)-{\mathsf{\Psi }}_l(w,w)+{\mathcal{P}}_l^{\circ }(w;{exp}_w^{-1}(u))\right)\notin -int{\mathbb{R}}_{+}^l\}.\hfill \end{array}$$

Notably, $\mathcal{G}(u)$ is a nonempty set for every $u\in \mathcal{F}$.

To prove the existence of a solution for WVVHVIP, we divide the proof into two parts:

(i)

In this part, we prove that $\mathcal{G}$ is a KKM map. On the contrary, we suppose that there exists a finite set $\{w_1,w_2,\cdots ,w_k\}\subseteq \mathcal{F}$ such that for some $\tilde{u}\in \mathrm{conv}\left(\{w_1,w_2,\cdots ,w_k\}\right)$ we have

$$\tilde{u}\notin \bigcup _{j=1}^k\mathcal{G}(w_j).$$

This implies that for every $j=1,\cdots ,k,$ and for every ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I})$, the following inclusion relation holds:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w_j)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w_j)-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w_j)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w_j)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},w_j)-{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w_j))\right)\in -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

Hence, for every $i\in \mathcal{I}$, $j\in \{1,2,\cdots ,k\}$, and for every ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I})$, we have

$$\begin{array}{c}\hfill {\langle {\xi }_i^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w_j)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_i(\tilde{u},w_j)-{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})+{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w_j))<0.\end{array}$$

(3)

Let us consider the following set, which is defined as follows:

$$\begin{array}{cc}\hfill \mathcal{H}:=\{w\in \mathcal{F}|& \forall {\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I}):\hfill \\ & {\langle {\xi }_i^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})+{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))<0\}.\hfill \end{array}$$

Notably, $\mathcal{H}$ is a nonempty subset of $\mathcal{F}$. From the given hypotheses, $\mathcal{H}$ is a geodesic convex set. Therefore, $\tilde{u}\in \mathcal{H},$ which implies that

$$0>{\langle {\xi }_i^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(\tilde{u})\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})+{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(\tilde{u}))=0,$$

for every ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I}),$ which is a contradiction. Therefore, $\mathcal{G}$ is a KKM map.

(ii)

In this part, we show that $\mathcal{G}(u)$ is a closed set-valued map for every $u\in \mathcal{F}.$

Let $w\in \mathcal{F}$ and ${\{w_k\}}_{k=1}^{\infty }\subseteq \mathcal{G}(u)$ such that $w_k\to w$ as $k\to \infty .$ Then there exist ${\xi }_{ik}^{\ast }\in {\partial }_c{h}_i(w_k)(i\in \mathcal{I}),$ satisfying:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_{1k}^{\ast }-\sigma ,{exp}_{w_k}^{-1}(u)\rangle }_{w_k}+{\mathsf{\Psi }}_1(w_k,u)-{\mathsf{\Psi }}_1(w_k,w_k)+{\mathcal{P}}_1^{\circ }(w_k;{exp}_{w_k}^{-1}(u)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_{lk}^{\ast }-\sigma ,{exp}_{w_k}^{-1}(u)\rangle }_{w_k}+{\mathsf{\Psi }}_l(w_k,u)-{\mathsf{\Psi }}_l(w_k,w_k)+{\mathcal{P}}_l^{\circ }(w_k;{exp}_{w_k}^{-1}(u))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(4)

Employing continuity of ${\mathsf{\Psi }}_i(i\in \mathcal{I}),$ upper semicontinuity of ${\mathcal{P}}_i^{\circ }(i\in \mathcal{I})$ and Lemma 4, we have

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_1(w,u)-{\mathsf{\Psi }}_1(w,w)+{\mathcal{P}}_1^{\circ }(w;{exp}_w^{-1}(u)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_l(w,u)-{\mathsf{\Psi }}_l(w,w)+{\mathcal{P}}_l^{\circ }(w;{exp}_w^{-1}(u))\right)\notin -int{\mathbb{R}}_{+}^l,\hfill \end{array}$$

(5)

where ${\xi }_i^{\ast }\in {\partial }_c{h}_i(w)(i\in \mathcal{I}).$ Therefore, $w\in \mathcal{G}(u).$

Moreover, $\mathcal{G}(u)\subseteq \mathcal{F}$ and $\mathcal{F}$ is a compact set implies that $\mathcal{G}(u)$ is a bounded set for every $u\in \mathcal{F}$. In view of (ii), $\mathcal{G}(u)$ is a closed set for every $u\in \mathcal{F}$. Therefore, $\mathcal{G}$ is a KKM mapping such that $\mathcal{G}(u)$ is a compact set for every $u\in \mathcal{F}$. From Lemma 6, there exists $\tilde{u}\in \mathcal{F}$ such that $\tilde{u}\in \bigcap _{u\in \mathcal{F}}\mathcal{G}(u).$ Therefore, for every $u\in \mathcal{F}$ there exists ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I})$ such that

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(u)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},u)-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(u)),\dots ,\right.\hfill \\ \hfill & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(u)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},u)-{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(u))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(6)

Hence, $\tilde{u}\in \mathcal{F}$ is a solution of WVVHVIP. This completes the proof. □

Remark 6. 

Theorem 1 generalizes Theorem 3.5 derived by Jayswal et al. [44] from mixed vector variational inequality problem to vector variational-hemivariational inequality problem, namely WVVHVIP. Moreover, Theorem 1 is applicable to a broader class of vector variational inequality problems and hemivariational inequality problems as the proof of Theorem 1 has not utilized the monotonicity assumption as employed in prior works (see, for instance, [35,44]).

In the following example, we illustrate the significance of Theorem 1.

Example 2. 

Let us consider the Hadamard manifold ${\mathcal{P}}_{+}^2$ as considered in Example 1.

Let $\mathcal{F}:=\{U\in {\mathcal{P}}_{+}^2|U=\mathrm{Exp}(c_{1}I),-1\le c_1\le 1\}$ be a nonempty, compact, and geodesic convex subset of ${\mathcal{P}}_{+}^2$. Moreover, let $h_1,h_2:\mathcal{F}\to \mathbb{R}$ be defined as follows:

$$h_1(U):=|ln\det U|\mathrm{and}{h}_2(U):=ln{u}_1+ln{u}_4,$$

where $U=\left[\begin{array}{cc}{u}_1& u_2\\ u_2& u_4\end{array}\right]\in \mathcal{F}.$ The Clarke subdifferentials of $h_1$ and $h_2$ are given by

$${\partial }_c{h}_1(U)=\left\{\begin{array}{cc}U,\hfill & \det U>1,\hfill \\ (2t-1)I\hfill & \det U=1,t\in [0,1],\hfill \\ -U,\hfill & \det U<1,\hfill \end{array}\right.\mathrm{and}{\partial }_c{h}_2(U)=\{U\}.$$

Define ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2:\mathcal{F}\times \mathcal{F}\to \mathbb{R}$ as follows:

$${\mathsf{\Psi }}_1(U,W):=2ln\det UW\mathrm{and}{\mathsf{\Psi }}_2(U,W):={\displaystyle \frac{\det U}{2}}+ln\det W.$$

Notably, ${\mathsf{\Psi }}_i(U,·)(i=1,2)$ are geodesic convex functions on $\mathcal{F}$ for every $U\in \mathcal{F}.$

The functions ${\mathcal{P}}_1,{\mathcal{P}}_2:{\mathcal{P}}_{+}^2\to \mathbb{R}$ are given by:

$${\mathcal{P}}_1(U):=|\det U-1|\mathrm{and}{\mathcal{P}}_2(U):=\det U.$$

The Clarke generalized directional derivatives of ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ are given by

$${\mathcal{P}}_1^{\circ }(U;{exp}_U^{-1}(W))=\left\{\begin{array}{cc}\det U(ln\det W-ln\det U),\hfill & \det U>1,\hfill \\ |ln\det W-ln\det U|,\hfill & \det U=1,\hfill \\ \det U(-ln\det W+ln\det U),\hfill & \det U<1.\hfill \end{array}\right.$$

$${\mathcal{P}}_2^{\circ }(U;{exp}_U^{-1}(W))=\{\det U(ln\det W-ln\det U)\}.$$

Consider the following nonsmooth weak vector variational-hemivariational inequality problem (WVVHVIP1): Find $U\in \mathcal{F}$ such that for any $W\in \mathcal{F},$ there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(U)(i=1,2),$ satisfying:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_U^{-1}(W)\rangle }_U+{\mathsf{\Psi }}_1(U,W)-{\mathsf{\Psi }}_1(U,U)+{\mathcal{P}}_1^{\circ }(U;{exp}_U^{-1}(W)),\right.\hfill \\ & \left.{\langle {\xi }_2^{\ast }-\sigma ,{exp}_U^{-1}(W)\rangle }_U+{\mathsf{\Psi }}_2(U,W)-{\mathsf{\Psi }}_2(U,U)+{\mathcal{P}}_2^{\circ }(U;{exp}_U^{-1}(W))\right)\notin -int{\mathbb{R}}_{+}^2,\hfill \end{array}$$

where $\sigma =\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right].$

It can be verified that for every $U\in \mathcal{F}$ and for every $i\in \mathcal{I}=\{1,2\},$ the following set

$$\begin{array}{cc}\hfill B_i& :=\left\{W\in \mathcal{F}|\forall {\xi }_i\in {\partial }_c{h}_i(U):{\langle {\xi }_i-\sigma ,{exp}_U^{-1}(W)\rangle }_u+{\mathsf{\Psi }}_i(U,W)-{\mathsf{\Psi }}_i(U,U)\right.\hfill \\ & \left.+{\mathcal{P}}_i^{\circ }(U;{exp}_U^{-1}(W))<0\right\},\hfill \end{array}$$

is a geodesic convex set. Moreover, ${\mathsf{\Psi }}_i(U,·)(i\in \mathcal{I})$ is a geodesic convex function for every $U\in \mathcal{F}.$ Therefore, all the hypotheses of Theorem 1 are satisfied, which concludes that there exists $\widehat{U}\in \mathcal{F}$, such that $\widehat{U}$ is a solution of WVVHVIP1. Furthermore, one can verify that $U=\mathrm{Exp}(-I)$ is a solution of WVVHVIP1.

In the following theorem, we establish the uniqueness of the solution to WVVHVIP under generalized geodesic strong monotonicity assumptions.

Theorem 2. 

