On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces

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1 Computers and Mathematics with Applications 53 (2007) On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces Y.X. Tian a, S.S. Chang b,, J.L. Huang b a College of Computers, Chongqing Post Telecommunications University, Chongqing, , China b Department of Mathematics, Yibin University, Yibin, Sichuan , China Received 20 July 2006; received in revised form 6 October 2006; accepted 11 October 2006 Abstract The purpose of this paper is to introduce the concept of non-self asymptotically quasi-nonexpansive-type mappings and to construct a iterative sequence to converge to a common fixed point for a finite family of non-self asymptotically quasinonexpansive-type mappings in Banach spaces. The results presented in this paper improve and extend the corresponding results in Alber, Chidume and Zegeye [Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications (2006) Article ID10673], Ghosh and Debnath [M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 207 (1997) ], Liu [Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive type mappings, Journal of Mathematical Analysis and Applications 259 (2001) 1 37; Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications 259 (2001) 18 24; Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications 266 (2002) ], Petryshyn [W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 43 (1973) ], Quan and Chang [J. Quan, S.S. Chang, X.J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications V (2006) Article ID 70830], Shahzad and Udomene [N. Shahzad, A. Udomene, Approximating common fixed point of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2006) Article ID 18909] Xu [B.L. Xu, M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002) ], Zhang [S.S. Zhang, Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Mathematicae Applicatae Sinica 24 (2001) ] and Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis and Optimization 23 (2002) ]. c 2007 Elsevier Ltd. All rights reserved. Keywords: Non-self asymptotically quasi-nonexpansive-type mapping; Asymptotically nonexpansive mappings; Non-self asymptotically nonexpansive mapping; Iterative sequence with mean errors; Common fixed point Corresponding author. Tel.: addresses: tianyx@cqupt.edu.cn (Y.X. Tian), sszhang 1@yahoo.com.cn (S.S. Chang), jialinh2880@163.com (J.L. Huang) /$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi: /j.camwa

2 1848 Y.X. Tian et al. / Computers and Mathematics with Applications 53 (2007) Introduction and preliminaries Throughout this paper, we assume that E is a real Banach space, C is a nonempty closed convex subset of E and F(T ) is the set of fixed points of mapping T. Definition 1.1. Let T : C C be a mapping. (1) T is said to be nonexpansive, if T x T y x y for every x, y C; (2) T is said to be asymptotically nonexpansive [1,12] if there exists a sequence {k n } [1, ) with k n 1 as n such that T n x T n y k n x y, x, y C; n 1. Definition 1.2. Let E be a real Banach space and C be a nonempty subset of E. (1) A mapping P from E onto C is said to be a retraction, if P 2 = P; (2) If there exists a continuous retraction P : E C such that P x = x, x C, then the set C is said to be a retract of E. (3) In particular, if there exists a nonexpansive retraction P : E C such that Px = x, x C, then the set C is said to be a nonexpansive retract of E. Next we introduce the following concepts for non-self mapping: Definition 1.3. Let E be a real Banach space, C be a nonempty nonexpansive retract of E and P be the nonexpansive retraction from E onto C. Let T : C E be a non-self mapping. (1) T is said to be a non-self asymptotically nonexpansive mapping [4], if there exists a sequence {k n } [1, ) with lim n k n = 1 such that T (PT ) n 1 x T (PT ) n 1 y k n x y x, y C, n 1 (2) T is said to be a non-self asymptotically quasi-nonexpansive mapping, if F(T ) and there exists a sequence {k n } [1, ) with lim n k n = 1 such that T (PT ) n 1 x p k n x p x C, p F(T ) n 1; (3) T is said to be a non-self asymptotically nonexpansive-type mapping, if lim sup{ sup [ T (PT ) n 1 x T (PT ) n 1 y x y ]} 0; n x,y C (4) T is said to be a non-self asymptotically quasi-nonexpansive-type mapping, if F(T ) and lim sup{sup[ T (PT ) n 1 x p x p ]} 0 p F(T ). n x C Remark. It follows from Definition 1.3 that (a) if T : C E is a non-self asymptotically nonexpansive mapping, then T is a non-self asymptotically nonexpansive-type mapping; (b) if T : C E is a non-self asymptotically quasi-nonexpansive mapping, then T is a non-self asymptotically quasi-nonexpansive-type mapping (c) If F(T ) is nonempty and T is non-self asymptotically nonexpansive-type mapping, then T is a non-self asymptotically quasi-nonexpansive-type mapping. Definition 1.4. Let E be a real Banach space and C be a nonempty closed convex subset of E which is also a nonexpansive retract of E with a retraction P. Let T 1, T 2,..., T N : C E be non-self asymptotically quasinonexpansive-type mappings. Let x 1 C be any given point. Then the sequence {x n } defined by

