Weak and Strong Convergence Theorems for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings

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1 Int. J. Nonlinear Anal. Appl. 3 (2012) No. 1, 9-16 ISSN: (electronic) Weak and Strong Convergence Theorems for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings P. Yatakoat, S. Suantai Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand. (Communicated by M. Eshaghi Gordji) Abstract In this paper, we introduce and study a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasi-nonexpansive nonself-mappings in Banach spaces. Several strong and weak convergence theorems of the proposed iteration are established. The main results obtained in this paper generalize and refine some known results in the current literature. Keywords: Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings, Common Fixed Points, Weak and Strong Convergence MSC: 47H09, 47H Introduction Let X be a real Banach space, C a nonempty closed convex subset of X and T a self-mapping of C. The fixed point set of T is denote by F (T ). Definition 1.1. The mapping T is said to be; (i) Nonexpansive if T x T y x y for all x, y C; (ii) Quasi-nonexpansive if F (T ) and T x p x p for all x, y C and p F (T ); (iii) Asymptotically nonexpansive if there exists a sequence {r n } in [0, ) with lim n r n = 0 and T n x T n y (1 + r n ) x y, for all x, y C and n 1; Corresponding author addresses: p_yatakoat@npu.ac.th (P. Yatakoat ), scmti005@chiangmai.ac.th (S. Suantai ) Received: January 2011 Revised: June 2012

2 10 Yatakoat, Suantai (iv) Asymptotically quasi-nonexpansive if F (T ) and there exists a sequence {r n } in [0, ) with lim n r n = 0 and T n x p (1 + r n ) x p, for all x C, p F (T ) and n 1; (v) Aeneralized quasi-nonexpansive if F (T ) and there exists a sequences {s n } in [0, ) with s n 0 as n such that T n x p x p + s n, for all x C and p F (T ) and n 1; (iv) Aeneralized asymptotically quasi-nonexpansive if F (T ) and there exist two sequences {r n } and {s n } in [0, ) with r n 0 and s n 0 as n such that T n x p (1+r n ) x p +s n, for all x C, p F (T ) and n 1. (vii) 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 C and n 1. From the above definition (1.1) it follows that (i) a nonexpansive mapping is a generalized asymptotically quasi-nonexpansive; (ii) a quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive; (iii) an asymptotically nonexpansive mapping with nonempty fixed points set is a generalized asymptotically quasi-nonexpansive; (iv) an asymptotically quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive; (v) a generalized quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive. The concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [1] as an important generalization of asymptotically nonexpansive mappings. Recently, Deng and Liu [2] generalized the concept of generalized asymptotically quasi-nonexpansive self-mappings defined by Shahzad and Zegeye [3] to the case of nonself-mappings. Those mappings are defined as follows ; Definition 1.2. (see [1],[4]) Let X be a real Banach space and C a nonempty closed convex subset of X and let P : X C be the nonexpansive retraction of X onto C. A nonself-mappins T : C X is said to be (i) Asymptotically nonexpansive if there exists a sequence {r n } in [0, ) with lim n r n = 0 such that T (P T ) n 1 x T (P T ) n 1 y (1 + r n ) x y, for all x, y C and n 1; (i) Asymptotically quasi-nonexpansive if F (T ) and there exists a sequence {r n } in [0, ) with lim n r n = 0 such that T (P T ) n 1 x p (1 + r n ) x p, for all x C, p F (T ) and n 1; (iii) Generalized asymptotically quasi-nonexpansive [5] if F (T ) and there exist two sequences {r n } and {s n } in [0, ) with r n 0 and s n 0 as n such that T (P T ) n 1 x p (1 + r n ) x p + s n, for all x C, p F (T ) and n 1. (iv) Uniformly L-Lipschitzian if there exists a constant L > 0 such that T (P T ) n 1 x T (P T ) n 1 y L x y, for all x, y C and n 1.

