Research Article Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings

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1 Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID , 11 pages doi: /2008/ Research Article Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings Chao Wang and Jinghao Zhu Department of Applied Mathematics, Tongji University, Shanghai , China Correspondence should be addressed to Chao Wang, Received 1 April 2008; Revised 12 June 2008; Accepted 19 July 2008 Recommended by Simeon Reich We introduce a new three-step iterative scheme with errors. Several convergence theorems of this scheme are established for common fixed points of nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our theorems improve and generalize recent known results in the literature. Copyright q 2008 C. Wang and J. Zhu. 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. 1. Introduction Let K be a nonempty closed convex subset of real normed linear space E. Recall that a mapping T : K K is called asymptotically nonexpansive if there exists a sequence{r n } 0,, with lim r n 0 such that T n x T n y 1 r n x y, for all x, y K and n 1. Moreover, it is uniformly L-Lipschitzian if there exists a constant L > 0 such that T n x T n y L x y, for all x, y K and each n 1. Denote and define by F T {x K : Tx x} the set of fixed points of T. Suppose F T /. A mapping T is called asymptotically quasi-non-expansive if there exists a sequence {r n } 0,, with lim r n 0 such that T n x p 1 r n x p, for all x, y K, p F T,andn 1. It is clear from the above definitions that an asymptotically nonexpansive mapping must be uniformly L-Lipschitzian as well as asymptotically quasi-non-expansive, but the converse does not hold. Iterative technique for asymptotically nonexpansive self-mapping in Hilbert spaces and Banach spaces including Mann-type and Ishikawa-type iteration processes has been studied extensively by many authors; see, for example, 1 6. Recently, Chidume et al. 7 have introduced the concept of nonself asymptotically nonexpansive mappings, which is the generalization of asymptotically nonexpansive mappings. Similarly, the concept of nonself asymptotically quasi-non-expansive mappings

2 2 Fixed Point Theory and Applications can also be defined as the generalization of asymptotically quasi-non-expansive mappings and nonself asymptotically nonexpansive mappings. These mappings are defined as follows. Definition 1.1. Let K be a nonempty closed convex subset of real normed linear space E, let P : E K be the nonexpansive retraction of E onto K, andlett : K E be a nonself mapping. i T is said to be a nonself asymptotically nonexpansive mapping if there exists a sequence {r n } 0,, with lim r n 0 such that T PT n 1 x T PT n 1 y 1 rn x y, 1.1 for all x, y K and n 1. ii T is said to be a nonself uniformly L-Lipschitzian mapping if there exists a constant L>0such that T PT n 1 x T PT n 1 y L x y, 1.2 for all x, y K and n 1. iii T is said to be a nonself asymptotically quasi-non-expansive mapping if F T / and there exists a sequence {r n } 0,, with lim r n 0 such that T PT n 1 x p 1 rn x p, 1.3 for all x, y K, p F T,andn 1. By studying the following iteration process Mann-type iteration : x 1 K, x n 1 P (( 1 α n ) xn α n T PT n 1 x n ), n 1, 1.4 where {α n } 0, 1, Chidume et al. 7 obtained many convergence theorems for the fixed points of nonself asymptotically nonexpansive mapping T. Later on, Wang 8 generalized the iteration process 1.4 as follows Ishikawa-type iteration : x 1 K, x n 1 P ( n 1yn ) 1 α n xn α n T 1 PT1, y n P (( 1.5 ) ( ) 1 β n xn β n T 2 PT2, n 1 where T 1,T 2 : K E are nonself asymptotically nonexpansive mappings and {α n }, {β n } 0, 1. Also, he got several convergence theorems of the iterative scheme 1.5 under proper conditions. In 2000, Noor 9 first introduced a three-step iterative sequence and studied the approximate solutions of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. Glowinski and Tallec 10 showed that the three-step iterative schemes perform better than the Mann-type and Ishikawa-type iterative schemes. On the other hand, Xu and Noor 11 introduced and studied a three-step scheme to approximate fixed points of asymptotically nonexpansive mappings in Banach spaces. Cho et al. 12 and Plubtieng et al. 13 extended the work of Xu and Noor to the threestep iterative scheme with errors, and gave weak and strong convergence theorems for asymptotically nonexpansive mappings in Banach spaces.