Let for every $i\in \mathcal{I}$, ${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}$ be a geodesic convex function on $\mathcal{F}$ for every $u\in \mathcal{F}$. Further, we assume that the following assumptions hold:

(i)

The set-valued vector field $\bigcup _{i\in \mathcal{I}}{\partial }_c{h}_i$ is geodesic strongly monotone with respect to σ having a positive constant $m_h^{\sigma }.$

(ii)

There exists a constant $m_{\mathsf{\Psi }}>0$ such that

$${\mathsf{\Psi }}_i(u_1,w_2)-{\mathsf{\Psi }}_i(u_1,w_1)+{\mathsf{\Psi }}_p(u_2,w_1)-{\mathsf{\Psi }}_p(u_2,w_2)\le m_{\mathsf{\Psi }}\parallel {exp}_{u_2}^{-1}(u_1){\parallel }_{u_2}{\parallel {exp}_{w_2}^{-1}(w_1)\parallel }_{w_2},$$

for every $u_1,u_2,w_1,w_2\in \mathcal{F}$ and $i,p\in \mathcal{I}.$

(iii)

There exists a constant $m_{\mathcal{P}}>0$ such that

$${\mathcal{P}}_i^{\circ }(w_1;{exp}_{w_1}^{-1}(w_2))+{\mathcal{P}}_p^{\circ }(w_2;{exp}_{w_2}^{-1}(w_1))\le m_{\mathcal{P}}{\parallel {exp}_{w_2}^{-1}(w_1)\parallel }_{w_2}^2,$$

for all $w_1,w_2\in \mathcal{F}$ and $i,p\in \mathcal{I}.$

Furthermore, we assume that $m_h^{\sigma }>m_{\mathsf{\Psi }}+m_{\mathcal{P}}.$ Then WVVHVIP has a unique solution.

Proof. 

In view of Theorem 1, there exists at least one solution of WVVHVIP. On the contrary, we suppose that $u_1^{\ast }$ and $u_2^{\ast }$ in $\mathcal{F}$ are two distinct solutions of WVVHVIP. Therefore, there exist $j,k\in \mathcal{I}$, ${\xi }_k^{\ast }\in {\partial }_c{h}_k(u_1^{\ast })$, and ${\zeta }_j^{\ast }\in {\partial }_c{h}_j(u_2^{\ast })$ such that the following inequalities hold:

$$\begin{array}{cc}& {\langle {\xi }_k^{\ast }-\sigma ,{exp}_{u_1^{\ast }}^{-1}(u_2^{\ast })\rangle }_{u_1^{\ast }}+{\mathsf{\Psi }}_k(u_1^{\ast },u_2^{\ast })-{\mathsf{\Psi }}_k(u_1^{\ast },u_1^{\ast })+{\mathcal{P}}_k^{\circ }(u_1^{\ast };{exp}_{u_1^{\ast }}^{-1}(u_2^{\ast }))\ge 0,\end{array}$$

(7)

$$\begin{array}{cc}& {\langle {\zeta }_j^{\ast }-\sigma ,{exp}_{u_2^{\ast }}^{-1}(u_1^{\ast })\rangle }_{u_2^{\ast }}+{\mathsf{\Psi }}_j(u_2^{\ast },u_1^{\ast })-{\mathsf{\Psi }}_j(u_2^{\ast },u_2^{\ast })+{\mathcal{P}}_j^{\circ }(u_2^{\ast };{exp}_{u_2^{\ast }}^{-1}(u_1^{\ast }))\ge 0.\end{array}$$

(8)

On adding (7) and (8) we get

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_k(u_1^{\ast },u_2^{\ast })-{\mathsf{\Psi }}_k(u_1^{\ast },u_1^{\ast })+{\mathsf{\Psi }}_j(u_2^{\ast },u_1^{\ast })& -{\mathsf{\Psi }}_j(u_2^{\ast },u_2^{\ast })+{\mathcal{P}}_k^{\circ }(u_1^{\ast };{exp}_{u_1^{\ast }}^{-1}(u_2^{\ast }))+{\mathcal{P}}_j^{\circ }(u_2^{\ast };{exp}_{u_2^{\ast }}^{-1}(u_1^{\ast }))\hfill \\ & \ge -{\langle {\xi }_k^{\ast }-\sigma ,{exp}_{u_1^{\ast }}^{-1}(u_2^{\ast })\rangle }_{u_1^{\ast }}-{\langle {\zeta }_j^{\ast }-\sigma ,{exp}_{u_2^{\ast }}^{-1}(u_1^{\ast })\rangle }_{u_2^{\ast }}.\hfill \end{array}$$

Therefore, from the given hypotheses (i)-(iii), we obtain the following inequality

$$(m_h^{\sigma }-m_{\mathcal{P}}-m_{\mathsf{\Psi }})\left|\right|{exp}_{u_1^{\ast }}^{-1}(u_2^{\ast }){\left|\right|}_{u_1^{\ast }}^2\le 0.$$

(9)

In view of the given hypotheses, we have

$$(m_h^{\sigma }-m_{\mathcal{P}}-m_{\mathsf{\Psi }})>0.$$

Then from (9), it follows that $u_1^{\ast }=u_2^{\ast }$. This completes the proof. □

4. Gap Functions for WVVHVIP

In this section, we formulate gap functions and regularized gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of a solution to WVVHVIP. In addition, we formulate the Moreau-Yosida type regularized gap function for WVVHVIP.

In the following definition, we introduce the definition of a gap function for WVVHVIP.

Definition 8. 

A function $\mathsf{\Pi }:\mathcal{F}\to \mathbb{R}$ is called a gap function for WVVHVIP, if it satisfies the following conditions:

(i)

$\mathsf{\Pi }(u)\ge 0,\mathrm{for}\mathrm{all}u\in \mathcal{F}.$

(ii)

$\tilde{u}\in \mathcal{F}$ is a solution of WVVHVIP if and only if $\mathsf{\Pi }(\tilde{u})=0.$

Consider a function $\mathsf{\Pi }:\mathcal{F}\to \mathbb{R}$, which is defined as follows:

$$\begin{array}{c}\hfill \mathsf{\Pi }(u):=\underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{u,w}{\xi }_i^{\ast }-L_{u,w}\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,w)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))\},\end{array}$$

(10)

where $u\in \mathcal{F}$ and ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u)(i\in \mathcal{I}).$

In the following theorem, we prove that the function $\mathsf{\Pi }$ defined in (10) is a gap function for WVVHVIP.

Theorem 3. 

Let for every $i\in \mathcal{I}$ and $u\in \mathcal{F}$, ${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}$ be a geodesic convex function on $\mathcal{F}$. Then the function Π is a gap function for WVVHVIP.

Proof. 

From the definition of $\mathsf{\Pi }$ for every $u\in \mathcal{F}$, there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u)(i\in \mathcal{I}),$ such that

$$\begin{array}{ccc}\hfill \mathsf{\Pi }(u)& =\underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{u,w}{\xi }_i^{\ast }-L_{u,w}\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,w)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))\}\hfill & \\ \hfill & \ge \underset{i\in \mathcal{I}}{min}\left\{{\langle {\xi }_i^{\ast }-\sigma ,{exp}_u^{-1}(u)\rangle }_u+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,u)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(u))\right\}=0.\hfill \end{array}$$

It follows that for any $u\in \mathcal{F}$, we have

$$\mathsf{\Pi }(u)\ge 0.$$

(11)

Let $\mathsf{\Pi }(\tilde{u})=0$ for some $\tilde{u}\in \mathcal{F}$. This implies that there exists ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})$ for every $i\in \mathcal{I}$ such that the following condition holds:

$$\begin{array}{c}\hfill \underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{\tilde{u},v}{\xi }_i^{\ast }-L_{\tilde{u},v}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\}=0.\end{array}$$

It follows that for every $w\in \mathcal{F}$, we have

$$\underset{i\in \mathcal{I}}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right\}\le 0.$$

Therefore, for any $w\in \mathcal{F}$ there exists some $p\in \mathcal{I}$ such that the following inequality holds:

$${\langle L_{\tilde{u},w}{\xi }_p^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\le 0.$$

(12)

Applying the parallel transport from w to $\tilde{u}$ in (12), we have

$${\langle {\xi }_p^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})+{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\ge 0.$$

Therefore, we have

$$\begin{array}{ccc}\hfill ({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathsf{\Psi }}_1& (\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w)),\dots ,\hfill & \\ \hfill {\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},w)& -{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w)))\notin -int{\mathbb{R}}_{+}^l,\hfill \end{array}$$

which claims that $\tilde{u}$ is a solution of WVVHVIP.

For the converse part, we assume that $\tilde{u}\in \mathcal{F}$ is a solution of WVVHVHIP. Then, for every $w\in \mathcal{F},$ there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I})$, satisfying:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},w)-{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(13)

This implies that for any $w\in \mathcal{F}$ there exists some $p\in \mathcal{I}$ such that

$${\langle {\xi }_p^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})+{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\ge 0.$$

(14)

Applying parallel transport from $\tilde{u}$ to w in (14), we infer that

$${\langle L_{\tilde{u},w}{\xi }_p^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\le 0.$$

Hence, for every $w\in \mathcal{F}$, we get the following inequality

$$\begin{array}{c}\hfill \underset{i\in \mathcal{I}}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right\}\le 0.\end{array}$$

(15)

On taking supremum over $w\in \mathcal{F}$ in (15) we have

$$\begin{array}{cc}\hfill \mathsf{\Pi }(\tilde{u})& =\underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right\}\hfill \\ \hfill & \le 0.\hfill \end{array}$$

(16)

From (11) and (16), it follows that

$$\mathsf{\Pi }(\tilde{u})=0.$$

This completes the proof. □

Remark 7. 

1.

If $\sigma =0$, ${\mathsf{\Psi }}_i(u,w)={\mathsf{\Psi }}_i(w),\mathrm{for}\mathrm{all}u,w\in \mathcal{F},$ and ${\mathcal{P}}_i^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in \mathcal{F},\mathrm{for}\mathrm{all}i\in \mathcal{I},$ then Theorem 3 reduces to Theorem 3.3 deduced by Jayswal et al. [44].

2.

If ${\mathbb{H}}^n={\mathbb{R}}^n,\sigma =0,{\mathsf{\Psi }}_i(u,w)=0,\mathrm{for}\mathrm{all}u,w\in \mathcal{F},$${\partial }_c{h}_i(u)=F_i(u),\mathrm{for}\mathrm{all}u\in \mathcal{F},$ where $F_i:{\mathbb{R}}^n\to {\mathbb{R}}^l(i\in \mathcal{I})$ is a vector-valued map, and ${\mathcal{P}}_i^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in \mathcal{F},i\in \mathcal{I}$, then Theorem 3 reduces to Theorem 4.1 established by Charitha et al. [34].

3.

