An iterative method for fixed point problems and variational inequality problems

Transcription

1 Mathematical Communications 12(2007), An iterative method for fixed point problems and variational inequality problems Muhammad Aslam Noor, Yonghong Yao, Rudong Chen and Yeong-Cheng Liou Abstract. In this paper, we present an iterative method for fixed point problems and variational inequality problems. Our method is based on the so-called extragradient method and viscosity approximation method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping. Key words: variational inequality, fixed point, monotone mapping, nonexpansive mapping AMS subject classifications: 49J30, 47H10, 47H17 Received March 12, 2007 Accepted April 27, Introduction Let C be a closed convex subset ofa real Hilbert space H. A mapping A of C into H is called monotone if Ax Ay, x y 0, for all x, y C. A is called α-inverse-strongly-monotone ifthere exists a positive real number α such that Ax Ay, x y α Ax Ay 2, for all x, y C. It is clear that an α-inverse-strongly-monotone mapping A is monotone and Lipschitz continuous. The research is supported by the Higher Education Commission, Pakistan, through research grant: No: I-28/HEC/HRD/2005/90. The research is also supported by the grant NSC E Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, noormaslam@hotmail.com Department of Mathematics, Tianjin Polytechnic University, Tianjin , China, yuyanrong@tjpu.edu.cn Department of Mathematics, Tianjin Polytechnic University, Tianjin , China, chenrd@tjpu.edu.cn Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan, simplex liou@hotmail.com

2 122 M. A. Noor, Y. Yao, R. Chen and Y.-C. Liou It is well known that the variational inequality problem VI(A,C) is to find x C such that VI(A,C): Ax, y x 0, for all y C (see [1-3]). The variational inequality has been extensively studied in the literature; see, e.g., [9-20] and the references therein. A mapping S of C into itselfis called nonexpansive if Sx Sy x y, for all x, y C. We denote by F (S) the set offixed points ofs. Recall that a mapping f : C C is called contractive ifthere exists a constant β (0, 1) such that f(x) f(y) β x y, x, y C. For finding an element of F (S) VI(A, C) under the assumption that a set C H is closed and convex, a mapping S of C into itselfis nonexpansive and a mapping A of C into H is α-inverse-strongly-monotone, Takahashi and Toyoda [4] introduced the following iterative scheme: x n+1 = α n x n +(1 α n )SP C (x n λ n Ax n ), (1) for every n =0, 1, 2,,whereP C is the metric projection of H onto C, x 0 = x C, {α n } is a sequence in (0, 1), and {λ n } is a sequence in (0, 2α). They showed that, if F (S) VI(A, C) is nonempty, then the sequence {x n } generated by (1) converges weakly to some z F (S) VI(A, C). Recently, Nadezhkina and Takahashi [12] introduced a so-called extragradient method motivated by the idea ofkorpelevich [5] for finding a common element ofthe set offixed points ofa nonexpansive mapping and the set ofsolutions ofa variational inequality problem. They obtained the following weak convergence theorem. Theorem NT ([12]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a monotone, k-lipschitz continuous mapping and S : C C be a nonexpansive mapping such that F (S) VI(A, C). Letthe sequences {x n }, {y n } be generated by x 0 = x H, y n = P C (x n λ n Ax n ), (2) x n+1 = α n x n +(1 α n )SP C (x n λ n Ay n ) n 0, where {λ n } [a, b] for some a, b (0, 1/k) and {α n } [c, d] for some c, d (0, 1). Then the sequences {x n }, {y n } converge weakly to the same point P F (S) V I(A,C) (x 0 ). Recently, inspired by Nadezhkina and Takahashi s results, Zeng and Yao [13] introduced another iterative scheme for finding a common element of the set of fixed points ofa nonexpansive mapping and the set ofsolutions ofa variational inequality problem. They obtained the following strong convergence theorem. Theorem ZY1 ([13]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a monotone, k-lipschitz continuous mapping

