In this paper, we study a multi-step iterative scheme with errors involving N nonexpansive mappings in the Banach space. Some weak and strong convergence theorems for approximation of common fixed points of nonexpansive mappings are proved using this iteration scheme. The results extend and improve the corresponding results of [1]

F. Gu

Z. He

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Mathematical Communications 11(2006), 47-54 47Multi-step iterative process with errors forcommon fixed points of a finite family ofnonexpansive mappings∗Feng Gu†and Zhenhua He‡Abstract. In this paper, we study a multi-step iterative schemewith errors involving N nonexpansive mappings in the Banach space.Some weak and strong convergence theorems for approximation of com-mon fixed points of nonexpansive mappings are proved using this itera-tion scheme. The results extend and improve the corresponding resultsof [1].Key words: nonexpansive mapping, multi-step iteration processwith errors, common fixed pointsAMS subject classifications: 47H05, 47H09, 49M05Received January 17, 2006 Accepted May 26, 20061. Introduction and preliminariesLet K be a nonempty convex subset of a normed linear space E, and let {Ti}Ni=1 beN self-maps of K. Khan and Fukhar-ud-din [1] introduced the following iterativescheme.The sequence{xn} defined byx1 = x ∈ K,xn+1 = anSyn + bnxn + cnun,yn = a′nTxn + b′nxn + c′nvn, ∀n ≥ 1.(1.1)where {an}, {bn}, {cn}, {a′n}, {b′n} and {c′n} are six real sequences in [0,1] with0 < δ ≤ an, a′n ≤ 1 − δ < 1, an + bn + cn = a′n + b′n + c′n = 1, and {un}, {vn} arebounded sequences in K.∗The present studies were supported by the Natural Science Foundation of Zhejiang Province(Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Key TeacherCreating Capacity Fund of Heilongjiang General College (1053G015) and the Starting Foundationof Scientific Research of Hangzhou Teacher’s College.†Institute of Applied Mathematics and Department of Mathematics, Hangzhou Teacher’s Col-lege, Hangzhou, Zhejiang 310 036, China, e-mail: gufeng99@sohu.com‡Institute of Applied Mathematics and Department of Mathematics, Hangzhou Teacher’s Col-lege, Hangzhou, Zhejiang 310 036, China, e-mail: zhenhuahe@126.com48 F.Gu and Z.HeIf cn = c′n = 0, then the above scheme means thatx1 = x ∈ K,xn+1 = anSyn + bnxn,yn = a′nTxn + b′nxn, ∀n ≥ 1.(1.2)where {an}, {bn}, {a′n} and {b′n} are four real sequences in [0,1] satisfying an+bn =a′n + b′n = 1. This scheme has been studied by Das and Debata [2] and Takahashiand Tamura [3].Now, we further generalize the scheme given in (1.1) as follows.The sequence {xn} is defined byx1 = x ∈ K,xn+1 = a(1)n T1x(1)n + b(1)n xn + c(1)n u(1)n ,x(1)n = a(2)n T2x(2)n + b(2)n xn + c(2)n u(2)n ,x(2)n = a(3)n T3x(3)n + b(3)n xn + c(3)n u(3)n ,...x(N−2)n = a(N−1)n TN−1x(N−1)n + b(N−1)n xn + c(N−1)n u(N−1)n ,x(N−1)n = a(N)n TNx(N)n + b(N)n xn + c(N)n u(N)n ,x(N)n = xn for all n ≥ 1.(1.3)The scheme is expressed in a compact form asx(i−1)n = a(i)n Tix(i)n + b(i)n xn + c(i)n u(i)n , for all n ≥ 1, i ∈ I, (1.4)where I = {1, 2, 3, · · · , N}, xn+1 = x(0)n , {a(i)n }, {b(i)n } and {c(i)n } are three realsequences in [0,1] with 0 < δ ≤ a(i)n ≤ 1 − δ < 1, a(i)n + b(i)n + c(i)n = 1, and {u(i)n }and {v(i)n } are two bounded sequences in K.Since their introduction nonexpansive mappings have been extensively studiedby many authors in different frames of work. The purpose of this paper is to studythe weak and strong convergence of a multi-step iteration scheme (1.4) for N nonex-pansive mappings in a uniformly convex Banach space. The results presented in thispaper extend and improve the corresponding results of [1] from two nonexpansivemappings to a family of