A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES

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1 A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES FENG GU AND JING LU Received 18 January 2006; Revised 22 August 2006; Accepted 23 August 2006 The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002). Copyright 2006 F. Gu and J. Lu. 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 and preliminaries Throughout this paper we assume that E is a real Banach space and T : E E is a mapping. We denote by F(T) andd(t) the set of fixed points and the domain of T, respectively. Recall that E is said to satisfy Opial condition [11], if for each sequence {x n } in E, the condition that the sequence x n x weakly implies that liminf x n x < liminf x n y (1.1) for all y E with y x. It is well known that (see, e.g., Dozo [9]) inequality (1.1) is equivalent to limsup x n x < limsup x n y. (1.2) Definition 1.1. Let D be a closed subset of E and let T : D D be a mapping. (1) T is said to be demiclosed at the origin, if for each sequence {x n } in D, the conditions x n x 0 weakly and Tx n 0stronglyimplyTx 0 = 0. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 82738, Pages 1 11 DOI /FPTA/2006/82738

2 2 A new composite implicit iterative process (2) T is said to be semicompact, if for any bounded sequence {x n } in D such that x n Tx n 0(n ), then there exists a subsequence {x ni } {x n } such that x ni x D. (3) T is said to be nonexpansive,if Tx Ty x y,foralln 1forallx, y D. Let E be a Hilbet space, let K be a nonempty closed convex subset of E, andlet {T 1,T 2,...,T N } : K K be N nonexpansive mappings. In 2001, Xu and Ori [19] introduced the following implicit iteration process {x n } defined by x n = α n x n 1 +(1 α n )T n(modn) x n, n 1, (1.3) where x 0 K is an initial point, {α n } n 1 is a real sequence in (0,1) and proved the weakly convergence of the sequence {x n } defined by (1.3) to a common fixed point p F = Ni=1 F(T i ). Recently concerning the convergence problems of an implicit (or nonimplicit) iterative process to a common fixed point for a finite family of asymptotically nonexpansive mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces have been considered by several authors (see, e.g., Bauschke [1], Chang and Cho [3], Chang et al. [4], Chidume et al. [5], Goebel and Kirk [6], Górnicki [7], Halpern [8], Lions [10], Reich [12], Rhoades [13], Schu [14], Shioji and Takahashi [15], Tan and Xu [16, 17], Wittmann [18], Xu and Ori [19], and Zhou and Chang [20]). In this paper, we introduce the following new implicit iterative sequence {x n } with errors: x 1 = α 1 x 0 + β 1 T 1 ( α1 x 0 + β ) 1 T 1 x 1 + γ 1 v 1 + γ1 u 1, x 2 = α 2 x 1 + β 2 T 2 ( α2 x 1 + β ) 2 T 2 x 2 + γ 2 v 2 + γ2 u 2,. x N = α N x N 1 + β N T N ( αn x N 1 + β ) N T N x N + γ N v N + γn u N, x N+1 = α N+1 x N + β N+1 T 1 ( αn+1 x N + β ) N+1 T 1 x N+1 + γ N+1 v N+1 + γn+1 u N+1,. x 2N = α 2N x 2N 1 + β 2N T N ( α2n x 2N 1 + β ) 2N T N x 2N + γ 2N v 2N + γ2n u 2N, x 2N+1 = α 2N+1 x 2N + β 2N+1 T 1 ( α2n+1 x 2N + β ) 2N+1 T 1 x 2N+1 + γ 2N+1 v 2N+1 + γ2n+1 u 2N+1,. (1.4) for a finite family of nonexpansive mappings {T i } N i=1 : K K, where{α n }, {β n }, {γ n }, { α n }, { β n },and{ γ n } are six sequences in [0, 1] satisfying α n + β n + γ n = α n + β n + γ n = 1 for all n 1, x 0 is a given point in K,aswellas{u n } and {v n } are two bounded sequences

