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Convergence theorems of a new iteration for two nonexpansive mappings
Journal of Inequalities and Applications volume 2014, Article number: 82 (2014)
Abstract
The purpose of this paper is to introduce the following new general implicit iteration scheme for approximating the common fixed points of a pair of nonexpansive mappings in a uniformly convex Banach space: for any {x}_{0}\in C, the iterative process \{{x}_{n}\} defined by {x}_{n}={a}_{n}{x}_{n1}+{b}_{n}T{y}_{n}+{c}_{n}S{x}_{n}, {y}_{n}={a}_{n}^{\prime}{x}_{n1}+{b}_{n}^{\prime}{x}_{n}+{c}_{n}^{\prime}S{x}_{n1}+{d}_{n}^{\prime}T{x}_{n}, where \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{a}_{n}^{\prime}\}, \{{b}_{n}^{\prime}\}, \{{c}_{n}^{\prime}\}, \{{d}_{n}^{\prime}\} are seven sequences of real numbers satisfying {a}_{n}+{b}_{n}+{c}_{n}=1, {a}_{n}^{\prime}+{b}_{n}^{\prime}+{c}_{n}^{\prime}+{d}_{n}^{\prime}=1, and T,S:C\to C are two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme.
1 Introduction
Let C be a nonempty subset of a real Banach space E. A mapping T of C into itself is called nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel holds for all x,y\in C. We first recall the following two iterative processes due to Ishikawa [1] and Mann [2], respectively.
(I) Let C be a nonempty convex subset of E and let T:C\to C be a mapping. For any given {x}_{0}\in C the sequence \{{x}_{n}\} defined by
is called the Ishikawa iteration sequence, where \{{a}_{n}\} and \{{b}_{n}\} are two real sequences in [0,1] satisfying some conditions. In particular, if {b}_{n}=0 for all n\ge 0, then \{{x}_{n}\} defined by
is called the Mann iteration sequence.
In [3], Liu introduced the concepts of Ishikawa and Mann iterative processes with errors as follows.
(II) For a nonempty subset C of a Banach space E and a mapping T:C\to C, the sequence \{{x}_{n}\} defined by
where \{{u}_{n}\} and \{{v}_{n}\} are two summable sequences in E. In particular, if {b}_{n}=0, {v}_{n}=0, the sequence \{{x}_{n}\} is called the Mann iteration sequence with errors.
Unfortunately, the definitions of Liu, which depend on the convergence of the error terms, are against the randomness of errors. Xu [4] studied the following new iteration process.
(III) Let C be a nonempty convex subset of E and let T:C\to C be a mapping. For any given {x}_{0}\in C the sequence \{{x}_{n}\} defined by
is called the Ishikawa iteration sequence with errors. Here \{{u}_{n}\} and \{{v}_{n}\} are two bounded sequences in C and \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{a}_{n}^{\prime}\}, \{{b}_{n}^{\prime}\}, \{{c}_{n}^{\prime}\} are six sequences in [0,1] satisfying the following conditions: {a}_{n}+{b}_{n}+{c}_{n}=1, {a}_{n}^{\prime}+{b}_{n}^{\prime}+{c}_{n}^{\prime}=1, n\ge 1. In particular, if {b}_{n}^{\prime}=0, {c}_{n}^{\prime}=0, the sequence \{{x}_{n}\} is called the Mann iteration sequence with errors. Chidume and Moore [5] studied the above schemes in 1999.
A generalization of Mann and Ishikawa iterative schemes was given by Das and Debata [6] and Takahashi and Tamura [7]. This scheme dealt with two mappings: {x}_{0}\in C
Recently Khan and Fukhar [8] considered the above iterative process with bounded errors.
In the present paper, we consider the following scheme:
where \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{a}_{n}^{\prime}\}, \{{b}_{n}^{\prime}\}, \{{c}_{n}^{\prime}\}, \{{d}_{n}^{\prime}\} are seven sequences of real numbers in [0,1] satisfying {a}_{n}+{b}_{n}+{c}_{n}=1, {a}_{n}^{\prime}+{b}_{n}^{\prime}+{c}_{n}^{\prime}+{d}_{n}^{\prime}=1, and T,S:C\to C are two nonexpansive mappings. Recently, some authors discuss similar issues (for example, please refer to [9–11]).
Approximating fixed points is an important subject in the theory of nonexpansive mappings and its applications in numerous applied areas. One is the convergence of iteration schemes constructed through nonexpansive mappings. In this paper, we study the iterative scheme given in (1) for weak and strong convergence for a pair of nonexpansive mappings in a uniformly convex Banach space. Before our discussions, we first recall the following definitions.
A Banach space E is said to satisfy Opial’s condition if whenever \{{x}_{n}\} is a sequence in E which converges weakly to x, then
It is well known that every Hilbert space satisfies the Opial condition (see for example [12]).
A mapping T is said to be semicompact (see, e.g., [13]) if for any sequence {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} in C such that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0, there exists a subsequence {\{{x}_{{n}_{j}}\}}_{j=1}^{\mathrm{\infty}} of {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} such that {\{{x}_{{n}_{j}}\}}_{j=1}^{\mathrm{\infty}} converges strongly to some u\in C.
A mapping T with domain D(T) and range R(T) in E is said to be demiclosed at a point p\in E if whenever \{{x}_{n}\} is as sequence in D(T) such that \{{x}_{n}\} converges weakly to x\in D(T) and \{T{x}_{n}\} converges strongly to p, then Tx=p.
We shall make use of the following results.
Lemma 1.1 [14]
Let \{{s}_{n}\}, \{{t}_{n}\} be two sequences of nonnegative real numbers satisfying

