Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 20, Article ID 635030, 3 pages doi:0.55/20/635030 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 23@rediffmail.com Received 8 December 200; Accepted 23 December 200 Academic Editor: Qamrul Hasan Ansari Copyright q 20 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 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 2. 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,2 E resp., CB E is the family of nonempty resp., nonempty closed and bounded subsets of E, andh, is the Housdorff metric on CB E defined by H P, Q max { sup x P d x, Q, sup d ( P, y )}, y Q 2. where d x, Q inf y Q d x, y and d P, y inf x P d x, y. The normalized duality mapping J : E 2 E is defined by J x { f E : x, f x f, f x, x E }. 2.2 Definition 2.. Let E be a real Banach space. Let η : E E E; g,a : E E be the singlevalued mapping, and let M : E 2 E 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 ii the mapping g is said to be strictly accretive if g x g ( y ),j ( x y ) 0, x, y E, 2.4 and the equality hold if and only if x y, iii the mapping g is said to be k-strongly accretive k 0, if for any x, y E, there exists j x y J x y such that g x g ( y ),j ( x y ) k x y 2, 2.5 iv the mapping A is said to be r-strongly η-accretive, if there exists a constant r > 0suchthat A x A ( y ),j ( η ( x, y )) r x y 2, x, y E, 2.6 v the mapping M is said to be m-relaxed η-accretive, if there exists a constant m> 0suchthat u v, j ( η ( x, y )) m x y 2, x, y E, u M x, v M ( y ). 2.7
Fixed Point Theory and Applications 3 Definition 2.2. Let A : E E, η : E E E be the single-valued mappings. Then, a set-valued mapping M : E 2 E is called A, η -accretive if M is m-relaxed η- accretive and A ρm E E, for every ρ>0. Proposition 2.3 see 8, 9. Let E be a real Banach space, and let J : E 2 E be the normalized duality mapping. Then, for any x, y E x y 2 x 2 2 y, j ( x y ), j ( x y ) J ( x y ). 2.8 Definition 2.4. Let A : E E, W : E E E, andletn : E E E E E be the mappings. Then, i the mapping A is said to be Lipschitz continuous with constant λ A if A x A ( y ) λa x y, x, y E, 2.9 ii the mapping W is said to be Lipschitz continuous in the first argument with constant λ W if W x, W x 2, λ W x x 2, x,x 2 E. 2.0 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 β i if N,...,xi,... N (,...,y i,... ) βi xi y i, xi,y i E. 2. Definition 2.5. Let A : E E be a strictly η-accretive mapping, and let M : E 2 E be an A, η -accretive mapping. Then, the resolvent operator J ρ,a η,m : E E is defined by J ρ,a η,m u ( A ρm ) u, u E. 2.2 Proposition 2.6 see 0. 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 2 E be an A, η accretive mapping. Then the resolvent operator J ρ,a η,m : E E is τ/ r ρm -Lipschitz continuous, that is, J ρ,a η,m ρ,a ( ) τ x Jη,M y x y, x, y E, 2.3 r ρm where ρ 0,r/m is a constant.
4 Fixed Point Theory and Applications Example 2.7. Let E R, A x x, M y y,andη x, y x y for all x, y 0 E. Then, M is η-accretive. Example 2.8. Let M, : E E 2 E be r-strongly η-accretive in the first argument. Then, M is m-relaxed η-accretive for m,r r 2,forr>0.680. Let T i : E CB E, i, 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 2 E is A, η -accretive mapping in the first argument. We consider the following problem. Find u E, w i T i u, i, 2,...,, a B u, x C u, and y D u such that 0 N w,w 2,... W ( x, y ) m a M ( g u m a,u ). 2.4 The problem 2.4 is called variational-like inclusions problem. Special Cases i If W 0, m 0, then problem 2.4 reduces to the problem of finding u E, w i T i u, i, 2,..., such that 0 N w,w 2,... M ( g u,u ). 2.5 Problem 2.5 is introduced and studied by Wang. ii If W 0, m 0, N,... N,, then problem 2.4 reduces to a problem considered by Chang, et al. 2, 3 that is, find u H, w T u, w 2 T 2 u such that 0 N w,w 2 M ( g u,u ). 2.6 It is now clear that for a suitable choice of maps involved in the formulation of problem 2.4, we can drive many known variational inclusions considered and studied in the literature. In connection with problem 2.4, we consider the following resolvent equation problem. Find z, u E, w i T i u, i, 2,..., ; a B u, x C u, y D u such that N w,w 2,... W ( x, y ) m a ρ R ρ,a η,m,u z 0, 2.7 where ρ is a constant and R ρ,a ρ,a ρ,a ρ,a I A J,whereA J z A J η,m,u η,m,u η,m,u and I is the identity mapping. Equation 2.7 is called the resolvent equation problem. η,m,u z