nonexpansive mappings.To proceed in this direction, we first recall the following definitions.A Banach space E is said to satisfy the Opial’s condition if whenever {xn} is asequence in E which converges weakly to x, thenlim supn→∞‖xn − x‖ < lim supn→∞‖xn − y‖ for all y ∈ E,with y = xA mapping T : K → E is called demiclosed with respect to y ∈ E if for eachsequence {xn} in K and each x ∈ E, xn ⇀ x, and Txn → y imply that x ∈ K andTx = y.Multi-step iterative process with errors 49In the sequel we shall need the following lemmas.Lemma 1.1 [Schu[4]]. Suppose that E is a uniformly convex Banach spaceand 0 < p ≤ tn ≤ q < 1 for all positive integers n. Also suppose that {xn} and {yn}are two sequences of E such that lim supn→∞ ‖xn‖ ≤ d, lim supn→∞ ‖yn‖ ≤ d andlimn→∞ ‖tnxn +(1− tn)yn‖ = d hold for some d ≥ 0, then limn→∞ ‖xn − yn‖ = 0.Lemma 1.2 [Tan and Xu[5]]. Let {sn} and {tn} be two nonnegative sequencessatisfyingsn+1 ≤ sn + tn for all n ≥ 1.If∑∞n=1 tn <∞, then limn→∞ sn exists.Lemma 1.3 [Browder[6]]. Let E be a uniformly convex Banach space satis-fying the Opial’s condition and let K be a nonempty closed convex subset of E. LetT be a nonexpansive mapping of K into itself, then I−T is demiclosed with respectto zero.2. Main resultsIn this section, let F (T ) denote the set of all fixed points of T .Lemma 2.1. Let E be a normed space and K its nonempty bounded convexsubset. Let {Ti}Ni=1 : K → K be N nonexpansive mappings and let {xn} be thesequence as defined in (1.4) with∑∞n=1 c(i)n < ∞, i ∈ I = {1, 2, 3, · · · , N}. IfF = ∩Ni=1F (Ti) = ∅, then limn→∞ ‖xn − x∗‖ exist for all x∗ ∈ F = ∩∞i=1F (Ti).Proof. Since K is bounded, there exists M > 0, such that ‖xn − u(i)n ‖ ≤M, for all i ∈ I. Assume that F = ∩∞i=1F (Ti) = ∅. Let x∗ ∈ F = ∩∞i=1F (Ti).Then||xn+1 − x∗|| = ||a(1)n T1x(1)n + b(1)n xn + c(1)n u(1)n − x∗||= ||a(1)n (T1x(1)n − x∗ + c(1)n (u(1)n − xn))+(1− a(1)n )(xn − x∗ + c(1)n (u(1)n − xn))||≤ a(1)n ||T1x(1)n − x∗|| + (1− a(1)n )||xn − x∗|| + c(1)n ||u(1)n − xn||≤ a(1)n ||x(1)n − x∗||+ (1− a(1)n )||xn − x∗||+ c(1)n M= a(1)n ||a(2)n T2x(2)n + b(2)n xn + c(2)n u(2)n − x∗||+(1− a(1)n )||xn − x∗||+ c(1)n M= a(1)n ||a(2)n (T2x(2)n − x∗ + c(2)n (u(2)n − xn))+(1− a(2)n )(xn − x∗ + c(2)n (u(2)n − xn))||+(1− a(1)n )||xn − x∗||+ c(1)n M≤ a(1)n (a(2)n ||x(2)n − x∗||+ (1 − a(2)n )||xn − x∗||+ c(2)n M)+(1− a(1)n )||xn − x∗||+ c(1)n M= a(1)n a(2)n ||x(2)n − x∗||+ a(1)n (1− a(2)n )||xn − x∗||+(1− a(1)n )||xn − x∗||+ a(1)n c(2)n M + c(1)n M50 F.Gu and Z.He= a(1)n a(2)n ||x(2)n − x∗|| + (1− a(1)n a(2)n )||xn − x∗|| + a(1)n c(2)n M + c(1)n M≤ a(1)n a(2)n (a(3)n ||x(3)n − x∗|| + (1− a(3)n )||xn − x∗|| + c(3)n M)+(1− a(1)n a(2)n )||xn − x∗|| + a(1)n c(2)n M + c(1)n M= a(1)n a(2)n a(3)n ||x(3)n − x∗|| + (1− a(1)n a(2)n a(3)n )||xn − x∗||+a(1)n a(2)n c(3)n M + a(1)n c(2)n M + c(1)n M· · · · · · · · · · · · · · · · · · · · · · · ·≤ a(1)n a(2)n · · · a(N)n ||x(N)n − x∗|| + (1− a(1)n a(2)n · · ·a(N)n )||xn − x∗||+a(1)n a(2)n · · ·a(N−1)n c(N)n M + a(1)n a(2)n · · · a(N−2)n c(N−1)n M+ · · ·+ a(1)n c(2)n M + c(1)n M≤ a(1)n a(2)n · · · a(N)n ||x(N)n − x∗|| + (1− a(1)n a(2)n · · ·a(N)n )||xn − x∗|| +MΣNi=1c(i)n= ||xn − x∗|| +MΣNi=1c(i)n .Since Σ∞n=1c(i)n <∞, hence, using Lemma 1.2, we have thatlimn→∞ ||xn − x∗|| exists for each x∗ ∈ F = ∩Ni=1F (Ti).This completes the proof of Lemma 2.1. ✷Lemma 2.2. Let E be a uniformly convex Banach space and K its nonemptybounded convex subset. Let {Ti}Ni=1 : K → K be nonexpansive mappings, and{xn} the sequence as defined in (1.4) with∑∞n=1 c(i)n < ∞, i = 1, 2, 3, · · · , N . IfF = ∩Ni=1F (Ti) = ∅, then limn→∞ ‖Tixn − xn‖ = 0 for all i ∈ I.Proof. By Lemma 2.1, limn→∞ ||xn−x∗|| exists. Suppose limn→∞ ||xn−x∗|| = dfor some