3 An iterative method 123 and S : C C be a nonexpansive mapping such that F (S) VI(A, C). Letthe sequences {x n }, {y n } be generated by x 0 = x H, y n = P C (x n λ n Ax n ), (3) x n+1 = α n x 0 +(1 α n )SP C (x n λ n Ay n ) n 0, where {λ n } and {α n } satisfy the conditions: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) {α n } (0, 1), n=0 α n =, lim n α n =0. Then the sequences {x n } and {y n } converge strongly to the same point P F (S) VI(A,C) (x 0 ) provided lim x n+1 x n =0. (4) n Further, very recently, Zeng and Yao [14] introduced an extragradient-like approximation method basing on the extragradient method and viscosity approximation method and obtained the following very interesting result. Theorem ZY2 ([14]). Let C be a nonempty closed convex subset of a real Hilbert space. Let f : C C be a contractive mapping, A : C H be a monotone, L-Lipschitz continuous mapping and S : C C be a nonexpansive mapping such that F (S) VI(A, C). Let{x n }, {y n } be the sequence generated by x 0 = x C, y n =(1 γ n )x n + P C (x n λ n Ax n ), (5) x n+1 =(1 α n β n )x n + α n f(y n )+β n SP C (x n λ n Ay n ) n 0, where {λ n } is a sequence in (0, 1) with n=0 λ n <, and{α n }, {β n }, {γ n } are three sequences in [0, 1] satisfying the conditions: (i) α n + β n 1 for all n 0; (ii) lim n α n =0, n=0 α n = ; (iii) 0 < lim inf n β n lim sup n β n < 1. Then the sequences {x n }, {y n } converge strongly to q = P F (S) VI(A,C) f(q) if and only if {Ax n } is bounded and lim inf Ax n,y x n 0, y C. (6) n Remark 1. (1) The authors think that all of the above results are very interesting and important. (2) We note that the iterative scheme (2) in Theorem NT has only weak convergence. The iterative scheme (3) in Theorem ZY1 has strong convergence but imposed the assumption (4) on the sequence {x n }. The iterative scheme (5) has strong convergence but also imposed the assumption (6) on the sequence {x n }.

4 124 M. A. Noor, Y. Yao, R. Chen and Y.-C. Liou This natural bring us the following question? Question 2. Could we construct an iterative algorithm to approximate the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping without any assumptions on the sequence {x n }? In this paper, motivated by the iterative schemes (2), (3) and (5), we introduced an new iterative method for finding a common element of the set of fixed points ofa nonexpansive mapping and the set ofsolutions ofthe variational inequality problem for monotone mapping. We obtain a strong convergence theorem under the some mild conditions. 2. Preliminaries Let H be a real Hilbert space with inner product, and norm and let C be a closed convex subset of H. It is well known that, for any x H, thereexists unique y 0 C such that x y 0 =inf{ x y : y C}. We denote y 0 by P C x,wherep C is called the metric projection of H onto C. The metric projection P C of H onto C has the following basic properties: (i) P C x P C y x y for all x, y H, (ii) x y, P C x P C y P C x P C y 2 for every x, y H, (iii) x P C x, y P C x 0 for all x H, y C, (iv) x y 2 x P C x 2 + y P C x 2 for all x H, y C. Such properties of P C will be crucial in the proofofour main results. Let A be a monotone mapping of C into H. In the context ofthe variational inequality problem, it is easy to see from (iv) that x VI(A, C) x = P C (x λax ), λ >0. A set-valued mapping T : H 2 H is called monotone if, for all x, y H, f Tx and g Ty imply x y, f g 0. A monotone mapping T : H 2 H is maximal ifits graph G(T ) is not properly contained in the graph ofany other monotone mapping. It is known that a monotone mapping T is maximal if and only if, for (x, f) H H, x y, f g 0 for every (y, g) G(T ) implies f Tx.LetA be a monotone mapping of C into H and let N C v be the normal cone to C at v C; i.e., N C v = {w H : v u, w 0, u C}. Define { Av + NC v, if v C, Tv =, if v/ C.