Fixed Point Theory and Applications 5 In support of problem 2.7, we have the following example. Example 2.9. Let us suppose that E R, T i u i, i, i, 2,...,, C u {π/2}, B u 0,,andD u {}. We define for w i T i u, i, 2,...,, a B u, x C u and y D u. i N w,w 2,... min{, sin w, sin w 2,...}, ii m a sin a cos a, iii W x, y xy, iv A x x, for all x R, v M,x, for all x R, Then, for ρ, it is easy to check that the resolvent equation problem 2.7 is satisfied. 3. An Iterative Algorithm and Convergence Result We mention the following equivalence between the problem 2.4 and a fixed point problem which can be easily proved by using the definition of resolvent operator. Lemma 3.. Let u, a, x, y, w,w 2,... where u E, w i T i u, i, 2,...,, a B u, x C u, andy D u, is a solution of 2.4 if and only if it is a solution of the following equation: g u m a J ρ,a η,m,u [ A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }]. 3. Now, we show that the problem 2.4 is equivalent to a resolvent equation problem. Lemma 3.2. Let u E, w i T i u, i, 2,...,, a B u, x C u, y D u, then the following are equivalent: i u, a, x, y, w,w 2,... is a solution of variational inclusion problem 2.4, ii z, u, a, x, y, w,w 2,... is a solution of the problem 2.7, where z A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }, g u m a J ρ,a η,m,u [ A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }]. 3.2
6 Fixed Point Theory and Applications Proof. Let u, a, x, y, w,w 2,... be a solution of the problem 2.4,thenbyLemma 3.,itis a solution of the problem g u m a J ρ,a η,m,u [ A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }], 3.3 using the fact that ( ) R ρ,a η,m,u I A J ρ,a, η,m,u R ρ,a [ ( ) { η,m,u z Rρ,A η,m,u A g u m a ρ N w,w 2,... W ( x, y ) m a }] ( )) [A ( ) { g u m a ρn w,w 2,... W ( x, y ) m a }] ( I A J ρ,a η,m,u A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a } ( [ ( ) { A A g u m a ρ N w,w 2,... W ( x, y ) m a }]) J ρ,a η,m,u A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a } A ( g u m a ), 3.4 which implies that N w,w 2,... W ( x, y ) m a ρ R ρ,a η,m,u z 0, 3.5 with z A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }, 3.6 that is, z, u, a, x, y, w,w 2,... is a solution of problem 2.7. Conversly, let z, u, a, x, y, w,w 2,... be a solution of problem 2.7, then ρ { N w,w 2,... W ( x, y ) m a } R ρ,a η,m,u z, ρ { N w,w 2,... W ( x, y ) m a } [ ] A J ρ,a η,m,u z z, 3.7 from 3.2 and 3.7, wehave ρ { N w,w 2,... W ( x, y ) m a } [ ( ( ) { A A g u m a ρ N w,w 2,... W ( x, y ) m a })] J ρ,a η,m,u 3.8 [ A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }],
Fixed Point Theory and Applications 7 which implies that g u m a J ρ,a η,m,u [ A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a }], 3.9 that is, u, a, x, y, w,w 2,... is a solution of 2.4. We now invoke Lemmas 3. and 3.2 to suggest the following iterative algorithm for solving resolvent equation problem 2.7. Algorithm 3.3. For a given z 0,u 0 E, w 0 i T i u 0, i, 2,...,, a 0 B u 0, x 0 C u 0,and y 0 D u 0.Let z A ( g u 0 m a 0 ) { ) ρ N (w 0,w0 2,... W ( ) } x 0,y 0 m a0. 3.0 Take z,u E such that g u m a J ρ,a η,m,u z. 3. Since for each i, w 0 i T i u 0, a 0 B u 0,x 0 C u 0,andy 0 D u 0 by Nadler s theorem 4 there exist w i T i u, a B u,x C u,andy D u such that w 0 i w i H T i u 0,T i u, a 0 a H B u 0,B u, x 0 x H C u 0,C u, y0 y H D u0,d u, 3.2 where H is the Housdorff metric on CB E. Let z 2 A ( g u m a ) { ) ρ N (w,w 2,... W ( ) } x,y m a, 3.3 and take any u 2 E such that g u 2 m a 2 J ρ,a η,m,u 2 z 2. 3.4 Continuing the above process inductively, we obtain the following.
