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