3 C. Wang and J. Zhu 3 Inspired and motivated by these facts, a new class of three-step iterative schemes with errors, for three nonself asymptotically quasi-non-expansive mappings, is introduced and studied in this paper. This scheme can be viewed as an extension for 1.4, 1.5, and others. This scheme is defined as follows. Let K be a nonempty convex subset of real normed linear space X,letP : E K be the nonexpansive retraction of E onto K,andletT 1,T 2,T 3 : K E be three nonself asymptotically quasi-non-expansive mappings. Compute the sequences{x n }, {y n },and{z n } by x 1 K, x n 1 P ( n 1yn ) α n T 1 PT1 β n x n γ n w n, y n P ( α n 1zn ) nt 2 PT2 β nx n γ nv n, z n P α n 1xn ) nt 3 PT 3 β nx n γ nu n, n where {α n }, {α n}, {α n}, {β n }, {β n}, {β n}, {γ n }, {γ n}, and{γ n} are real sequences in 0, 1 with α n β n γ n α n β n γ n α n β n γ n 1, and {u n }, {v n },and{w n } are bounded sequences in K. Remark 1.2. i If T 1 T 2 T 3 : T, γ n γ n γ n 0, and α n α n 0, then scheme 1.6 reduces to the Mann-type iteration 1.4. ii If T 2 T 3, γ n γ n γ n 0, and α n 0, then scheme 1.6 reduces to the Ishikawatype iteration 1.5. iii If T 1,T 2, and T 3 are three self-asymptotically nonexpansive mappings, then scheme 1.6 reduces to the three-step iteration with errors defined by 12, 13, and others. The purpose of this paper is to study the iterative sequences 1.6 to converge to a common fixed point of three nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our results extend and improve the corresponding results in 5, 7, 8, 11 13, and many others. 2. Preliminaries and lemmas In this section, we first recall some well-known definitions. A real Banach space E is said to be uniformly convex if the modulus of convexity of E: { } x y δ E ε inf 1 : x y 1, x y ε > 0, for all 0 <ε 2 i.e., δ E ε is a function 0, 2 0, 1. AsubsetK of E is said to be a retract if there exists continuous mapping P : E K such that Px x, for all x K, and every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E E is said to be a retraction if P 2 P. A mapping T : K E with F T / is said to satisfy condition A see 14 if there exists a nondecreasing function f : 0, 0, with f 0 0, for all r 0,, such that x Tx f ( d ( x, F T )), 2.2 for all x K, where d x, F T inf{ x x : x F T }. We modify this condition for three mappings T 1,T 2,T 3 : K E as follows. Three mappings T 1,T 2,T 3 : K E, where K is a subset of E, are said to satisfy condition B if there