If ${\mathbb{H}}^n={\mathbb{R}}^n$, $\sigma =0$, $\mathcal{I}=\{1\},{\mathsf{\Psi }}_1(u,w)=0,\mathrm{for}\mathrm{all}u,w\in \mathcal{F},$${\mathcal{P}}_1^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in \mathcal{F},$ then ${exp}_u^{-1}(w)=w-u,\mathrm{for}\mathrm{all}u,w\in \mathcal{F}.$ Moreover, if ${\partial }_c{h}_1(u)=F(u)$, for all $u\in \mathcal{F}$, where $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a vector-valued function, then Theorem 3 reduces to Lemma 2.1 derived by Yamashita and Fukushima [30].

4.

It is worth noting that Hadamard manifolds, in general, represent a nonlinear space. For instance, ${exp}_u^{-1}(w)\ne -{exp}_w^{-1}(u)$ for any $u,w\in {\mathbb{H}}^n.$ However, in the framework of Euclidean space $u-w=-(w-u),\mathrm{for}\mathrm{all}u,w\in {\mathbb{R}}^n.$ Therefore, the techniques that have been successfully employed in the context of linear spaces cannot be applied to the optimization problems defined on Hadamard manifolds. This limitation underscores the significant challenges associated with the development of optimization techniques in the framework of Hadamard manifolds.

Now, we provide a non-trivial example to illustrate the significance of Theorem 3.

Example 3. 

Let us consider the same sets ${\mathcal{P}}_{+}^2,\mathcal{F}$ and functions $h_i,{\mathsf{\Psi }}_i,{\mathcal{P}}_i(i=1,2)$ as defined in Example 2. Now, we define a function $\mathsf{\Pi }:\mathcal{F}\to \mathbb{R}$ for $\sigma =\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ as follows:

$$\begin{array}{cc}\hfill \mathsf{\Pi }(U):& =\underset{w\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{U,W}{\xi }_i^{\ast }-L_{U,W}\sigma ,{exp}_W^{-1}(U)\rangle }_W+{\mathsf{\Psi }}_i(U,U)-{\mathsf{\Psi }}_i(U,W)\right.\hfill \\ & \left.-{\mathcal{P}}_i^{\circ }(U;{exp}_U^{-1}(W))\right\}\hfill \end{array}$$

Let $U\in \mathcal{F}$, such that $1<detU\le e^2$. Then, the function Π for $\sigma =-I$ is given by:

$$\begin{array}{cc}\hfill \mathsf{\Pi }(U)& =\underset{W\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{U,W}{\xi }_i^{\ast }-L_{U,W}\sigma ,{exp}_W^{-1}(U)\rangle }_W+{\mathsf{\Psi }}_i(U,U)-{\mathsf{\Psi }}_i(U,W)\right.\hfill \\ & \left.-{\mathcal{P}}_i^{\circ }(U;{exp}_U^{-1}(W))\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\underset{W\in \mathcal{F}}{sup}min\left\{-{〈{\xi }_1^{\ast }-\sigma ,{exp}_U^{-1}(W)〉}_U+{\mathsf{\Psi }}_1(U,U)-{\mathsf{\Psi }}_1(U,W)-{\mathcal{P}}_1^{\circ }(U;{exp}_U^{-1}(W)),\right.\hfill \\ & \left.-{〈{\xi }_2^{\ast }-\sigma ,{exp}_U^{-1}(W)〉}_U+{\mathsf{\Psi }}_2(U,U)-{\mathsf{\Psi }}_2(U,W)-{\mathcal{P}}_2^{\circ }(U;{exp}_U^{-1}(W))\right\}\hfill \\ \hfill & =\underset{W\in \mathcal{F}}{sup}min\left\{-{〈U+I,{\displaystyle \frac{ln\frac{\det W}{\det U}}{2}}U〉}_U+2ln{\displaystyle \frac{\det U}{\det W}}+\det U\left(ln{\displaystyle \frac{\det U}{\det W}}\right),\right.\hfill \\ & -\left.{〈U+I,{\displaystyle \frac{ln\frac{\det W}{\det U}}{2}}U〉}_U+ln{\displaystyle \frac{\det U}{\det W}}+\det U\left(ln{\displaystyle \frac{\det U}{\det W}}\right)\right\}\hfill \\ \hfill & =\min \left\{(1+{(\det U)}^{-{\displaystyle \frac{1}{2}}}+2+\det U)(ln\det U+2),(1+{(\det U)}^{-{\displaystyle \frac{1}{2}}}+1)(ln\det U+2)\right\}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

For $U\in \mathcal{F}$, such that $\det U=1,$ the function Π is given by:

$$\begin{array}{cc}\hfill \mathsf{\Pi }(U)& =\underset{W\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{U,W}{\xi }_i^{\ast }-L_{U,W}\sigma ,{exp}_W^{-1}(U)\rangle }_W+{\mathsf{\Psi }}_i(U,U)-{\mathsf{\Psi }}_i(U,W))\right.\hfill \\ & \left.-{\mathcal{P}}_i^{\circ }(U;{exp}_U^{-1}W)\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\underset{W\in \mathcal{F}}{sup}min\left\{-{〈{\xi }_1^{\ast }-\sigma ,{exp}_U^{-1}(W)〉}_u+{\mathsf{\Psi }}_1(U,U)-{\mathsf{\Psi }}_1(U,W)-{\mathcal{P}}_1^{\circ }(U;{exp}_U^{-1}(W)),\right.\hfill \\ & \left.-{〈{\xi }_2^{\ast }-\sigma ,{exp}_U^{-1}(W)〉}_U+{\mathsf{\Psi }}_2(U,U)-{\mathsf{\Psi }}_2(U,W)-{\mathcal{P}}_2^{\circ }(U;{exp}_U^{-1}(W))\right\}\hfill \\ \hfill & =\underset{W\in \mathcal{F}}{sup}min\left\{-〈(2t-1)I+I,{\displaystyle \frac{ln\det W}{2}}I〉-2ln\det W-|ln\det W|,\right.\hfill \\ & -\left.〈2I,{\displaystyle \frac{ln\det W}{2}}I〉-ln\det W-ln\det W\right\}\hfill \\ \hfill & =\min \left\{4t-2+4-2,8\right\},0\le t\le 1\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Let $U\in \mathcal{F}$, such that $\det U\in [e^{-2},1).$ Then, we have

$$\begin{array}{cc}\hfill \mathsf{\Pi }(U)& =\underset{W\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{U,W}{\xi }_i^{\ast }-L_{U,W}\sigma ,{exp}_W^{-1}(U)\rangle }_W+{\mathsf{\Psi }}_i(U,U)-{\mathsf{\Psi }}_i(U,W)\right.\hfill \\ & \left.-{\mathcal{P}}_i^{\circ }(U;{exp}_U^{-1}(W))\right\}\hfill \\ \hfill & =\underset{W\in \mathcal{F}}{sup}min\left\{-{〈{\xi }_1^{\ast }-\sigma ,{exp}_U^{-1}(W)〉}_U+{\mathsf{\Psi }}_1(U,U)-{\mathsf{\Psi }}_1(U,W)-{\mathcal{P}}_1^{\circ }(U;{exp}_U^{-1}(W)),\right.\hfill \\ & \left.-{〈{\xi }_2^{\ast }-\sigma ,{exp}_U^{-1}(W)〉}_U+{\mathsf{\Psi }}_2(U,U)-{\mathsf{\Psi }}_2(U,W)-{\mathcal{P}}_2^{\circ }(U;{exp}_U^{-1}(W))\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\underset{W\in \mathcal{F}}{sup}min\left\{-〈-U+I,{\displaystyle \frac{ln\frac{\det W}{\det U}}{2}}U〉+2ln{\displaystyle \frac{\det U}{\det W}}-\det U\left(ln{\displaystyle \frac{\det U}{\det W}}\right),\right.\hfill \\ & -\left.〈U+I,{\displaystyle \frac{ln\frac{\det W}{\det U}}{2}}U〉+ln{\displaystyle \frac{\det U}{\det W}}+\det U\left(ln{\displaystyle \frac{\det U}{\det W}}\right)\right\}\hfill \\ \hfill & =\min \left\{(-1+{(\det U)}^{-{\displaystyle \frac{1}{2}}}+2-\det U)(ln\det U+2),\right.\hfill \\ & \left.({(\det U)}^{-{\displaystyle \frac{1}{2}}}+1+1+\det U)(ln\det U+2)\right\}\hfill \\ \hfill & =\left\{\left(1+{(\det U)}^{-{\displaystyle \frac{1}{2}}}-\det U\right)(ln\det U+2),({(\det U)}^{-{\displaystyle \frac{1}{2}}}+\det U+2)(ln\det U+2)\right\}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Notably, $\mathsf{\Pi }(U)=0$ if and only if $U=\mathrm{Exp}(-I).$ Therefore, Π is a gap function for WVVHVIP1.

Now, we introduce a regularized gap function for WVVHVIP. Let $\gamma >0$ be a fixed parameter and a function ${\vartheta }^{\gamma }:\mathcal{F}\to \mathbb{R}$ be defined as follows:

$$\begin{array}{c}\hfill {\vartheta }^{\gamma }(u):=\underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{u,w}{\xi }_i^{\ast }-L_{u,w}\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,w)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))\\ \hfill -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(u)\parallel }_w^2\},\end{array}$$

(17)

where $u\in \mathcal{F}$ and ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u)(i\in \mathcal{I}).$

In the following theorem, we establish that the function ${\vartheta }^{\gamma }$ defined in (17) is a gap function for WVVHVIP.

Theorem 4. 

Let ${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}(i\in \mathcal{I})$ be geodesic convex functions on $\mathcal{F}$ for every $u\in \mathcal{F}$. Then, for $\gamma >0$, the function ${\vartheta }^{\gamma }$ is a gap function for WVVHVIP.

Proof. 