(a)
If {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}<\mathrm{\infty}, then {lim}_{n\to \mathrm{\infty}}{s}_{n} exists.

(b)
If {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}<\mathrm{\infty} and \{{s}_{n}\} has a subsequence converging to zero, then {lim}_{n\to \mathrm{\infty}}{s}_{n}=0.
Lemma 1.2 [15]
Let E be a uniformly convex Banach space satisfying Opial’s condition and let C be a nonempty closed convex subset of E. Let T be a nonexpansive mapping of C into itself. Then IT is demiclosed with respect to zero.
Lemma 1.3 [16]
Suppose that E is a uniformly convex Banach space and 0<p\le {t}_{n}\le q<1 for all positive integers n. Also suppose that \{{x}_{n}\} and \{{y}_{n}\} are two sequences of E such that {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\parallel {x}_{n}\parallel \le r, {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\parallel {y}_{n}\parallel \le r and {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\parallel {t}_{n}{x}_{n}+(1{t}_{n}){y}_{n}\parallel =r hold for some r\ge 0. Then {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0.
2 Main results
We first give the following key lemma.
Lemma 2.1 Let E be a real uniformly convex Banach space and C its nonempty convex subset. Let T,S:C\to C be nonexpansive mappings. Let \{{x}_{n}\} be the sequence as defined in (1) with the following conditions:

(1)
{a}_{n}\to 0, {a}_{n}^{\prime}\to 0, {b}_{n}^{\prime}\to 0, as n\to \mathrm{\infty};

(2)
{b}_{n},{c}_{n},{c}_{n}^{\prime},{d}_{n}^{\prime}\in [\delta ,1\delta ] for some \delta \in (0,1);

(3)
{c}_{n}^{\prime}+{d}_{n}^{\prime}\le \gamma for some \gamma \in (0,1).
If F:=F(T)\cap F(S)\ne \mathrm{\varnothing}, then we have

(i)
{lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists for all p\in F and \{{x}_{n}\}, \{T{x}_{n}\} and \{S{x}_{n}\} are all bounded;

