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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 635030, 13 pages doi:10.1155/2011/635030 Research Article Variational-Like Inclusions and Resolvent Equations Involving Infinite Family of Set-Valued Mappings Rais Ahmad and Mohd Dilshad Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Correspondence should be addressed to Rais Ahmad, raisain 123@rediffmail.com Received 18 December 2010; Accepted 23 December 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 R Ahmad and M Dilshad This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We study variational-like inclusions involving infinite family of set-valued mappings and their equivalence with resolvent equations It is established that variational-like inclusions in real Banach spaces are equivalent to fixed point problems This equivalence is used to suggest an iterative algorithm for solving resolvent equations Some examples are constructed Introduction The important generalization of variational inequalities, called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance and applied sciences, and so forth; see, for example 1–7 and references theirin The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important The resolvent operator technique is used to establish an equivalence between variational inequalities and resolvent equations The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities inclusions and related optimization problems In this paper, we established a relationship between variational-like inclusions and resolvent equations We propose an iterative algorithm for computing the approximate solutions which converge to exact solution of considered resolvent equations Some examples are constructed 2 Fixed Point Theory and Applications Formulation and Preliminaries Throughout the paper, unless otherwise specified, we assume that E is a real Banach space with its norm · , E∗ is the topological dual of E, ·, · is the pairing between E and E∗ , d is the metric induced by the norm · , 2E resp., CB E is the family of nonempty resp., nonempty closed and bounded subsets of E, and H ·, · is the Housdorff metric on CB E defined by H P, Q max sup d x, Q , sup d P, y x∈P where d x, Q infy∈Q d x, y and d P, y ∗ mapping J : E → 2E is defined by J x f ∈ E∗ : x, f , 2.1 y∈Q infx∈P d x, y The normalized duality x · f , x , ∀x ∈ E f 2.2 Definition 2.1 Let E be a real Banach space Let η : E × E → E; g, A : E → E be the singlevalued mapping, and let M : E → 2E be a set-valued mapping Then, i the mapping g is said to be accretive if g x −g y ,j x −y ≥ 0, ∀x, y ∈ E, 2.3 ∀x, y ∈ E, 2.4 ii the mapping g is said to be strictly accretive if g x −g y ,j x −y ≥ 0, and the equality hold if and only if x y, iii the mapping g is said to be k-strongly accretive k ∈ 0, there exists j x − y ∈ J x − y such that g x − g y ,j x −y ≥k x−y if for any x, y ∈ E, , 2.5 iv the mapping A is said to be r-strongly η-accretive, if there exists a constant r > such that ≥r x−y A x − A y , j η x, y , ∀x, y ∈ E, 2.6 v the mapping M is said to be m-relaxed η-accretive, if there exists a constant m > such that u − v, j η x, y ≥ −m x − y , ∀x, y ∈ E, u ∈ M x , v ∈ M y 2.7 Fixed Point Theory and Applications Definition 2.2 Let A : E → E, η : E × E → E be the single-valued mappings Then, a set-valued mapping M : E → 2E is called A, η -accretive if M is m-relaxed ηaccretive and A ρM E E, for