4 4 Fixed Point Theory and Applications exist a real number α>0and a nondecreasing function f : 0, 0, with f 0 0, for all r 0,, such that x T1 x αf d x, F or x T2 x αf d x, F or x T3 x αf d x, F, 2.3 for all x K, where F F T 1 F T 2 F T 3 /. Note that condition B reduces to condition A when T 1 T 2 T 3 and α 1. A mapping T : K E is said to be semicompact if, for any sequence {x n } in K such that x n Tx n 0 n, there exists subsequence {x nj } of {x n } such that {x nj } converges strongly to x K. Next we state the following useful lemmas. Lemma 2.1 see 5. Let {a n }, {b n }, and {c n } be sequences of nonnegative real numbers satisfying the inequality a n 1 1 c n an b n, n If n 1 c n < and n 1 b n <,thenlim a n exists. Lemma 2.2 see 15. Let E be a real uniformly convex Banach space and 0 k t n q<1, for all positive integer n 1. Suppose that {x n } and {y n } are two sequences of E such that lim sup x n r, lim sup y n r, and lim t n x n 1 t n y n r hold, for some r 0; thenlim x n y n Main results In this section, we will prove the strong convergence of the iteration scheme 1.6 to a common fixed point of nonself asymptotically quasi-non-expansive mappings T 1,T 2,andT 3. We first prove the following lemmas. Lemma 3.1. Let K be a nonempty closed convex subset of a real normed linear space E.LetT 1,T 2,T 3 : K E be nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that n 1 r i n <, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with n 1 γ n <, n 1 γ n <, and n 1 γ n <.IfF F T 1 F T 2 F T 3 /,thenlim x n p exists, for all p F. Proof. Let p F. Since {u n }, {v n },and{w n } are bounded sequences in K, therefore there exists M>0such that { M max sup un p, sup vn p, sup wn p }. 3.1 n 1 n 1 n 1 Let r n max{r 1 n,r 2 n,r 3 n } and k n max{γ n,γ n,γ n}. Then n 1 r n < and n 1 k n <. By 1.6, we have xn 1 p [ ] P αn T 1 PT n 1 1 yn β n x n γ n w n P p αn T 1 PT n 1 1 yn β n x n γ n w n α n β n γ n p [ α n T1 PT n 1 1 yn p ] ( β n xn p ) ( γ n wn p ) 3.2 α n 1 rn yn p βnxn p knwn p, yn p [ P α n T 2 PT n 1 2 zn β nx n γ nv ] n P p α nt 2 PT n 1 2 zn β nx n γ nv n ( α n β n γ n ) p α n 1 rn zn p β xn n p knvn p 3.3,

5 C. Wang and J. Zhu 5 and similarly, we also have zn p α n 1 rn xn p β n xn p knun p. 3.4 Substituting 3.4 into 3.3, weobtain yn p [ α n 1 rn α n 1 rn xn p β n xn p knun p ] β n xn p knvn p α nα ) 2 n xn p ) α n β n( 1 rn xn p β xn n p α nk n 1 rn u n p k n v n p ( 1 β n γ n ) ) α 2 n x n p ( 1 β n γ n ) β n 1 rn x n p β n x n p ( k n 1 rn u n p k n v n p 3.5 ( 1 β n γ n )( α n β n) ) 2 xn p β n xn p mn ( 1 β n) ) 2 xn p β n ) 2 xn p mn ( 1 r n ) 2 xn p mn, where m n k n 2 r n M. Since n 1 r n < and n 1 k n <, then n 1 m n <. Substituting 3.5 into 3.2, we have x n 1 p α n )[( 1 r 2 n ) x n p m n ] βn x n p γ n w n p [ α n ) 3 βn ] x n p α n ) mn γ n w n p ( α n β n ) ) 3 xn p ) mn k n wn p 3.6 ( 1 r n ) 3 xn p ) mn k n M ( 1 c n ) xn p bn, where c n 1 r n 3 1andb n 1 r n m n k n M. Since n 1 r n and n 1 m n <, then n 1 c n < and n 1 b n lim x n p exists. This completes the proof. <, n 1 k n <, <. It follows from Lemma 2.1 that Lemma 3.2. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.Let T 1,T 2,T 3 : K E be uniformly L-Lipschitzian nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that n 1 r i n <, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with n 1 γ n <, n 1 γ n <, and n 1 γ n <, whereα n,α n, and α n are three sequences in ε, 1 ε,forsomeε>0. IfF F T 1 F T 2 F T 3 /, then lim x n T 1 x n lim x n T 2 x n lim x n T 3 x n Proof. For any p F, bylemma 3.1, we see that lim x n p exists. Assume lim x n p a, for some a 0. For all n 1, let r n max{r 1 n,r 2 n,r 3 n } and k n max{γ n,γ n,γ n}.