(ii)
{lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}S{x}_{n}\parallel.
Proof For any p\in F, we have
However, as the proof of the above inequality, it follows that
Thus from (2) and (3), it is easy to check
which implies that
and the limit {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists for all p\in F. Furthermore \{{x}_{n}\} is bounded and \{S{x}_{n}\} and \{T{x}_{n}\} are both bounded also. Now suppose {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel =c for some c\ge 0. By the inequalities (3) and (5), we have
From the iterative process (1), we have
Since {a}_{n}\to 0, \{{x}_{n}\}, \{S{x}_{n}\} are both bounded and (6), it follows from Lemma 1.3 that
Next, by the inequality (2) and (5), we have
which means
Taking liminf on both sides in the above inequality and by (6) and {b}_{n}\in [\delta ,1\delta ] for some \delta \in (0,1), we have
which yields
It is easy to see that
Since {a}_{n}^{\prime}\to 0, {b}_{n}^{\prime}\to 0, \{S{x}_{n}\} and \{T{x}_{n}\} are both bounded, then by Lemma 1.3 we have
Moreover, by the iterative process (1) again, we get
which, by {a}_{n}\to 0 as n\to \mathrm{\infty} and (8), implies that
Hence from (8) and (13), we have
as n\to \mathrm{\infty}. Note that
then it follows from the above inequality that
That is to say,
which, from (8), (12), (14), and {a}_{n}^{\prime}\to 0, means that
The proof of the lemma is completed. □
Now we give the weak convergence first.
Theorem 2.2 Let E be a uniformly convex Banach space satisfying Opial’s condition and C, S, T, and \{{x}_{n}\} be as taken in Lemma 2.1. If F:=F(T)\cap F(S)\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges weakly to a common fixed point of S and T.
Proof Let p\in F. As the proof of Lemma 2.1, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists. Since E is uniformly convex, every bounded subset of E is weakly compact, so that there exists a subsequence \{{x}_{{n}_{k}}\} of the bounded sequence \{{x}_{n}\} such that \{{x}_{{n}_{k}}\} converges weakly to a point q\in C. Therefore, it follows from (ii) in Lemma 2.1 that
By Lemma 1.2, we know IT is demiclosed, then it is easy to see that q\in F(T). With a similar proof, it follows that q\in F(S) also. Next we prove uniqueness. Suppose that this is not true, then there must exist a subsequence \{{x}_{{n}_{j}}\}\subset \{{x}_{n}\} such that \{{x}_{{n}_{j}}\} converges weakly to another {q}^{\ast}\in C and {q}^{\ast}\ne q. Then by the same method given above, we can also prove {q}^{\ast}\in F(T)\cap F(S).
Because we have proved that for any p\in F, the limit {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists, we can let
By the Opial condition of E, we have
This is a contradiction, hence q={q}^{\ast}. This implies that \{{x}_{n}\} converges weakly to a common fixed point of S and T. □
Next we give several strong convergence results.
Theorem 2.3 Let E be a uniformly convex Banach space and C, \{{x}_{n}\} be as taken in Lemma 2.1. If one of the nonexpansive mappings T and S is semicompact and F:=F(T)\cap F(S)\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges strongly to a common fixed point of S and T.
Proof Since one of T and S is semicompact and by (ii) in Lemma 2.1, then there exists a subsequence {\{{x}_{{n}_{j}}\}}_{j=1}^{\mathrm{\infty}} of the sequence {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} such that {\{{x}_{{n}_{j}}\}}_{j=1}^{\mathrm{\infty}} converges strongly to u. Since C is closed, u\in C. Continuity of S and T gives \parallel S{x}_{{n}_{j}}Su\parallel \to 0 and \parallel T{x}_{{n}_{j}}Tu\parallel \to 0 as {n}_{j}\to \mathrm{\infty}. Then by Lemma 2.1,
This yields u\in F. By Lemma 2.1 again, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists for all p\in F, therefore \{{x}_{n}\} must itself converge to u\in F. This completes the proof. □
Recall that a mapping T:C\to C, where C is a subset of E, is said to satisfy condition (A) if there exists a nondecreasing function f:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with f(0)=0, f(r)>0 for all r\in (0,\mathrm{\infty}) such that \parallel xTx\parallel \ge f(d(x,F(T))) for all x\in C where d(x,F(T))=inf\{\parallel x{x}^{\ast}\parallel :{x}^{\ast}\in F(T)\}.
In [17], Senter and Dotson approximated fixed points of a nonexpansive mapping T by Mann iteration. Recently Maiti and Ghosh [18] and Tan and Xu [14] considered the approximation of fixed points of a nonexpansive mapping T by Ishikawa iteration under the same condition (A) which is weaker than the requirement that T is semicompact. Khan and Fukhar [8] modified this condition for two mappings S,T:C\to C as follows.
Let C be a subset of a Banach space E. Two mappings S,T:C\to C are said to satisfy condition ({A}^{\prime}) if there exists a nondecreasing function f:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with f(0)=0, f(r)>0 for all r\in (0,\mathrm{\infty}) such that \frac{1}{2}(\parallel xTx\parallel +\parallel xSx\parallel )\ge f(d(x,F)) for all x\in K, where d(x,F)=inf\{\parallel x{x}^{\ast}\parallel :{x}^{\ast}\in F=F(T)\cap F(S)\}.