3 F. Gu and J. Lu 3 in K, which can be written in the following compact form: x n = α n x n 1 + β n T n(modn) y n + γ n u n, y n = α n x n 1 + β n T n(modn) x n + γ n v n, n 1. (1.5) Especially, if {T i } N i=1 : K K are N nonexpansive mappings, {α n }, {β n }, {γ n } are three sequences in [0, 1], and x 0 is a given point in K, then the sequence {x n } defined by x n = α n x n 1 + β n T n(modn) x n 1 + γ n u n, n 1 (1.6) is called the explicit iterative sequence for a finite family of nonexpansive mappings {T i } N i=1. The purpose of this paper is to study the weak and strong convergence of iterative sequence {x n } defined by (1.5) and(1.6) to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper not only generalized and extend the corresponding results of Chang and Cho [3], Xu and Ori [19], and Zhou and Chang [20], but also in the case of γ n = γ n = 0or β n = γ n = 0arealsonew. In order to prove the main results of this paper, we need the following lemmas. Lemma 1.2 [2]. Let E be a uniformly convex Banach space, let K be a nonempty closed convex subset of E, andlett : K K be a nonexpansive mapping with F(T). Then I T is semiclosed at zero, that is, for each sequence {x n } in K, if {x n } converges weakly to q K and {(I T)x n } converges strongly to 0, then (I T)q = 0. Lemma 1.3 [17]. Let {a n } and {b n } be two nonnegative real sequences satisfying the following condition: a n+1 a n + b n for all n n 0,wheren 0 is some nonnegative integer. If n=0 b n <, then lim a n exists. If in addition {a n } has a subsequence which converges strongly to zero, then lim a n = 0. Lemma 1.4 [14]. Let E be a uniformly convex Banach space, let b and c be two constants with 0 <b<c<1.supposethat{t n } is a sequence in [b,c] and {x n } and {y n } are two sequence in E such that lim t n x n +(1 t n )y n =d, limsup x n d, andlimsup y n d hold for some d 0, then lim x n y n =0. Lemma 1.5. Let E be a real Banach space, let K beanonemptyclosedconvexsubsetofe, and let {T 1,T 2,...,T N } : K K be N nonexpansive mappings with F = N i=1 F(T i ). Let {u n } and {v n } betwoboundedsequencesink, andlet{α n }, {β n }, {γ n }, { α n }, { β n }, and { γ n } be six sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = α n + β n + γ n = 1,foralln 1; (ii) τ = sup{β n : n 1} < 1; (iii) n=1 γ n <, n=1 γ n <. If {x n } is the implicit iterative sequence defined by (1.5), then for each p F = N i=1 F(T i ) the limit lim x n p exists.

4 4 A new composite implicit iterative process Proof. Since F = N n=1 F(T i ),foranygivenp F,itfollowsfrom(1.5)that x n p = ( 1 βn γ n ) xn 1 + β n T n(modn) y n + γ n u n p ( 1 β n γ n ) x n 1 p + β n T n(modn) y n p + γ n u n p = ( 1 β n γ n ) x n 1 p + β n T n(modn) y n T n(modn) p + γ n u n p ( 1 β n ) x n 1 p + β n y n p + γ n u n p. (1.7) Again it follows from (1.5)that y n p = ( 1 β n γ n ) xn 1 + β n T n(modn) x n + γ n v n p ( 1 β n γ n ) x n 1 p + β n T n(modn) x n p + γ n v n p = ( 1 β n γ n ) x n 1 p + β n T n(modn) x n T n(modn) p + γ n v n p ( 1 β n ) x n 1 p + β n x n p + γ n v n p. Substituting (1.8)into(1.7), we obtain that x n p ( 1 βn β n ) x n 1 p + β n β n x n p + β n γ n v n p + γ n u n p. (1.8) (1.9) Simplifying we have ( 1 βn β n ) x n p ( 1 βn β n ) x n 1 p + σ n, (1.10) where σ n = β n γ n v n p + γ n u n p. By condition (iii) and the boundedness of the sequences {β n }, { u n p }, and{ v n p }, wehave n=1 σ n <. From condition (ii) we know that β n β n β n τ<1 andso 1 β n β n 1 τ>0; (1.11) hence, from (1.10) wehave x n p x n 1 p + σ n 1 τ = x n 1 p + b n, (1.12) where b n = σ n /(1 τ)with i=1 b n <. Taking a n = x n 1 p in inequality (1.12), we have a n+1 a n + b n,foralln 1, and satisfied all conditions in Lemma 1.3. Therefore the limit lim x n p exists. This completes the proof of Lemma Main results We are now in a position to prove our main results in this paper.