8 Fixed Point Theory and Applications For any z 0,u 0 E, w 0 i T i u 0, i, 2,...,, a 0 B u 0, x 0 C u 0, and y 0 D u 0. Compute the sequences {z n }, {u n }, {w n i }, i, 2,...,, {a 0}, {x 0 }, {y 0 } by the following iterative schemes: i g u n m a n J ρ,a η,m,u n z n, 3.5 ii a n B u n, a n a n H B u n,b u n, 3.6 iii x n C u n, x n x n H C u n,c u n, 3.7 iv y n D u n, y n y n H D u n,d u n, 3.8 v for each i, w n i T i u n, w n i w n H T i u n,t i u n, 3.9 i vi z n A ( g u n m a n ) ρ { N ( w n,wn 2,...) W ( x n,y n ) m an }, 3.20 where ρ>0 is a constant and n 0,, 2,... Theorem 3.4. Let E be a real Banach space. Let T i,b,c,d : E CB E be H-Lipschitz continuous mapping with constants δ i,α,t,γ, respectively. Let N E E E E E be Lipschitz continuous with constant β i,leta, g, m : E E be Lipschitz continuous with constants λ A,λ g,λ m, 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 λ W and λ W2, respectively. Let M : E E 2 E be A, η -accretive mapping in the first argument such that the following holds for μ>0: J ρ,η M,u n z n J ρ,η M,u n z n μ u n u n. 3.2 Suppose there exists a ρ>0 such that λ A λ g λ m α ( λ A ρ ) ρ β i δ i ρ ( λ W t λ W2 γ ) < ( r ρm ) τ i ( λ 2 mα 2 μ 2 2k ), m < r ρ,λ2 mα 2 < 2k μ 2. 3.22 Then, there exist z, u, E, a B E, andx C E, y D E, and w i T i u that satisfy resolvent equation problem 2.7. The iterative sequences {z n }, {u n }, {a n }{x n }, {y n }, and {w n i },i, 2,...,, n 0,,... generated by Algorithm 3.3 converge strongly to z, u, a, x, y, w i, respectively.
Fixed Point Theory and Applications 9 Proof. From Algorithm 3.3, wehave z n z n A ( g u n m a n ) ρ { N ( w n,wn 2,...) W ( x n,y n ) m an } [A ( g u n m a n ) { ( ) ρ N w n,w n 2,... W ( ) x n,y n m an }] ( A g un m a n ) ( A ( g u n m a n ) ρn ( ( w n,wn 2,...) N w n,w n 2,...) 3.23 ρ W ( xn,y n ) W ( xn,y n ) ρ m an m a n. By using the Lipschitz continuty of A, g, andm with constants λ A,λ g,andλ m, respectively, and by Algorithm 3.3, wehave A ( g un m a n ) ( A ( g u n m a n )) λ A g un g u n λa m a n m a n λ A λ g u n u n λ A λ m a n a n λ A λ g u n u n λ A λ m H B u n,b u n 3.24 λ A λ g u n u n λ A λ m α u n u n ( λ A λ g λ A λ m α ) u n u n. Since N is Lipschitz continuous in all the arguments with constant β i,i, 2,..., respectively, and using H-Lipschitz continuity of T i s with constant δ i,wehave N ( ( w n,wn 2,...) N w n,w n 2,...) N ( ( ) ( ) w n,wn 2,...) N w n,w n 2,... N w n,w n 2,... N ( ( ) w n,wn 2,...) N w n,w n 2,... N β w n ( w n ),w n 2,... N w n β 2 wn ( w n,w n 2 wn 2 2,...)