every ρ > ∗ Proposition 2.3 see 8, Let E be a real Banach space, and let J : E → 2E be the normalized duality mapping Then, for any x, y ∈ E x y ≤ x 2 y, j x y , ∀j x y ∈J x Definition 2.4 Let A : E → E, W : E × E → E, and let N : E∞ mappings Then, y 2.8 E × E × E · · · → E be the i the mapping A is said to be Lipschitz continuous with constant λA if ≤ λA x − y , A x −A y ∀x, y ∈ E, 2.9 ii the mapping W is said to be Lipschitz continuous in the first argument with constant λW1 if W x1 , · − W x2 , · ≤ λW1 x1 − x2 , ∀x1 , x2 ∈ E 2.10 Similarly, we can define Lipschitz continuity in the second argument iii the mapping N is said to be Lipschitz continuous in the ith argument with constant βi if N ·, , xi , − N ·, , yi , ≤ βi xi − yi , ∀xi , yi ∈ E 2.11 Definition 2.5 Let A : E → E be a strictly η-accretive mapping, and let M : E → 2E be an ρ,A A, η -accretive mapping Then, the resolvent operator Jη,M : E → E is defined by ρ,A Jη,M u A ρM −1 u, ∀u ∈ E 2.12 Proposition 2.6 see 10 Let E be a real Banach space, and let η : E × E → E be τ-Lipschitz continuous; let A : E → E be an r-strongly η-accretive mapping, and let M : E → 2E be an A, η ρ,A accretive mapping Then the resolvent operator Jη,M : E → E is τ/ r − ρm -Lipschitz continuous, that is, ρ,A ρ,A Jη,M x − Jη,M y where ρ ∈ 0, r/m is a constant ≤ τ x−y , r − ρm ∀x, y ∈ E, 2.13 Fixed Point Theory and Applications √ x, M y Example 2.7 Let E Ê, A x Then, M is η-accretive √ y, and η x, y √ √ x − y for all x, y ≥ ∈ E Example 2.8 Let M ·, · : E × E → 2E be r-strongly η-accretive in the first argument Then, M is m-relaxed η-accretive for m ∈ 1, r r , for r > 0.6180 Let Ti : E → CB E , i 1, 2, , ∞ be an infinite family of set-valued mappings, and let N : E∞ E×E×E · · · → E be a nonlinear mapping Let η, W : E×E → E; A, g, m : E → E be single-valued mappings, let and B, C, D:E → CB E be set-valued mappings Suppose that M ·, · : E × E → 2E is A, η -accretive mapping in the first argument We consider the following problem Find u ∈ E, wi ∈ Ti u , i 1, 2, , ∞, a ∈ B u , x ∈ C u , and y ∈ D u such that ∈ N w1 , w2 , − W x, y ma M g u − m a ,u 2.14 The problem 2.14 is called variational-like inclusions problem Special Cases i If W 0, m 0, then problem 2.14 reduces to the problem of finding u ∈ E, wi ∈ Ti u , i 1, 2, , ∞ such that ∈ N w1 , w2 , M g u ,u 2.15 Problem 2.15 is introduced and studied by Wang 11 ii If W 0, m 0, N ·, N ·, · , then problem 2.14 reduces to a problem considered by Chang, et al 12, 13 that is, find u ∈ H, w1 ∈ T1 u , w2 ∈ T2 u such that ∈ N w1 , w2 M g u ,u 2.16 It is now clear that for a suitable choice of maps involved in the formulation of problem 2.14 , we can drive many known variational inclusions considered and studied in the literature In connection with problem 2.14 , we consider the following resolvent equation problem Find z, u ∈ E, wi ∈ Ti u , i 1, 2, , ∞; a ∈ B u , x ∈ C u , y ∈ D u such that N w1 , w2 , − W x, y ρ,A ma ρ,A ρ,A ρ−1 Rη,M ·,u z ρ,A 0, 2.17 ρ,A I − A Jη,M ·,u , where A Jη,M ·,u z A Jη,M ·,u where ρ is a constant and Rη,M ·,u and I is the identity mapping Equation 2.17 is called the resolvent equation problem z Fixed Point Theory and Applications In support of problem 2.17 , we have the following example −i, i , i 1, 2, , ∞, C u {π/2}, B u Example 2.9 Let us suppose that E Ê, Ti u 0, , and D u {1} We define for wi ∈ Ti u , i 1, 2, , ∞, a ∈ B u , x ∈ C u and y ∈ D u i N w1 , w2 , sin−1 