A theoretical view of existence results by using fixed point theory for quasi-variational inequalities

Abstract

In this paper, we present new existence results for the quasi-variational inequality problem (QV I) in reflexive Banach spaces using the fixed point method with quasi-monotonicity and local upper sign-continuity assumptions. These results improve upon previous ones which only required a weaker monotonicity condition and did not impose compactness on the involved set.

Keywords:

• Quasi-variational inequality
• fixed point
• quasi-monotonicity
• local upper sign-continuity

Mathematics Subject Classifications:

• 58E35
• 47J20

1. Introduction

The variational inequality problem and its applications are important research topics in modern non-linear programming. It is widely used in a variety of fields, including engineering mechanics and economics. The Stampacchia variational inequality (V I) is to find $\varsigma \in \mathrm{{\rm Y}}$ such that $\mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma ):⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,\mathrm{\forall }\varrho \in \mathrm{{\rm Y}},$where Υ is a nonempty closed convex set and λ is a set-valued mapping.

Let Ω be a reflexive Banach space with a topological dual ${\mathrm{\Omega }}^{\ast }$, a nonempty subset $\mathrm{{\rm Y}}\subseteq \mathrm{\Omega }$, set-valued maps $\eta :\mathrm{{\rm Y}}\to 2^{\mathrm{{\rm Y}}}$ and $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$. The quasi-variational inequality (QV I) is to find $\varsigma \in \eta (\varsigma )$ such that $\mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma ):⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,\mathrm{\forall }\varrho \in \eta (\varsigma ),$we denote $QVI(\lambda ,\eta )$ the solution set for the preceding problem.

The QV I problem is an important generalization of the V I problem. Compared with the fact that the constraint set Υ in the V I is a fixed set, the constraint set $\eta (\varsigma )$ in the QV I depends on variables ς, which makes QV I can be applied to more complex models such as generalized Nash equilibrium problems in economics and contact problems with deformation in mechanics.

The fixed point theorem is used in any branch of applied science for obtaining existence results and uniqueness results. For example for models of fractional differential equations (see, e.g. [1–3] and the references therein.) The fixed point theory is also an important method in proving the existence of solutions to variational inequalities and related problems. Up to now, the QV I problem has been extensively studied in the setting of both finite and infinite-dimensional spaces. One can see [4–14] and the references therein. Noor [4] proved the existence of solutions to QV I in Hilbert spaces under some monotonicity assumptions by using projection operators and showed the equivalence between QV I and fixed point problems. Jadamb et al. [5] used the relationship between the QV I and the minimization problem to obtain the existence of solutions to the QV I. However, some existing existence results for the QV I problem in infinite-dimensional spaces presented in [6–9, 14] are deduced by applying the same method as the one used in [7], which requires six slightly complex assumptions. Meanwhile, it is worth pointing out that the results for the existence of solutions to QV I problem often depend on the monotonicity of the involved mappings and the compactness of sets.

Motivated by the mentioned above, the goal of this paper is to establish the existence of solutions to QV I with weaker assumptions on the monotonicity of the involved mappings and without requiring compactness of the sets. Our approach is based on the idea that the solutions of QV I are fixed points of the solution map of a perturbed variational inequality. In comparison to existing results such as [7, 11], our proof is simpler and does not require additional assumptions to achieve the same goal, only requiring weaker monotonicity assumptions and removing the compactness assumption on the sets.

2. Existence results for the QV I problem

In this section, we derive some existence results for QV I in reflexive Banach spaces based on the equivalency of QV I and fixed point problems under quasi-monotone and local upper sign-continuous mapping conditions.

2.1. Variational inequality problems

The variational inequality (V IP) problems that will be used in this section are introduced as follows.

Given a reflexive Banach space Ω with its topological dual ${\mathrm{\Omega }}^{\ast }$, a nonempty closed and convex subset $\mathrm{{\rm Y}}\subseteq \mathrm{\Omega }$, two set-valued maps $\eta :\mathrm{{\rm Y}}\to 2^{\mathrm{{\rm Y}}}$ and $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$, we will consider the following concepts of solutions to the QV I problems:

• QV I solutions $QVI(\lambda ,\eta )=\{\varsigma \in \eta (\varsigma )\mid \mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma ):⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,\mathrm{\forall }\varrho \in \eta (\varsigma )\}.$

• Star QV I solutions $QV{I}^{\ast }(\lambda ,\eta )=\{\varsigma \in \eta (\varsigma )\mid \mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma )\mathrm{\setminus }\{0\}:⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,\mathrm{\forall }\varrho \in \eta (\varsigma )\}.$It is clearly that $QV{I}^{\ast }(\lambda ,\eta )\subseteq QVI(\lambda ,\eta )$.