3 Weak and Strong Convergence Theorems for a Finite Family...3 (2012) No. 1, If T is self-mapping, then P becomes the identity mapping, so that (i) (iii) of Definition 1.2 reduce to (iii), (iv) and (vi) of Definition 1.1, respectively. In this paper, we introduced a new iteration process for a finite family {T i : i = 1, 2, 3,..., m} of generalized asymptotically quasi-nonexpansive nonself-mappings as follows: Let X be a real Banach space, C a nonempty closed convex subset of X and P : X C a nonexpansive retraction of X onto C, and let T i : C X (i = 1, 2, 3,..., m) be nonself-mappings. Let {x n } be a sequence defined by x 0 C, x n+1 = S n x n, n 1 (1.1) where S n = P (α 0n I + α 1n T 1 (P T 1 ) n 1 + α 2n T 2 (P T 2 ) n 1 + α 3n T 3 (P T 3 ) n α mn T m (P T m ) n 1 ) with α in [0, 1] for i = 1, 2, 3,..., m and m i=0 α in = 1. The main purpose of this paper is to prove strong convergence theorems of the iterative scheme (1.1) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive nonself-mappings in real Banach spaces. 2. Preliminaries and lemmas In this section, we give some definitions and lemmas used in the main results. A subset C of X is said to be retract of X if there exists a continuous mappings P : X C such that P (x) = x for all x C. A mappings T : C X with F (T ) is said to be; (i) Demiclosed at 0 if for each sequence {x n } converging weakly to x and {T x n } converging strongly to 0, we have T x = 0; (ii) semi-compact if for each sequence {x n } with lim n x n T x n = 0, there exists a subsequence {x nk } of {x n } such that x nk p; (iii) completely continuous if for every bounded sequence {x n } C, there is a subsequence {x nk } such that {T x nk } is convergent. A Banach space X is said to satisfy Opial s property (see [6])if for each x X and each sequence {x n } weakly converges to x, the following condition holds for all x y: lim inf n x n x < lim inf n x n y. Lemma 2.1. [7] Let the sequences {a n }, {δ n } and {c n } of real numbers satisfy: a n+1 (1 + δ n )a n + c n, where a n 0, δ n 0, c n 0 for all n = 1, 2, 3,... and n=1 δ n <, n=1 c n <. Then (i) lim n a n exists; (ii) if lim inf n a n = 0, then lim n a n = 0. Lemma 2.2. [8] Let X be a Banach space which satisfy Opial s property and let {x n } be a sequence in X. Let x, y X be such that lim n x n x and lim n x n y exists. If {x nk } and {x mk } are subsequences of {x n } which converge weakly to x and y, then x = y.

4 12 Yatakoat, Suantai Lemma 2.3. [9] Let X be a uniformly convex Banach. Then there exists a continuous strictly increasing convex function g : [0, ) [0, ) with g(0) = 0 such that for each m N and j {1, 2, 3,..., m}, α i x i 2 α i x 2 α m i m 1 ( α i g( x j x i )), for all x i B r (0) and α i [0, 1] for all i = 1, 2, 3,..., m with m α i = Main Results The aim of this section is to establish weak and strong convergence of the iterative scheme (1.1) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive nonselfmappings in a Banach space under some appropriate conditions. Lemma 3.1. Let C be a nonempty closed convex subset of a real Banach space X, and T i : C X, (i = 1, 2, 3,..., m) be family of generalized asymptotically quasi-nonexpansive nonself-mappings with the sequence {r in }, {s in } [0, ) and p i F (T i ), i = 1, 2,..., m. Suppose that F = m F (T i), and the iterative sequence {x n }, is defined by (1.1). Assume that n=1 r in < and n=1 s in <. Then we get (i) there exists two sequences {δ n }, {c n } in [0, ) such that n=1 δ n <, x n+1 p (1 + δ n ) x n p + c n for all p F (T ) and n 1; n=1 c n < and (ii) there exist L, D > 0 such that x n+k p L x n p + D, for all p F (T ) and n, k N. Proof. (i) Let p F and r n = max 1 i k {r in }, s n = max 1 i k {s in } for all n. Since n=1 r in < and n=1 s in < for all i = 1, 2, 3,..., m, we obtain that n=1 r n < and n=1 s n <. For i = 1, 2, 3,..., m, we have x n+1 p = S n x n p = P (α 0n I + α 1n T 1 (P T 1 ) n 1 + α 2n T 2 (P T 2 ) n α mn T m (P T m ) n 1 )x n P (p) α 0n x n p + α 1n T 1 (P T 1 ) n 1 x n p + α 2n T 2 (P T 2 ) n 1 x n p α mn T m (P T m ) n 1 x n p α 0n x n p + α 1n ((1 + r 1n ) x n p + s 1n ) +... where δ n = mr n and c n = ms n. +α 2n ((1 + r 2n ) x n p + s 2n ) + α mn ((1 + r mn ) x n p + s mn ) (α 0n + α 1n (1 + r n ) + α 2n (1 + r n ) α mn (1 + r n )) x n p +s 1n + s 2n + s 3n s mn (1 + mr n ) x n p + ms n = (1 + δ n ) x n p + c n (3.1)