a ii m a iii W x, y min{−1, sin w1 , sin w2 , }, cos−1 a, xy, x − 1, for all x ∈ Ê, iv A x v M ·, x 1, for all x ∈ Ê, Then, for ρ 1, it is easy to check that the resolvent equation problem 2.17 is satisfied An Iterative Algorithm and Convergence Result We mention the following equivalence between the problem 2.14 and a fixed point problem which can be easily proved by using the definition of resolvent operator Lemma 3.1 Let u, a, x, y, w1 , w2 , where u ∈ E, wi ∈ Ti u , i 1, 2, , ∞, a ∈ B u , x ∈ C u , and y ∈ D u , is a solution of 2.14 if and only if it is a solution of the following equation: g u m a ρ,A Jη,M ·,u A g u − m a − ρ N w1 , w2 , − W x, y ma 3.1 Now, we show that the problem 2.14 is equivalent to a resolvent equation problem Lemma 3.2 Let u ∈ E, wi ∈ Ti u , i following are equivalent: i u, a, x, y, w1 , w2 , ii z, u, a, x, y, w1 , w2 , 1, 2, , ∞, a ∈ B u , x ∈ C u , y ∈ D u , then the is a solution of variational inclusion problem 2.14 , is a solution of the problem 2.17 , where z g u ma A g u −m a ρ,A − ρ N w1 , w2 , − W x, y Jη,M ·,u A g u − m a m a − ρ N w1 , w2 , − W x, y , 3.2 m a Fixed Point Theory and Applications Proof Let u, a, x, y, w1 , w2 , a solution of the problem g u be a solution of the problem 2.14 , then by Lemma 3.1, it is ρ,A Jη,M ·,u A g u − m a m a − ρ N w1 , w2 , − W x, y ma , 3.3 using the fact that ρ,A Rη,M ·,u ρ,A Rη,M ·,u z ρ,A I − A Jη,M ·,u , ρ,A Rη,M ·,u A g u − m a ρ,A I − A Jη,M ·,u A g u −m a − ρ N w1 , w2 , − W x, y − ρ N w1 , w2 , − W x, y ρ,A A g u −m a − ρN w1 , w2 , − W x, y A g u −m a − A Jη,M ·,u A g u − m a m a m a ma − ρ N w1 , w2 , − W x, y − ρ N w1 , w2 , − W x, y ma ma −A g u −m a , 3.4 which implies that N w1 , w2 , − W x, y ma ρ−1 Rη,M ·,u z ρ,A 0, 3.5 ma , 3.6 with z A g u −m a − ρ N w1 , w2 , − W x, y that is, z, u, a, x, y, w1 , w2 , is a solution of problem 2.17 Conversly, let z, u, a, x, y, w1 , w2 , be a solution of problem 2.17 , then ρ N w1 , w2 , − W x, y ρ N w1 , w2 , − W x, y m a m a ρ,A −Rη,M ·,u z , ρ,A A Jη,M ·,u z 3.7 − z, from 3.2 and 3.7 , we have ρ N w1 , w2 , − W x, y ρ,A ma A Jη,M ·,u A g u − m a − A g u −m a − ρ N w1 , w2 , − W x, y − ρ N w1 , w2 , − W x, y m a m a , 3.8 Fixed Point Theory and Applications which implies that g u m a ρ,A Jη,M ·,u A g u − m a that is, u, a, x, y, w1 , w2 , − ρ N w1 , w2 , − W x, y ma , 3.9 is a solution of 2.14 We now invoke Lemmas 3.1 and 3.2 to suggest the following iterative algorithm for solving resolvent equation problem 2.17 Algorithm 3.3 For a given z0 , u0 ∈ E, wi0 ∈ Ti u0 , i y0 ∈ D u0 Let z1 A g u0 − m a0 1, 2, , ∞, a0 ∈ B u0 , x0 ∈ C u0 , and 0 − ρ N w1 , w2 , − W x0 , y0 m a0 3.10 Take z1 , u1 ∈ E such that g u1 m a1 ρ,A Jη,M ·,u1 z1 3.11 Since for each i, wi0 ∈ Ti u0 , a0 ∈ B u0 , x0 ∈ C u0 , and y0 ∈ D u0 by Nadler’s theorem 14 there exist wi1 ∈ Ti u1 , a1 ∈ B u1 , x1 ∈ C u1 , and y1 ∈ D u1 such that wi0 − wi1 ≤ H Ti u0 , Ti u1 , a0 − a1 ≤ H B u0 , B u1 , 3.12 x0 − x1 ≤ H C u0 , C u1 , y0 − y1 ≤ H D u0 , D u1 , where H is the Housdorff metric on CB E Let z2 A g u1 − m a1 1 − ρ N w1 , w2 , − W x1 , y1 m a1 , 3.13 and take any u2 ∈ E such that g u2 m a2 ρ,A Jη,M ·,u2 z2 Continuing the above process inductively, we obtain the following 3.14 Fixed Point Theory and Applications 1, 2, , ∞, a0 ∈ B u0 , x0 ∈ C u0 , and For any z0 , u0 ∈ E, wi0 ∈ Ti u0 , i y0 ∈ D u0 Compute the sequences {zn }, {un }, {win }, i 1, 2, , ∞, {a0 }, {x0 }, {y0 } by the