3 Y.X. Tian et al. / Computers and Mathematics with Applications 53 (2007) x n+1 = P[(1 a n1 b n1 )x n + a n1 T 1 (PT 1 ) n 1 y n1 + b n1 u n1 ] y n1 = P[(1 a n2 b n2 )x n + a n2 T 2 (PT 2 ) n 1 y n2 + b n2 u n2 ]. n 1 (1.1) y nn 2 = P[(1 a nn 1 b nn 1 )x n + a nn 1 T N 1 (PT N 1 ) n 1 y nn 1 + b nn 1 u nn 1 ] y nn 1 = P[(1 a nn b nn )x n + a nn T N (PT N ) n 1 x n + b nn u nn ] is called the N-step iterative sequence with errors of T 1, T 2,..., T N, where {a ni } n=1, {b ni} n=1, i = 1, 2,..., N are real sequences in [0, 1] satisfying the conditions a ni + b ni 1, n 1, i = 1, 2,..., N, and {u ni } n=1, i = 1, 2,..., N are bounded sequences in C. Concerning the convergence problems of various iterative sequences to converge to a fixed point, common fixed points for (self or non-self) nonexpansive and asymptotically nonexpansive mappings have been studied by many authors (see, for example, [1 8,13 16] and the references therein). The purpose of this paper is to study the iterative sequence (1.1) to converge to a common fixed point for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces. The results presented in this paper improve and extend the corresponding results in Alber, Chidume and Zegeye [1], Ghosh and Debnath [2], Liu [3 5], Petryshyn [6], Quan and Chang [7], Shahzad and Udomene [8], Xu [9], Zhang [10] and Zhou and Chang [11]. The following lemma will be needed in proving our main results. Lemma 1.1 ([8]). Let {a n }, {b n } be nonnegative real sequences satisfying the following conditions: a n+1 a n + b n, n 1, where n=1 b n <. Then lim n a n exists. 2. Main results Theorem 2.1. Let E be a real Banach space and C be a nonempty closed convex subset of E which is also a nonexpansive retract of E with a retraction P. Let T 1, T 2,..., T N : C E be non-self asymptotically quasi-nonexpansive-type mappings and let the common fixed point set F := N n=1 F(T i ) of T 1, T 2,..., T N be nonempty and closed. Let {u ni } n=1, i = 1, 2,..., N be bounded sequences in C, {a ni} n=1, i = 1, 2,..., N and {b ni } n=1, i = 1, 2,..., N be sequences in [0, 1] satisfying the following conditions: (i) a ni + b ni 1, n 1, i = 1, 2,..., N; (ii) n=1 b ni <, i = 1, 2,..., N; (iii) n=1 a ni <, i = 1, 2,..., N. Then the sequence {x n } defined by (1.1) converges strongly to a common fixed point of T 1, T 2,..., T N, if and only if the following condition is satisfied: lim inf n d(x n, F) = 0, where d(x n, F) is the distance of x n to the set F. Proof. The necessity of condition (2.1) is obvious. Next we prove the sufficiency of condition (2.1). For any given p F, since {u ni } n=1, i = 1, 2,..., N are bounded sequences in C, let M = sup u ni p. n 1,i=1,2,...,N Since T 1, T 2,..., T N : C E are non-self asymptotically quasi-nonexpansive-type mappings, for any given ε > 0, there exists a positive integer n 0 such that for all n n 0 and u F sup { T i (PT i ) n 1 x u x u } < ε. (2.2) x C,i=1,2,...,N (2.1)