5 An iterative method 125 Then T is maximal monotone and 0 Tv ifand only ifv VI(C, A) (see[6]). Now, we introduce several lemmas for our main results in this paper. The first lemma is an immediate consequence ofan inner product. The second lemma is well known demiclosedness principle. Lemma 2.1. In a real Hilbert space H, there holds the inequality x + y 2 x 2 +2 y, x + y, x, y H. Lemma 2.2. Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset of C of a real Hilbert space H. If F (S), then I S is demiclosed; that is, whenever {x n } is a sequence in C weakly converging to some x C and the sequence {(I S)x n } strongly converges to some y, it follows that (I S)x = y. HereI is the identity operator of H. Lemma 2.3. ([7]) Let {x n } and {y n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0, 1] with 0 < lim inf n β n lim sup n β n < 1. Suppose x n+1 =(1 β n )y n + β n x n for all integers n 0 and lim sup n ( y n+1 y n x n+1 x n ) 0. Then,lim n y n x n =0. Lemma 2.4. ([8]) Assume {a n } is a sequence of nonnegative real numbers such that a n+1 (1 σ n )a n + δ n, where {σ n } is a sequence in (0, 1) and {δ n } is a sequence such that (1) n=1 σ n = ; (2) lim sup n δ n /σ n 0 or n=1 δ n <. Then lim n a n =0. 3. Main results In this section, we will state and prove our main results. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone, L-Lipschitz continuous mapping of C into H, S be a nonexpansive mapping of C into itself such that F (S) VI(A, C) and f : C C be a contraction. For given x 0 C arbitrary, let the sequences {x n }, {y n }, {z n } be generated by z n = P C (x n λ n Ax n ), y n =(1 γ n )x n + γ n P C (x n λ n Az n ), (7) x n+1 =(1 α n )x n + α n S[β n f(x n )+(1 β n )y n ], where {λ n } is a sequence in (0, 1) with lim n λ n =0,and{α n }, {β n }, {γ n } are three sequences in [0, 1] satisfying the conditions: (i) lim n β n =0, n=0 β n =, (ii) 0 < lim inf n α n lim sup n α n < 1;

6 126 M. A. Noor, Y. Yao, R. Chen and Y.-C. Liou (iii) lim n (γ n+1 γ n )=0. Then the sequences {x n }, {y n }, {z n } converge strongly to the same point q = P F (S) VI(A,C) f(q). Proof. We divide the proofinto several steps. Step 1. {x n } is bounded. Let x F (S) VI(A, C), then x = P C (x λ n Ax ). Put t n = P C (x n λ n Az n ). Substituting x by x n λ n Az n and y by x in (iv), we have t n x 2 x n λ n Az n x 2 x n λ n Az n t n 2 = x n x 2 2λ n Az n,x n x + λ 2 n Az n 2 x n t n 2 +2λ n Az n,x n t n λ 2 n Az n 2 = x n x 2 +2λ n Az n,x t n x n t n 2 (8) = x n x 2 x n t n 2 +2λ n Az n Ax,x z n +2λ n Ax,x z n +2λ n Az n,z n t n. Using the fact that A is monotonic and x is a solution ofthe variational inequality problem VI(A,C), we have Az n Ax,x z n 0 and Ax,x z n 0. (9) It follows from (8) and (9) that t n x 2 x n x 2 x n t n 2 +2λ n Az n,z n t n = x n x 2 (x n z n )+(z n t n ) 2 +2λ n Az n,z n t n = x n x 2 x n z n 2 2 x n z n,z n t n (10) z n t n 2 +2λ n Az n,z n t n = x n x 2 x n z n 2 z n t n 2 +2 x n λ n Az n z n,t n z n. Substituting x by x n λ n Ax n and y by t n in (iii), we have x n λ n Ax n z n,t n z n 0. (11) It follows that x n λ n Az n z n,t n z n = x n λ n Ax n z n,t n z n + λ n Ax n λ n Az n,t n z n (12) λ n Ax n λ n Az n,t n z n λ n L x n z n t n z n.