Another classical concept of variational inequality, which plays an important role in variational analysis is the Minty quasi-variational inequality (MQV I). We denote the solution set for the MQV I by $MQVI(\lambda ,\eta )$. $MQVI(\lambda ,\eta )=\{\varsigma \in \eta (\varsigma )\mid ⟨{\varrho }^{\ast },\varrho -\varsigma ⟩\ge 0,\mathrm{\forall }\varrho \in \eta (\varsigma ),\mathrm{\forall }{\varrho }^{\ast }\in \lambda (\varrho )\}.$Let us recall the most classical concepts of solution of Stampacchia variational inequality (V I). Let Ω and ${\mathrm{\Omega }}^{\ast }$ be a Banach space and its topological dual, η be a nonempty subset of Ω, and $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$ be a set-valued mapping. The solutions to the V I problems are described as follows.

• V I solutions $\begin{array}{rl}S(\lambda ,\eta )& =\{\varsigma \in \eta \mid \mathrm{\forall }\varrho \in \eta ,\mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma ):⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0\};\\ S(\lambda \mathrm{\setminus }\{0\},\eta )& =\{\varsigma \in \eta \mid \mathrm{\forall }\varrho \in \eta ,\mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma )\mathrm{\setminus }\{0\}:⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0\}.\end{array}$

• Strong V I solutions $S_{str}(\lambda ,\eta )=\{\varsigma \in \eta \mid \mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma ),\mathrm{\forall }\varrho \in \eta :⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0\}.$

• Star V I solutions $S^{\ast }(\lambda ,\eta )=\{\varsigma \in \eta \mid \mathrm{\exists }{\varsigma }^{\ast }\in \lambda (\varsigma )\mathrm{\setminus }\{0\},\mathrm{\forall }\varrho \in \eta :⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0\}.$We denote by $M(\lambda ,\eta )$ the collection of solutions to the Minty variational inequality (MV I) problems $M(\lambda ,\eta )=\{\varsigma \in \eta \mid \mathrm{\forall }\varrho \in \eta ,\mathrm{\forall }{\varrho }^{\ast }\in \lambda (\varrho ):⟨{\varrho }^{\ast },\varrho -\varsigma ⟩\ge 0\}.$Finally, we call $\varsigma \in \eta$ a localized solution of the MV I if there is a neighbourhood U of ς such that $\varsigma \in M(\lambda ,\eta \cap U)$. The set of these local solutions is denoted by $LM(\lambda ,\eta )$. Clearly, $M(\lambda ,\eta )\subseteq LM(\lambda ,\eta )$.

2.2. Preliminaries

Let Ω be a reflexive Banach space, ${\mathrm{\Omega }}^{\ast }$ be its dual space, and $\eta \subset \mathrm{\Omega }$ be a closed nonempty set.

Definition 2.1

A mapping $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$ with nonempty values is called

1. quasi-monotone on η if, for every $\varsigma ,\varrho \in \eta$ and for all ${\varsigma }^{\ast }\in \lambda (\varsigma ),{\varrho }^{\ast }\in \lambda (\varrho )$, $⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩>0⟹⟨{\varrho }^{\ast },\varrho -\varsigma ⟩\ge 0,$

2. pseudo-monotone on η if, for every $\varsigma ,\varrho \in \eta$ and for all ${\varsigma }^{\ast }\in \lambda (\varsigma ),{\varrho }^{\ast }\in \lambda (\varrho )$, $⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0⟹⟨{\varrho }^{\ast },\varrho -\varsigma ⟩\ge 0,$

3. properly quasi-monotone on η if, for all ${\varsigma }_1,{\varsigma }_2,\dots {\varsigma }_n\in \eta$ and all $\varsigma \in co\{{\varsigma }_1,{\varsigma }_2,\dots {\varsigma }_n\}$, there is $i\in \{1,2,\dots n\}$ such that $\mathrm{\forall }{{\varsigma }_i}^{\ast }\in \lambda ({\varsigma }_i),⟨{{\varsigma }_i}^{\ast },{\varsigma }_i-\varsigma ⟩\ge 0.$

Clearly, we have the following implications: pseudo-monotonicity ⇒ properly quasi-monotonicity ⇒ quasi-monotonicity.