0 Fixed Point Theory and Applications w n β i i w n i i β i H T i u n,t i u n i β i δ i u n u n, n 0,, 2,... i 3.25 Since W is a Lipschitz continuous in both the arguments with constant λ W,λ W2 respectively, and C and D are H-Lipschitz continuous with constant t and γ, respectively, we have ( ) ( ) W xn,y n W xn,y n λw2yn y n λw x n x n λ W2 γ u n u n λ W t u n u n 3.26 ( λ W t λ W2 γ ) u n u n. Combining 3.24, 3.25,and 3.26 with 3.23, wehave z n z n ( λ A λ g λ A λ m α ) u n u n ρ β i δ i u n u n i ρ ( λ W t λ W2 γ ) u n u n ρλ m α u n u n 3.27 [ (λa λ g λ m α ( λ A ρ )) ρ β i δ i ρ ( λ W t λ W2 γ )] u n u n. By using Proposition 2.3 and k-strong accretiveness of g,wehave u n u n 2 i m a n J ρ,η M,u n z n m a n J ρ,η M,u n z n [ g u n u n ( g u n u n )] 2 m a n m a n 2 J ρ,η M,u n z n J ρ,η M,u n z n 2 2 g u n u n ( g u n u n ),j un u n m a n m a n 2 J ρ,η M,u n z n J ρ,η M,u n z n J ρ,η M,u n z n J ρ,η 2 g u n u n ( g u n u n ),j un u n, M,u n z n 2
Fixed Point Theory and Applications u n u n 2 λ 2 m α2 u n u n 2 J ρ,η M,u n z n J ρ,η M,u n z n 2 u n u n 2 u n u n 2 u n u n J ρ,η M,u n z n J ρ,η M,u n z n 2 2 g u n u n ( g u n u n ),j un u n λ 2 mα 2 u n u n 2 μ 2 u n u n 2 ( ) τ 2 z n z n 2 2k u n u n 2, r ρm ( ) ( ) τ 2 λ 2 m α2 μ 2 2k u n u n 2 z n z n 2, r ρm ( τ/ ( r ρm )) 2 [ ( λ 2 m α 2 μ 2 2k )] z n z n 2, τ ( r ρm ) [ ( λ 2 m α 2 μ 2 2k )] z n z n. 3.28 Using 3.28, 3.27 becomes z n z n [ λa λ g λ m α ( λ A ρ ) ρ i β iδ i ρ ( λ W t λ W2 γ )] τ ( r ρm ) ( λ 2 mα 2 μ 2 2k ) z n z n, 3.29 that is, z n z n θ z n z n, where θ [ λa λ g λ m α ( λ A ρ ) ρ i β iδ i ρ ( λ W t λ W2 γ )] τ ( r ρm ) ( λ 2 mα 2 μ 2 2k ). 3.30 From 3.22, wehaveθ <, and consequently {z n } is a Cauchy sequence in E. SinceE is a Banach space, there exists z E such that z n z. From 3.28, we know that {u n } is also a Cauchy sequence in E. Therefore, there exists u E such that u n u. Since the mappings T i s, B, C and D are H-Lipschitz continuous, it follows from 3.6 3.9 of Algorithm 3.3 that {a n }, {x n }, {y n },and{w n i } are also Cauchy sequences. We can assume that w n i w i,a n a, x n x, andy n y.
2 Fixed Point Theory and Applications Now, we prove that w i T i u. Infact,sincew n i d w i,t i u wi w n i d ( w n i,t i u ) T i u i and { wi w n i max sup d ( q 2,T i u ), sup d ( ) } T i u,q q 2 T i u n q T i u wi w n i H Ti u n,t i u wi w n i δi u n u n 0 n, 3.3 which implies that d w i,t i u 0. As T i u CB E, wehavew i T i u,i, 2,.... Finally, by the continuity of A, g, m, N, andw and by Algorithm 3.3, it follows that z n A ( g u n m a n ) ρ { N ( w n,wn 2,...) W ( x n,y n ) m a }, z A ( g u m a ) ρ { N w,w 2,... W ( x, y ) m a } n, J ρ,η M,u n z n g u n m a n g u m a J ρ,η M,u z n. From 3.32, andlemma 3.2, it follows that N w,w 2,... W ( x, y ) m a ρ [ ( )] z A J ρ,η M,u z 0, N w,w 2,... W ( x, y ) 3.33 m a ρ R ρ,η M,u z 0, 3.32 that is, z, u, a, x, y, w,w 2,... is a solution of resolvent equation poblem 2.7. 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. 63, no., pp. 295 308, 2005. 2 R. Ahmad and Q. H. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Applied Mathematics Letters, vol. 3, no. 5, pp. 23 26, 2000. 3 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 Applications, vol. 7, no. 6, pp. 807 828, 2002. 4 R. P. Agarwal, Y. J. Cho, and N. J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Applied Mathematics Letters, vol. 3, no. 6, pp. 9 24, 2000. 5 S.-S Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publishers, Huntington, NY, USA, 2002. 6 J.-Y. Chen, N.-C. Wong, and J.-C. Yao, Algorithm for generalized co-complementarity problems in Banach spaces, Computers & Mathematics with Applications, vol. 43, no. -2, pp. 49 54, 2002.
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