7 An iterative method 127 By (10) and (12), we obtain t n x 2 x n x 2 x n z n 2 z n t n 2 +2λ n L x n z n t n z n x n x 2 x n z n 2 z n t n 2 +λ n L( x n z n 2 + z n t n 2 ) (13) x n x 2 +(λ n L 1) x n z n 2 +(λ n L 1) z n t n 2. Since λ n 0asn, there exists a positive integer N 0 such that λ n L when n N 0. It follows from (13) that t n x x n x. From (7), we have y n x = (1 γ n )(x n x )+γ n (t n x ) (1 γ n ) x n x + γ n x n x = x n x. Write W n = β n f(x n )+(1 β n )y n,thenwehave W n x = β n (f(x n ) x )+(1 β n )(y n x ) β n f(x n ) x +(1 β n ) y n x β n f(x n ) f(x ) + β n f(x ) x +(1 β n ) x n x ββ n x n x + β n f(x ) x +(1 β n ) x n x = β n f(x ) x +[1 (1 β)β n ] x n x. From (7), we have x n+1 x = (1 α n )(x n x )+α n (SW n Sx ) (1 α n ) x n x + α n W n x (1 α n ) x n x + α n β n f(x ) x d + α n [1 (1 β)β n ] x n x =[1 (1 β)α n β n ] x n x 1 +(1 β)α n β n 1 β f(x ) x max{ x 0 x 1, 1 β f(x ) x }. Therefore, {x n } is bounded. Hence {t n }, {St n }, {Ax n } and {Az n } are also bounded. Step 2. lim n x n+1 x n =0. For all x, y C, weget (I λ n A)x (I λ n A)y x y + λ n Ax Ay (1 + Lλ n ) x y. (14)

8 128 M. A. Noor, Y. Yao, R. Chen and Y.-C. Liou By (7) and (14), we have t n+1 t n = P C (x n+1 λ n+1 Az n+1 ) P C (x n λ n Az n ) (x n+1 λ n+1 Az n+1 ) (x n λ n Az n ) = (x n+1 λ n+1 Ax n+1 ) (x n λ n+1 Ax n ) +λ n+1 (Ax n+1 Az n+1 Ax n )+λ n Az n (x n+1 λ n+1 Ax n+1 ) (x n λ n+1 Ax n ) (15) +λ n+1 ( Ax n+1 + Az n+1 + Ax n )+λ n Az n (1 + λ n+1 L) x n+1 x n + λ n+1 ( Ax n+1 + Az n+1 + Ax n )+λ n Az n. Put u n = SW n,thenx n+1 =(1 α n )x n + α n u n. We can obtain u n+1 u n = SW n+1 SW n W n+1 W n = β n+1 f(x n+1 )+(1 β n+1 )y n+1 β n f(x n ) (1 β n )y n (16) β n+1 ( f(x n+1 ) + y n+1 )+β n ( f(x n ) + y n ) + y n+1 y n We note that y n+1 y n = (1 γ n+1 )x n+1 + γ n+1 t n+1 (1 γ n )x n γ n t n = (1 γ n+1 )(x n+1 x n )+(γ n γ n+1 )x n +γ n+1 (t n+1 t n )+(γ n+1 γ n )t n (17) (1 γ n+1 ) x n+1 x n + γ n+1 γ n x n +γ n+1 t n+1 t n + γ n+1 γ n t n Combining (15) and (17), we have y n+1 y n x n+1 x n + γ n+1 γ n ( x n + t n )+λ n Az n +λ n+1 (L x n+1 x n + Ax n+1 + Az n+1 + Ax n ), this together with (16) implies that u n+1 u n x n+1 x n β n+1 ( f(x n+1 ) + y n+1 )+β n ( f(x n ) + y n ) + γ n+1 γ n ( x n + t n )+λ n Az n +λ n+1 (L x n+1 x n + Ax n+1 + Az n+1 + Ax n ). It follows that lim sup( u n+1 u n x n+1 x n ) 0. n Hence by Lemma 2.3, weobtain u n x n 0asn.Consequently, lim n x n+1 x n = lim n α n u n x n =0.