5 F. Gu and J. Lu 5 Theorem 2.1. Let E be a real Banach space, let K be a nonempty closed convex subset of E,andlet{T 1,T 2,...,T N } : K K be N nonexpansive mappings with F = N i=1 F(T i ) (the set of common fixed points of {T 1,T 2,...,T N }). Let {u n } and {v n } be two bounded sequences in K, andlet{α n }, {β n }, {γ n }, { α n }, { β n },and{ γ n } be six sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = α n + β n + γ n = 1,foralln 1; (ii) τ = sup{β n : n 1} < 1; (iii) n=1 γ n <, n=1 γ n <. Then the implicit iterative sequence {x n } defined by (1.5)convergesstronglytoacommon fixed point p F = N i=1 F(T i ) if and only if liminf d(x n,f) = 0. (2.1) Proof. The necessity of condition (2.1)isobvious. Next we prove the sufficiency of Theorem 2.1. Foranygivenp F, itfollowsfrom (1.12)inLemma 1.5 that x n p x n 1 p + b n n 1, (2.2) where b n = σ n /(1 τ)with n=1 b n <.Hence,wehave d ( x n,f ) d ( x n 1,F ) + b n n 1. (2.3) It follows from (2.3) andlemma 1.3 that the limit lim d(x n,f) exists. By condition (2.1), we have lim d(x n,f) = 0. Next we prove that the sequence {x n } is a Cauchy sequence in K. In fact, for any positive integers m and n, from(2.2), it follows that x n+m p x n+m 1 p + b n+m x n+m 2 p + b n+m 1 + b n+m x n p + n+m i=n+1 b i x n p + i=n+1 b i. (2.4) Since lim d(x n,f) = 0and n=1 b n <, foranygivenɛ > 0, there exists a positive integer n 0 such that d(x n,f) < ɛ/8, i=n+1 b i < ɛ/2, for all n n 0. Therefore there exists p 1 F such that x n p 1 < ɛ/4, for all n n 0. Consequently, for any n n 0 and for all m 1, from (2.4), we have x n+m x n x n+m p 1 + x n p 1 2 x n p 1 + i=n+1 b i < 2 ɛ 4 + ɛ 2 = ɛ. (2.5) This implies that {x n } is a Cauchy sequence in K. By the completeness of K, wecan assume that lim x n = x K. Moreover, since the set of fixed points of a nonexpansive mapping is closed, so is F;thusx F from lim d(x n,f) = 0, that is, x is a common fixed point of T 1,T 2,...,T N. This completes the proof of Theorem 2.1.

6 6 A new composite implicit iterative process Theorem 2.2. Let E be a real Banach space, let K be a nonempty closed convex subset of E,andlet{T 1,T 2,...,T N } : K K be N nonexpansive mappings with F = N i=1 F(T i ) (the set of common fixed points of {T 1,T 2,...,T N }). Let {u n } be a bounded sequence in K, and let {α n }, {β n },and{γ n } be three sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = 1,foralln 1; (ii) τ = sup{β n : n 1} < 1; (iii) n=1 γ n <. Then the explicit iterative sequence {x n } defined by (1.6)convergesstronglytoacommon fixed point p F = N i=1 F(T i ) if and only if liminf d(x n,f) = 0. Proof. Taking β n = γ n = 0, for all n 1 in Theorem 2.1, then the conclusion of Theorem 2.2 can be obtained from Theorem 2.1 immediately. This completes the proof of Theorem 2.2. Theorem 2.3. Let E be a real uniformly convex Banach space satisfying Opial condition, let K beanonemptyclosedconvexsubsetofe, andlet{t 1,T 2,...,T N } : K K be N nonexpansive mappings with F = N i=1 F(T i ).Let{u n } and {v n } be two bounded sequences in K,andlet{α n }, {β n }, {γ n }, { α n }, { β n },and{ γ n } be six sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = α n + β n + γ n = 1,foralln 1; (ii) 0 <τ 1 = inf{β n : n 1} sup{β n : n 1}=τ 2 < 1; (iii) β n 0(n ); (iv) n=1 γ n <, n=1 γ n <. Then the implicit iterative sequence {x n } defined by (1.5) converges weakly to a common fixed point of {T 1,T 2,...,T N }. Proof. First, we prove that lim x n T n(modn)+j x n = 0, j = 1,2,...,N. (2.6) Let p F.Putd = lim x n p.itfollowsfrom(1.5)that x n p = ( 1 βn )[ xn 1 p + γ n ( un x n 1 )] + β n [ Tn(modN) y n p + γ n ( un x n 1 )] d, n. (2.7) Again since lim x n p exists, so {x n } is a bounded sequence in K. Byvirtueof condition (iv) and the boundedness of sequences {x n } and {u n } we have limsup ( ) x n 1 p + γ n un x n 1 limsup x n 1 p +limsup γ n u n x n 1 = d, p F. (2.8)