d ≥ 0. Now,||x(i−1)n − x∗|| = ||a(i)n Tix(i)n + b(i)n xn + c(i)n u(i)n − x∗||= ||a(i)n (Tix(i)n − x∗ + c(i)n (u(i)n − xn))+(1− a(i)n )(xn − x∗ + c(i)n (u(i)n − xn))||≤ a(i)n ||Tix(i)n − x∗||+ (1 − a(i)n )||xn − x∗|| + c(i)n ||u(i)n − xn||≤ a(i)n ||x(i)n − x∗||+ (1 − a(i)n )||xn − x∗|| + c(i)n M= a(i)n ||a(i+1)n Ti+1x(i+1)n +b(i+1)n xn+c(i+1)n u(i+1)n −x∗||+(1− a(i)n )||xn−x∗||+c(i)n M≤ a(i)n (a(i+1)n ||x(i+1)n − x∗||+ (1− a(i+1)n )||xn − x∗|| + c(i+1)n M)+(1− a(i)n )||xn − x∗|| + c(i)n M= a(i)n a(i+1)n ||x(i+1)n − x∗|| + a(i)n (1− a(i+1)n )||xn − x∗||+(1− a(i)n )||xn − x∗|| + a(i)n c(i+1)n M + c(i)n M= a(i)n a(i+1)n ||x(i+1)n −x∗||+ (1−a(i)n a(i+1)n )||xn−x∗||+a(i)n c(i+1)n M+c(i)n M· · · · · · · · · · · · · · · · · ·≤ a(i)n a(i+1)n · · · a(N)n ||x(N)n − x∗||+ (1 − a(i)n a(i+1)n · · · a(N)n )||xn − x∗||+MΣNk=ic(k)n= ||xn − x∗|| +MΣNk=1c(k)n .Multi-step iterative process with errors 51Taking limsup on both sides in the above inequality, we havelim supn→∞||x(i−1)n − x∗|| ≤ d. (2.1)Next, consider||Tix(i)n − x∗ + c(i)n (u(i)n − xn)|| ≤ ||Tix(i)n − x∗|| + c(i)n ||u(i)n − xn||≤ ||x(i)n − x∗|| + c(i)n MTaking limsup on both sides in the above inequality and then using (2.1), we havethatlim supn→∞||Tix(i)n − x∗ + c(i)n (u(i)n − xn)|| ≤ d for each i ∈ I. (2.2)Also,||xn − x∗ + c(i)n (u(i)n − xn)|| ≤ ||xn − x∗||+ c(i)n ||u(i)n − xn||≤ ||xn − x∗||+ c(i)n Mgives thatlim supn→∞||xn − x∗ + c(i)n (u(i)n − xn)|| ≤ d for each i ∈ I. (2.3)Further, limn→∞ ||xn+1 − x∗|| = d means thatlimn→∞ ||a(1)n (T1x(1)n − x∗ + c(1)n (u(1)n − xn)) + (1− a(1)n )(xn − x∗ + c(1)n (u(1)n − x∗))|| = dHence, applying Lemma 1.1, we get thatlimn→∞ ||T1x(1)n − xn|| = 0Next,||xn − x∗|| ≤ ||xn − T1x(1)n || + ||T1x(1)n − x∗|| ≤ ||xn − T1x(1)n ||+ ||x(1)n − x∗||.It follows from (2.1) thatd ≤ lim infn→∞ ||x(1)n − x∗|| ≤ lim supn→∞||x(1)n − x∗|| ≤ dThat is,limn→∞ ||x(1)n − x∗|| = dNow limn→∞ ||x(1)n − x∗|| = d can be expressed aslimn→∞ ||a(2)n (T2x(2)n − x∗ + c(2)n (u(2)n − xn)) + (1− a(2)n )(xn − x∗c(2)n (u(2)n − x∗))|| = dMoreover, by (2.2) and (2.3) we have thatlim supn→∞||T2x(2)n − x∗ + c(2)n (u(2)n − xn)|| ≤ d,52 F.Gu and Z.Heandlim supn→∞||xn − x∗ + c(2)n (u(2)n − x∗)|| ≤ dSo again by Lemma 1.1, we obtain thatlimn→∞ ||T2x(2)n − xn|| = 0Now,||xn − x∗|| ≤ ||xn − T2x(2)n || + ||T2x(2)n − x∗|| ≤ ||xn − T2x(2)n ||+ ||x(2)n − x∗||.From (2.1) it follows thatd ≤ lim infn→∞ ||x(2)n − x∗|| ≤ lim supn→∞||x(2)n − x∗|| ≤ dThat islimn→∞ ||x(2)n − x∗|| = dUsing the same method, we get thatlimn→∞ ||T3x(3)n − xn|| = 0, limn→∞ ||x(3)n − x∗|| = dlimn→∞ ||T4x(4)n − xn|| = 0, limn→∞ ||x(4)n − x∗|| = d· · · · · · · · · · · ·limn→∞ ||TNx(N)n − xn|| = 0, limn→∞ ||x(N)n − x∗|| = di.e.limn→∞ ||Tix(i)n − xn|| = 0, limn→∞ ||x(i)n − x∗|| = d for all i ∈ INow, observe that||xn − Tixn|| ≤ ||xn − Tix(i)n || + ||Tix(i)n − Tixn||≤ ||xn − Tix(i)n || + ||x(i)n − xn||= ||xn − Tix(i)n || + ||a(i+1)n Ti+1x(i+1)n + b(i+1)n xn + c(i+1)n u(i+1)n − xn||= ||xn − Tix(i)n || + ||a(i+1)n (Ti+1x(i+1)n − xn) + c(i+1)n (u(i+1)n − xn)||≤ ||xn − Tix(i)n || + a(i+1)n ||Ti+1x(i+1)n − xn||+ c(i+1)n MHencelimn→∞ ||xn − Tixn|| = 0 for all i ∈ I.This completes the proof of Lemma 2.2. ✷Theorem 2.1. Let E be a uniformly convex Banach space satisfying theOpial’s condition and let K,{Ti}Ni=1 and {xn} be taken as in Lemma 2.1. If F =∩Ni=1F (Ti) = ∅, then {xn} converges weakly to a common fixed point of {Ti}Ni=1.Multi-step iterative process with errors 53Proof. Let x∗ ∈ F = ∩Ni=1F (Ti), then as proved in Lemma 2.1, limn→∞ ||xn −x∗|| exists. Now we prove that {xn} has a unique weak subsequential limit inF = ∩Ni=1F (Ti). To prove this, let z1 and z2 be weak limits of the subsequences{xni} and {xnj} of {xn}, respectively. By