Definition 2.2

A set-valued mapping with nonempty values $\lambda :\mathrm{\Omega }\to Y$, where Y is a topological vector space.The mapping λ is said to be

1. lower semi-continuous at ${\varsigma }_0\in dom\lambda$ if, for any $\{{\varsigma }_n\}\subseteq \mathrm{\Omega }$ converging to ${\varsigma }_0$ and any ${\varrho }_0\in \lambda ({\varsigma }_0)$, there exists $\{{\varrho }_n\}\subseteq Y$ converging to ${\varrho }_0$, with regard to the proposed topology on Y, and such that ${\varrho }_n\in \lambda ({\varsigma }_n)$ for any n;

2. upper semi-continuous at ${\varsigma }_0\in dom\lambda$ if, for any neighbourhood V of ${\lambda }_{(}{\varsigma }_0)$, there exists a neighbourhood U of ${\varsigma }_0$ such that $\lambda (U)\subset V$;

3. closed at ${\varsigma }_0\in dom\lambda$ if, for any sequence $({\varsigma }_n,{\varrho }_n)\in Gr\lambda$ converging to $({\varsigma }_0,{\varrho }_0)$, then $({\varsigma }_0,{\varrho }_0)\in Gr\lambda$.

The following semi-continuity refinements were introduced in [13, 15], which are made up of some kind of directional semi-continuity and are ideal for studying quasi-monotone or pseudo-monotone set-valued operators.

Definition 2.3

Let $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$ and $\mathrm{\forall }t\in ]0,1[,\underset{{{\varsigma }_t}^{\ast }\in \lambda ({\varsigma }_t)}{inf}⟨{{\varsigma }_t}^{\ast },\varrho -\varsigma ⟩\ge 0⟹\underset{{\varsigma }^{\ast }\in \lambda (\varsigma )}{inf}⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,$for any $\varsigma ,\varrho \in \eta$, where ${\varsigma }_t=t\varsigma +(1-t)\varrho$ and η is a convex subset of Ω. Then λ is called lower sign-continuous on η.

Also for any $\varsigma ,\varrho \in \eta$, such that $\mathrm{\forall }t\in ]0,1[,\underset{{{\varsigma }_t}^{\ast }\in \lambda ({\varsigma }_t)}{inf}⟨{{\varsigma }_t}^{\ast },\varrho -\varsigma ⟩\ge 0⟹\underset{{\varsigma }^{\ast }\in \lambda (\varsigma )}{sup}⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,$where ${\varsigma }_t=t\varsigma +(1-t)\varrho$. Then λ is called upper sign-continuous on η.

The λ is called locally upper sign-continuous at ς if there exists a convex neighbourhood $V_{\varsigma }$ of ς and an upper sign-continuous sub-map ${\mathrm{\Phi }}_{\varsigma }:V_{\varsigma }\to 2^{{\mathrm{\Omega }}^{\ast }}$ with nonempty convex ${\omega }^{\ast }$ compact values such that $\mathrm{\forall }v\in V_{\varsigma },{\mathrm{\Phi }}_{\varsigma }(v)\subset \lambda (v)\mathrm{\setminus }\{0\}.$

It follows from the definition that any lower sign-continuous map is also an upper sign-continuous map. If λ is upper (or lower) semi-continuous, or even upper (or lower) hemi-continuous, then λ is upper (or lower) sign-continuous. However, in general, the upper semi-continuity of λ does not entail its lower sign-continuity (see [15]). Although it may look sophisticated, the concept of locally upper sign-continuous is essentially a very weak regularity-type notion and it is important in proving the existence results for variational inequalities (see [16] and the references therein).

Proposition 2.4

[16]

Assume η be a nonempty, convex subset of a Banach space Ω and let $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$ be quasi-monotone. Then one of the following claims is true:

(i)	λ is properly quasi-monotone,
(ii)	$LM(\lambda ,\eta )\ne \mathrm{\varnothing }$.

Furthermore, in both situations, if η is weakly compact, $LM(\lambda ,\eta )\ne \mathrm{\varnothing }$.