6 6 Fixed Point Theory and Applications Then, n 1 r n < and n 1 k n <.From 3.5, we have yn p 2 1 rn xn p mn. 3.8 Taking lim sup on both sides in 3.8,since n 1 r n < and n 1 m n <,weobtain lim sup yn p lim sup xn p lim xn p a 3.9 so that lim sup T1 PT 1 n 1 y n p lim sup 1 rn yn p lim sup yn p a. Next consider n 1yn T1 PT1 p γ n wn T1 n 1yn PT1 p knwn. Since lim k n 0, we have n 1yn lim supt 1 PT1 p γ n wn a In addition, This implies that xn p γ n ( wn ) xn p kn wn. lim sup xn p γ n wn a Further, observe that a lim xn p lim αn n 1yn T 1 PT1 β n x n γ n w n p lim n 1yn α n T 1 PT1 1 α n xn γ n x n γ n w n 1 α n p αn p lim αn n 1yn T 1 PT1 α n p α n γ n w n α n γ n x n 1 α n xn ( 1 α n ) p γn x n γ n w n α n γ n w n α n γ n x n 3.15 lim α n [ T1 ) n 1yn p γ n ( wn )] ( 1 αn )[ xn p γ n ( wn )]. By Lemma 2.2, 3.12, 3.14, and 3.15, we have n 1yn lim T 1 PT

7 C. Wang and J. Zhu 7 Next we will prove that lim T 2 PT 2 n 1 z n 0. Since xn p T1 n 1yn PT1 T1 n 1yn PT1 p T1 ) n 1yn ) yn p and lim T 1 PT 1 n 1 y n 0 lim r n,weobtain a lim x n p lim infy n p. Thus, it follows from 3.10 and 3.18 that lim yn p a. On the other hand, from 3.4, we have zn p [ ] α n 1 rn β xn n p knun p 1 r n x n p k n u n p. By boundedness of the sequence {u n } and by lim r n lim k n 0, we have lim supz n p lim supx n p a so that lim sup n 1zn T2 PT2 p lim sup 1 rn zn p a Next consider n 1zn ) T2 PT2 p γ n( vn T2 n 1zn PT2 p knvn Thus, we have lim sup n 1zn ) T2 PT2 p γ n( vn a, ) x n p γ n( vn x n p k n v n This implies that ) lim sup xn p γ n( vn a Note that a lim yn p lim α n 1zn n T 2 PT2 β nx n γ nv n p lim α n 1zn n[ T2 PT2 p γ n( )] [ vn 1 α n xn p γ n ] vn. 3.26

8 8 Fixed Point Theory and Applications It follows from Lemma 2.2, 3.24, and 3.25 that lim n 1zn T2 PT2 0. Similarly, by using the same argument as in the proof above, we obtain lim n 1xn T3 PT Hence, n 1yn lim T 1 PT1 lim n 1zn T 2 PT2 lim n 1xn T 3 PT3 0, 3.29 and this implies that xn 1 αnt1 n 1yn PT1 knwn 0 as n Since T 1 is uniformly L-Lipschitzian mapping, then we have T 1 T1 T 1 ) n 1yn T1 ) n 1yn L xn y n T1 ) n 1yn L xn α n 1zn nt 2 PT2 β nx n γ nv n T1 n 1yn PT1 Lα n 1zn n T2 PT2 Lknvn T1 n 1yn PT1 0 as n, 3.31 xn T 1 x n xn 1 x n xn 1 T 1 ) nxn 1 T1 ) nxn 1 T 1 ) nxn T1 ) nxn T 1 x n x n 1 x n 1 T 1 ) nxn 1 L x n 1 L T 1. It follows from 3.30, 3.31,and 3.32 that lim x n T 1 x n 0. Next consider T T2 T 2 ) n 1zn T2 ) n 1zn L xn z n T2 n 1zn PT2 Lα n 1xn n T3 PT3 Lknun T2 n 1zn PT2 0 as n, 3.34 xn T 2 x n xn 1 x n xn 1 T 2 ) nxn 1 T2 ) nxn 1 T 2 ) nxn T2 ) nxn T 2 x n xn 1 xn 1 T 2 ) nxn 1 L xn 1 L T