Lemma 2.2, limn→∞ ‖Tixn − xn‖ = 0and I − Ti is demiclosed with respect to zero by Lemma 1.3, therefore we obtainTiz1 = z1, i.e., z1 ∈ F = ∩Ni=1F (Ti). Again in the same way, we can prove thatz2 ∈ F = ∩Ni=1F (Ti). Next, we prove the uniqueness. For this suppose that z1 = z2,then by the Opial’s conditionlimn→∞ ||xn − z1|| = limni→∞ ||xni − z1|| < limni→∞ ||xni − z2||= limn→∞ ||xn − z2|| = limnj→∞ ||xnj − z2||< limnj→∞||xnj − z1|| = limn→∞ ||xn − z1||This is a contradition. Hence {xn} converges weakly to a point in F = ∩Ni=1F (Ti).This completes the proof of Theorem 2.1. ✷Now we will prove a strong convergence theorem.Two mappings S, T :K → K, where K is a subset of E, are said to satisfycondition (A) if there exists a nondecreasing function f : [0,∞) → [0,∞) withf(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that 12 (||x−Tx||+ ||x−Sx||) ≥ f(d(x, F ))for allx ∈ K where d(x, F ) = inf{||x− x∗|| : x∗ ∈ F = F (S) ∩ F (T )}.Khan and Fukhar-ud-din [1] approximated a common fixed point of two nonex-pansive mappings S and T by iterating scheme (1.1). We modify this condition formappings {Ti}Ni=1 : K → K as follow:N mappings {Ti}Ni=1 : K → K where K is a subset of E, are said to satisfycondition (A′) if there exists a nondecreasing function f : [0,∞) → [0,∞) withf(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that 1N ΣNi=1||x − Tix|| ≥ f(d(x, F )) forall x ∈ K where d(x, F ) = inf{||x− x∗|| : x∗ ∈ ∩Ni=1F (Ti)}.Note that condition (A′) reduces to condition (A) when N = 2. We shall usecondition (A′) to study the strong convergence of {xn} defined in (1.4).Theorem 2.2. Let E be a uniformly convex Banach space and let K, {xn} betaken as in Lemma 2.1. Let {Ti}Ni=1 : K → K be nonexpansive mappings satisfyingcondition (A′). If F = ∩Ni=1F (Ti) = ∅, then {xn} converges strongly to a commonfixed point of {Ti}Ni=1.Proof. By Lemma 2.1, suppose that limn→∞ ||xn − x∗|| = d for all x∗ ∈ F =∩Ni=1F (Ti). If d = 0, there is nothing to prove. Assume d > 0, by Lemma 2.2limn→∞ ||Tixn − xn|| = 0, i ∈ I. Moreover,||xn+1 − x∗|| ≤ ||xn − x∗||+MΣNi=1c(i)n ,gives thatinfx∗∈F||xn+1 − x∗|| ≤ infx∗∈F||xn − x∗||+MΣNi=1c(i)n .That is,d(xn+1, F ) ≤ d(xn, F ) +MΣNi=1c(i)ngives that limn→∞ d(xn, F ) exists by virtue of Lemma 1.2. Now by condition (A′),limn→∞ f(d(xn, F )) = 0. Since f is a nondecreasing function and f(0) = 0, there-fore limn→∞ d(xn, F ) = 0. Now we can take a subsequence {xnj} of {xn} and a54 F.Gu and Z.Hesequence {pj} ⊂ F such that ||xnj − pj || < 2−j. Then following the method ofproof of Tan and Xu [5], we get that {pj} is a Cauchy sequence in F and so itconverges. Let pj → p. Since F is closed, therefore p ∈ F and then xnj → p. Aslimn→∞ ||xn − x∗|| exists, xn → p ∈ F = ∩∞i=1F (Ti). This completes the proof ofTheorem 2.2. ✷Remark 2.1. Theorems 2.1-2.2 extend and improve the corresponding resultsof [1].References[1] S.H.Khan, H.Fukhar-ud-din, Weak and strong convergence of a schemewith errors for two nonexpansive mappings, Nonlinear Analysis 61(2005), 1295-1301.[2] G.Das, J. P.Debata, Fixed points of quasi-nonexpansive mappings, IndianJ. Pure Appl. Math. 17(1986), 1263-1269.[3] W.Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansivemappings, J. Convex Analysis 5(1998), 45-58.[4] J. Schu, Weak and strong convergence to fixed points of asymptotically nonex-pansive mappings, Bull. Austral. Math. Soc. 43(1991), 153-159.[5] K.K.Tan, H.K.Xu, Approximating fixed points of nonexpansive mappingsby the Ishikawa iteration process, J. Math. Anal. Appl. 178(1993), 301-308.[6] F.E.Brower, Nonlinear operators and nonlinear equations of evolution inBanach spaces, Proceedings of the Symposium on Pure Mathematics, Vol.18,AMS, Providence, RI, 1976.

