Existence Theorems for Quasivariational Inequality Problem on Proximally Smooth Sets

and Applied Analysis 3 2. Main Results In this paper, we are interested in the following classes of mappings. Definition 8. Let T : H → H be a mapping.Then T is called (a) β-strongly monotone if there exists β > 0 such that ⟨T (x) − T (y) , x − y⟩ ≥ β 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ H, (12) (b) ξLipschitz if there exists ξ > 0 such that 󵄩󵄩󵄩󵄩T (x) − T (y) 󵄩󵄩󵄩󵄩 ≤ ξ 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 , ∀x, y ∈ H. (13) That is, in otherword, wewillmake the following assumption. Assumption (A). Let T : H → H and C : H → [Cl(H)] r be mappings. (i) T is a β-strongly monotone and a ξ-Lipschitz singlevalued mapping; (ii) C is a κ-Lipschitz set-valued mapping; (iii) there is ω ∈ [0, 1) such that 󵄩󵄩󵄩󵄩 Proj C(x) (z) − Proj C(y) (z) 󵄩󵄩󵄩󵄩 ≤ ω 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 , ∀x, y, z ∈ H. (14) Remark 9. LetH = (−∞,∞) andK = [a, b]∪[c, d], for some positive real numbers a, b, c, dwith a < b < c < d. If we define C : H → 2 H by C(x) = m(x) + K, where m : H → H is a mapping defined by m(x) = kx for k ∈ R, then we see that C is a max{d − b, c − a}-Lipschitz mapping, and Assumption (A) (iii) is satisfied with a constant 2(max{d − b, c − a}). This means that Assumptions (A) (ii) and (iii) are independent. The following remark is very useful in order to prove our results. Before seeing that, for the sake of simplicity, let us make a notation: for each r ∈ (0,∞) and s ∈ (0, r), we will write t s := r/(r − s). Remark 10. Let r, β, ξ, ω, and δ be five positive real numbers such that β ∈ (0, ξ), δ ∈ (0, r√β2 − ξ2(1 − (1 − ω)2)/(1 −ω)2) andω ∈ [0, 1−ξr√ξ2 − β2/(ξr−βδ)). If h : [1, ⬦] → [0,∞) is a function defined by


Introduction and Preliminaries
Variational inequality theory is a branch of the mathematics which is important, and it was also the inspiration for researchers to find new works, both in terms of mathematics and applications such as in economics, physical, biological, and engineering science, and other applied sciences.In 1973, Bensoussan et al. [1] introduced and studied the concept of the quasivariational inequality, which is a generalized form of the classical variational inequality that introduced by Stampacchia [2].Later, many researchers proposed and analyzed the concept of the generalized quasivariational inequality; see, for example [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].It is worth mentioning that the quasivariational inequality problem is of interest to study, since in many important problems the considered set also depends upon the solutions explicitly or implicitly.In fact, the concept of the quasi variational inequality has been applied in many fields such as in economics and transportation equilibrium, control and optimization theory, mathematical programming, and game theory.
In the early period of the research, it should be pointed out that almost all the results regarding the existence and iterative schemes for solving those variational inequalities problems are being considered in the convexity setting.This is because they need the convexity assumption for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping.However, in fact, the convexity assumption may not be required because it may be well defined even if the considered set is nonconvexs (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.).However, it may be from the practical point of view one may see that the nonconvex problems are more useful than convex case.Consequently, now many researchers are paying attention to many nonconvex cases.
Let  :  →  be a mapping and let  :  → C() be a set-valued mapping, where C() is a family of all nonempty closed subsets of .In this paper, we are interested in the following problem: find  * ∈ ( * ) such that where   () () is denoted for the proximal normal cone of () of .The problem of type (1) was introduced by Bounkhel et al. [19].In such a paper, they proposed some iterative algorithms for finding a solution of type (1), when the considered mapping is a set-valued mapping, while, in this paper, we will provide sufficient conditions for the existence of a solution of such a problem (1).To do this, let us start by recalling some basic concepts and useful results that will be needed in this work.
Let  be a real Hilbert space equipped with norm ‖ ⋅ ‖ and inner product ⟨⋅, ⋅⟩.Let 2  be denoted for the class of all nonempty subsets of  and C() denoted for the family of all nonempty closed subsets of .