7 F. Gu and J. Lu 7 It follows from (1.8) and condition (iii) that limsup ( ) T n(modn) y n p + γ n un x n 1 limsup y n p +limsupγ n u n x n 1 = limsup y n p {( ) limsup 1 β n x n 1 p + β n x n p + γ n v n p } limsup = d, p F. ( 1 β n ) x n 1 p +limsup β n x n p +limsup γ n v n p (2.9) Therefore, from condition (ii), (2.7) (2.9), and Lemma 1.4 we know that lim T n(modn) y n x n 1 = 0. (2.10) From (1.5)and(2.10)wehave [ ] ( ) x n x n 1 = β n Tn(modN) y n x n 1 + γn un x n 1 β n T n(modn) y n x n 1 + γ n u n x n 1 (2.11) 0, n, which implies that and so lim x n x n 1 = 0 (2.12) n lim x n x n+j = 0 j = 1,2,...,N. (2.13) On the other hand, we have x n T n(modn) x n x n x n 1 + x n 1 T n(modn) y n + T n(modn) y n T n(modn) x n. (2.14) Now, we consider the third term on the right-hand side of (2.14). From (1.5)wehave T n(modn) y n T n(modn) x n y n x n = αn ( xn 1 x n ) + β n ( Tn(modN) x n x n ) + γn ( vn x n ) α n x n 1 x n + β n T n(modn) x n x n + γ n v n x n. Substituting (2.15)into(2.14), we obtain that x n T n(modn) x n ( 1+ αn ) x n x n 1 + x n 1 T n(modn) y n + β n T n(modn) x n x n + γ n v n x n. (2.15) (2.16)

8 8 A new composite implicit iterative process Hence,by virtue of conditions (iii), (iv), (2.10), (2.12) and the boundedness of sequences { T n(modn) x n x n } and { v n x n } we have lim x n T n(modn) x n = 0. (2.17) Therefore, from (2.13)and(2.17), for any j = 1,2,...,N,wehave x n T n(modn)+j x n x n x n+j + x n+j T n(modn)+j x n+j + T n(modn)+j x n+j T n(modn)+j x n 2 x n x n+j + x n+j T n(modn)+j x n+j 0, n. (2.18) That is, (2.6)holds. Since E is uniformly convex, every bounded subset of E is weakly compact. Again since {x n } is a bounded sequence in K, there exists a subsequence {x nk } {x n } such that {x nk } converges weakly to q K. Without loss of generality, we can assume that n k = i(modn), where i is some positive integer in {1,2,...,N}. Otherwise, we can take a subsequence {x nkj } {x nk } such that n kj = i(modn). For any l {1,2,...,N}, there exists an integer j {1,2,...,N} such that n k + j = l(modn). Hence, from (2.18)wehave lim x nk T l x nk = 0, l = 1,2,...,N. (2.19) k By Lemma 1.2,weknowthatq F(T l ). By the arbitrariness of l {1,2,...,N},weknow that q F = N j=1 F(T j ). Finally, we prove that {x n } converges weakly to q. In fact, suppose the contrary, then there exists some subsequence {x nj } {x n } such that {x nj } converges weakly to q 1 K and q 1 q. Then by the same method as given above, we can also prove that q 1 F = Nj=1 F(T j ). Taking p = q and p = q 1 and by using the same method given in the proof of Lemma 1.5, we can prove that the following two limits exist and lim x n q =d 1 and lim x n q 1 =d 2,whered 1 and d 2 are two nonnegative numbers. By virtue of the Opial condition of E,wehave d 1 = limsup x nk q < limsup x nk q 1 = d2 n k n k = limsup n j x nj q 1 < limsup n j x nj q = d1. (2.20) This is a contradiction. Hence q 1 = q. This implies that {x n } converges weakly to q. This completes the proof of Theorem 2.3. Theorem 2.4. Let E be a real uniformly convex Banach space satisfying Opial condition, let K beanonemptyclosedconvexsubsetofe, andlet{t 1,T 2,...,T N } : K K be N nonexpansive mappings with F = N i=1 F(T i ).Let{u n } be a bounded sequence in K, andlet