English

Multi-step iterative process with errors for common fixed points of a finite family of nonexpansive mappings

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Mathematical Communications 11(2006), 47-54 47Multi-step iterative process with errors forcommon fixed points of a finite family ofnonexpansive mappings∗Feng Gu† and Zhenhua He‡Abstract. In this paper, we study a multi-step iterative schemewith errors involving N nonexpansive mappings in the Banach space.Some weak and strong convergence theorems for approximation of com-mon fixed points of nonexpansive mappings are proved using this itera-tion scheme. The results extend and improve the corresponding resultsof [1].Key words: nonexpansive mapping, multi-step iteration processwith errors, common fixed pointsAMS subject classifications: 47H05, 47H09, 49M05Received January 17, 2006 Accepted May 26, 20061. Introduction and preliminariesLet K be a nonempty convex subset of a normed linear space E, and let {Ti}Ni=1 beN self-maps of K. Khan and Fukhar-ud-din [1] introduced the following iterativescheme.The sequence{xn} defined byx1 = x ∈ K,xn+1 = anSyn + bnxn + cnun,yn = a′nTxn + b′nxn + c′nvn, ∀n ≥ 1.(1.1)where {an}, {bn}, {cn}, {a′n}, {b′n} and {c′n} are six real sequences in [0,1] with0 < δ ≤ an, a′n ≤ 1 − δ < 1, an + bn + cn = a′n + b′n + c′n = 1, and {un}, {vn} arebounded sequences in K.∗The present studies were supported by the Natural Science Foundation of Zhejiang Province(Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Key TeacherCreating Capacity Fund of Heilongjiang General College (1053G015) and the Starting Foundationof Scientific Research of Hangzhou Teacher’s College.†Institute of Applied Mathematics and Department of Mathematics, Hangzhou Teacher’s Col-lege, Hangzhou, Zhejiang 310 036, China, e-mail: gufeng99@sohu.com‡Institute of Applied Mathematics and Department of Mathematics, Hangzhou Teacher’s Col-lege, Hangzhou, Zhejiang 310 036, China, e-mail: zhenhuahe@126.com48 F.Gu and Z.HeIf cn = c′n = 0, then the above scheme means thatx1 = x ∈ K,xn+1 = anSyn + bnxn,yn = a′nTxn + b′nxn, ∀n ≥ 1.(1.2)where {an}, {bn}, {a′n} and {b′n} are four real sequences in [0,1] satisfying an+bn =a′n + b′n = 1. This scheme has been studied by Das and Debata [2] and Takahashiand Tamura [3].Now, we further generalize the scheme given in (1.1) as follows.The sequence {xn} is defined byx1 = x ∈ K,xn+1 = a(1)n T1x(1)n + b(1)n xn + c(1)n u(1)n ,x(1)n = a(2)n T2x(2)n + b(2)n xn + c(2)n u(2)n ,x(2)n = a(3)n T3x(3)n + b(3)n xn + c(3)n u(3)n ,...x(N−2)n = a(N−1)n TN−1x(N−1)n + b(N−1)n xn + c(N−1)n u(N−1)n ,x(N−1)n = a(N)n TNx(N)n + b(N)n xn + c(N)n u(N)n ,x(N)n = xn for all n ≥ 1.(1.3)The scheme is expressed in a compact form asx(i−1)n = a(i)n Tix(i)n + b(i)n xn + c(i)n u(i)n , for all n ≥ 1, i ∈ I, (1.4)where I = {1, 2, 3, · · · , N}, xn+1 = x(0)n , {a(i)n }, {b(i)n } and {c(i)n } are three realsequences in [0,1] with 0 < δ ≤ a(i)n ≤ 1 − δ < 1, a(i)n + b(i)n + c(i)n = 1, and {u(i)n }and {v(i)n } are two bounded sequences in K.Since their introduction nonexpansive mappings have been extensively studiedby many authors in different frames of work. The purpose of this paper is to studythe weak and strong convergence of a multi-step iteration scheme (1.4) for N nonex-pansive mappings in a uniformly convex Banach space. The results presented in thispaper extend and improve the corresponding results of [1] from two nonexpansivemappings to a family of nonexpansive mappings.To proceed in this direction, we first recall the following definitions.A Banach space E is said to satisfy the Opial’s condition if whenever {xn} is asequence in E which converges