New Iterative Algorithm for Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces

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1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 20, HIKARI Ltd, New Iterative Algorithm for Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces Fuhai Wang Department of Mathematics and Physics North China Electric Power University Baoding , China Minjiang Chen and Jianmin Song School of Mathematics and Sciences Shijiazhuang University of Economics Shijiazhuang , China Copyright c 2014 Fuhai Wang, Minjiang Chen and Jianmin Song. 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. Abstract In this paper, we introduce a new iterative scheme with a countable family of nonexpansive mappings for variational inequality problem in a Hilbert space and prove a strong convergence theorem for the iterative scheme. Mathematics Subject Classification: 4705; 47H09; 47J25; 47N10 Keywords: Nonexpansive mappings; Hilbert spaces; Strong convergence; Variational inequality 1. Introduction Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let F : H H be a nonlinear operator. The variational inequality problem on F is to find a point x C such that F (x ),x x 0, x C.

2 996 Fuhai Wang, Minjiang Chen and Jianmin Song The variational inequality problem is denoted by VI(F, C) [5]. F is called to be κ-lischitzian and η-strongly monotone if there exist constants κ, η > 0 such that Fx Fy κ x y, for all x, y H, and Fx Fy,x y η x y 2, for all x, y H, respectively. It is well known that if F is strongly monotone and Lipschitzian on C, then VI(F, C) has a unique solution. An important problem is how to find a solution of VI(F, C). It is known that the VI(F, C) is equivalent to the fixed point equation [16] where P C is the projection from H onto C; i.e., u = P C (u μf (u )), (1.1) P C x = min x y, x H, y C and where μ>0 is an arbitrarily fixed constant. So, the fixed point methods can be implemented to find a solution of the VI(F, C). The fixed point formulation (1.1) involves the projection P C, which may not be easy to compute, due to the complexity of the convex set C. Hence, in order to reduce the complexity probably caused by the projection P C, Yamada [16] introduced the hybrid steepest-descent method for solving the VI(F, C) by replacing P C with a nonexpansive mapping T. Recall that a mapping T : H H is called nonexpansive if for all x, y H, one holds Tx Ty x y. The set of fixed points of T is denoted by Fix(T ). More precisely, Yamada [16] gave the following iterative scheme: u 0 H, u n+1 = Tu n λ n+1 μf (Tu n ), n 0, (1.2) where T : H H is a nonexpansive mapping, F is η-strongly monotone and κ-lipschitzian on C = F (T ), {λ n } is a sequence in (0, 1) and μ is a fixed number with 0 <μ<2η/κ 2. Yamada [16] proved that if the sequence {λ n } satisfies the conditions: (L1) lim λ n =0, (L2) λ n =, (L3) lim (λ n λ n+1 )/λ 2 n+1 =0, then {u n } converges strongly to the unique solution of VI(F, C). Since Yamada s hybrid steepest-descent method for solving variational inequalities [16], there are much research on this aspect; see, e.g., [15, 19, 20, 21].

3 Approximation of fixed points 997 In 2011, Buong and Duong [2] introduced a new iterative algorithm, based on a combination of thy hybrid steepest-descent method for variational inequalities with the Krasnosel skii-mann type algorithm for fixed point problems. Very recently Zhou and Wang [23] simplified the algorithm of Buong and Duong and proved a strong convergence theorem for the variational inequality problem and fixed point problem on finite nonexpansive mappings in Hilbert space. In this paper, motivated by the work of Zhou and Wang [23], we introduce a new iterative algorithm for solving the solution of variational inequality problem and prove the solution is the common fixed point of a countable family of nonexpansive mappings in Hilbert space. 1. Preliminaries Let H be a Hilbert space. Let T : H H be nonexpansive mapping and let F : H H be a κ-lipschitzian and η-strong monotone nonlinear operator. Lemma 2.1 ([16]). Assume λ (0, 1) and μ (0, 2η/κ 2 ). Define a mapping T λ : H H by T λ x = Tx λμf (Tx) x H. Then T λ x T λ y (1 λτ) x y for all x, y H, where τ = 1 1 μ(2η μκ2 ) (0, 1). Obviously, if T = I, then T λ = I λμf and by Lemma 2.1 we have T λ x T λ y = (I λμf )x (I λμf )y (1 λτ) x y for all x, y H. Lemma 2.2 ([7]). Let {s n }, {c n } be the sequences of nonnegative real numbers and let {a n } (0, 1). Suppose {b n } is a real number sequence such that s n+1 (1 a n )s n + b n + c n, n 0. Assume n=0 c n <. Then the following results hold: (1) If b n βa n where (β 0), then {s n } is a bounded sequence. (2) If we have b n a n = and lim sup 0, a n=0 n then lim s n =0. Let T : H H be a nonexpansive mapping. Let α (0, 1). Define the mapping S = αi +(1 α)t. Then the mapping S is said to be an averaged mapping with Fix(T ) = Fix(S). Furthermore, on a family of averaged mappings, we have the following conclusion. Lemma 2.3 ([1, 12]). Let {T i } N be the averaged mappings on H and assume that N Fix(T i ). Then N Fix(T i )=Fix(T 1...sT N ).