4 1850 Y.X. Tian et al. / Computers and Mathematics with Applications 53 (2007) Since {x n } and {y ni } C, i = 1, 2,..., N 1, for any n n 0 we have T 1 (PT 1 ) n 1 y n1 u y n1 u < ε T 2 (PT 2 ) n 1 y n2 u y n2 u < ε. u F. (2.3) T N 1 (PT 1 ) n 1 y nn 1 u y nn 1 u < ε T N (PT N ) n 1 x n u x n u < ε. Hence for any n n 0, it follows from (1.1) and (2.3) that x n+1 p = P[(1 a n1 b n1 )x n + a n1 T 1 (PT 1 ) n 1 y n1 + b n1 u n1 ] P(p) (1 a n1 b n1 )x n + a n1 T 1 (PT 1 ) n 1 y n1 + b n1 u n1 p (1 a n1 b n1 ) x n p + a n1 T 1 (PT 1 ) n 1 y n1 p + b n1 u n1 p (1 a n1 ) x n p + a n1 { T 1 (PT 1 ) n 1 y n1 p y n1 p } + a n1 y n1 p + b n1 u n1 p (1 a n1 ) x n p + a n1 ε + a n1 y n1 p + b n1 M. (2.4) Next we consider the third term on the right-hand side. From (1.1) and for n n 0 we have y n1 p = P[(1 a n2 b n2 )x n + a n2 T 2 (PT 2 ) n 1 y n2 + b n2 u n2 ] P(p) Similarly, we can prove that = (1 a n2 b n2 )x n + a n2 T 2 (PT 2 ) n 1 y n2 + b n2 u n2 p (1 a n2 b n2 ) x n p + a n2 { T 2 (PT 2 ) n 1 y n2 p y n2 p } a n2 y n2 p + b n2 u n2 p (1 a n2 ) x n p + a n2 ε + a n2 y n2 p + b n2 M. (2.5) y ni p (1 a ni+1 ) x n p + a ni+1 ε + a ni+1 y ni+1 p + b ni+1 M, i = 1, 2,..., N 2; (2.6) y nn 1 p (1 a nn ) x n p + a nn ε + b nn x n p + b nn M = x n p + a nn ε + b nn M. (2.7) From (2.6) and (2.7) we have y nn 2 p (1 a nn 1 ) x n p + a nn 1 ε + a nn 1 y nn 1 p + b nn 1 M, (1 a nn 1 ) x n p + a nn 1 εa nn 1 { x n p + a nn ε + b nn M} + b nn 1 M x n p + (a nn + a nn 1 )ε + (b nn + b nn 1 )M (2.8) By induction, we can prove that for any i = 1, 2,..., N 1, ( ) ( ) i 1 i 1 y nn i p x n p + a nn j ε + b nn j M. (2.9) In particular, taking i = N 1 in (2.9) we have ( ) N 2 y n1 p x n p + a nn j ε + Therefore from (2.4) and (2.10) we have ( N 2 x n+1 p (1 a n1 ) x n p + a n1 ε + a n1 { x n p + b nn j ) M. (2.10) ( N 2 a nn j ) ε + ( N 2 b nn j ) M } + b n1 M = x n p + A n, n n 0. (2.11)

5 Y.X. Tian et al. / Computers and Mathematics with Applications 53 (2007) where A n = ( N j=1 a nj )ε + ( N j=1 b nj )M, n 1. It follows from conditions (i) (iii) that A n <. n=1 By the arbitrariness of p F, from (2.11) we have inf x n+1 p inf x n p + A n, n n 0, p F p F and so we have d(x n+1, F) d(x n, F) + A n, n n 0. (2.12) By Lemma 1.1, the limit lim n d(x n, F) exists. Again, by the condition lim inf n d(x n, F) = 0 we have lim d(x n, F) = 0. n (2.13) Next we prove that the sequence {x n } defined by (1.1) is a Cauchy sequence in C. Indeed, for any n n 0, any m 1 and any p F, from (2.11) we have x n+m p x n+m 1 p + A n+m 1 x n+m 2 p + (A n+m 1 + A n+m 2 ) Hence for n n 0, m 1, n+m 1 x n p + A k. (2.14) x n+m x n x n+m p + x n p n+m 1 2 x n p + A k. (2.15) By the arbitrariness of p F, we have x n+m x n 2d(x n, F) + A k, n n 0. Since n=1 A n < and d(x n, F) 0 (as n ), for any given ε > 0, there exists a positive integer n 1 n 0 such that for any n n 1 we have d(x n, F) < ε 4 ; Therefore we have x n+m x n < ε, and so for any m 1 A k < ε 2. lim x n+m x n = 0. n (2.16) (2.17) This shows that {x n } is a Cauchy sequence in C. Since C is a closed subset of E, and so it is complete. Hence there exists p C such that x n p (n ). Finally, we prove that p F. Suppose the contrary: p F. Since F is a closed set, d(p, F) > 0. Hence for any p F, we have p p p x n + x n p. This implies that d(p, F) p x n + d(x n, F). (2.18)