In view of the definition of ${\vartheta }^{\gamma }(u),$ there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u)(i\in \mathcal{I})$ such that

$$\begin{array}{cc}\hfill {\vartheta }^{\gamma }(u)& =\underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\left\{{\langle L_{u,w}{\xi }_i^{\ast }-L_{u,w}\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,w)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(u)\parallel }_w^2\right\}\hfill \\ \hfill & \ge \underset{i\in \mathcal{I}}{min}\left\{{\langle {\xi }_i^{\ast }-\sigma ,{exp}_u^{-1}(u)\rangle }_u+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,u)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(u))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_u^{-1}(u)\parallel }_u^2\right\}\hfill \\ & =0.\hfill \end{array}$$

This implies that

$${\vartheta }^{\gamma }(u)\ge 0,\mathrm{for}\mathrm{all}u\in \mathcal{F}.$$

(18)

Let ${\vartheta }^{\gamma }(\tilde{u})=0$ for some $\tilde{u}\in \mathcal{F}$. This implies that there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I})$ such that

$$\begin{array}{cc}\hfill & \underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\hfill \\ & -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\}=0.\hfill \end{array}$$

(19)

That is, for every $w\in \mathcal{F}$, we have

$$\begin{array}{cc}\hfill & \underset{i\in \mathcal{I}}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\right\}\le 0.\hfill \end{array}$$

(20)

Therefore, for every $w\in \mathcal{F},$ there exists at least one index $p\in \mathcal{I}$ such that

$$\begin{array}{cc}\hfill {〈L_{\tilde{u},w}{\xi }_p^{\ast }-L_{\tilde{u},v}\sigma ,{exp}_w^{-1}(\tilde{u})〉}_w+{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_p(\tilde{u},w)& -{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\hfill \\ & -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\le 0.\hfill \end{array}$$

(21)

Applying parallel transport from w to $\tilde{u}$ in (21), we get the following inequality

$${\langle {\xi }_p^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})+{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\ge -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2.$$

(22)

For any $\tilde{w}\in \mathcal{F}$ and $\tau \in (0,1),$$w=w_{\tau }:={exp}_{\tilde{u}}(\tau {exp}_{\tilde{u}}^{-1}(\tilde{w}))\in \mathcal{F}$ by using geodesic convexity assumptions on $\mathcal{F}$. Therefore, by employing geodesic convexity assumptions of ${\mathsf{\Psi }}_i(i\in \mathcal{I})$ and from (21)- (22), it follows that there exists some $k\in \mathcal{I}$ such that the following inequalities hold:

$$\begin{array}{cc}\hfill \tau \langle {\xi }_k^{\ast }-\sigma ,& {exp}_{\tilde{u}}^{-1}(\tilde{w}){\rangle }_{\tilde{u}}+\tau {\mathsf{\Psi }}_k(\tilde{u},\tilde{w})-\tau {\mathsf{\Psi }}_k(\tilde{u},\tilde{u})+\tau {\mathcal{P}}_k^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(\tilde{w}))\hfill \\ \hfill & \ge {\langle {\xi }_k^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w_{\tau })\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_k(\tilde{u},w_{\tau })-{\mathsf{\Psi }}_k(\tilde{u},\tilde{u})+{\mathcal{P}}_k^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w_{\tau }))\hfill \\ \hfill & \ge -{\displaystyle \frac{1}{2\gamma }}\parallel {exp}_{\tilde{u}}^{-1}(w_{\tau }){\parallel }_{\tilde{u}}^2=-{\displaystyle \frac{{\tau }^2}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(\tilde{w})\parallel }_{\tilde{u}}^2.\hfill \end{array}$$

(23)

Hence, we obtain the following inequality for every $\tilde{w}\in \mathcal{F}$,

$${\langle {\xi }_k^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(\tilde{w})\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_k(\tilde{u},\tilde{w})-{\mathsf{\Psi }}_k(\tilde{u},\tilde{u})+{\mathcal{P}}_k^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(\tilde{w}))\ge -{\displaystyle \frac{\tau }{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(\tilde{w})\parallel }_{\tilde{u}}^2.$$

(24)

Let $\tau \downarrow 0$ in (24) we get the following inequality

$${\langle {\xi }_p^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(\tilde{w})\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_p(\tilde{u},\tilde{w})-{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})+{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(\tilde{w}))\ge 0,\mathrm{for}\mathrm{all}\tilde{w}\in \mathcal{F}.$$

That is,

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(\tilde{w})\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},\tilde{w})-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(\tilde{w})),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(\tilde{w})\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},\tilde{w})-{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(\tilde{w}))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(25)

Therefore, $\tilde{u}$ is a solution of WVVHVIP.

For the converse part, we assume that there exists a solution $\tilde{u}\in \mathcal{F}$ of WVVHVIP. This implies that for every $w\in \mathcal{F},$ there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I})$ such that the following condition holds:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},w)-{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(26)

Equivalently, for every $w\in \mathcal{F}$, there exists some $p\in \mathcal{I},$ such that we have

$${\langle {\xi }_p^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})+{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))+{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\ge 0.$$

(27)

Applying parallel transport from $\tilde{u}$ to w in (27) we get

$$\begin{array}{cc}\hfill {\langle L_{\tilde{u},w}{\xi }_p^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))& -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\hfill \\ & \le 0.\hfill \end{array}$$

(28)

From (28) it follows that

$$\begin{array}{cc}\hfill {\vartheta }^{\gamma }(\tilde{u})& =\underset{w\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\hfill \\ & -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\}\hfill \\ & \le 0.\hfill \end{array}$$

(29)

From (18) and (29) we have

$${\vartheta }^{\gamma }(\tilde{u})=0.$$

This completes the proof. □

Remark 8. 

1.

If ${\mathbb{H}}^n={\mathbb{R}}^n,{\mathsf{\Psi }}_i(u,w)=0,\mathrm{for}\mathrm{all}u,w\in \mathcal{F},i\in \mathcal{I},$${\partial }_c{h}_i(u)=F_i(u),\mathrm{for}\mathrm{all}u\in \mathcal{F},i\in \mathcal{I}$, where $F_i:{\mathbb{R}}^n\to {\mathbb{R}}^l(i\in \mathcal{I})$ is a vector-valued map, and ${\mathcal{P}}_i^{\circ }(u;·)=0,\mathrm{for}\mathrm{all}u\in \mathcal{F}(i\in \mathcal{I})$, then Theorem 4 reduces to Theorem 4.2 deduced by Charitha et al. [34].

2.

The results derived in Theorem 4 generalize the corresponding results established in Theorem 3.1 deduced by Fukushima [32] from the Euclidean space to an even more general space, namely Hadamard manifolds, as well as generalize them from variational inequality problem to a broader category of vector variational-hemivariational inequality problems, in particular, WVVHVIP.

In the following example, we illustrate the significance of Theorem 4.

Example 4. 

Consider the following Riemannian manifold (see, [61]):

$${\mathbb{H}}^1={\mathbb{R}}_{++}:=\{u\in \mathbb{R}|u>0\}.$$

The Riemannian manifold ${\mathbb{H}}^1$ is endowed with the following Riemannian metric (see, [41]):

$$\mathcal{G}(u)={\displaystyle \frac{1}{u^2}},\mathrm{for}\mathrm{all}u\in {\mathbb{H}}^1.$$

Further, ${\mathbb{H}}^1$ is a Hadamard manifold and the tangent space at every $u\in {\mathbb{H}}^1$ is the set of real numbers. That is, $T_u{\mathbb{H}}^1=\mathbb{R}$ (see, [61]). The exponential map is defined by:

$${exp}_u(\nu ):=u{e}^{\left(\nu /u\right)},\mathrm{for}\mathrm{all}\nu \in T_u{\mathbb{H}}^1.$$

Moreover, the inverse of the exponential map is given by:

$${exp}_u^{-1}(w):=uln\left({\displaystyle \frac{w}{u}}\right),\mathrm{for}\mathrm{all}w\in {\mathbb{H}}^1.$$

Let $\mathcal{F}:=\{u\in {\mathbb{H}}^1|u=e^{\tau },\tau \in [-2,2]\}$ be a nonempty, compact, and geodesic convex subset of ${\mathbb{H}}^1$. Further, let $h_1,h_2:\mathcal{F}\to \mathbb{R}$ be defined as follows:

$$h_1(u):=\left\{\begin{array}{cc}2u^2+u,\hfill & u\ge 2,\hfill \\ u,\hfill & u<2,\hfill \end{array}\right.\mathrm{and}{h}_2(u):=\left\{\begin{array}{cc}2u^2,\hfill & u\ge 2,\hfill \\ 2u,\hfill & u<2.\hfill \end{array}\right.$$

The Clarke subdifferentials of $h_1$ and $h_2$ are given by:

$${\partial }_c{h}_1(u)=\left\{\begin{array}{cc}4u^3+u^2,\hfill & u>2,\hfill \\ ,\hfill & u=2,\hfill \\ u^2,\hfill & u<2,\hfill \end{array}\right.\mathrm{and}{\partial }_c{h}_2(u)=\left\{\begin{array}{cc}4u^3,\hfill & u>2,\hfill \\ ,\hfill & u=2,\hfill \\ 2u^2,\hfill & u<2.\hfill \end{array}\right.$$

Let the functions ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2:\mathcal{F}\times \mathcal{F}\to \mathbb{R}$ be defined as follows:

$${\mathsf{\Psi }}_1(u,w):=2lnuw\mathrm{and}{\mathsf{\Psi }}_2(u,w):=2u+lnw.$$

Notably, the functions ${\mathsf{\Psi }}_i(u,·)(i=1,2)$ are geodesic convex functions on $\mathcal{F}$.

The functions ${\mathcal{P}}_1,{\mathcal{P}}_2:{\mathbb{H}}^1\to \mathbb{R}$ are defined as follows:

$${\mathcal{P}}_1(u):=|u-2|\mathrm{and}{\mathcal{P}}_2(u):=u+2.$$

The generalized directional derivative of functions ${\mathcal{P}}_1,{\mathcal{P}}_2$ are given by:

$${\mathcal{P}}_1^{\circ }(u;\nu )=\left\{\begin{array}{cc}\nu ,\hfill & u>2,\hfill \\ \left|\nu \right|,\hfill & u=2,\hfill \\ -\nu ,\hfill & u<2,\hfill \end{array}\right.\mathrm{and}{\mathcal{P}}_2^{\circ }(u;\nu )=\nu ,\mathrm{for}\mathrm{all}u\in \mathcal{F},\nu \in T_u{\mathbb{H}}^1.$$

Let $\gamma =5$ and $\sigma =-1.$ Now, we define a function ${\vartheta }^{\gamma }:\mathcal{F}\to \mathbb{R}$ as follows:

$$\begin{array}{cc}\hfill {\vartheta }^{\gamma }(\tilde{u})& :=\underset{w\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{\tilde{u},w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(u;{exp}_{\tilde{u}}^{-1}(w))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\right\}\hfill \end{array}$$