Theorem 2.5

[17], Kluge's Fixed Point Theorem

Let Ω be a reflexive Banach space, ${\mathrm{\Omega }}^{\ast }$ be its dual space, $\mathrm{{\rm Y}}\subset \mathrm{\Omega }$ be a nonempty convex and closed set and $\eta :\mathrm{{\rm Y}}\to 2^{{\mathrm{{\rm Y}}}^{\ast }}$ be a set-valued map such that for every point $\varsigma \in \mathrm{{\rm Y}}$, $\eta (\varsigma )$ is a nonempty convex and closed set and η has a weakly closed graph. If $\eta (\varsigma )$ is a bounded set, then there exists an ${\varsigma }_0\in \mathrm{{\rm Y}}$ such that ${\varsigma }_0\in \eta ({\varsigma }_0)$.

2.3. Main result

Lemma 2.6

Assume Ω and ${\mathrm{\Omega }}^{\ast }$ be a Banach space and its dual space respectively, $\eta \subseteq \mathrm{\Omega }$ be a nonempty convex set and $\lambda :\eta \to 2^{{\mathrm{\Omega }}^{\ast }}$ be a set-valued mapping. If for every $\varsigma \in \eta$, there exists a convex neighbourhood $V_{\varsigma }$ of ς and an upper sign-continuous map $S_{\varsigma }:V_{\varsigma }\cap \eta \to 2^{{\mathrm{\Omega }}^{\ast }}$ with nonempty $w\ast$-compact convex values such that $S_{\varsigma }(\varrho )\subset \lambda (\varrho )\mathrm{\setminus }\{0\}$ holds for any $\varrho \in V_{\varsigma }\cap \eta$, then $LM(\lambda ,\eta )\subseteq S(\lambda \mathrm{\setminus }\{0\},\eta )=S^{\ast }(\lambda ,\eta )$.

Proof.

Assume $\varsigma \in LM(\lambda ,\eta )$, there is a neighbourhood U of ς such that $\varsigma \in M(\lambda ,\eta \cap U)$, i.e. there exists $\varsigma \in \eta \cap U$ such that $⟨{\varrho }^{\ast },\varrho -\varsigma ⟩\ge 0,\mathrm{\forall }\varrho \in \eta \cap U,\mathrm{\forall }{\varrho }^{\ast }\in \lambda (\varrho ).$There is a convex neighbourhood $V_{\varsigma }$ of ς and an upper sign-continuous map $S_{\varsigma }:V_{\varsigma }\cap \eta \to 2^{{\mathrm{\Omega }}^{\ast }}$ with nonempty $w^{\ast }$-compact convex values such that $\varsigma \in \eta \cap V_{\varsigma }\cap U$ and $S_{\varsigma }(\overline{\varrho })\subseteq \lambda (\overline{\varrho })\mathrm{\setminus }\{0\}\subseteq \lambda (\overline{\varrho })$ for any $\overline{\varrho }\in \eta \cap V_{\varsigma }$, then $⟨{\overline{\varrho }}^{\ast },\overline{\varrho }-\varsigma ⟩\ge 0,\mathrm{\forall }\overline{\varrho }\in \eta \cap U\cap V_{\varsigma },\mathrm{\forall }{\overline{\varrho }}^{\ast }\in S_{\varsigma }(\overline{\varrho }),$i.e. $\varsigma \in M(S_{\varsigma },\eta \cap U\cap V_{\varsigma })$ by the definition of $LM(\lambda ,\eta )$.

Let $\varrho \in \eta \cap V_{\varsigma }$, there is a $\tilde{\varrho }\in ]\varsigma ,\varrho ]$ such that $[\varsigma ,\tilde{\varrho }]\subset \eta \cap U\cap V_{\varsigma }$, $\eta \cap V_{\varsigma }$ is convex and U is an absorbing set, then $\underset{u\in [\varsigma ,\tilde{\varrho }]}{inf}\underset{u^{\ast }\in S_{\varsigma }(u)}{inf}⟨u^{\ast },u-\varsigma ⟩\ge 0.$The map $S_{\varsigma }:V_{\varsigma }\cap \eta \to 2^{{\mathrm{\Omega }}^{\ast }}$ is upper sign-continuous with nonempty $w^{\ast }$-compact convex values, then $\begin{array}{rl}& \underset{{\varsigma }^{\ast }\in S_{\varsigma }(\varsigma )}{sup}⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,\\ & \underset{\varrho \in \eta \cap V_{\varsigma }}{inf}\underset{{\varsigma }^{\ast }\in S_{\varsigma }(\varsigma )}{max}⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0,\end{array}$which implies for any $\varrho \in \eta \cap V_{\varsigma }$ there is ${\varsigma }^{\ast }\in S_{\varsigma }(\varsigma )\subseteq \lambda (\varsigma )\mathrm{\setminus }\{0\}$ such that$⟨{\varsigma }^{\ast },\varrho -\varsigma ⟩\ge 0$.