Abstract and Applied Analysis
For each  ∈ 2  , the usual distance function on  to  is denoted by (⋅, ), that is: Let  ∈ 2  and  ∈ .A point  ∈  is called the closest point or the projection of  onto  if The set of all such closest points is denoted by Proj  (), that is: Further, for each  ∈ , the proximal normal cone to  at  is given by The following is called the proximal normal inequality; the proof of this characterization can be found in [20].
Lemma 1 (see [19]).Let  be a closed subset of a Hilbert space .Then We recall also [21] that the Clarke normal cone is given by where co[] means the closure of the convex hull of .It is clear that one always has    () ⊂ (, ).The converse is not true in general.Note that (, ) is always a closed and convex cone and that    () is always a convex cone but may be nonclosed (see [20,21]).
In 1995, Clarke et al. [22] have introduced and studied a new class of nonconvex sets, which are called proximally smooth sets.This class of proximally smooth sets has played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions.Subsequently, the proximally smooth sets have been proposed by many researchers.In recent years, Bounkhel et al. [19], Cho et al. [23], Moudafi [24], Noor [25], Noor et al. [26], Petrot [27], and Pang et al. [28] have considered both variational inequalities and equilibrium problems in the context of proximally smooth sets.They suggested and analyzed some projection type iterative algorithms by using the prox-regular technique and auxiliary principle technique.Here, we will take the following characterization proved in [20] as the definition of proximally smooth sets.Note that the original definition was given in terms of the differentiability of the distance function (see [22,29]).
Definition 2. For a given  ∈ (0, +∞], a subset  of  is said to be uniformly prox-regular with respect to , say, uniformly -prox-regular set, if for all  ∈  and for all 0 ̸ =  ∈    (), one has For the case of  = ∞, the uniform -prox-regularity  is equivalent to the convexity of  (see [22]).Moreover, it is known that the class of uniformly prox-regular sets is sufficiently large to include the class -convex sets,  1,1 submanifolds (possibly with boundary) of , the images under a  1,1 diffeomorphism of convex sets, and many other nonconvex sets; see [20,29].
From now on, we will denote [Cl()]  for the class of all uniformly -prox-regular closed subset of , where  ∈ (0, +∞] is fixed positive real number.Also, for each  ∈ (0, +∞], we write Remark 3. If  :  →  and  :  → [Cl()]  are mappings, then the problem of type ( 1) is equivalent to the following problem: find  * ∈ ( * ) such that see [19].This means, in particular, that the problem (1) contains the well-known Stampacchia's variational inequality, as a special case.
The following lemma which summarizes some important consequences of the uniformly prox-regularity sets is needed in the sequel.The proof of this result can be found in [22,29].Lemma 4. Let  ∈ (0, +∞] and  be a nonempty closed subset of .If  is uniformly -prox-regular set, then the following holds. (i) For all  ∈   , Proj  () ̸ = 0.
(iii) The proximal normal cone is closed as a set-valued mapping.
The following definition and lemma are also needed, in order to obtain our main results.Definition 6.A set-valued mapping  :  → 2  is said to be -Lipschitz if there exists  > 0 such that Lemma 7 (see [19]).Let  ∈ (0, +∞] and let  :  → 2 That is, in other word, we will make the following assumption. Assumption (A).Let  :  →  and  :  → [Cl()]  be mappings.
(i)  is a -strongly monotone and a -Lipschitz singlevalued mapping; (ii)  is a -Lipschitz set-valued mapping; The following remark is very useful in order to prove our results.Before seeing that, for the sake of simplicity, let us make a notation: for each  ∈ (0, ∞) and  ∈ (0, ), we will write   := /( − ).
Remark 10.Let , , , , and  be five positive real numbers for all  ∈ [1, ⬦], where ⬦ = (1 − )/√ 2 −  2 , then ℎ is an increasing continuous function on its domain.Moreover, we can check that / 2 is an element of the range of ℎ.
Theorem 11.Let  :  →  be a single-valued mapping and let  :  → [Cl()]  be a set-valued mapping.Assume that Assumption (A) holds and the following control conditions are satisfied: , where  = sup{‖()‖ :  ∈ }; , where ℎ and  are defined as Remark 10.
for each  ∈ N.This implies Write  := ((1 + )/(1 + )).From the previous argument, we see that the sequence {  } also has a property that for all  ∈ N.
Next, we will show that {  } is a convergent sequence, and its limit is nothing but a solution of the problem (1).Now, by using the Assumption (A) (iii) and Lemma 4 (ii), we have Meanwhile, since  is a -strongly monotone and -Lipschitz mapping, we see that Let then we see that Assumptions (A) (ii) and (iii) are satisfied with a constant zero.In this case, Theorem 11 is reduced to a result which was presented by Petrot [27].
It is well known that if  is a closed convex set, then it is -prox-regular set for every  > 0 (see [22]).Using this fact, and by careful of the proof of Theorem 11, one can see that control conditions (i) and (iii) of Theorem 11 can be omitted.So, we have the following results immediately.

Conclusion
In this paper, we provide some conditions for the existence theorems of the quasivariational inequality problem on a class of nonconvex sets.In fact, there are two constraints on the assumptions of considered mapping  in the main Theorem 11, that is, (i)  is a strongly monotone and Lipschitz mapping, (ii) the range of mapping  is a bounded set.
Hence, these lead to some natural questions in the future works for relaxing these constraints.At this point, we desire that the results presented here will be useful for those researchers, because this paper may also be extended and generalized for considering the mutlivalued and set-valued extended general variational inequalities problems.

Finally, in viewCorollary 17 .
of Remark 15 and Corollary 16, we can obtain the following classical result immediately.Let  be a real Hilbert space, let  be a closed convex subset of , and let  :  →  be a single-valued mapping.If  is a strongly monotone and Lipschitz singlevalued mapping, then the problem (1) has a solution.