5 Weak and Strong Convergence Theorems for a Finite Family...3 (2012) No. 1, (ii) If t 0 then 1 + t e t. Thus, from part (i) and for n, k N, we have x n+k p (1 + δ n+k 1 ) x n+k 1 p + c n+k 1 exp{δ n+k 1 } x n+k 1 p + c n+k 1 exp{δ n+k 1 }[(1 + δ n+k 2 ) x n+k 2 p + c n+k 2 ] + c n+k 1 exp{δ n+k 1 }exp{δ n+k 2 } x n+k 2 p + exp{δ n+k 1 }c n+k 2 + c n+k 1. k 1 k 1 k 1 exp{ δ n+i } x n p + exp{ δ n+i } δ n+i (3.2) i=0 Setting L = exp{ δ i} and D = L c i, we obtain x n+k p L x n p + D. Thus (ii) is satisfied. Lemma 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X, and T i : C X, (i = 1, 2, 3,..., m) be family of uniformly L i -Lipschitzian and generalized asymptotically quasi-nonexpansive nonself-mappings with the sequence {r in }, {s in } [0, ) and p i F (T i ), i = 1, 2,..., m. Suppose that F = m F (T i), and lim inf n α 0n α in > 0, i = 1, 2, 3,..., m and {x n } is defined by (1.1) such that n=1 r in < and n=1 s in <. Then we get (i) lim n x n p exists, p F (T ); i=0 i=0 (ii) lim n x n T i x n = 0, for each i = 1, 2,..., m. Proof. (i) By lemmas 2.1 and 3.1(i), we obtain that lim n x n p exists. (ii) From (i), we have that {x n } is bounded. For each i = 1, 2, 3,..., m, we have T i (P T i ) n 1 x n p (1 + r in ) x n p + s in (1 + r n ) x n p + s n It follows that {T i (P T i ) n 1 x n p} is bounded i = 1, 2, 3,..., m. Put r = sup{ T i (P T i ) n 1 x n p : 1 i m, n N} + sup{ x n p : n N}. Let 1 i m. By Lemma 2.3, there is a continuous strictly increasing convex function g : [0, ) [0, ) with g(0) = 0 such that α i x i 2 α i x 2 α m i m 1 ( α i g( x j x i )), (3.3) for all x i B r (0) and α i [0, 1] for all i = 1, 2, 3,..., m with m α i = 1. By (3.3), we have x n+1 p 2 = α 0n (x n p) + α 1n (T 1 (P T 1 ) n 1 x n p) α mn (T m (P T m ) n 1 x n p) 2 α 0n x n p 2 + α 1n ((1 + r 1n ) x n p + s 1n )

6 14 Yatakoat, Suantai It follows that +α mn ((1 + r mn ) x n p + s mn ) 2 α m 0 m ( α i g( x n T i (P T i ) n 1 x n )) α 0n x n p α in x n p α in rn x 2 n p α in r n x n p 2 α in (1 + r n )s n x n p α in s 2 n α m 0 m ( α i g( x n T i (P T i ) n 1 x n )) x n p α in r n x n p 2 + α in (1 + r n )s n x n p + α in rn x 2 n p 2 α in s 2 n α m 0 m ( α i g( x n T i (P T i ) n 1 x n )). m α 0 m ( α i g( x n T i (P T i ) n 1 x n )) ( x n+1 p 2 x n p 2 ) α in rn x 2 n p α in s 2 n. α in r n x n p 2 α in (1 + r n )s n x n p Since lim n x n p exists,lim n r n = 0 = lim n s n and lim inf n α 0n α in > 0 for each i = 1, 2, 3,..., m, it follows that lim n g( x n T i (P T i ) n 1 x n ) = 0. Since g is continuous strictly increasing with g(0) = 0, we can conclude that lim x n T i (P T i ) n 1 x n = 0, i = 1, 2, 3,..., m. (3.4) n For each n N, we have x n+1 T 1 (P T 1 ) n 1 x n = α 0n x n + α 1n T 1 (P T 1 ) n 1 x n + α 2n T 2 (P T 2 ) n 1 x n +... From (3.4) and (3.5), we have +α mn T m (P T m ) n 1 x n T 1 (P T 1 ) n 1 x n α 0n x n T 1 (P T 1 ) n 1 x n + α 2n T 2 (P T 2 ) n 1 x n T 1 (P T 1 ) n 1 x n α mn T m (P T m ) n 1 x n T 1 (P T 1 ) n 1 x n α 0n x n T 1 (P T 1 ) n 1 x n + α 2n T 2 (P T 2 ) n 1 x n x n +α 2n x n T 1 (P T 1 ) n 1 x n α mn T m (P T m ) n 1 x n x n + α mn x n T 1 (P T 1 ) n 1 x n. (3.5) x n+1 T 1 (P T 1 ) n 1 x n o as n (3.6)