For any $z\in \eta$, let $z_t=\varsigma +t(z-\varsigma )$, there is t such that $z_t\in \eta \cap V_{\varsigma }$, $\varsigma \in \eta \cap V_{\varsigma }$ and $\eta \cap V_{\varsigma }$ is convex, then $⟨{\varsigma }^{\ast },z_t-\varsigma ⟩=t⟨{\varsigma }^{\ast },z-\varsigma ⟩\ge 0,\mathrm{\forall }z\in \eta ,$which implies $\varsigma \in S(\lambda \mathrm{\setminus }\{0\},\eta )=S^{\ast }(\lambda ,\eta )$ by Sion Minimax theorem.

Lemma 2.7

Let Ω and ${\mathrm{\Omega }}^{\ast }$ be a Banach space and its dual space respectively, $\eta \subseteq \mathrm{\Omega }$ be a nonempty convex set, and $\lambda :\eta \to 2^{{\mathrm{\Omega }}^{\ast }}$ be a quasi-monotone set-valued mapping. Suppose that the following conditions are satisfied:

(i)	for some $\rho >0$ and any $\varsigma \in \eta \mathrm{\setminus }\overline{B}(0,\rho )$, there exists $\varrho \in \eta$ with $\Vert \varrho \Vert <\Vert \varsigma \Vert$ such that $\mathrm{\forall }{\varsigma }^{\ast }\in \lambda (\varsigma ),⟨{\varsigma }^{\ast },\varsigma -\varrho ⟩\ge 0,$where $\overline{B}(0,\rho )$ is a closed ball with zero as the centre and ρ as the radius;
(ii)	there exists ${\rho }^{\mathrm{\prime }}>\rho$ such that $\eta \cap \overline{B}(0,{\rho }^{\mathrm{\prime }})$ is nonempty weakly compact;
(iii)	for each $\varsigma \in \eta$, there exists a convex neighbourhood $V_{\varsigma }$ of ς and an upper sign-continuous operator $S_{\varsigma }:V_{\varsigma }\cap \eta \to 2^{{\mathrm{\Omega }}^{\ast }}$ with nonempty, convex, $w\ast$-compact values satisfying $S_{\varsigma }(\varrho )\subseteq \lambda (\varrho )\mathrm{\setminus }\{0\},\mathrm{\forall }\varrho \in V_{\varsigma }\cap \eta .$

Then $S^{\ast }(\lambda ,\eta )=S(\lambda \mathrm{\setminus }\{0\},\eta )\ne \mathrm{\varnothing }$.

Proof.

Let ${\eta }_{{\rho }^{\mathrm{\prime }}}=\eta \cap \overline{B}(0,{\rho }^{\mathrm{\prime }})$ be a weakly compact and nonempty convex set. According to Proposition 2.4, $LM(\lambda ,{\eta }_{{\rho }^{\mathrm{\prime }}})\ne \mathrm{\varnothing }$ holds. By Lemma 2.6, we have $\mathrm{\varnothing }\ne LM(\lambda ,{\eta }_{{\rho }^{\mathrm{\prime }}})\subseteq S(\lambda \mathrm{\setminus }\{0\},{\eta }_{{\rho }^{\mathrm{\prime }}})=S^{\ast }(\lambda ,{\eta }_{{\rho }^{\mathrm{\prime }}}).$Choose ${\varsigma }_0\in S^{\ast }(\lambda ,{\eta }_{{\rho }^{\mathrm{\prime }}})$, then there is ${{\varsigma }_0}^{\ast }\in \lambda ({\varsigma }_0)\mathrm{\setminus }\{0\}$ such that (1) $⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩\ge 0,\mathrm{\forall }\varrho \in {\eta }_{{\rho }^{\mathrm{\prime }}}.$(1)