Let $\tilde{u}\in \mathcal{F}$ such that $\tilde{u}\in (2,e^2]$, then the function ${\vartheta }^5$ is given by:

$$\begin{array}{cc}\hfill {\vartheta }^5(\tilde{u})& =\underset{w\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\underset{w\in \mathcal{F}}{sup}min\left\{-{〈{\xi }_1^{\ast }+1,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right.\hfill \\ & -{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2,\hfill \\ & -{〈{\xi }_2^{\ast }+1,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{\mathsf{\Psi }}_2(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_2(\tilde{u},w)-{\mathcal{P}}_2^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\hfill \\ & \left.-{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\underset{w\in \mathcal{F}}{sup}min\left\{-〈4{\tilde{u}}^3+{\tilde{u}}^2+1,\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)〉+ln{\tilde{u}}^2-ln\tilde{u}w-\tilde{u}ln\left({\displaystyle \frac{w}{u}}\right)-{\displaystyle \frac{1}{10}}{\left|ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)\right|}^2,\right.\hfill \\ & \left.-〈4{\tilde{u}}^3+1,\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)〉+2(ln\tilde{u}-lnw)-\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)-{\displaystyle \frac{1}{10}}{\left|ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)\right|}^2\right\}\hfill \\ \hfill & =min\left\{(4{\tilde{u}}^4+{\tilde{u}}^3+2\tilde{u}+2)(ln\tilde{u}+2)-{\displaystyle \frac{1}{10}}{(ln\tilde{u}+2)}^2,\right.\hfill \\ \hfill & \left.(4{\tilde{u}}^4+2\tilde{u}+1)(ln\tilde{u}+2)-{\displaystyle \frac{1}{10}}{(ln\tilde{u}+2)}^2\right\}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Now, we have

$$\begin{array}{cc}\hfill {\vartheta }^5(2)& =\underset{w\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{2,w}{\xi }_i^{\ast }-\sigma ,{exp}_w^{-1}2\rangle }_w+{\mathsf{\Psi }}_i(2,2)-{\mathsf{\Psi }}_i(2,w)-{\mathcal{P}}_i^{\circ }(2;{exp}_2^{-1}(w))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\right\}\hfill \\ \hfill & =\underset{w\in \mathcal{F}}{sup}min\left\{-〈{\xi }_1^{\ast }+1,2ln{\displaystyle \frac{w}{2}}〉+2ln{\displaystyle \frac{2}{w}}-2\left|ln{\displaystyle \frac{2}{w}}\right|-{\displaystyle \frac{1}{10}}{\left|ln{\displaystyle \frac{2}{w}}\right|}^2,\right.\hfill \\ \hfill & -\left.〈{\xi }_2^{\ast }+1,2ln{\displaystyle \frac{w}{2}}〉+ln{\displaystyle \frac{2}{w}}+2ln{\displaystyle \frac{2}{w}}-{\displaystyle \frac{1}{10}}{\left|ln{\displaystyle \frac{2}{w}}\right|}^2\right\}\hfill \\ \hfill & =min\left\{\left(2{\xi }_1^{\ast }+2-{\displaystyle \frac{1}{10}}(2ln2)\right)(2ln2),\right.\hfill \\ & \left.\left(2{\xi }_2^{\ast }+2+6-{\displaystyle \frac{1}{10}}(2ln2)\right)2ln2\right\}({\xi }_1^{\ast }\in [4,36],{\xi }_2^{\ast }\in [8,32])\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Moreover, for any $\tilde{u}\in \mathcal{F}$ such that $\tilde{u}\in [e^{-2},2)$, the function ${\vartheta }^5$ is given by:

$$\begin{array}{cc}\hfill {\vartheta }^5(\tilde{u})& =\underset{w\in \mathcal{F}}{sup}\underset{1\le i\le 2}{min}\left\{{\langle L_{\tilde{u},w}{\xi }_i^{\ast }-L_{2,w}\sigma ,{exp}_w^{-1}(\tilde{u})\rangle }_w+{\mathsf{\Psi }}_i(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_i(\tilde{u},w)-{\mathcal{P}}_i^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\right\}\hfill \end{array}$$

$$\begin{array}{cc}& =\underset{w\in \mathcal{F}}{sup}min\left\{-{〈{\xi }_1^{\ast }+1,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right.\hfill \\ & -{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2,\hfill \\ & -{〈{\xi }_2^{\ast }+1,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{\mathsf{\Psi }}_2(\tilde{u},\tilde{u})-{\mathsf{\Psi }}_2(\tilde{u},w)-{\mathcal{P}}_2^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\hfill \\ & \left.-{\displaystyle \frac{1}{10}}{\parallel {exp}_w^{-1}(\tilde{u})\parallel }_w^2\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\underset{w\in \mathcal{F}}{sup}min\left\{-〈{\tilde{u}}^2,\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)〉+2ln{\tilde{u}}^2-2ln\tilde{u}w+\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)-{\displaystyle \frac{1}{2}}{\left|ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)\right|}^2,\right.\hfill \\ & -\left.〈{\tilde{u}}^2,\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)〉+2(ln\tilde{u}-lnw)-\tilde{u}ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)-{\displaystyle \frac{1}{2}}{\left|ln\left({\displaystyle \frac{w}{\tilde{u}}}\right)\right|}^2\right\}\hfill \\ \hfill & =min\left\{({\tilde{u}}^3+2)(ln\tilde{u}+2)-{\displaystyle \frac{1}{10}}{(ln\tilde{u}+2)}^2,(2{\tilde{u}}^3+2\tilde{u}+1)(ln\tilde{u}+2)-{\displaystyle \frac{1}{10}}{(ln\tilde{u}+2)}^2\right\}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Further, it can be verified that ${\vartheta }^{\gamma }(\tilde{u})=0$ if and only if $\tilde{u}=e^{-2}$ is a solution of the following nonsmooth weak vector variational-hemivariational inequality problem (WVVHVIP2): Find $u\in \mathcal{F}$ such that there exist ${\xi }_i\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I}=\{1,2\})$, satisfying:

$$\begin{array}{cc}\hfill & \left({〈{\xi }_1-\sigma ,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(u;{exp}_{\tilde{u}}^{-1}(w)),\right.\hfill \\ & \left.{〈{\xi }_2-\sigma ,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{\mathsf{\Psi }}_2(\tilde{u},w)-{\mathsf{\Psi }}_2(\tilde{u},\tilde{u})+{\mathcal{P}}_2^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right)\notin -int{\mathbb{R}}_{+}^2,\mathrm{for}\mathrm{all}w\in \mathcal{F}.\hfill \end{array}$$

In the following lemma, we prove the lower semicontinuity of the gap function ${\vartheta }^{\gamma }$ defined in (24).

Lemma 7. 

Let for every $i\in \mathcal{I},$ the function ${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}$ be a geodesic convex function on $\mathcal{F}$ for every $u\in \mathcal{F}$. Then, ${\vartheta }^{\gamma }$ is a lower semicontinuous function.

Proof. 

Consider a function ${\widehat{\vartheta }}^{\gamma }:\mathcal{F}\times \mathcal{F}\to \mathbb{R}$, which is defined as follows:

$$\begin{array}{cc}\hfill {\widehat{\vartheta }}^{\gamma }(u,w):=\underset{i\in \mathcal{I}}{min}\{{\langle L_{u,w}{\xi }_i^{\ast }-L_{u,w}\sigma ,{exp}_w^{-1}(u)\rangle }_w+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,w)& -{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))\hfill \\ & -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_w^{-1}(u)\parallel }_w^2\},\hfill \end{array}$$

(30)

where ${\xi }_i^{\ast }\in {\partial }_c{h}_i(u)(i\in \mathcal{I}).$

Since ${\mathsf{\Psi }}_i(i\in \mathcal{I})$ and $-{\mathcal{P}}_i^{\circ }(i\in \mathcal{I})$ are continuous and lower semicontinuous functions on $\mathcal{F}$, respectively, therefore, the function $\widehat{\vartheta }(·,w)$ defined in (30) is a lower semicontinuous function on $\mathcal{F}$ for every $w\in \mathcal{F}.$

Notably,

$${\vartheta }^{\gamma }(u)=\underset{w\in \mathcal{F}}{sup}\widehat{\vartheta }(u,w).$$

(31)

Let ${\{u_j\}}_{j=1}^{\infty }\subseteq \mathcal{F}$ be a sequence such that $u_j\to u$ as $j\to \infty .$ Therefore, from (31), it follows that

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim\; inf}{\vartheta }^{\gamma }(u_j)& =\underset{j\to \infty }{lim\; inf}\underset{w\in \mathcal{F}}{sup}{\widehat{\vartheta }}^{\gamma }(u_j,w)\hfill \\ \hfill & \ge \underset{j\to \infty }{lim\; inf}{\widehat{\vartheta }}^{\gamma }(u_j,\tilde{u}),\tilde{u}\in \mathcal{F},\hfill \\ \hfill & \ge {\widehat{\vartheta }}^{\gamma }(u,\tilde{u}).\hfill \end{array}$$

(32)

On taking supremum in (32), we have

$$\underset{j\to \infty }{lim\; inf}{\vartheta }^{\gamma }(u_j)\ge \underset{\tilde{u}\in \mathcal{F}}{sup}{\widehat{\vartheta }}^{\gamma }(u,\tilde{u})={\vartheta }^{\gamma }(u),$$

which claims that the function ${\vartheta }^{\gamma }$ is a lower semicontinuous function on $\mathcal{F}$. □

Now, we define the Moreau-Yosida regularization of ${\vartheta }^{\gamma }$ in the setting of Hadamard manifolds.

Let $\gamma ,\zeta >0$ be two fixed parameters and ${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }:\mathcal{F}\to \mathbb{R}$ be Moreau-Yosida regularization of ${\vartheta }^{\gamma }$, which is defined as follows:

$${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(u):=\underset{w\in \mathcal{F}}{inf}\left\{{\vartheta }^{\gamma }(w)+\zeta {\parallel {exp}_w^{-1}(u)\parallel }_w^2\right\}.$$

(33)

In the following theorem, we prove that the function ${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }$ defined in (33) is a gap function for WVVHVIP.

Theorem 5. 

Let for every $i\in \mathcal{I}$ and $u\in \mathcal{F}$, ${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}$ be a geodesic convex function on $\mathcal{F}$. Then, for $\gamma ,\zeta >0,$${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }$ is a gap function for WVVHVIP.

Proof. 