following iterative schemes: i g un ρ,A m an Jη,M ·,un zn , ii an ∈ B un , iii xn ∈ C un , xn − xn iv yn ∈ D un , yn − yn v for each i, win ∈ Ti un , vi zn an − an ≤ H B un , B un 1 A g un − m an where ρ > is a constant and n 3.15 , ≤ H C un , C un win − win , ≤ H D un , D un 3.16 3.17 , ≤ H Ti un , Ti un n n − ρ N w1 , w2 , − W xn , yn 3.18 , 3.19 m an , 3.20 0, 1, 2, Theorem 3.4 Let E be a real Banach space Let Ti , B, C, D : E → CB E be H-Lipschitz continuous mapping with constants δi , α, t, γ , respectively Let N E∞ E × E × E · · · → E be Lipschitz continuous with constant βi , let A, g, m : E → E be Lipschitz continuous with constants λA , λg , λm , respectively, and let A be r-strongly η-accretive mapping Suppose that η,W : E×E → E are mappings such that η is Lipschitz continuous with constant τ and W is Lipschitz continuous in both the argument with constant λW1 and λW2 , respectively Let M : E × E → 2E be A, η -accretive mapping in the first argument such that the following holds for μ > 0: ρ,η ρ,η JM ·,un zn − JM ·,un−1 zn ≤ μ un − un−1 3.21 Suppose there exists a ρ > such that ∞ λA λg λm α λA ρ ρ βi δi ρ λ W1 t λ W2 γ i r − ρm < τ 1− λ2 α2 m μ2 − 2k , r m < , λ2 α2 < ρ m 3.22 2k − μ Then, there exist z, u, ∈ E, a ∈ B E , and x ∈ C E , y ∈ D E , and wi ∈ Ti u that satisfy resolvent equation problem 2.17 The iterative sequences {zn }, {un }, {an } {xn }, {yn }, 1, 2, , ∞, n 0, 1, generated by Algorithm 3.3 converge strongly to and {win }, i z, u, a, x, y, wi , respectively Fixed Point Theory and Applications Proof From Algorithm 3.3, we have zn − zn A g un − m an n n − ρ N w1 , w2 , − W xn , yn m an − A g un−1 − m an−1 n−1 n−1 − ρ N w1 , w2 , − W xn−1 , yn−1 ≤ A g un − m an m an−1 3.23 − A g un−1 − m an−1 n n n−1 n−1 ρ N w1 , w2 , − N w1 , w2 , ρ W xn , yn − W xn−1 , yn−1 ρ m an − m an−1 By using the Lipschitz continuty of A, g, and m with constants λA , λg , and λm , respectively, and by Algorithm 3.3, we have A g un − m an − A g un−1 − m an−1 ≤ λA g un − g un−1 λA m an − m an−1 ≤ λA λg un − un−1 λA λm an − an−1 ≤ λA λg un − un−1 λA λm H B un , B un−1 ≤ λA λg un − un−1 λA λm α un − un−1 λA λg 3.24 λA λm α un − un−1 Since N is Lipschitz continuous in all the arguments with constant βi , i respectively, and using H-Lipschitz continuity of Ti ’s with constant δi , we have n n n−1 n−1 N w1 , w2 , − N w1 , w2 , n n n−1 n N w1 , w2 , − N w1 , w2 , n−1 n N w1 , w2 , n n n−1 n ≤ N w1 , w2 , − N w1 , w2 , n−1 n n−1 n−1 N w1 , w2 , − N w1 , w2 , n n−1 ≤ β1 w1 − w1 n n−1 β2 w2 − w2 ··· ··· ··· 1, 2, , 10 Fixed Point Theory and Applications ≤ ∞ βi win − win−1 i ≤ ∞ βi H Ti un , Ti un−1 i ≤ ∞ βi δi un − un−1 , n 0, 1, 2, i 3.25 Since W is a Lipschitz continuous in both the arguments with constant λW1 , λW2 respectively, and C and D are H-Lipschitz continuous with constant t and γ , respectively, we have ≤ λW2 yn − yn−1 λW1 xn − xn−1 ≤ λW2 γ un − un−1 W xn , yn − W xn−1 , yn−1 λW1 t un − un−1 λ W1 t λ W2 γ 3.26 un − un−1 Combining 3.24 , 3.25 , and 3.26 with 3.23 , we have zn − zn ≤ λA λg ∞ λA λm α un − un−1 ρ βi δi un − un−1 i un − un−1 ρ λ W1 t λ W2 γ λA λg λm α λA ρλm α un − un−1 3.27 ∞ ρ ρ βi δi ρ λ W1 t un − un−1 λ W2 γ i By using Proposition 2.3 and k-strong accretiveness of g, we have un − un−1 m an ρ,η JM ·,un zn − m an−1 ρ,η − JM ·,un−1 zn−1 − g un − un − g un−1 − un−1 ≤ m an − m an−1 ρ,η ρ,η JM ·,un zn − JM ·,un−1 zn−1 2 − g un − un − g un−1 − un−1 , j un − un−1 ≤ m an − m an−1 ρ,η ρ,η JM ·,un zn − JM ·,un−1 zn ρ,η ρ,η JM ·,un−1 zn − JM ·,un−1 zn−1 − g un − un − g un−1 − un−1 , j un − un−1 , Fixed Point Theory and Applications un − un−1 11 ρ,η ≤ λ2 α2 un − un−1 m ρ,η JM ·,un zn − JM ·,un−1 zn ρ,η ρ,η JM ·,un−1 zn − JM ·,un−1 zn−1 2 − g un − un − g un−1 − un−1 , j un − un−1 ≤ λ2 α2 un − un−1 m τ r − ρm un − un−1 un − un−1 ≤ λ2 α2 m ≤ un − un−1 ≤ μ2 − 2k λ2 α2 m zn − zn−1 μ2 τ r − ρm 2 zn − zn−1 , zn − zn−1 , − 2k τ r − ρm − 2k un − un−1 , un − un−1 τ/ r − ρm 1− μ2 un − un−1 − λ2 α2 m μ2 − 2k zn − zn−1 3.28 Using 3.28 , 3.27 becomes zn λA λg − zn ≤ λm α λA ρ ρ r − ρm that is, zn ∞ i βi δi − λ2 α2 m ρ λ W1 t λ W2 γ τ μ2 − 2k zn − zn−1 , 3.29 − zn ≤ θ zn − zn−1 , where θ λA λg λm α λA ρ r − ρm ρ 1− ∞ i βi δi λ2 α2 m ρ λ W1 t μ2 − 2k λ W2 γ τ 3.30 From 3.22 , we have θ < 1, and consequently {zn } is a Cauchy sequence in E Since E is a Banach space, there exists z ∈ E such that zn → z From 3.28 , we know that {un } is also a Cauchy sequence in E Therefore, there exists u ∈ E such that un → u Since the mappings Ti ’s, B, C and D are H-Lipschitz continuous, it follows from 3.16 – 3.19 of Algorithm 3.3 that {an }, {xn }, {yn }, and {win } are also Cauchy sequences We can assume that win → wi , an → a, xn → x, and yn → y 12 Fixed Point Theory and Applications Now, we prove that wi ∈ Ti u In fact, since win ∈ Ti ui and ≤ wi − win d win , Ti u ≤ wi − win d wi , Ti u max wi − win ≤ wi − win sup d q2 , Ti u , sup d Ti u , q1 q2 ∈Ti un q1 ∈Ti u 3.31 H Ti un , Ti u δi un − un−1 −→ n −→ ∞ , which implies that d wi , Ti u As Ti u ∈ CB E , we have wi ∈ Ti u , i 1, 2, ∞ Finally, by the continuity of A, g, m, N, and W and by Algorithm 3.3, it follows that zn −→ z ρ,η JM ·,un zn A g un − m an A g u −m a n n − ρ N w1 , w2 , − W xn , yn − ρ N w1 , w2 , − W x, y g un − m an −→ g u − m a ρ,η JM ·,u z m a , n −→ ∞ , m a 3.32 n −→ ∞ From 3.32 , and Lemma 3.2, it follows that N w1 , w2 , − W x, y ma N w1 , w2 , − W x, y that is, z, u, a, x, y, w1 , w2 , ρ−1 z − A JM ·,u z ρ,η ma ρ−1 RM ·,u z ρ,η 0, 3.33 0, is a solution of resolvent equation poblem 2.17 Acknowledgment This work is supported by Department of Science and Technology, Government of India, under Grant no SR/S4/MS: 577/09 References R Ahmad, A H Siddiqi, and Z Khan, “Proximal point algorithm for generalized multivalued nonlinear quasi-variational-like inclusions in Banach spaces,” Applied Mathematics and Computation, vol 163, no 1, pp 295–308, 2005 R Ahmad and Q H Ansari, “An iterative algorithm for generalized nonlinear variational inclusions,” Applied Mathematics Letters, vol 13, no 5, pp 23–26, 2000 R P Agarwal, N J Huang, and Y J Cho, “Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings,” Journal of Inequalities and 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vol 30, pp 475–488, 1996  ... an infinite family of set-valued mappings, and let N : E∞ E×E×E · · · → E be a nonlinear mapping Let η, W : E×E → E; A, g, m : E → E be single-valued mappings, let and B, C, D:E → CB E be set-valued. .. “The infinite family of generalized set-valued quasi-variation inclusions in Banach spaces,” Acta Analysis Functionalis Applicata, pp 1009–1327, 2008 12 S S Chang, ? ?Set-valued variational inclusions. .. of resolvent operator Lemma 3.1 Let u, a, x, y, w1 , w2 , where u ∈ E, wi ∈ Ti u , i 1, 2, , ∞, a ∈ B u , x ∈ C u , and y ∈ D u , is a solution of 2.14 if and only if it is a solution of