1. If $\parallel {\varsigma }_0\parallel <{\rho }^{\mathrm{\prime }}$, for each $\varrho \in \eta$ there is $t\in (0,1)$ such that ${\varsigma }_0+t(\varrho -{\varsigma }_0)\in {\eta }_{{\rho }^{\mathrm{\prime }}}$. From (1(1) $⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩\ge 0,\mathrm{\forall }\varrho \in {\eta }_{{\rho }^{\mathrm{\prime }}}.$(1) ), it follows that $⟨{{\varsigma }_0}^{\ast },{\varsigma }_0+t(\varrho -{\varsigma }_0)-{\varsigma }_0⟩\ge 0,\mathrm{\forall }\varrho \in \eta ⟹⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩\ge 0,\mathrm{\forall }\varrho \in \eta .$This implies ${\varsigma }_0\in S^{\ast }(\lambda ,\eta )=S(\lambda \mathrm{\setminus }\{0\},\eta )$.

2. If $\parallel {\varsigma }_0\parallel ={\rho }^{\mathrm{\prime }}>\rho$, then ${\varsigma }_0\in \eta \mathrm{\setminus }\overline{B}(0,\rho )$. By assumption (i), there exists ${\varrho }_0\in \eta$ satisfying $\Vert {\varrho }_0\Vert <\Vert {\varsigma }_0\Vert ={\rho }^{\mathrm{\prime }}$ such that for any ${{\varsigma }_0}^{\ast }\in \lambda ({\varsigma }_0)\mathrm{\setminus }\{0\}\subset \lambda ({\varsigma }_0)$, we have (2) $⟨{{\varsigma }_0}^{\ast },{\varsigma }_0-{\varrho }_0⟩\ge 0.$(2) From (1(1) $⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩\ge 0,\mathrm{\forall }\varrho \in {\eta }_{{\rho }^{\mathrm{\prime }}}.$(1) ) and (2(2) $⟨{{\varsigma }_0}^{\ast },{\varsigma }_0-{\varrho }_0⟩\ge 0.$(2) ), we have (3) $⟨{{\varsigma }_0}^{\ast },{\varrho }_0-{\varsigma }_0⟩=0.$(3) For any $\varrho \in \eta$, there is $t\in [0,1]$ such that $t\varrho +(1-t){\varrho }_0\in {\eta }_{{\rho }^{\mathrm{\prime }}}$. From (1(1) $⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩\ge 0,\mathrm{\forall }\varrho \in {\eta }_{{\rho }^{\mathrm{\prime }}}.$(1) ) and (3(3) $⟨{{\varsigma }_0}^{\ast },{\varrho }_0-{\varsigma }_0⟩=0.$(3) ), it follows that $\begin{array}{rl}& ⟨{{\varsigma }_0}^{\ast },t\varrho +(1-t){\varrho }_0-{\varsigma }_0⟩\\ & =t⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩+(1-t)⟨{{\varsigma }_0}^{\ast },{\varrho }_0-{\varsigma }_0⟩\\ & =t⟨{{\varsigma }_0}^{\ast },\varrho -{\varsigma }_0⟩\\ & \ge 0.\end{array}$Then $⟨{\varsigma }_0^{\ast },\varrho -{\varsigma }_0⟩\ge 0$, this entails ${\varsigma }_0\in S^{\ast }(\lambda ,\eta )=S(\lambda \mathrm{\setminus }\{0\},\eta )$.

Theorem 2.8

Let Ω be a reflexive Banach space, ${\mathrm{\Omega }}^{\ast }$ its dual space, $\mathrm{{\rm Y}}\subseteq \mathrm{\Omega }$ be a convex nonempty closed set. Suppose that the following conditions are satisfied:

