α-Well-Posedness for Mixed Quasi Variational-Like Inequality Problems

and Applied Analysis 3 It is worth noting that if α 0, then the definitions of α-well-posedness, αwell-posedness in the generalized sense, L-α-well-posedness, and L-α-well-posedness in the generalized sense for MQVLI , respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for MQVLI in 20 . We also note that Definition 2.2 generalizes and extends α-well-posedness and α-well-posedness in the generalized sense of variational inequalities in 10 which are related to the continuously differentiable gap function of variational inequalities introduced by Fukushima 21 . In order to investigate the α-well-posedness for MQVLI , we need the following definitions. We recall the notion of Mosco convergence 22 . A sequence Hn n of subsets of E Mosco converges to a set H if H lim inf n Hn w − lim sup n Hn, 2.4 where lim infnHn andw− lim supnHn are, respectively, the Painlevé-Kuratowski strong limit inferior and weak limit superior of a sequence Hn n, that is, lim inf n Hn { y ∈ E : ∃yn ∈ Hn, n ∈ N, with yn → y } , w − lim sup n Hn { y ∈ E : ∃nk ↑ ∞, nk ∈ N, ∃ynk ∈ Hnk , k ∈ N, with ynk ⇀ y } , 2.5 where “⇀” means weak convergence, and “→ ” means strong convergence. If H lim infnHn, we call the sequence Hn n of subsets of E Lower Semi-Mosco convergent to a set H . It is easy to see that a sequence Hn n of subsets of E Mosco converges to a set H implies that the sequence Hn n also Lower Semi-Mosco converges to the set H , but the converse is not true in general. We will use the usual abbreviations usc and lsc for “upper semicontinuous” and “lower semicontinuous”, respectively. For any x, y ∈ E, x, y will denote the line segment {tx 1 − t y : t ∈ 0, 1 }, while x, y and x, y are defined analogously. We will frequently use s,w, and w∗ to denote, respectively, the norm topology on E, the weak topology on E, and the weak∗ topology on E. Given a convex set K, a multivalued map F : K → 2E will be called upper hemicontinuous, if its restriction on any line segment x, y ⊆ K is usc with respect to the w∗ topology on E∗. F : K → 2E will be called η-monotone if, for any x, y ∈ K, for all u ∈ F x , v ∈ F y , 〈u − v, η x, y 〉 ≥ 0. We refer the reader to 23, 24 for basic facts about multivalued maps. Lemma 2.5 see 25 . Let Hn n be a sequence of nonempty subsets of a Banach space E such that i Hn is convex for every n ∈ N; ii H0 ⊆ lim infnHn; iii there existsm ∈ N such that intn≥m Hn / ∅. Then, for every u0 ∈ intH0, there exists a positive real number δ such that intB u0, δ ⊆ Hn, ∀n ≥ m, 2.6 4 Abstract and Applied Analysis where B u0, δ is a closed ball with a center u0 and radius δ. If E is a finite dimensional space, then assumption (iii) can be replaced by iii ′ intH0 / ∅. The following lemmas play important role in this paper. Lemma 2.6. Let E be a real separable Banach space with the dual E∗, let S0 be a nonempty convex subset of E, and let F : S0 → 2E be a set-valued map with nonempty, weakly∗ compact convex valued, η-monotone, and upper hemicontinuous. Let η : S0 × S0 → E be a single-valued map with η x, x 0, for all x ∈ S0, and let f : S0 → R be a convex lsc function. Assume that the map y → 〈u, η x, y 〉 is concave for each u, x ∈ F S0 × S0 and usc. If S1 is a convex subset of S0 with the property that, for each x ∈ S0 and each y ∈ S1, x, y ⊆ S1, then for each x0 ∈ S0, the following conditions are equivalent: i There exists u0 ∈ F x0 , such that for all y ∈ S0, 〈u0, η x0, y 〉 f x0 −f y − α/2 ‖x0− y‖2 ≤ 0, ii for all y ∈ S1, there exists v ∈ F y , such that 〈v, η x0, y 〉 f x0 −f y − α/2 ‖x0 − y‖2 ≤ 0. Proof. According to the η-monotonicity of F, i ⇒ ii is obvious. Next prove ii ⇒ i . Suppose that ii holds. Given any y ∈ S1, let yn 1/n y 1 − 1/n x0, for n ∈ N. By the assumptions of S1, yn ∈ S1 for each n ∈ N. It follows from the condition ii that for each n ∈ N, there exists vn ∈ F yn such that 〈 vn, η ( x0, yn )〉 f x0 − f ( yn ) − α 2 ∥ ∥x0 − yn ∥ ∥2 ≤ 0. 2.7