9 F. Gu and J. Lu 9 {α n }, {β n },and{γ n } be three sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = 1, n 1; (ii) 0 <τ 1 = inf{β n : n 1} sup{β n : n 1}=τ 2 < 1; (iii) n=1 γ n <. Then the explicit iterative sequence {x n } defined by (1.6) converges weakly to a common fixed point of {T 1,T 2,...,T N }. Proof. Taking β n = γ n = 0, for all n 1 in Theorem 2.3, then the conclusion of Theorem 2.4 can be obtained from Theorem 2.3 immediately. This completes the proof of Theorem 2.4. Theorem 2.5. Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E,andlet{T 1,T 2,...,T N } : K K be N nonexpansive mappings with F = Ni=1 F(T i ) and there exists an T j,1 j N, which is semicompact (without loss of generality, assume that T 1 is semicompact). Let{u n } and {v n } be two bounded sequences in K, andlet{α n }, {β n }, {γ n }, { α n }, { β n },and{ γ n } be six sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = α n + β n + γ n = 1,foralln 1; (ii) 0 <τ 1 = inf{β n : n 1} sup{β n : n 1}=τ 2 < 1; (iii) β n 0(n ); (iv) n=1 γ n <, n=1 γ n <. Then the implicit iterative sequence {x n } defined by (1.5)convergesstronglytoacommon fixed point of {T 1,T 2,...,T N } in K. Proof. For any given p F = N i=1 F(T i ), by the same method as given in proving Lemma 1.5 and (2.19), we can prove that lim x n p = d, (2.21) where d 0 is some nonnegative number, and lim x nk T l x nk = 0, k l = 1,2,...,N. (2.22) Especially, we have lim x nk T 1 x nk = 0. (2.23) k By the assumption, T 1 is semicompact; therefore it follows from (2.23) that there exists a subsequence {x nki } {x nk } such that x nki x K.Hencefrom(2.22)wehavethat x T l x = lim x nki T l x nki = 0 l = 1,2,...,N, (2.24) ki which implies that x F = N i=1 F(T i ). Take p = x in (2.21), similarly we can prove that lim x n x =d 1,whered 1 0 is some nonnegative number. From x nki x we know that d 1 = 0, that is, x n x. This completes the proof of Theorem 2.5.

10 10 A new composite implicit iterative process Theorem 2.6. Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, andlet{t 1,T 2,...,T N } : K K be N nonexpansive mappings with F = N i=1 F(T i ) and there exists an T j,1 j N, which is semicompact (without loss of generality, assume that T 1 is semicompact). Let {u n } be a bounded sequence in K,andlet {α n }, {β n },and{γ n } be three sequences in [0,1] satisfying the following conditions: (i) α n + β n + γ n = 1,foralln 1; (ii) 0 <τ 1 = inf{β n : n 1} sup{β n : n 1}=τ 2 < 1; (iii) n=1 γ n <. Then the explicit iterative sequence {x n } defined by (1.6)convergesstronglytoacommon fixed point of {T 1,T 2,...,T N } in K. Proof. Taking β n = γ n = 0, for all n 1 in Theorem 2.5, then the conclusion of Theorem 2.6 can be obtained from Theorem 2.5 immediately. This completes the proof of Theorem 2.6. Remark 2.7. Theorems improve and extend the corresponding results in Chang and Cho [3, Theorem 3.1] and Zhou and Chang [20, Theorem 3], and the implicit iterative process {x n } defined by (1.3) is replaced by the more general implicit or explicit iterative process {x n } defined by (1.5)or(1.6). Remark 2.8. Theorems generalize and improve the main results of Xu and Ori [19] in the following aspects. (1) The class of Hilbert spaces is extended to that of Banach spaces satisfying Opial s or semicompactness condition. (2) The implicit iterative process {x n } defined by (1.3)isreplacedbythemoregeneral implicit or explicit iterative process {x n } defined by (1.5)or(1.6). Remark 2.9. The iterative algorithm used in this paper is different from those in [1, 8, 10, 14, 18]. Acknowledgments The present studies were supported by the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), and the Scientific Research Foundation from Zhejiang Province Education Committee ( ). References [1] H. H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, Journal of Mathematical Analysis and Applications 202 (1996), no. 1, [2] F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. 18, Part 2, Chicago, Ill., 1968), American Mathematical Society, Rhode Island, 1976, pp [3] S.-S. Chang and Y. J. Cho, The implicit iterative processes for asymptotically nonexpansive mappings, Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday. Vol. 1, 2, Kluwer Academic, Dordrecht, 2003, pp

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