(i)	the mapping $\eta :\mathrm{{\rm Y}}\to 2^{\mathrm{{\rm Y}}}$ is lower semi-continuous with nonempty convex values and has closed graph, $int\eta (\varsigma )\ne \mathrm{\varnothing }$ for any $\varsigma \in \mathrm{{\rm Y}}$;
(ii)	the mapping $\lambda :\mathrm{\Omega }\to 2^{{\mathrm{\Omega }}^{\ast }}$ is quasi-monotone and locally upper sign-continuous, that is, for any ${\varsigma }_n\to \varsigma ,{\varrho }_n\to \varrho$, the following implication holds true: (4) $\underset{n}{lim inf}\underset{{\varrho }_n^{\ast }\in \lambda ({\varrho }_n)}{sup}⟨{\varrho }_n^{\ast },{\varsigma }_n-{\varrho }_n⟩\le 0\Rightarrow \underset{{\varrho }^{\ast }\in \lambda (\varrho )}{sup}⟨{\varrho }^{\ast },\varsigma -\varrho ⟩\le 0;$(4)
(iii)	for some $\rho >0$ and any $\varsigma \in \mathrm{{\rm Y}},\overline{\varsigma }\in \eta (\varsigma )\mathrm{\setminus }\overline{B}(0,\rho )$, there exists $\overline{\varrho }\in \eta (\varsigma )$ with $\Vert \overline{\varrho }\Vert <\Vert \overline{v}\Vert$ such that $⟨{\overline{\varsigma }}^{\ast },\overline{\varsigma }-\overline{\varrho }⟩\ge 0,\mathrm{\forall }{\overline{\varsigma }}^{\ast }\in \lambda (\overline{\varsigma });$
(iv)	there is ${\rho }^{\mathrm{\prime }}>\rho$ and $\varsigma \in \mathrm{{\rm Y}}$ such that the set $\eta (\varsigma )\cap \overline{B}(0,{\rho }^{\mathrm{\prime }})$ is nonempty convex and weakly compact.

Then at least one solution is granted by $QV{I}^{\ast }(\lambda ,\eta )$ and $QVI(\lambda ,\eta )\ne \mathrm{\varnothing }$.

Proof.

By the above conditions, for every $\varsigma \in \mathrm{{\rm Y}}$, the set $\eta (\varsigma )$ satisfying Lemma 2.7, then for every $\varsigma \in \mathrm{{\rm Y}}$, $S^{\ast }(\lambda ,\eta (\varsigma ))\ne \mathrm{\varnothing }.$Let ${\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}=\{\varsigma \in \mathrm{{\rm Y}}|\eta (v)\cap \overline{B}(0,{\rho }^{\mathrm{\prime }})\}$ be the set satisfying conditions (iv), then we have $\overline{B}(0,{\rho }^{\mathrm{\prime }})\supseteq \overline{{\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}}\supseteq {\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}\ne \mathrm{\varnothing },$and $\overline{{\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}}$ is weakly compact due to the weakly compactness of $\overline{B}(0,{\rho }^{\mathrm{\prime }})$.

We define a mapping ${\mathcal{S}}^{\ast }:\overline{{\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}}\to 2^{\overline{{\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}}}$ as ${\mathcal{S}}^{\ast }(\cdot )=S^{\ast }(\lambda ,\eta (\cdot ))$and we further show that the mapping ${\mathcal{S}}^{\ast }$ has a fixed point.

By Lemma 3.1(v) in [15], ${\mathcal{S}}^{\ast }(\varsigma )=S^{\ast }(\lambda ,\eta (\varsigma ))=M(\lambda ,\eta (\varsigma ))$ is available. By the definition of $M(\lambda ,\eta (\varsigma ))$, it is easy to verify that $M(\lambda ,\eta (\varsigma ))$ is a convex closed set, then ${\mathcal{S}}^{\ast }(\varsigma )$ has convex values and nonempty closed.

We will prove that graph GrM is a closed set, then the mapping $S^{\ast }$ is closed.

Let $({\mu }_n,{\varsigma }_n)\subset GrM$ and $({\mu }_n,{\varsigma }_n)\to (\mu ,\varsigma )$, since the mapping $\eta :\mathrm{{\rm Y}}\to 2^{\mathrm{{\rm Y}}}$ is closed and lower semi-continuous, then $\varsigma \in \eta (\mu )$. For any $\varrho \in \eta (\mu )$, there is N such that ${\varrho }_n\in \eta ({\mu }_n)$ and ${\varrho }_n\to \varrho$ when n>N.

