Existence of projected solutions for quasi-variational hemivariational inequality

: In this short article, we prove the existence of projected solutions to a class of quasi-variational hemivariational inequalities with non-self-constrained mapping, which generalizes the results of Allevi et al. ( Quasi-variational problems with non-self map on Banach spaces: Existence and applications , Nonlinear Anal. Real World Appl. 67 (2022), 103641, DOI: https://doi.org/10.1016/j.nonrwa.2022.103641.)


Introduction
The theory of variational inequalities was considered and studied by the Kinderlehrer and Stampacchia [1] and Fichera [2].Since then, it has been greatly developed in various aspects, both from the theoretical and from the applied point of view, because it can provide a powerful tool of investigating various mathematical problems [3,4].With the deepening of research, a generalized quasi-variational inequality was introduced by Chan and Pang [5].Recently, Zeng et al. [6] first studied a generalized nonlinear quasi-hemivariational inequality involving a multivalued map in a Banach space, and also, they developed a useful and elegant framework to examine the existence of solution for the considered inequality.The generalized nonlinear quasi-hemivariational inequality introduced in [6] can be a very powerful mathematical model to study various approximations of elastic contact problems with the constitutive law involving convex subdifferential inclusions, the multivalued version of the normal compliance contact condition with frictionless effect, and a frictional contact law with the slip-dependent coefficient of friction.
In general, the solution to the generalized quasi-variational inequality is described as [7,8].However, in some applications, the self-constrained mapping condition is not always satisfied and the classical existence theorem is impossible to apply (see [9] and references therein).The concept of projected solutions of generalized quasi-variational inequality with non-self-constrained mapping was introduced in [9].In [10], the existence of solutions to the generalized quasi-variational inequality where constraint set-valued mapping is not necessarily self-constrained mapping was obtained.The projected solutions of Ky Fan's quasi-equilibrium problem were studied in [11], where the variational inequality becomes a classic example.All the results in [9][10][11] are established in finite dimensional spaces, while [12] provides the existence of projected solutions for the generalized quasi-variational inequality on infinite dimensional Banach spaces.
As we know that the hemivariational inequalities [13][14][15] can be seen as an important generalization of variational inequalities, which is relatively recent and relates to non-convex energy functionals.If both convex and non-convex functions are involved, they are called variational hemivariational inequalities.Today, variational hemivariational inequalities have been studied (see [16,17] and references therein).Correspondingly, quasi-variational hemivariational inequalities have also been proposed for many application problems (see [18,19] and references therein).This presents an open problem in the study of quasi-variational hemivariational inequalities with non-self-constrained mapping.Hence, the purpose of this short article is to prove the existence of projected solutions to a class of quasi-variational hemivariational inequalities on infinite dimensional Banach space.
Assume that V is a real reflexive Banach space that continuously and compactly embeds into a Banach space X , and C is nonempty, convex, and closed subset of V .Denote by γ the embedding operator from V to X , and denote by → γ X V * : * * its adjoint operator.For simplicity, we write : 2 V * be the two set-valued maps.We consider the following quasi-variational hemivariational inequality.
Let us first introduce the following hypotheses on the data of Problem 3.1.
T u is a single-valued mapping, then Problem (1) is a special case of the following problem: find which was considered by [20].
This problem was studied by [21].
This problem was considered by [18], where the locally Lipschitz function is of particular interest in applications: Now, we introduce the concept of projected solutions for quasi-variational hemivariational inequality, Problem (1), on real Banach spaces.Definition 1.5.A point ∈ u C is said to be a projected solution of (1) if and only if there exists ∈ u V ¯such that: (i) u is a projection of u ¯on C; (ii) u ¯is a solution to the following problem: find where m belongs to V * and → K C : 2 V and → F V : 2 V * are the two set-valued maps.
We recall that if ∈ y V and ⊂ C V is a nonempty set, the projection of y on C is defined as ( ) P y C , i.e., Clearly, if K is a self-map, then the projected solutions coincide the classical ones.
This article is organized as follows.Section 2 introduces the concepts, definitions, and properties.In Section 3, we prove the existence of projected solutions of quasi-variational hemivariational inequality, Problem (1).

Preliminaries
Let X be a real Banach space equipped with the norm ‖ ‖ ⋅ X , and let ⟨ ⟩ ⋅ ⋅ , X be the duality pairing between X and its dual X*.The strong convergence and weak convergence in X are denoted by symbols → and ⇀ , respectively.
(ii) Stably ϕ-pesudomonotone with respect to a set ⊂ W X* if F and its translations + F y are ϕ-pesudomonotone for every ∈ y W.
Existence of projected solutions for QVHVI  3 Definition 2.3.Assume that : is proper, convex, and lower semicontinuous.The subdifferential of φ at ( ) ∈ u D φ is defined to be The Clarke generalized gradient (subdifferential) of f at u is a subset of the dual space X* given by where, for each max , : .

Main result
In this section, we prove the existence of projected solutions to a class of quasi-variational hemivariational inequalities with non-self-constrained mapping on a real Banach space.To this end, we need to use Lemma 2.6.Therefore, we prove that all conditions of Lemma 2.6 are available.First, we give some notations.Fixed ∈ u C, let us introduce the following parametric quasi-variational hemivariational inequality: find ( ) The following lemma is needed for our proof.
be the solution mapping of Problem (3), namely, u ¯: ¯is a solution to 3 .
The solution set of Problem (3) is nonempty; (ii) u ¯is a solution to (3) if and only if it solves the following inequality problem: find Proof.It is a direct consequence of [21, Lemma 3.3].□ In what follows, we will prove that S satisfies all conditions of Lemma 2.6.Proposition 3.2.Let ( ) K C be relatively compact.Under hypotheses ( ) H K , ( ) H J , ( ) H φ , and ( ) H F , S is upper semicontinuous with nonempty, convex, and compact values.