From Theorem 4, ${\vartheta }^{\gamma }(u)\ge 0,\mathrm{for}\mathrm{all}u\in \mathcal{F}.$ Therefore, from (33), we have

$${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(u)\ge 0,\mathrm{for}\mathrm{all}u\in \mathcal{F}.$$

(34)

Let $\tilde{u}$ be an arbitrary solution of WVVHVIP. This implies that ${\vartheta }^{\gamma }(\tilde{u})=0$. Therefore,

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(\tilde{u})& =\underset{w\in \mathcal{F}}{inf}\{{\vartheta }^{\gamma }(\tilde{w})+\zeta \parallel {exp}_w^{-1}(\tilde{u}){\parallel }_w^2\}\hfill \\ \hfill & \le {\vartheta }^{\gamma }(\tilde{u})+\zeta {\parallel {exp}_{\tilde{u}}^{-1}\tilde{u}\parallel }_{\tilde{u}}^2=0.\hfill \end{array}$$

(35)

From (34) and (35), it follows that

$${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(\tilde{u})=0.$$

Conversely, let $\tilde{u}\in \mathcal{F}$ be an element such that ${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(\tilde{u})=0.$ That is,

$$\underset{w\in \mathcal{F}}{inf}\{{\vartheta }^{\gamma }(w)+\zeta \parallel {exp}_w^{-1}(\tilde{u}){\parallel }^2\}=0.$$

Hence, it follows that there exists a minimizing sequence ${\{w_j\}}_{j=1}^{\infty }\subseteq \mathcal{F}$ such that

$$0\le {\vartheta }^{\gamma }(w_j)+\zeta \parallel {exp}_{w_j}^{-1}\widetilde{(u)}\parallel <{\displaystyle \frac{1}{j}}.$$

(36)

Moreover, ${\vartheta }^{\gamma }$ is a lower semicontinuous function on $\mathcal{F}$ such that ${\vartheta }^{\gamma }(u)\ge 0,\mathrm{for}\mathrm{all}u\in \mathcal{F}$. Therefore, as $j\to \infty ,$${\vartheta }^{\gamma }(w_j)\to 0$ and $\parallel {exp}_{w_j}^{-1}(\tilde{u})\parallel \to 0,$ which implies that $w_j\to \tilde{u}.$ From (36), it follows that

$$0\le {\vartheta }^{\gamma }(\tilde{u})\le \underset{j\to \infty }{lim\; inf}{\vartheta }^{\gamma }(w_j)=0.$$

(37)

Therefore, ${\vartheta }^{\gamma }(\tilde{u})=0,$ which implies that $\tilde{u}\in \mathcal{F}$ is a solution of WVVHVIP from Theorem 4. This completes the proof. □

Remark 9. 

Theorem 5 generalizes Theorem 2.4 deduced by Yamashita and Fukushima [30] from the Euclidean space setting to the framework of Hadamard manifolds, as well as from variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problem, namely, WVVHVIP.

5. Global Error Bounds for WVVHVIP

In this section, by employing the regularized gap functions we derive the global error bounds for the solution of WVVHVIP under generalized geodesic monotonicity assumptions and properties of Clarke subdifferential.

Theorem 6. 

Let for every $i\in \mathcal{I}$ and $u\in \mathcal{F}$, ${\mathsf{\Psi }}_i(u,·):\mathcal{F}\to \mathbb{R}$ be a geodesic convex function on $\mathcal{F}$. Further, we assume that the following assumptions hold:

(i)

The set-valued map $\bigcup _{i\in \mathcal{I}}{\partial }_c{h}_i$ is geodesic strongly monotone with respect to σ having a positive constant $m_h^{\sigma }.$

(ii)

There exists a constant $m_{\mathsf{\Psi }}>0$ such that

$${\mathsf{\Psi }}_i(u_1,w_2)-{\mathsf{\Psi }}_i(u_1,w_1)+{\mathsf{\Psi }}_p(u_2,w_1)-{\mathsf{\Psi }}_p(u_2,w_2)\le m_{\mathsf{\Psi }}\parallel {exp}_{u_2}^{-1}(u_1){\parallel }_{u_2}{\parallel {exp}_{w_2}^{-1}(w_1)\parallel }_{w_2},$$

for every $u_1,u_2,w_1,w_2\in \mathcal{F}$ and $i,p\in \mathcal{I}.$

(iii)

There exists a constant $m_{\mathcal{P}}>0,$ such that

$${\mathcal{P}}_i^{\circ }(w_1;{exp}_{w_1}^{-1}(w_2))+{\mathcal{P}}_p^{\circ }(w_2;{exp}_{w_2}^{-1}(w_1))\le m_{\mathcal{P}}{\parallel {exp}_{w_2}^{-1}(w_1)\parallel }_{w_2}^2,$$

for all $w_1,w_2\in \mathcal{F}$ and $i,p\in \mathcal{I}.$

Further, assume that $\tilde{u}\in \mathcal{F}$ is the unique solution of WVVHVIP and $\gamma >0$ is a fixed parameter, such that $m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}>{\displaystyle \frac{1}{2\gamma }}.$ Then, the following condition holds:

$$\parallel {exp}_{\tilde{u}}^{-1}{(w)\parallel }_{\tilde{u}}\le \sqrt{{\displaystyle \frac{{\vartheta }^{\gamma }(w)}{\left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma }\right)}}},\mathrm{for}\mathrm{all}w\in \mathcal{F}.$$

Proof. 

From the given hypotheses and Theorem 2, there exists a unique solution, namely $\tilde{u}$ of WVVHVIP in $\mathcal{F}$. This implies that for every $w\in \mathcal{F},$ there exist ${\xi }_i^{\ast }\in {\partial }_c{h}_i(\tilde{u})(i\in \mathcal{I}),$ satisfying:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathsf{\Psi }}_1(\tilde{u},\tilde{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w)),\dots ,\right.\hfill \\ & \left.{\langle {\xi }_l^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_l(\tilde{u},w)-{\mathsf{\Psi }}_l(\tilde{u},\tilde{u})+{\mathcal{P}}_l^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right)\notin -int{\mathbb{R}}_{+}^l.\hfill \end{array}$$

(38)

Equivalently, for every $w\in \mathcal{F}$ there exists some $p\in \mathcal{I}$ such that

$${\langle {\xi }_p^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_p(\tilde{u},w)-{\mathsf{\Psi }}_p(\tilde{u},\tilde{u})+{\mathcal{P}}_p^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\ge 0.$$

(39)

Let ${\zeta }_i\in {\partial }_c{h}_i(w).$ Then, from the definition of ${\vartheta }^{\gamma }(w)$ in (17), we have

$$\begin{array}{c}\hfill {\vartheta }^{\gamma }(w):=\underset{u\in \mathcal{F}}{sup}\underset{i\in \mathcal{I}}{min}\{{\langle L_{w,u}{\zeta }_i-L_{w,u}\sigma ,{exp}_u^{-1}(w)\rangle }_u+{\mathsf{\Psi }}_i(w,w)-{\mathsf{\Psi }}_i(w,u)-{\mathcal{P}}_i^{\circ }(w;{exp}_w^{-1}(u))\\ \hfill -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_u^{-1}w\parallel }_u^2\},\\ \hfill \ge \underset{i\in \mathcal{I}}{min}\{{\langle L_{w,\tilde{u}}{\zeta }_i-L_{w,\tilde{u}}\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_i(w,w)-{\mathsf{\Psi }}_i(w,\tilde{u})-{\mathcal{P}}_i^{\circ }(w;{exp}_w^{-1}(\tilde{u}))\\ \hfill -{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\}.\end{array}$$

(40)

From the given hypotheses (i)-(iii) and the fact that $\tilde{u}$ is a solution of WVVHVIP, it follows that for every $w\in \mathcal{F}$ there exists $k\in \mathcal{I}$ such that the following inequalities hold:

$$\begin{array}{cc}\hfill \langle L_{w,\tilde{u}}{\zeta }_i-& L_{w,\tilde{u}}\sigma ,{exp}_{\tilde{u}}^{-1}{(w)\rangle }_w+{\mathsf{\Psi }}_i(w,w)-{\mathsf{\Psi }}_i(w,\tilde{u})-{\mathcal{P}}_i^{\circ }(w;{exp}_w^{-1}(\tilde{u}))-{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\hfill \\ \hfill & ={〈L_{w,\tilde{u}}{\zeta }_i-L_{w,\tilde{u}}\sigma ,{exp}_{\tilde{u}}^{-1}(w)〉}_w-{〈{\xi }_k^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}+{〈{\xi }_k^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)〉}_{\tilde{u}}\hfill \\ & +{\mathsf{\Psi }}_i(w,w)-{\mathsf{\Psi }}_i(w,\tilde{u})-{\mathcal{P}}_i^{\circ }(w;{exp}_w^{-1}(\tilde{u}))-{\displaystyle \frac{1}{2\gamma }}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\hfill \\ \hfill & \ge 〈{\xi }_k^{\ast }-\sigma ,{exp}_{\tilde{u}}^{-1}(w)〉+{\mathsf{\Psi }}_k(\tilde{u},w)-{\mathsf{\Psi }}_k(\tilde{u},\tilde{u})+{\mathcal{P}}_k^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\hfill \\ & +\left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }}\right){\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2\hfill \\ \hfill & \ge \left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }}\right){\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2.\hfill \end{array}$$

(41)

From (39), (40) and (41), the following inequality holds:

$${\vartheta }^{\gamma }(w)\ge \left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }}\right){\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2,\mathrm{for}\mathrm{all}w\in \mathcal{F}.$$

Hence, we complete the proof. □

Remark 10. 

Theorem 6 generalizes Lemma 4.1 derived by Yamashita and Fukushima [30] from Euclidean space to an even more general space, namely Hadamard manifolds and from variational inequality problem to a more general problem, in particular, WVVHVIP.

In the following example, we illustrate the significance of Theorem 6.

Example 5. 

Consider the following Poincaré half-plane:

$${\mathbb{H}}^2:=\{u=(u_1,u_2)\in {\mathbb{R}}^2|u_2>0\}.$$

Furthermore, ${\mathbb{H}}^2$ is a Hadamard manifold of dimension 2 and has negative constant sectional curvature $-1$ (see, [62]). The tangent space at any point $u\in {\mathbb{H}}^2$ is $T_u{\mathbb{H}}^2={\mathbb{R}}^2$.

The Riemannian metric induced an inner product on $T_u{\mathbb{H}}^2$, for all $u\in {\mathbb{H}}^2$. That is,

$${\langle \nu ,\mu \rangle }_u=\langle \mathcal{G}(u)\nu ,\mu \rangle ,\mathrm{for}\mathrm{all}\nu ,\mu \in T_u{\mathbb{H}}^2={\mathbb{R}}^2,$$

where $\mathcal{G}(u)=\left[\begin{array}{cc}{\displaystyle \frac{1}{{(u_2)}^2}}& 0\\ 0& {\displaystyle \frac{1}{{(u_2)}^2}}\end{array}\right]$ and $\langle \mathcal{G}(u)\nu ,\mu \rangle$ represents the standard inner product.