6 1852 Y.X. Tian et al. / Computers and Mathematics with Applications 53 (2007) Letting n in (2.18) and noting (2.13), it gets d(p, F) 0. This is a contradiction. Hence p F. Theorem 2.1 is proved. Remark. It is easy to prove that if the mappings T 1, T 2,..., T N : C E in Theorem 2.1 are continuous, then the common fixed point set F of T 1, T 2,..., T N is closed. Theorem 2.2. Let E be a real Banach space and C be a nonempty closed convex subset of E which is also a nonexpansive retract of E with a retraction P. Let T 1, T 2,..., T N : C E be non-self asymptotically quasinonexpansive mappings with F := N i=1 F(T i ) being nonempty and closed. Let {u ni } n=1, i = 1, 2,..., N be bounded sequences in C, {a ni } n=1, i = 1, 2,..., N and {b ni} n=1, i = 1, 2,..., N be sequences in [0, 1] satisfying the following conditions: (i) a ni + b ni 1, n 1, i = 1, 2,..., N; (ii) n=1 b ni <, i = 1, 2,..., N; (iii) n=1 a ni <, i = 1, 2,..., N. Then the sequence {x n } defined by (1.1) converges strongly to a common fixed point of T 1, T 2,..., T N, if and only if the following condition is satisfied: lim inf n d(x n, F) = 0. Proof. Since T 1, T 2,..., T N : C E are non-self asymptotically quasi-nonexpansive mappings, by the definition they are non-self asymptotically quasi-nonexpansive-type mappings. The conclusion of Theorem 2.2 can be obtained from Theorem 2.1 immediately. Theorem 2.3. Let E be a real Banach space and C be a nonempty closed convex subset of E which is also a nonexpansive retract of E with a retraction P. Let T 1, T 2,..., T N : C E be non-self asymptotically nonexpansive mappings with F := N n=1 F(T i ). Let {u ni } n=1, i = 1, 2,..., N be N bounded sequences in C, {a ni } n=1, i = 1, 2,..., N and {b ni} n=1, i = 1, 2,..., N be sequences in [0, 1] satisfying the following conditions: (i) a ni + b ni 1, n 1, i = 1, 2,..., N; (ii) n=1 b ni <, i = 1, 2,..., N; (iii) n=1 a ni <, i = 1, 2,..., N. Then the sequence {x n } defined by (1.1) converges strongly to a common fixed point of T 1, T 2,..., T N, if and only if the following condition is satisfied: lim inf n d(x n, F) = 0. Proof. Since T 1, T 2,..., T N : C E are non-self asymptotically nonexpansive mappings, taking n = 1 in Definition 1.3(ii), we know that T 1, T 2,..., T N : C E are continuous non-self asymptotically nonexpansive mappings. Therefore the conclusion of Theorem 2.3 can be obtained from Theorem 2.1. Acknowledgment The authors would like to express their thanks to the referees for their helpful comments and suggestions. This research was supported by the Natural Science Foundation of Yibin University (No. 2005Z3). References [1] Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications (2006) Article ID [2] M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 207 (1997) [3] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive type mappings, Journal of Mathematical Analysis and Applications 259 (2001) 1 7. [4] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications 259 (2001)

7 Y.X. Tian et al. / Computers and Mathematics with Applications 53 (2007) [5] Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications 266 (2002) [6] W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 43 (1973) [7] J. Quan, S.S. Chang, X.J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications V (2006) 1 8. Article ID [8] N. Shahzad, A. Udomene, Approximating common fixed point of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2006) Article ID [9] B.L. Xu, M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002) [10] S.S. Zhang, Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Mathematicae Applicatae Sinica 24 (2001) [11] Y.Y. Zhou, S.S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis and Optimization 23 (2002) [12] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society 35 (1972) [13] S.S. Chang, J.K. Kim, S.M. Kang, Approximating fixed points of asymptotically quasi-nonexpansive type mappings by the Ishikawa iterative sequences with mixed errors, Dynamic Systems and Applications 13 (2004) [14] S.S. Chang, H.W.J. Lee, Y.J. Cho, On the convergence of finite steps iterative sequences for asymptotically nonexpansive mappings, Dynamics of Continuous, Discrete and Impulsive Systems. Series A. Mathematical Analysis 11 (2004) [15] C.E. Chidume, H. Zegeye, N. Shahzad, Convergence theorems for a common fixed point of a finite family of non-self nonexpansive mappings, Fixed Point Theory and Applications (2005) [16] K.K. Tan, H.K. Xu, Approximating fixed point of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications 178 (1993)