4 998 Fuhai Wang, Minjiang Chen and Jianmin Song Lemma 2.4 ([13]). Let {a n } be a sequence of nonnegative real numbers satisfying the following condition a n+1 (1 b n )a n + b n c n, where {b n } and {c n } are sequences of real numbers such that b n [0, 1], b n = and n=0 lim sup c n 0. Then lim a n =0. 3. Main results Theorem 3.1. Let H be a real Hilbert space and F : H H be an L-Lipschitz continuous and η-strongly monotone mapping. Let {T i } be a countable family of nonexpansive of H such that C = Fix(T i). Let {λ n }, {β n }, {α n } be sequences in (0, 1). Assume that {β n } is strictly decreasing and let β 0 =1and μ (0, 2η/L 2 ). Suppose that the following conditions hold: lim λ n =0, λ n =, λ n 1 lim λ n =1and β n β n+1 <. Then the sequence {x n } generated by the following manner: x 1 H, x n+1 =(I λ n μf )(β n x n + (β i 1 β i )Π i j=1u j x n ), n 1, (3.1) where U i = α i I +(1 α i )T i, converges strongly to the unique solution x of the variational inequality: F (x ),x x 0, x C. (3.2)

5 Proof. Approximation of fixed points 999 x n+1 p = (I λ n μf )(β n x n + = (I λ n μf )(β n x n + (I λ n μf )(β n x n + (1 τλ n ) β n x n + (β i 1 β i )Π i j=1 U ix n ) p (β i 1 β i )Π i j=1 U ix n ) (I λμf )p + λ n μf (p) (β i 1 β i )Π i j=1u i x n ) (I λμf )p + λ n μ F (p) (β i 1 β i )Π i j=1 U ix n p + λ n μ F (p) (1 τλ n )[β n x n p +(1 β n ) x n p ]+λ n μ F (p) μ =(1 τλ n ) x n p + τλ n F (p) τ max{ x 0 p, μ F (p) }. τ So, {x n } is bounded and so are {U i x n } for each i 1. Next we prove that x n+1 x n 0as. Let y n = β n x n + n (β i 1 β i )W i x n, where W i =Π i j=1 U j for each i 1. We rewrite (3.1) as follows: Then y n y n 1 = β n x n + x n+1 =(I λ n μf )y n, n 1, (3.3) n 1 (β i 1 β i )W i x n (β n 1 x n 1 + (β i 1 β i )W i x n 1 ) = β n (x n x n 1 )+(β n β n 1 )x n 1 + (β i 1 β i )(W i x n W i x n 1 ) +(β n 1 β n )W n x n 1 n 1 β n x n x n 1 + β n 1 β n x n 1 + (β i 1 β i ) W i x n W i x n 1 + β n 1 β n W n x n 1 n 1 β n x n x n 1 + β n 1 β n x n 1 + (β i 1 β i ) x n x n 1 + β n 1 β n W n x n 1 = β n x n x n 1 + β n 1 β n ( x n 1 + W n x n 1 )+(1 β n ) x n x n 1 = x n x n 1 + β n 1 β n M,

6 1000 Fuhai Wang, Minjiang Chen and Jianmin Song where M = sup n 1 { x n 1 + W n x n 1 }. Then x n+1 x n = (I λ n μf )y n (I λ n 1 μf )y n 1 = (I λ n μf )(y n y n 1 )+((I λ n μf ) (I λ n 1 μf ))y n 1 = (I λ n μf )(y n y n 1 )+(λ n 1 μf λ n μf )y n 1 = (I λ n μf )(y n y n 1 )+(λ n 1 λ n )μf y n (1 λ n τ) y n y n 1 + λ n 1 λ n μ Fy n (1 λ n τ)( x n x n 1 + β n 1 β n M)+ λ n 1 λ n μ Fy n (1 λ n τ) x n x n 1 + β n 1 β n M + λ n 1 λ n μm, where M = sup n 1 { Fy n }. By Lemma 2.2 we conclude that Since λ n 0asn, it follows from (3.3) that Furthermore, by (3.4) and (3.5) we get lim x n+1 x n =0. (3.4) x n+1 y n 0, as n. (3.5) lim x n y n =0. (3.6) Note that y n x n = n (β i 1 β i )(W i x n x n ) and {β n } is a strictly decreasing sequence. So, for each i 1, we have W i x n x n (β i 1 β i ) W i x n x n = y n x n 0, as n. (3.7) Next we prove that lim sup F (x ),x n+1 x 0. To prove this, we pick a subsequence {x ni } of {x n } such that lim sup F (x ),x n x = lim F (x ),x ni x. i Without loss of generality, we may further assume that x ni ˆx for some ˆx C. By (3.7) and demiclosed principle we get ˆx Fix(W i ) for each i 1. Furthermore, by the definition of W i and Lemma 2.3 we conclude that ˆx j=1fix(t j ). Since x is the unique solution of (3.2), we obtain lim sup F (x ),x x n = lim F (x ),x ˆx 0. (3.8) i Finally, we prove that x n x as n.