The exponential map ${exp}_u:T_u{\mathbb{H}}^2\to {\mathbb{H}}^2$ is:

If $\nu =0,$

$$\begin{array}{c}\hfill {exp}_u(\nu )=\left(u_1,u_2{e}^{{\displaystyle \frac{{\nu }_2}{u_2}}}\right).\end{array}$$

If ${\nu }_1\ne 0,$

$$\begin{array}{c}\hfill {exp}_u(\nu )=\left(u_1+{\displaystyle \frac{{\nu }_2}{{\nu }_1}}+\sqrt{1+{\left({\displaystyle \frac{{\nu }_2}{{\nu }_1}}\right)}^2}tanh\left(p_{{\nu }_1,{\nu }_2}(1)\right),u_2\sqrt{1+{\left({\displaystyle \frac{{\nu }_2}{{\nu }_1}}\right)}^2}{\displaystyle \frac{1}{cosh\left(q_{\left.{\nu }_1,{\nu }_2(1)\right)}\right)}}\right),\end{array}$$

where

$$\begin{array}{c}{p}_{{\nu }_1,{\nu }_2}(t)=\left\{\begin{array}{cc}t\sqrt{{\nu }_1^2+{\nu }_2^2}-arcsinh{\displaystyle \frac{{\nu }_2}{{\nu }_1}},\hfill & \mathrm{if}{\nu }_1>0,\hfill \\ -t\sqrt{{\nu }_1^2+{\nu }_2^2}-arcsinh{\displaystyle \frac{{\nu }_2}{{\nu }_1}},\hfill & \mathrm{if}{\nu }_1<0,\hfill \end{array}\right.\\ q_{{\nu }_1,{\nu }_2}(t)=\left\{\begin{array}{cc}t{\displaystyle \frac{\sqrt{{\nu }_1^2+{\nu }_2^2}}{u_2}}-arcsinh{\displaystyle \frac{{\nu }_2}{{\nu }_1}},\hfill & \mathrm{if}{\nu }_1>0,\hfill \\ -t{\displaystyle \frac{\sqrt{{\nu }_1^2+{\nu }_2^2}}{u_2}}-arcsinh{\displaystyle \frac{{\nu }_2}{{\nu }_1}},\hfill & \mathrm{if}{\nu }_1<0.\hfill \end{array}\right.\end{array}$$

The inverse of exponential map ${exp}_u^{-1}:{\mathbb{H}}^2\to T_u{\mathbb{H}}^2$ is given by:

$${exp}_u^{-1}(w)=\left\{\begin{array}{cc}\left(0,u_{2}ln{\displaystyle \frac{w_2}{u_2}}\right),\hfill & \mathrm{if}{w}_1=u_1,\hfill \\ {\displaystyle \frac{u_2}{a}}\left(arctanh{\displaystyle \frac{b-u_1}{a}}-arctanh{\displaystyle \frac{b-w_1}{a}}\right)\left(u_2,b-u_1\right),\hfill & \mathrm{if}{u}_1\ne w_1,\hfill \end{array}\right.$$

where

$$b={\displaystyle \frac{{(u_1)}^2+{(u_2)}^2-\left({(w_1)}^2+{(w_2)}^2\right)}{2\left(u_1-w_1\right)}},a=\sqrt{{\left(u_1-b\right)}^2+{(u_2)}^2}.$$

Let $\mathcal{F}:=\left\{u=(u_1,u_2)\in {\mathbb{H}}^2|u_1=0,-{\displaystyle \frac{1}{8}}\le ln{u}_2\le {\displaystyle \frac{1}{8}}\right\}$ be a nonempty, compact and geodesic convex subset of ${\mathbb{H}}^2$. Further, let $h_1,h_2:\mathcal{F}\to \mathbb{R}$ be defined as follows:

$$h_1(u):=\left|u_1\right|+4{(ln{u}_2)}^2+1,h_2(u):=4{(ln{u}_2)}^2+e.$$

The Clarke subdifferentials of $h_1$ and $h_2$ at $\widehat{u}=\left(0,e^{-{\displaystyle \frac{1}{8}}}\right)$ are given as follows:

$${\partial }_c{h}_1(\widehat{u})=\left\{\left(0,-{\displaystyle \frac{1}{8}}{e}^{-{\displaystyle \frac{1}{4}}}\right)\right\},{\partial }_c{h}_2(\widehat{u})=\left\{\left(0,-{\displaystyle \frac{1}{8}}{e}^{-{\displaystyle \frac{1}{4}}}\right)\right\}.$$

Let us define ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2:\mathcal{F}\times \mathcal{F}\to \mathbb{R}$ as follows:

$${\mathsf{\Psi }}_1(u,w):=2ln{u}_{2}ln{w}_2,{\mathsf{\Psi }}_2(u,w):=ln{u}_{2}ln{w}_2^2+{\displaystyle \frac{1}{e}}.$$

Let ${\mathcal{P}}_1,{\mathcal{P}}_2:{\mathbb{H}}^2\to \mathbb{R}$ be two real-valued functions defined as follows:

$${\mathcal{P}}_1(u):={(ln{u}_2)}^2,{\mathcal{P}}_2(u):=\left\{\begin{array}{cc}{(ln{u}_2)}^2,\hfill & u_1=0,\hfill \\ |u_2-1|,\hfill & u_1\ne 0.\hfill \end{array}\right.$$

Then, the generalized directional derivatives of ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ at $u\in \mathcal{F}$ in the direction ${exp}_u^{-1}(w)$ for any $w\in \mathcal{F}$ are given as:

$${\mathcal{P}}_1^{\circ }(u;{exp}_u^{-1}(w))=ln{u}_{2}ln\left({\displaystyle \frac{w_2}{u_2}}\right),{\mathcal{P}}_2^{\circ }(u;{exp}_u^{-1}(w))=ln{u}_{2}ln\left({\displaystyle \frac{w_2}{u_2}}\right).$$

Now, we consider the following nonsmooth weak vector variational-hemivariational inequality problem for $\sigma =(1,-5)$:

(WVVHVIP3): Find $\tilde{u}\in \mathcal{F}$, such that there exists ${\xi }_i\in {\partial }_c{h}_i(\tilde{u})$ for every $i\in \mathcal{I}=\{1,2\},$ satisfying:

$$\begin{array}{cc}\hfill & \left({\langle {\xi }_1-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_1(\tilde{u},w)-{\mathsf{\Psi }}_1(\tilde{u},\widehat{u})+{\mathcal{P}}_1^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w)),\right.\hfill \\ & \left.{\langle {\xi }_2-\sigma ,{exp}_{\tilde{u}}^{-1}(w)\rangle }_{\tilde{u}}+{\mathsf{\Psi }}_2(\tilde{u},w)-{\mathsf{\Psi }}_2(\tilde{u},\tilde{u})+{\mathcal{P}}_2^{\circ }(\tilde{u};{exp}_{\tilde{u}}^{-1}(w))\right)\notin -int{\mathbb{R}}_{+}^2,\hfill \end{array}$$

for every $w\in \mathcal{F}.$ It can be verified that $\tilde{u}=(0,e^{-{\displaystyle \frac{1}{8}}})$ is a unique solution of WVVHVIP3.

Notably, $\bigcup _{i\in \{1,2\}}{\partial }_c{h}_i$ is geodesic strongly monotone with respect to σ with a positive constant $m_h^{\sigma }=3.$ Moreover, all the hypotheses (ii)-(iii) of Theorem 6 are satisfied with $m_{\mathsf{\Psi }}=2$, and $m_{\mathcal{P}}={\displaystyle \frac{1}{7}}.$

Now, we define a function ${\vartheta }^{\gamma }$ corresponding to $\gamma =5,$ as follows:

$$\begin{array}{cc}\hfill {\vartheta }^5(u)& =\underset{w\in \mathcal{F}}{sup}\underset{i\in \{1,2\}}{min}\left\{{〈L_{u,w}{\xi }_i-L_{u,w}\sigma ,{exp}_w^{-1}(u)〉}_w+{\mathsf{\Psi }}_i(u,u)-{\mathsf{\Psi }}_i(u,w)-{\mathcal{P}}_i^{\circ }(u;{exp}_u^{-1}(w))\right.\hfill \\ & \left.-{\displaystyle \frac{1}{10}}\left|\right|{exp}_w^{-1}u{\left|\right|}_w^2\right\}\hfill \\ \hfill & =\underset{w\in \mathcal{F}}{sup}min\left\{-〈{\xi }_1-\sigma ,\left(0,u_{2}ln\left({\displaystyle \frac{w_2}{u_2}}\right)\right)〉+2ln{u}_{2}ln{u}_2-2ln{u}_{2}ln{w}_2\right.\hfill \\ & -ln{u}_{2}ln\left({\displaystyle \frac{w_2}{u_2}}\right)-{\displaystyle \frac{1}{10}}{\left(ln\left({\displaystyle \frac{w_2}{u_2}}\right)\right)}^2,\hfill \\ & -〈{\xi }_2-\sigma ,\left(0,u_{2}ln\left({\displaystyle \frac{w_2}{u_2}}\right)\right)〉+{\displaystyle \frac{1}{e}}+ln{u}_{2}ln{u}_2^2-ln{u}_{2}ln{w}_2^2-{\displaystyle \frac{1}{e}}\hfill \\ & \left.-ln{u}_{2}ln\left({\displaystyle \frac{w_2}{u_2}}\right)-{\displaystyle \frac{1}{10}}{\left(ln\left({\displaystyle \frac{w_2}{u_2}}\right)\right)}^2\right\}\hfill \\ \hfill =& min\left\{\left(\left({\displaystyle \frac{5+8u_{2}ln{u}_2}{u_2}}\right)+3ln{u}_2-{\displaystyle \frac{1}{10}}\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)\right)\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right),\right.\hfill \\ & \left.\left(\left({\displaystyle \frac{5+8u_{2}ln{u}_2}{u_2}}\right)+3ln{u}_2-{\displaystyle \frac{1}{10}}\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)\right)\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)\right\}\hfill \\ \hfill =& \left(\left({\displaystyle \frac{5+8u_{2}ln{u}_2}{u_2}}\right)+3ln{u}_2-{\displaystyle \frac{1}{10}}\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)\right)\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)\hfill \\ \hfill \ge & 0,\mathrm{for}\mathrm{all}u=(u_1,u_2)\in \mathcal{F}.\hfill \end{array}$$

Moreover, ${\vartheta }^{\gamma }(\tilde{u})=0$ if and only if $\tilde{u}=(0,e^{-{\displaystyle \frac{1}{8}}}).$ Therefore, ${\vartheta }^{\gamma }$ for $\gamma =5$ is a regularized gap function for WVVHVIP3. Furthermore, $m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }}>0$ and $\tilde{u}=(0,e^{-{\displaystyle \frac{1}{8}}})$ is the unique solution of WVVHVIP3. Therefore, all the hypotheses of Theorem 6 hold, leading to the following inequality:

$$\left|\right|{exp}_{\tilde{u}}^{-1}(w){\left|\right|}_{\tilde{u}}\le \sqrt{{\displaystyle \frac{{\vartheta }^{\gamma }(w)}{\left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma }\right)}}},\mathrm{for}\mathrm{all}w\in \mathcal{F}.$$

Equivalently,

$$\begin{array}{cc}& \sqrt{{\displaystyle \frac{{\vartheta }^{\gamma }(w)}{\left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma }\right)}}}\hfill \\ & =\sqrt{0.76\left(\left({\displaystyle \frac{5+8w_{2}ln{w}_2}{w_2}}\right)+3ln{w}_2-{\displaystyle \frac{1}{10}}\left(ln{w}_2+{\displaystyle \frac{1}{8}}\right)\right)\left(ln{w}_2+{\displaystyle \frac{1}{8}}\right)}\hfill \\ & \ge \left(ln{w}_2+{\displaystyle \frac{1}{8}}\right),\mathrm{for}\mathrm{all}w=(w_1,w_2)\in \mathcal{F}.\hfill \end{array}$$

Theorem 7. 