9 An iterative method 129 Noting that (15), we also have t n+1 t n 0asn. Step 3. lim n Sx n x n = lim n St n t n =0. Indeed, we observe that Sx n x n Sx n SW n + SW n x n x n W n + u n x n β n f(x n ) x n +(1 β n ) y n x n + u n x n. Noting that x n = P C (x n ), then we have t n x n = P C (x n λ n Az n ) P C x n λ n Az n 0, and hence y n x n = γ n (t n x n ) γ n t n x n 0. Thus from the last three inequalities we conclude that Sx n x n 0. At the same time, we have St n t n St n Sx n + Sx n x n + x n t n 2 x n t n + Sx n x n 0. Step 4. lim sup n f(q) q, x n q 0. Indeed, we pick a subsequence {x ni } of {x n } so that lim sup f(q) q, x n q = lim f(q) q, x ni q. (18) n i Without loss ofgenerality, let x ni ˆx C. Then (18) reduces to lim sup f(q) q, x n q = f(q) q, ˆx q. n In order to show f(q) q, ˆx q 0, it suffices to show that ˆx F (S) VI(A, C). Note that by Lemma 2.2 and Step 3, we have ˆx F (S). Now we claim that ˆx VI(A, C). Let { Av + NC v, if v C, Tv =, if v/ C. Then T is maximal monotone and 0 Tv ifand only ifv VI(C, A). Let (v, w) G(T ). Then we have w Tv = Av + N C v and hence w Av N C v.thereforewe have v y, w Av 0 for all y C. In particular, taking y = x ni,weget v ˆx, w = lim inf v x n i,w i lim inf v x n i,av i = lim inf [ v x n i,av Ax ni + v x ni,ax ni ] i lim inf v x n i,ax ni i lim inf v x n,ax n i 0

10 130 M. A. Noor, Y. Yao, R. Chen and Y.-C. Liou and so v ˆx, w 0. Since T is maximal monotone, we have ˆx T 1 0and hence ˆx VI(A, C). This shows that ˆx F (S) VI(A, C). Therefore, we derive f(q) q, ˆx q 0. Then we obtain lim sup n f(q) q, x n q 0. At the same time, it is easily seen that W n x n 0, it follows that lim sup f(q) q, W n q 0. (19) n Step 5. lim n x n q =0whereq = P F (S) VI(A,C) f(q). First we observe that f(x n ) f(q) W n q β x n q β n (f(x n ) q)+(1 β n )(y n q) β n β x n q f(x n ) q +(1 β n )β x n q y n q (20) β n β x n q f(x n ) q +(1 β n )β x n q 2 β n x n q f(x n ) q + β x n q 2. From (7), (20) and Lemma 2.1, weget x n+1 q 2 = (1 α n )(x n q)+α n (SW n q) 2 (1 α n ) x n q 2 + α n W n q 2 =(1 α n ) x n q 2 + α n β n (f(x n ) q)+(1 β n )(y n q) 2 (1 α n ) x n q 2 + α n [(1 β n ) 2 y n q 2 +2β n f(x n ) q, W n q ] (1 2α n β n ) x n q 2 +2α n β n f(x n ) f(q),w n q +2α n β n f(q) q, W n q + α n βn 2 y n q 2 (1 2α n β n ) x n q 2 +2α n β n f(x n ) f(q) W n q +2α n β n f(q) q, W n q + α n βn 2 y n q 2 [1 2(1 β)α n β n ] x n q 2 +2α n βn x 2 n q f(x n ) q +2α n β n f(q) q, W n q + α n βn 2 y n q 2 =[1 2(1 β)α n β n ] x n q (1 β)α n β n { 1 β f(q) q, W n q + β n 1 β x β n n q f(x n ) q + 2(1 β) y n q 2 } =(1 σ n ) x n q 2 + δ n, (21) where σ n =2(1 β)α n β n and δ n =2(1 β)α n β n { 1 1 β f(q) q, W n q + βn 1 β x n q f(x n ) q + βn 2(1 β) y n q 2 }. It is easily seen that lim sup n δ n /σ n 0. There fore, according to Lemma 2.4 we conclude from (21) that x n q 0. Further from y n x n 0and z n x n = P C (x n λ n Ax n ) P C (x n ) λ n Ax n 0,