Note that condition ({A}^{\prime}) reduces to condition (A) when S=T. We use condition ({A}^{\prime}) to study the strong convergence of \{{x}_{n}\} defined in (1).
Theorem 2.4 Let E be a uniformly convex Banach space and C, \{{x}_{n}\} be as taken in Lemma 2.1. Let T, S satisfy the condition ({A}^{\prime}) and F:=F(T)\cap F(S)\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges strongly to a common fixed point of S and T.
Proof By Lemma 2.1, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}^{\ast}\parallel exists for all {x}^{\ast}\in F=F(T)\cap F(S). Let it be c for some c\ge 0. If c=0, there is nothing to prove. Suppose c>0. By Lemma 2.1, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}S{x}_{n}\parallel. Moreover, from (4), we have \parallel {x}_{n}{x}^{\ast}\parallel \le \parallel {x}_{n1}{x}^{\ast}\parallel which gives
That is to say, d({x}_{n},F)\le d({x}_{n1},F) shows that {lim}_{n\to \mathrm{\infty}}d({x}_{n},F) exists by virtue of Lemma 1.1. Now by condition ({A}^{\prime}), {lim}_{n\to \mathrm{\infty}}f(d({x}_{n},F))=0. By the properties of f, therefore {lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0. Next we can take a subsequence \{{x}_{{n}_{j}}\} of \{{x}_{n}\} and \{{y}_{j}\}\subset F such that \parallel {x}_{{n}_{j}}{y}_{j}\parallel <{2}^{j}. Then following the method of proof of Tan and Xu [14], we find that \{{y}_{j}\} is a Cauchy sequence in F and so it converges. Let {y}_{j}\to y. Since F is closed, therefore y\in F and then {x}_{{n}_{j}}\to y. As {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}^{\ast}\parallel exists, {x}_{n}\to y\in F. This completes the proof. □
Remark 2.5 From the proof of the above results, it is easy to see that we can extend our theorems to the iterative process (1) with errors as follows: let C be a bounded closed convex subset of E and the sequence \{{x}_{n}\} be defined by
where \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{e}_{n}\}, \{{a}_{n}^{\prime}\}, \{{b}_{n}^{\prime}\}, \{{c}_{n}^{\prime}\}, \{{d}_{n}^{\prime}\}, \{{e}_{n}^{\prime}\}, are sequences in [0,1] with {a}_{n}+{b}_{n}+{c}_{n}+{e}_{n}=1, {a}_{n}^{\prime}+{b}_{n}^{\prime}+{c}_{n}^{\prime}+{d}_{n}^{\prime}+{e}_{n}^{\prime}=1 and T,S:C\to C are both nonexpansive mappings, \{{u}_{n}\},\{{v}_{n}\}\in C.
References
Ishikawa S: Fixed point and iteration of a nonexpansive mapping in a Banach spaces. Proc. Am. Math. Soc. 1976, 73: 65–71.
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S00029939195300548463
Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289
Xu YG: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224: 91–101. 10.1006/jmaa.1998.5987
Chidume CE, Moore C: Fixed point iteration for pseudocontractive maps. Proc. Am. Math. Soc. 1999,127(4):1163–1170. 10.1090/S0002993999050509
Das G, Debata JP: Fixed points of quasinonexpansive mappings. Indian J. Pure Appl. Math. 1986, 17: 1263–1269.
Takahashi W, Tamura T: Convergence theorems for a pair of nonexpansive mappings. J. Convex Anal. 1998,5(1):45–58.
Khan SH, Fukharuddin H: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Proc. Am. Math. Soc. 2005,61(8):1295–1301.
Kim GE: Weak and strong convergence for quasinonexpansive mappings in Banach spaces. Bull. Korean Math. Soc. 2012,49(4):799–813. 10.4134/BKMS.2012.49.4.799
Zhang F, Zhang H, Zhang YL: New iterative algorithm for two infinite families of multivalued quasinonexpansive mappings in uniformly convex Banach spaces. J. Appl. Math. 2013., 2013: Article ID 649537
Zhang JL, Su YF, Cheng QQ: Strong convergence theorems for a common fixed point of two countable families of relatively quasi nonexpansive mappings and applications. Abstr. Appl. Anal. 2012., 2012: Article ID 956950
Opial Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S000299041967117610
Sun ZH: Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings. J. Math. Anal. Appl. 2003, 286: 351–358. 10.1016/S0022247X(03)005377
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. 18. Proceedings of the Symposium on Pure Mathematics 1976.
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Senter HF, Dotson WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 1974,44(2):375–380. 10.1090/S00029939197403466088
Maiti M, Gosh MK: Approximating fixed points by Ishikawa iterates. Bull. Aust. Math. Soc. 1989, 40: 113–117. 10.1017/S0004972700003555
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The authors are very grateful to the referees for their critical and valuable comments, which allowed us to improve the presentation of this article.
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Hou, X., Du, H. Convergence theorems of a new iteration for two nonexpansive mappings. J Inequal Appl 2014, 82 (2014). https://doi.org/10.1186/1029242X201482
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DOI: https://doi.org/10.1186/1029242X201482
Keywords
 nonexpansive mapping
 common fixed points
 weak and strong convergence
 semicompact
 condition ({A}^{\prime})