9 C. Wang and J. Zhu 9 It follows from 3.30, 3.34,and 3.35 that lim xn T 2 x n Finally, we consider x n T 3 x n x n 1 x n x n 1 T 3 ( PT3 ) nxn 1 T 3 ( PT3 ) nxn 1 T 3 ( PT3 ) nxn T 3 ( PT3 ) nxn T 3 x n xn 1 xn 1 nxn 1 T 3 PT3 Lxn 1 LT3 n 1xn PT3. It follows from 3.29, 3.30,and 3.37 that lim xn T 3 x n Therefore, lim x n T 1 x n lim x n T 2 x n lim x n T 3 x n This completes the proof. Now, we give our main theorems of this paper. Theorem 3.3. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T 1,T 2,T 3 : K E be uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that n 1 r i n <, for all i 1, 2, 3, satisfying condition (B). n <, where Suppose that {x n } is defined by 1.6 with n 1 γ n <, n 1 γ n <, and n 1 γ α n,α n, and α n are three sequences in ε, 1 ε, forsomeε>0. IfF F T 1 F T 2 F T 3 /, then {x n } converges strongly to a common fixed point of T 1,T 2, and T 3. Proof. It follows from Lemma 3.2 that lim x n T 1 x n lim x n T 2 x n lim x n T 3 x n 0. Since T 1,T 2,andT 3 satisfy condition B, we have lim d x n,f 0. From Lemma 3.1 and the proof of Qihou 5, we can obtain that {x n } is a Cauchy sequence in K. Assume that lim x n p K. Since lim x n T 1 x n lim x n T 2 x n lim x n T 3 x n 0, by the continuity of T 1,T 2,andT 3, we have p F,thatis,p is a common fixed point of T 1,T 2,andT 3. This completes the proof. Corollary 3.4. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. LetT 1,T 2,T 3 : K E be nonself asymptotically nonexpansive mappings with sequences {r i n } such that n 1 r i n <, for all i 1, 2, 3, satisfying condition (B). Suppose that {x n } is defined by 1.6 with n 1 γ n <, n 1 γ n <, and n 1 γ n <, whereα n,α n, and α n are three sequences in ε, 1 ε, for some ε>0. IfF F T 1 F T 2 F T 3 /, then{x n } converges strongly to a common fixed point of T 1,T 2, and T 3. Proof. Since every nonself asymptotically nonexpansive mapping is uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive, the result can be deduced immediately from Theorem 3.3. This completes the proof.

10 10 Fixed Point Theory and Applications Theorem 3.5. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T 1,T 2,T 3 : K E be uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive <, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with n 1 γ n <, n 1 γ n <, and n 1 γ n <, whereα n,α n, and α n are three sequences in ε, 1 ε, for some ε>0. IfF F T 1 F T 2 F T 3 / and one of T 1,T 2, and T 3 is demicompact, then {x n } converges strongly to a common fixed point of T 1,T 2, and T 3. mappings with sequences {r i n } such that n 1 r i n Proof. Without loss of generality, we may assume that T 1 is demicompact. Since lim x n T 1 x n 0, there exists a subsequence {x nj } {x n } such that x nj x K. Hence, from 3.39, we have x T i x lim xnj T i x nj 0, i 1, 2, This implies that x F. By the arbitrariness of p F, fromlemma 3.1, and taking p x, similarly we can prove that lim xn x d, 3.41 where d 0 is some nonnegative number. From x nj x n x. This completes the proof. x, we know that d 0, that is, Corollary 3.6. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. LetT 1,T 2,T 3 : K E be nonself asymptotically nonexpansive mappings with sequences {r i n } such that n 1 r i n <, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with n 1 γ n <, n 1 γ n <, and n 1 γ n <, whereα n,α n, and α n are three sequences in ε, 1 ε, for some ε > 0. IfF F T 1 F T 2 F T 3 / and one of T 1,T 2, and T 3 is demicompact, then {x n } converges strongly to a common fixed point of T 1,T 2, and T 3. Acknowledgments The authors would like to thank the referee and the editor for their careful reading of the manuscript and their many valuable comments and suggestions. This paper was supported by the National Natural Science Foundation of China Grant no References 1 W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society, vol. 4, no. 3, pp , S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proceedings of the American Mathematical Society, vol. 59, no. 1, pp , K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp , J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp , Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp , N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasinonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, vol. 2006, Article ID 18909, 10 pages, C. E. Chidume, E. U. Ofoedu, and H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp , 2003.

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