11 An iterative method 131 we get y n q 0and z n q 0. This completes the proof. Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone, L-Lipschitz continuous mapping of C into H such that F (S) and f : C C be a contraction. For given x 0 C arbitrary, let the sequences {x n }, {y n }, {z n } be generated by z n = P C (x n λ n Ax n ), y n =(1 γ n )x n + γ n P C (x n λ n Az n ), x n+1 =(1 α n )x n + α n [β n f(x n )+(1 β n )y n ], where {λ n } is a sequence in (0, 1) with lim n λ n =0,and{α n }, {β n }, {γ n } are three sequences in [0, 1] satisfying the conditions: (i) lim n β n =0, n=0 β n =, (ii) 0 < lim inf n α n lim sup n α n < 1; (iii) lim n (γ n+1 γ n )=0. Then the sequences {x n }, {y n }, {z n } converge strongly to the same point q = P F (S) f(q). References [1] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, Journal ofmathematical Analysis and Applications 20(1967), [2] F. Liu, M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Analysis 6 (1998), [3] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, [4] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal ofoptimization Theory and Applications 118(2003), [5] G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody 12(1976), [6] R. T. Rockafellar, On the maximality of sums of nonliner monotone operators, Transactions ofthe American Mathematical Society 149(1970), [7] T. Suzuki, Strong convergence of Krasnoselskii and Mann s Type sequences for one-parameter nonexpansive semigroups without Bochner integrals, Journal of Mathematical Analysis and Applications 305(2005), [8] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, Journal ofmathematical Analysis and Applications 298(2004),

12 132 M. A. Noor, Y. Yao, R. Chen and Y.-C. Liou [9] J. C. Yao, Variationali inequalities with generalized monotone operators, Mathematics ofoperations Research 19 (1994), [10] J. C. Yao, O. Chadli, Pseudomonotone complementarity problems and variational inequalities, in:handbook of Generalized Convexity and Monotonicity, (J. P. Crouzeix, N. Haddjissas and S. Schaible, Eds.), Kluwer Academic, 2005, [11] L. C. Zeng, S. Schaible, J. C. Yao, Iterative algorithm for generalized setvalued strongly nonlinear mixed variational-like inequalities, Journal ofoptimization Theory and Applications 124(2005), [12] N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications 128(2006), [13] L. C. Zeng, J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese Journal ofmathematics 10(2006), [14] L. C. Ceng, J. C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., in press [15] M. Aslam Noor, Some algorithms for general monotone variational inequalities, Math. Computer Modelling. 29(1999), 1-9. [16] M. Aslam Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(2000), [17] M. Aslam Noor, Some developments in general variational inequalities, Appl. Math. Computation 151(2004), , [18] M. Aslam Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl. 331(2007), [19] M. Aslam Noor, Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Computation 187(2007), [20] M. Aslam Noor, A. Bnouhachem, On an iterative algorithm for general variational inequalities, Appl. Math. Computation, in press.