weakly to x, thenlim supn→∞‖xn − x‖ < lim supn→∞‖xn − y‖ for all y ∈ E,with y = xA mapping T : K → E is called demiclosed with respect to y ∈ E if for eachsequence {xn} in K and each x ∈ E, xn ⇀ x, and Txn → y imply that x ∈ K andTx = y.Multi-step iterative process with errors 49In the sequel we shall need the following lemmas.Lemma 1.1 [Schu[4]]. Suppose that E is a uniformly convex Banach spaceand 0 < p ≤ tn ≤ q < 1 for all positive integers n. Also suppose that {xn} and {yn}are two sequences of E such that lim supn→∞ ‖xn‖ ≤ d, lim supn→∞ ‖yn‖ ≤ d andlimn→∞ ‖tnxn +(1− tn)yn‖ = d hold for some d ≥ 0, then limn→∞ ‖xn− yn‖ = 0.Lemma 1.2 [Tan and Xu[5]]. Let {sn} and {tn} be two nonnegative sequencessatisfyingsn+1 ≤ sn + tn for all n ≥ 1.If∑∞n=1 tn <∞, then limn→∞ sn exists.Lemma 1.3 [Browder[6]]. Let E be a uniformly convex Banach space satis-fying the Opial’s condition and let K be a nonempty closed convex subset of E. LetT be a nonexpansive mapping of K into itself, then I−T is demiclosed with respectto zero.2. Main resultsIn this section, let F (T ) denote the set of all fixed points of T .Lemma 2.1. Let E be a normed space and K its nonempty bounded convexsubset. Let {Ti}Ni=1 : K → K be N nonexpansive mappings and let {xn} be thesequence as defined in (1.4) with∑∞n=1 c(i)n < ∞, i ∈ I = {1, 2, 3, · · · , N}. IfF = ∩Ni=1F (Ti) = ∅, then limn→∞ ‖xn − x∗‖ exist for all x∗ ∈ F = ∩∞i=1F (Ti).Proof. Since K is bounded, there exists M > 0, such that ‖xn − u(i)n ‖ ≤M, for all i ∈ I. Assume that F = ∩∞i=1F (Ti) = ∅. Let x∗ ∈ F = ∩∞i=1F (Ti).Then||xn+1 − x∗|| = ||a(1)n T1x(1)n + b(1)n xn + c(1)n u(1)n − x∗||= ||a(1)n (T1x(1)n − x∗ + c(1)n (u(1)n − xn))+(1− a(1)n )(xn − x∗ + c(1)n (u(1)n − xn))||≤ a(1)n ||T1x(1)n − x∗||+ (1− a(1)n )||xn − x∗||+ c(1)n ||u(1)n − xn||≤ a(1)n ||x(1)n − x∗||+ (1− a(1)n )||xn − x∗||+ c(1)n M= a(1)n ||a(2)n T2x(2)n + b(2)n xn + c(2)n u(2)n − x∗||+(1− a(1)n )||xn − x∗||+ c(1)n M= a(1)n ||a(2)n (T2x(2)n − x∗ + c(2)n (u(2)n − xn))+(1− a(2)n )(xn − x∗ + c(2)n (u(2)n − xn))||+(1− a(1)n )||xn − x∗||+ c(1)n M≤ a(1)n (a(2)n ||x(2)n − x∗||+ (1 − a(2)n )||xn − x∗||+ c(2)n M)+(1− a(1)n )||xn − x∗||+ c(1)n M= a(1)n a(2)n ||x(2)n − x∗||+ a(1)n (1− a(2)n )||xn − x∗||+(1− a(1)n )||xn − x∗||+ a(1)n c(2)n M + c(1)n M50 F.Gu and Z.He= a(1)n a(2)n ||x(2)n − x∗||+ (1− a(1)n a(2)n )||xn − x∗||+ a(1)n c(2)n M + c(1)n M≤ a(1)n a(2)n (a(3)n ||x(3)n − x∗||+ (1− a(3)n )||xn − x∗||+ c(3)n M)+(1− a(1)n a(2)n )||xn − x∗||+ a(1)n c(2)n M + c(1)n M= a(1)n a(2)n a(3)n ||x(3)n − x∗||+ (1− a(1)n a(2)n a(3)n )||xn − x∗||+a(1)n a(2)n c(3)n M + a(1)n c(2)n M + c(1)n M· · · · · · · · · · · · · · · · · · · · · · · ·≤ a(1)n a(2)n · · · a(N)n ||x(N)n − x∗||+ (1− a(1)n a(2)n · · ·a(N)n )||xn − x∗||+a(1)n a(2)n · · ·a(N−1)n c(N)n M + a(1)n a(2)n · · · a(N−2)n c(N−1)n M+ · · ·+ a(1)n c(2)n M + c(1)n M≤ a(1)n a(2)n · · · a(N)n ||x(N)n − x∗||+ (1− a(1)n a(2)n · · ·a(N)n )||xn − x∗||+MΣNi=1c(i)n= ||xn − x∗||+MΣNi=1c(i)n .Since Σ∞n=1c(i)n <∞, hence, using Lemma 1.2, we have thatlimn→∞ ||xn − x∗|| exists for each x∗ ∈ F = ∩Ni=1F (Ti).This completes the proof of Lemma 2.1. ✷Lemma 2.2. Let E be a uniformly convex Banach space and K its nonemptybounded convex subset. Let {Ti}Ni=1 : K → K be nonexpansive mappings, and{xn} the sequence as defined in (1.4) with∑∞n=1 c(i)n < ∞, i = 1, 2, 3, · · · , N . IfF = ∩Ni=1F (Ti) = ∅, then limn→∞ ‖Tixn − xn‖ = 0 for all i ∈ I.Proof. By Lemma 2.1, limn→∞ ||xn−x∗|| exists. Suppose limn→∞ ||xn−x∗|| = dfor some d ≥ 0. Now,||x(i−1)n − x∗|| = ||a(i)n Tix(i)n + b(i)n xn + c(i)n u(i)n − x∗||= ||a(i)n (Tix(i)n − x∗ + c(i)n (u(i)n − xn))+(1− a(i)n )(xn − x∗ + c(i)n (u(i)n − xn))||≤ a(i)n ||Tix(i)n − x∗||+ (1 − a(i)n )||xn − x∗||+ c(i)n ||u(i)n − xn||≤ a(i)n ||x(i)n − x∗||+ (1 − a(i)n )||xn − x∗||+ c(i)n M= a(i)n ||a(i+1)n Ti+1x(i+1)n +b(i+1)n xn+c(i+1)n u(i+1)n −x∗||+(1− a(i)n )||xn−x∗||+c(i)n M≤ a(i)n (a(i+1)n ||x(i+1)n − x∗||+ (1− a(i+1)n )||xn − x∗||+ c(i+1)n M)+(1− a(i)n )||xn − x∗||+ c(i)n M= a(i)n a(i+1)n ||x(i+1)n − x∗||+ a(i)n (1− a(i+1)n )||xn − x∗||+(1− a(i)n )||xn − x∗||+ a(i)n c(i+1)n M + c(i)n M= a(i)n a(i+1)n ||x(i+1)n −x∗||+ (1−a(i)n a(i+1)n )||xn−x∗||+a(i)n c(i+1)n M+c(i)n M· · · · · · · · · · · · · · · · · ·≤ a(i)n a(i+1)n · · · a(N)n ||x(N)n − x∗||+ (1 − a(i)n a(i+1)n · · · a(N)n )||xn − x∗||+MΣNk=ic(k)n= ||xn − x∗||+MΣNk=1c(k)n .Multi-step iterative process with errors 51Taking limsup on both sides in the above inequality, we havelim supn→∞||x(i−1)n − x∗|| ≤ d. (2.1)Next, consider||Tix(i)n − x∗ + c(i)n (u(i)n − xn)|| ≤ ||Tix(i)n − x∗||+ c(i)n ||u(i)n − xn||≤ ||x(i)n − x∗||+ c(i)n MTaking limsup on both sides in the above inequality and then using (2.1), we havethatlim supn→∞||Tix(i)n − x∗ + c(i)n (u(i)n − xn)|| ≤ d for each i ∈ I. (2.2)Also,||xn − x∗ + c(i)n (u(i)n − xn)|| ≤ ||xn − x∗||+ c(i)n ||u(i)n − xn||≤ ||xn − x∗||+ c(i)n Mgives thatlim supn→∞||xn − x∗ + c(i)n (u(i)n − xn)|| ≤ d for each i ∈ I. (2.3)Further, limn→∞ ||xn+1 − x∗|| = d means thatlimn→∞ ||a(1)n (T1x(1)n − x∗ + c(1)n (u(1)n − xn)) + (1− a(1)n )(xn − x∗ + c(1)n (u(1)n − x∗))|| = dHence, applying Lemma 1.1, we get thatlimn→∞ ||T1x(1)n − xn|| = 0Next,||xn − x∗|| ≤ ||xn − T1x(1)n ||+ ||T1x(1)n − x∗|| ≤ ||xn − T1x(1)n ||+ ||x(1)n − x∗||.It follows from (2.1) thatd ≤ lim infn→∞ ||x(1)n − x∗|| ≤ lim supn→∞||x(1)n − x∗|| ≤ dThat is,limn→∞ ||x(1)n − x∗|| = dNow limn→∞ ||x(1)n − x∗|| = d can be expressed aslimn→∞ ||a(2)n (T2x(2)n − x∗ + c(2)n (u(2)n − xn)) + (1− a(2)n )(xn − x∗c(2)n (u(2)n − x∗))|| = dMoreover, by (2.2) and (2.3) we have thatlim supn→∞||T2x(2)n − x∗ + c(2)n (u(2)n − xn)|| ≤ d,52 F.Gu and Z.Heandlim supn→∞||xn − x∗ + c(2)n (u(2)n − x∗)|| ≤ dSo again by Lemma 1.1, we obtain thatlimn→∞ ||T2x(2)n − xn|| = 0Now,||xn − x∗|| ≤ ||xn − T2x(2)n ||+ ||T2x(2)n − x∗|| ≤ ||xn − T2x(2)n ||+ ||x(2)n − x∗||.From (2.1) it follows thatd ≤ lim infn→∞ ||x(2)n − x∗|| ≤ lim supn→∞||x(2)n − x∗|| ≤ dThat islimn→∞ ||x(2)n − x∗|| = dUsing the same method, we get thatlimn→∞ ||T3x(3)n − xn|| = 0, limn→∞ ||x(3)n − x∗|| = dlimn→∞ ||T4x(4)n − xn|| = 0, limn→∞ ||x(4)n − x∗|| = d· · · · · · · · · · · ·limn→∞ ||TNx(N)n − xn|| = 0, limn→∞ ||x(N)n − x∗|| = di.e.limn→∞ ||Tix(i)n − xn|| = 0, limn→∞ ||x(i)n − x∗|| = d for all i ∈ INow, observe that||xn − Tixn|| ≤ ||xn − Tix(i)n ||+ ||Tix(i)n − Tixn||≤ ||xn − Tix(i)n ||+ ||x(i)n − xn||= ||xn − Tix(i)n ||+ ||a(i+1)n Ti+1x(i+1)n + b(i+1)n xn + c(i+1)n u(i+1)n − xn||= ||xn − Tix(i)n ||+ ||a(i+1)n (Ti+1x(i+1)n − xn) + c(i+1)n (u(i+1)n − xn)||≤ ||xn − Tix(i)n ||+ a(i+1)n ||Ti+1x(i+1)n − xn||+ c(i+1)n MHencelimn→∞ ||xn − Tixn|| = 0 for all i ∈ I.This completes the proof of Lemma 2.2. ✷Theorem 2.1. Let E be a uniformly convex Banach space satisfying theOpial’s condition and let K,{Ti}Ni=1 and {xn} be taken as in Lemma 2.1. If F =∩Ni=1F (Ti) = ∅, then {xn} converges weakly to a common fixed point of {Ti}Ni=1.Multi-step iterative process with errors 53Proof. Let x∗ ∈ F = ∩Ni=1F (Ti), then as proved in Lemma 2.1, limn→∞ ||xn −x∗|| exists. Now we prove that {xn} has a unique weak subsequential limit inF = ∩Ni=1F (Ti). To prove this, let z1 and z2 be weak limits of the subsequences{xni} and {xnj} of {xn}, respectively. By Lemma 2.2, limn→∞ ‖Tixn − xn‖ = 0and I − Ti is demiclosed with respect to zero by Lemma 1.3, therefore we obtainTiz1 = z1, i.e., z1 ∈ F = ∩Ni=1F (Ti). Again in the same way, we can prove