7 Approximation of fixed points 1001 By virtue of (3.3) and Lemma 2.1, we have x n+1 x 2 = (I λ n μf )y n x 2 = (I λ n μf )y n (I λ n μf )x λ n μf (x ) 2 = (I λ n μf )y n (I λ n μf )x 2 +2λ n μ F (x ), (I λ n μf )x (I λ n μf )y n + λ 2 nμ 2 F (x ) 2 (1 τλ n ) x n x 2 +2λ n μ F (x ),x x n +2λ n μ F (x ) x n y n +2λ 2 n μ2 ( F (x ) + F (y n ) )+λ 2 n μ2 F (x ) 2 =(1 τλ n ) x n x 2 +2λ n μ F (x ),x x n +2λ n μ F (x ) x n y n + λ 2 n μ2 (2 F (x ) +2 F (y n ) + F (x ) 2 ) (1 τλ n ) x n x 2 + λ n τc n, where c n = μ ( 2 F (x ),x x n +2 F (x ) x n y n + λ n μl ) τ and L = sup(2 F (x ) +2 F (y n ) + F (x ) 2 ) is a fixed constant. n 1 From (3.6)-(3.8), we know that lim sup c n 0. By Lemma 2.4, we conclude that x n x as n. This completes the proof. Corollary 3.1. Let H be a real Hilbert space and F : H H be an L-Lipschitz continuous and η-strongly monotone mapping. Let {T i } N be a family of nonexpansive of H such that C = N Fix(T i). Let {λ n }, {β n} and {α n } be sequences in (0, 1). Assume that μ (0, 2η/L 2 ). Suppose that the following conditions hold: lim λ λ n 1 n =0, λ n =, lim =1and β n β n+1 <. λ n Then the sequence {x n } generated by the following manner: x 1 H, x 2 =(I λ 1 μf )(β 1 x 1 +(1 β 1 )U i x 1 ), x 3 =(I λ 2 μf )(β 2 x 2 +(1 β 1 )U 1 x 2 +(β 1 β 2 )U 1 U 2 x 2 ),. x N+1 =(I λ N μf )(β N x N +(1 β 1 )U 1 x N +(β 1 β 2 )U 1 U 2 x N +...s +(β N 1 β N )U 1 U 2...sU N x N ), x n+1 =(I λ n μf )(β n x n +(1 β 1 )U 1 x n +(β 1 β 2 )U 1 U 2 x n +...s +(β N 2 β N 1 )U 1 U 2...sU N 1 x n +(β N 1 β n )U 1 U 2...sU N x n ), n >N,

8 1002 Fuhai Wang, Minjiang Chen and Jianmin Song where U i = α i I +(1 α i )T i (i =1, 2,...s, N), converges strongly to the unique solution x of the variational inequality: F (x ),x x 0, x C. Corollary 3.2. Let H be a real Hilbert space and F : H H be an L- Lipschitz continuous and η-strongly monotone mapping. Let T be a nonexpansive of H such that C = Fix(T ). Let {λ n } and {β n } be sequences in (0, 1). Assume that μ (0, 2η/L 2 ). Suppose that the following conditions hold: lim λ n =0, λ n =, λ n 1 lim λ n =1and β n β n+1 <. Then the sequence {x n } generated by the following manner: x 1 H, x n+1 =(I λ n μf )(β n x n +(1 β n )Ux n ), n 1, where U = αi +(1 α)t and 0 <α<1, converges strongly to the unique solution x of the variational inequality: F (x ),x x 0, x C. Acknowledgements This work is supported by the Scientific Research Fund of Hebei Provincial Education Department (Grant No. Z ), Multi-level Advanced Mathematics Research Fund (Grant No. GH ) and HeiBei Education Department (Grant No ). References [1] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20 (2004) [2] D. Buong, L.T. Duong, An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces, J. Optim. Theory Appl. 151 (2011) [3] F. Cianciaruso, G. Marino, L. Muglia, Y. Yao, On a two-steps algorithm for hierarchical fixed points problems and variational inequalities, Journal Inequality Application, 2009 (2009), doi: /2009/208692, 13 pages, Article ID [4] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, [5] D. Kinderlehrer and G. Stampacchia, An introduction to Variational inequalities and their applications, Academic Press, New York, NY, [6] P.E. Mainge, A. Moudafi, Strong convergence ofan iterative method for hierarchical fixed point problems, Pacific Journal Optimization, vol. 3, pp , [7] P.E. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, Journal Mathematics Analysis and Application, vol. 325, pp , [8] G. Marino, H.K. Xu, Explicit hierarchical fixed point approach to variational inequalities, Journal of Optimization Theory and Applications, vol. 149, pp , [9] G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal Mathematics Analysis and Application, vol. 318, pp , 2006.

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