7 Weak and Strong Convergence Theorems for a Finite Family...3 (2012) No. 1, From (3.4) and (3.6), where n x n+1 x n x n+1 T 1 (P T 1 ) n 1 x n + T 1 (P T 1 ) n 1 x n x n 0. (3.7) Since T i is uniformly L i -Lipschitzian, we have x n T i x n x n+1 x n + x n+1 T i (P T i ) n 1 x n+1 + T i (P T i ) n 1 x n+1 T i (P T i ) n 1 x n + T i (P T i ) n 1 x n T i x n x n+1 x n + x n+1 T i (P T i ) n 1 x n+1 + L x n+1 x n + L (P T i ) n 1 x n x n x n+1 x n + x n+1 T i (P T i ) n 1 x n+1 + It follows from (3.4), (3.6) and (3.7) that x n T i x n = 0 This completes the proof. L x n+1 x n + L T i (P T i ) n 1 x n x n. (3.8) Theorem 3.3. Under the hypotheses of Lemma 3.2, assume that one of T i is completely continuous. Then the iterative sequence {x n } defined by (1.1) converges strongly to a common fixed point of the family {T i : i = 1, 2, 3,..., m}. Proof. Suppose that T i0 is completely continuous for some i 0 {1, 2,..., m}. Since {x n } is bounded, {x n } has a subsequence {x nk } such that T i0 x nk p. By Lemma 3.2 (ii), we have lim n x n T i x n = 0, i = 1, 2,..., m. It follows that x nk p x nk T i0 x nk + T i0 x nk p 0. Thus x nk p. By the continuity of T i, we have p T i p = lim k x nk T i x nk = 0, i = 1, 2,..., m. Hence p F. By Lemma 3.2 (i), we have that lim n x n p exists. This implies that lim n x n p = 0. Theorem 3.4. Under the hypotheses of Lemma 3.2, assume that one of T i is semi-compact. Then the iterative sequence {x n } defined by (1.1) converges strongly to a common fixed point of the family {T i : i = 1, 2, 3,..., m}. Proof. Suppose that T i0 is semi-compact for some i 0 {1, 2,..., m}. By Lemma 3.2 (ii), we have lim n x n T i x n = 0, i = 1, 2,..., m. Since {x n } is bounded and T i0 is semi-compact, {x n } has a convergent subsequence {x nk } such that x nk p. By the continuity of T i, we have p T i p = lim k x nk T i x nk = 0, i = 1, 2,..., m. Hence p F. By Lemma 3.2 (i), we have that lim n x n p exists. This implies that lim n x n p = 0. Theorem 3.5. Under the hypotheses of Lemma 3.2, assume that (I T i ) is demiclosed at 0, for each i = 1, 2,..., m. Then the iterative sequence {x n } defined by (1.1) converges weakly to a common fixed point of the family {T i : i = 1, 2, 3,..., m}.

8 16 Yatakoat, Suantai Proof. Let p F. By Lemma 3.2 (i), we have lim n x n p exists, and hence {x n } is bounded. Since a uniformly convex Banach space is reflexive, there exists a subsequence {x nk } of {x n } converging weakly q 1 C. By Lemma 3.2 (ii), lim n x n T i x n = 0. Since (I T i ) is demiclosed at 0, for each i = 1, 2,..., m, we obtain that T i q 1 = q 1. That is, q 1 F. Next, we show that {x n } converges weakly to q 1. Take another subsequence {x nk } of {x n } converging weakly to some q 2 C. Again, as above, we can conclude that q 2 F. By Lemma 3.2, we obtain that q 1 = q 2. This show that {x n } converges weakly to a common fixed point of the family {T i : i = 1, 2,..., m}. Remark 3.6. If {T i : C C} m i is a finite family of self-mappings, then the mapping S n in (1.1) is reduced to S n = α 0n I + α 1n T 1 + α 2n T 2 + α 3n T α mn T m by Cholamjiak and Suantai [9]. So results obtained in the paper generalized those in [9]. 4. Acknowledgments This research was supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, the Graduate School of Chiang Mai University and the Thailand Research Fund. References [1] C. E. Chidume, E. U. Ofoedu and H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 280 (2003) [2] L. Deng and Q. Liu, Scheme for nonself generalized asymptotically quasi-nonexpansive mappings, Appl. Math. Comput., 205 (2008) [3] N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpanisve mappings in Banach spaces, Fixed Point Theory Appl., article ID 18909, 10 pages, [4] C. Wang, J. Zhu and L. An, New iterative schemes dor asymptotically quasi-nonexpansive nonself-mappings in Banach spaces, Comput. Math. Appl., 233 (2010) [5] N. Shahzad and H. Zegeye, Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Appl. Math. Comput., 189 (2007) [6] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967) [7] Z. H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings, J. Math. Anal. Appl., 286 (2003) [8] S. Suantai, Weak and Strong convergence criteria of noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005) [9] W. Cholamjiak and S. Suantai, Weak and Strong convergence theorems for a finite family og generalized asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 60 (2010) [10] S. Thianwan, Common fixed point of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach spaces, Comput. Math. Appl., 224 (2009) [11] A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed point Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008) 1 11.