thatz2 ∈ F = ∩Ni=1F (Ti). Next, we prove the uniqueness. For this suppose that z1 = z2,then by the Opial’s conditionlimn→∞ ||xn − z1|| = limni→∞ ||xni − z1|| < limni→∞ ||xni − z2||= limn→∞ ||xn − z2|| = limnj→∞ ||xnj − z2||< limnj→∞||xnj − z1|| = limn→∞ ||xn − z1||This is a contradition. Hence {xn} converges weakly to a point in F = ∩Ni=1F (Ti).This completes the proof of Theorem 2.1. ✷Now we will prove a strong convergence theorem.Two mappings S, T :K → K, where K is a subset of E, are said to satisfycondition (A) if there exists a nondecreasing function f : [0,∞) → [0,∞) withf(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that 12 (||x−Tx||+ ||x−Sx||) ≥ f(d(x, F ))for allx ∈ K where d(x, F ) = inf{||x− x∗|| : x∗ ∈ F = F (S) ∩ F (T )}.Khan and Fukhar-ud-din [1] approximated a common fixed point of two nonex-pansive mappings S and T by iterating scheme (1.1). We modify this condition formappings {Ti}Ni=1 : K → K as follow:N mappings {Ti}Ni=1 : K → K where K is a subset of E, are said to satisfycondition (A′) if there exists a nondecreasing function f : [0,∞) → [0,∞) withf(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that 1NΣNi=1||x − Tix|| ≥ f(d(x, F )) forall x ∈ K where d(x, F ) = inf{||x− x∗|| : x∗ ∈ ∩Ni=1F (Ti)}.Note that condition (A′) reduces to condition (A) when N = 2. We shall usecondition (A′) to study the strong convergence of {xn} defined in (1.4).Theorem 2.2. Let E be a uniformly convex Banach space and let K, {xn} betaken as in Lemma 2.1. Let {Ti}Ni=1 : K → K be nonexpansive mappings satisfyingcondition (A′). If F = ∩Ni=1F (Ti) = ∅, then {xn} converges strongly to a commonfixed point of {Ti}Ni=1.Proof. By Lemma 2.1, suppose that limn→∞ ||xn − x∗|| = d for all x∗ ∈ F =∩Ni=1F (Ti). If d = 0, there is nothing to prove. Assume d > 0, by Lemma 2.2limn→∞ ||Tixn − xn|| = 0, i ∈ I. Moreover,||xn+1 − x∗|| ≤ ||xn − x∗||+MΣNi=1c(i)n ,gives thatinfx∗∈F||xn+1 − x∗|| ≤ infx∗∈F||xn − x∗||+MΣNi=1c(i)n .That is,d(xn+1, F ) ≤ d(xn, F ) +MΣNi=1c(i)ngives that limn→∞ d(xn, F ) exists by virtue of Lemma 1.2. Now by condition (A′),limn→∞ f(d(xn, F )) = 0. Since f is a nondecreasing function and f(0) = 0, there-fore limn→∞ d(xn, F ) = 0. Now we can take a subsequence {xnj} of {xn} and a54 F.Gu and Z.Hesequence {pj} ⊂ F such that ||xnj − pj || < 2−j. Then following the method ofproof of Tan and Xu [5], we get that {pj} is a Cauchy sequence in F and so itconverges. Let pj → p. Since F is closed, therefore p ∈ F and then xnj → p. Aslimn→∞ ||xn − x∗|| exists, xn → p ∈ F = ∩∞i=1F (Ti). This completes the proof ofTheorem 2.2. ✷Remark 2.1. Theorems 2.1-2.2 extend and improve the corresponding resultsof [1].References[1] S.H.Khan, H.Fukhar-ud-din, Weak and strong convergence of a schemewith errors for two nonexpansive mappings, Nonlinear Analysis 61(2005), 1295-1301.[2] G.Das, J. P.Debata, Fixed points of quasi-nonexpansive mappings, IndianJ. Pure Appl. Math. 17(1986), 1263-1269.[3] W.Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansivemappings, J. Convex Analysis 5(1998), 45-58.[4] J. Schu, Weak and strong convergence to fixed points of asymptotically nonex-pansive mappings, Bull. Austral. Math. Soc. 43(1991), 153-159.[5] K.K.Tan, H.K.Xu, Approximating fixed points of nonexpansive mappingsby the Ishikawa iteration process, J. Math. Anal. Appl. 178(1993), 301-308.[6] F.E.Brower, Nonlinear operators and nonlinear equations of evolution inBanach spaces, Proceedings of the Symposium on Pure Mathematics, Vol.18,AMS, Providence, RI, 1976.

Multi-step iterative process with errors for common fixed points of a finite family of nonexpansive mappings

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