Let $\gamma >0$ and $\zeta >0$ be fixed parameters, such that $m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}>{\displaystyle \frac{1}{2\gamma }}$. Furthermore, we assume that all the hypotheses of Theorem 6 are satisfied. Then, for each $w\in \mathcal{F}$, the following inequality holds:

$$\sqrt{{\displaystyle \frac{{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{\zeta }}}\le {\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}\le \sqrt{{\displaystyle \frac{2{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{min\{m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma },\zeta \}}}}.$$

Proof. 

From the given hypotheses and Theorem 2, there exists a unique solution of WVVHVIP in $\mathcal{F}$, namely $\tilde{u}$.

In view of (33), we have

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)& =\underset{\tilde{w}\in \mathcal{F}}{inf}\left\{{\vartheta }^{\gamma }(\tilde{w})+\zeta {\parallel {exp}_{\tilde{w}}^{-1}(w)\parallel }_{\tilde{w}}^2\right\}\hfill \\ & \le {\vartheta }^{\gamma }(\tilde{u})+\zeta \left|\right|{exp}_{\tilde{u}}^{-1}(w){\left|\right|}_{\tilde{u}}\hfill \\ & =\zeta \left|\right|{exp}_{\tilde{u}}^{-1}(w){\left|\right|}_{\tilde{u}}^2.\hfill \end{array}$$

This implies that

$$\left|\right|{exp}_{\tilde{u}}^{-1}(w){\left|\right|}_{\tilde{u}}\ge \sqrt{{\displaystyle \frac{{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{\zeta }}},\mathrm{for}\mathrm{all}w\in \mathcal{F}.$$

(42)

Let w be an arbitrary element of $\mathcal{F}.$ Then, we obtain the following inequalities:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)& =\underset{\tilde{w}\in \mathcal{F}}{inf}\left\{{\vartheta }^{\gamma }(\tilde{w})+\zeta {\parallel {exp}_{\tilde{w}}^{-1}(w)\parallel }_{\tilde{w}}^2\right\}\hfill \\ \hfill & \ge \underset{\tilde{w}\in \mathcal{F}}{inf}\left\{\left(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }}\right)\parallel {exp}_{\tilde{w}}^{-1}\tilde{u}{\parallel }_{\tilde{w}}^2+\zeta {\parallel {exp}_{\tilde{w}}^{-1}(w)\parallel }_{\tilde{w}}^2\right\}\hfill \\ \hfill & \ge min\left\{m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }},\zeta \right\}\underset{\tilde{w}\in \mathcal{F}}{inf}\{\parallel {exp}_{\tilde{w}}^{-1}\tilde{u}{\parallel }_{\tilde{w}}^2+\parallel {exp}_{\tilde{w}}^{-1}(w){\parallel }_{\tilde{w}}^2\}\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}min\left\{m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }},\zeta \right\}{\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}^2.\hfill \end{array}$$

(43)

From (42) and (43), the following conclusion holds:

$$\sqrt{{\displaystyle \frac{{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{\zeta }}}\le {\parallel {exp}_{\tilde{u}}^{-1}(w)\parallel }_{\tilde{u}}\le \sqrt{{\displaystyle \frac{2{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{min\{m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma },\zeta \}}}},\mathrm{for}\mathrm{all}w\in \mathcal{F}.$$

Hence, we complete the proof. □

Remark 11. 

If ${\mathbb{H}}^n={\mathbb{R}}^n,$ then ${exp}_u^{-1}(w)=w-u$, for all $u,w\in {\mathbb{H}}^n.$ Further, if $\mathcal{I}=\{1\},$${\mathcal{P}}_1^{\circ }(u,·)=0,{\partial }_c{h}_1(u)=F(u),$ for all $u\in {\mathbb{H}}^n$, where $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a vector-valued function, and ${\mathsf{\Psi }}_1(u,w)=0,$ for all $u,w\in \mathcal{F},$ then Theorem 7 reduces to Theorem 4.3 deduced by Yamashita and Fukushima [30].

Now, we provide a non-trivial example to demonstrate the significance of Theorem 7.

Example 6. 

Consider ${\mathbb{H}}^2,\mathcal{F},h_i,{\mathsf{\Psi }}_i,{\mathcal{P}}_i(i=1,2)$, as considered in Example 5. Then, the function ${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }:\mathcal{F}\to \mathbb{R}$ corresponding to $\gamma =5$ and $\zeta ={\displaystyle \frac{1}{7}}$ is defined as follows:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_{\vartheta ,5,{\displaystyle \frac{1}{7}}}(w)& :=\underset{\tilde{w}\in \mathcal{F}}{inf}\left\{{\vartheta }^5(\tilde{w})+{\displaystyle \frac{1}{7}}{∥ln\left({\displaystyle \frac{w_2}{{\tilde{w}}_2}}\right)∥}^2\right\}\hfill \\ \hfill & =\underset{\tilde{w}\in \mathcal{F}}{inf}\left\{\left(\left({\displaystyle \frac{5+8{\tilde{w}}_{2}ln{\tilde{w}}_2}{{\tilde{w}}_2}}\right)+3ln{\tilde{w}}_2-{\displaystyle \frac{1}{10}}\left(ln{\tilde{w}}_2+{\displaystyle \frac{1}{8}}\right)\right)\left(ln{\tilde{w}}_2+{\displaystyle \frac{1}{8}}\right)\right.\hfill \\ & \left.+{\displaystyle \frac{1}{7}}{\left(ln{w}_2-ln{\tilde{w}}_2\right)}^2\right\}\hfill \\ \hfill & ={\displaystyle \frac{1}{7}}{\left(ln{w}_2+{\displaystyle \frac{1}{8}}\right)}^2.\hfill \end{array}$$

Moreover, ${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(\tilde{u})=0$ if and only if $\tilde{u}=(0,e^{-{\displaystyle \frac{1}{8}}}).$ Therefore, ${\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }$ for $\gamma =5,\zeta ={\displaystyle \frac{1}{7}}$ is a Moreau-Yosida regularized gap function for WVVHVIP3.

From Example 5, $m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-{\displaystyle \frac{1}{2\gamma }}>0.$ Therefore, all the hypotheses of Theorem 7 are satisfied, which leads to the following inequality:

$$\sqrt{{\displaystyle \frac{{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{\zeta }}}\le \left|\right|{exp}_{\tilde{u}}^{-1}(w){\left|\right|}_{\tilde{u}}\le \sqrt{{\displaystyle \frac{2{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{min\{(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma }),\zeta \}}}},\mathrm{for}\mathrm{all}w\in \mathcal{F}.$$

Equivalently, for every $w=(w_1,w_2)\in \mathcal{F}$, the following conditions hold:

$$\begin{array}{c}\hfill \sqrt{{\displaystyle \frac{{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{\zeta }}}=\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)=\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right),\\ \hfill \sqrt{{\displaystyle \frac{2{\mathsf{\Theta }}_{\vartheta ,\gamma ,\zeta }(w)}{min\{(m_h^{\sigma }-m_{\mathsf{\Psi }}-m_{\mathcal{P}}-\frac{1}{2\gamma }),\zeta \}}}}=2\left(ln{u}_2+{\displaystyle \frac{1}{8}}\right)\ge \left(ln{u}_2+{\displaystyle \frac{1}{8}}\right).\end{array}$$

6. Conclusions and Future Research Directions

In this paper, we have investigated a class of nonsmooth weak vector variational-hemivariational inequality problems, namely, WVVHVIP in the setting of Hadamard manifolds. By employing an analogous to the KKM lemma, we have established the existence of solutions for WVVHVIP without using any monotonicity assumptions. Moreover, we have established the uniqueness of the solution to WVVHVIP under generalized geodesic strong monotonicity assumptions. We have formulated several gap functions for WVVHVIP, in particular, Auslender, regularized, and Moreau-Yosida type regularized gap functions. Subsequently, we have established necessary and sufficient conditions for the existence of a solution to the considered vector variational-hemivariational inequality problem. Furthermore, the global error bounds for the solution of WVVHVIP have been derived in terms of the regularized gap functions under generalized geodesic strong monotonicity assumptions of Clarke subdifferentials. Several non-trivial examples have been provided in the Hadamard manifold setting to demonstrate the significance of the established results.

The results derived in this paper extend and generalize several well-known results existing in the literature. In particular, the results derived in this article generalize the corresponding results established by Chen and Huang [10] and Jayswal et al. [44] from nonsmooth weak vector variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problems in the framework of Hadamard manifolds. Moreover, the results derived in this paper generalize the corresponding results established in [35] from variational-hemivariational inequality problems to WVVHVIP, which belongs to a broader category of vector variational-hemivariational inequality problems, as well as generalize them from the setting of finite-dimensional reflexive Banach space to the framework of Hadamard manifolds. Furthermore, the results established in the present article generalize the corresponding results derived in [30,31,32] from the Euclidean space setting to the framework of Hadamard manifolds and from variational inequality problems to WVVHVIP. In addition, several results related to gap functions and global error bounds for WVVHVIP generalize the corresponding results derived by Charitha et al. [34] from vector variational inequality problems to nonsmooth weak vector-variational hemivariational inequality problems and generalize them from the framework of Euclidean space to the setting of Hadamard manifolds.

The results derived in the present article can be applied to various real-life problems in the domains of mechanics and engineering, such as optimal control and frictional contact problems (see, [23,24]). Moreover, it is significant to observe that all the results derived in this paper use the linear property of the inner product in tangent spaces. However, several variational inequalities involve bifunctions (see, for instance, [20,50]), which may not necessarily be linear, then the results derived in the present article may no longer be applicable for nonsmooth vector variational-hemivariational inequalities involving bifunctions. We intend to address these limitations in our future research works.

The results established in the present article suggest various potential avenues for future research work. For instance, investigating gap functions and regularized gap functions for mixed vector quasi-variational-hemivariational inequality problems would be an interesting research problem. Furthermore, in view of the work of Oyewole [63], we would like to develop subgradient extragradient-type algorithms for WVVHVIP in our future research work.