As ${\varsigma }_n\in M(\lambda ,\eta ({\mu }_n))$, there exists ${\varsigma }_n\in \eta ({\mu }_n)$ such that $⟨{\varrho }^{\ast },{\varsigma }_n-\varrho ⟩\le 0,\mathrm{\forall }\varrho \in \eta ({\mu }_n),\mathrm{\forall }{\varrho }^{\ast }\in \lambda (\varrho ).$Then for each ${\varrho }_n^{\ast }\in \lambda ({\varrho }_n)$, $⟨{\varrho }_n^{\ast },{\varsigma }_n-{\varrho }_n⟩\le 0$ due to ${\varrho }_n\in \eta ({\mu }_n)$ for n large enough. By (4(4) $\underset{n}{lim inf}\underset{{\varrho }_n^{\ast }\in \lambda ({\varrho }_n)}{sup}⟨{\varrho }_n^{\ast },{\varsigma }_n-{\varrho }_n⟩\le 0\Rightarrow \underset{{\varrho }^{\ast }\in \lambda (\varrho )}{sup}⟨{\varrho }^{\ast },\varsigma -\varrho ⟩\le 0;$(4) ), it follows that $⟨{\varrho }^{\ast },\varsigma -\varrho ⟩\le 0,\mathrm{\forall }\varrho \in \eta (\mu ),\mathrm{\forall }{\varrho }^{\ast }\in \lambda (\varrho ),$which implies $(\mu ,\varsigma )\subset GrM$. Then $S^{\ast }$ is closed and so $S^{\ast }$ is weakly closed.

It is obvious that the mapping ${\mathcal{S}}^{\ast }$ is weakly closed on the weakly compact set $\overline{{\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}}$. We can deduce that ${\mathcal{S}}^{\ast }$ has a fixed point on $\overline{{\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}}$ by Theorem 2.5 (Kluge's Fixed Point Theorem). Further, we have that $\mathrm{\varnothing }\ne S^{\ast }(\lambda ,\eta ({\mathrm{{\rm Y}}}_{{\rho }^{\mathrm{\prime }}}))\subseteq S^{\ast }(\lambda ,\eta (\mathrm{{\rm Y}}))\subseteq S_{str}(\lambda ,\eta (\mathrm{{\rm Y}})).$Thus $QVI(\lambda ,\eta )\supseteq QV{I}^{\ast }(\lambda ,\eta )\ne \mathrm{\varnothing }$. So at least one solution is granted by $QV{I}^{\ast }(\lambda ,\eta )$ and $QVI(\lambda ,\eta )\ne \mathrm{\varnothing }$.

3. Conclusion

Fixed point theory is a commonly used tool in pure and applied mathematics, particularly in the fields of existence and uniqueness. The Banach contraction principle is a well-known source of inspiration for metric fixed point theory. This principle shows the unifying power of functional analysis methods, as well as the practical applications of fixed point theorems in various areas. It has become a fundamental and powerful tool in non-linear analysis due to its wide range of applications in establishing the existence and uniqueness of solutions for various types of differential equations, integral equations, matrix equations, functional equations, and variational inequalities. In this paper, we investigate the existence of solutions to the quasi-variational inequality problem (QV I) in reflexive Banach spaces through fixed point methodologies. To do so, we establish the equivalence relationship between the existence of solutions to (QV I) and the existence of a fixed point for a certain mapping. By using weaker monotonicity assumptions and removing the compact assumption on the sets being considered, we are able to obtain similar results to those found in [7]. This paper is therefore a generalization of the variational inequalities problem using fixed point methodologies. We also propose some potential areas for future research. Most contractive conditions in fixed point theory involve two metric terms: the distance between two points, $d(x,y)$, and the distance between their images, $d(Tx,Ty)$, under the self-mapping. These terms are often used in various types of inequalities. Understanding more about these inequalities, such as the trapezoid and Newton-type inequalities (see [18–20]), is crucial for generalizing existing contractions and proving the existence of fixed points in general metric spaces. Additionally, we can strive to establish the relationship between generalized V I problems and the obtained fixed point theorem, adding to the relevant theories on the existence of solutions to variational inequalities.

Acknowledgments

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Conceptualization, Min Wang, Mi Zhou, Yeliz Karaca; formal analysis, Min Wang; investigation, Yeliz Karaca, Mati ur Rahman; writing original draft preparation, Min Wang, Mi Zhou; writing review and editing, Min Wang, Yeliz Karaca, Mati ur Rahman, Mi Zhou. All authors read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

No data were used to support this study.

Additional information

Funding

This work is supported by the scientific research start-up project of Mianyang Teachers' College [grant number QD2019A08]. Mi Zhou is partially supported by the High Level Project of Hainan Provincial Natural Science Foundation (Grant No. 621RC602) and Key Special Project of University of Sanya (Grant No.USY22XK-04).