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Convergence and stability of modified Ishikawa iteration sequence with errors
Fixed Point Theory and Applications volume 2014, Article number: 224 (2014)
Abstract
We show some stability and convergence theorems of the modified Ishikawa iterative sequence with errors for a strongly successively pseudocontractive and strictly asymptotically pseudocontractive mapping in a real Banach space. Additionally, we prove that if T is a uniformly Lipschitzian strongly accretive mapping, the modified Ishikawa iteration sequence with errors converges strongly to the unique solution of the equation Tx=f. The main results of this paper improve and extend the known results in the current literature.
MSC:47H09, 47H10, 47J25.
1 Introduction
Developments in fixed point theory reflect that the iterative construction of fixed points is proposed and vigorously analyzed for various classes of maps in different spaces. The class of pseudocontractive mappings in their relation with iteration procedures has been studied by several researchers under suitable conditions; for more details, see [1–3] and the references therein. Also, the class of nonexpansive mappings via iteration methods has extensively been studied in this regard; see Tan and Xu [4]. The class of strongly pseudocontractive mappings has been studied by many researchers (see [5–7]) under certain conditions. Stability results established in metric space, normed linear space, and Banach space settings are available in the literature. There are several authors whose contributions are of colossal value in the study of stability of the fixed point iterative procedures: Imoru and Olatinwo [8], Olatinwo [9], Haghi et al. [10], Olatinwo and Postolache [11]. Reich and Zaslavski [12] in Chapter 2 established the existence and uniqueness of a fixed point for a generic mapping, convergence of the iterates of a nonexpansive mapping, stability of the fixed point under small perturbations of a mapping, convergence of Krasnosel’skiiMann iterations of nonexpansive mappings, generic power convergence of order preserving mappings, and existence and uniqueness of positive eigenvalues and eigenvectors of orderpreserving linear operators. They also studied the convergence of iterates of nonexpansive mappings in the presence of computational errors in this chapter. Harker and Hicks in [13] showed how a stability sequence could arise in practice and demonstrated the importance of investigating the stability of various iterative sequences for some kinds of nonlinear mappings.
The purpose in this paper is to study the modified Ishikawa iteration sequence with errors converging strongly to a fixed point of the uniformly Lipschitzian strongly successively pseudocontractive mapping under the lack of some conditions. On the other hand, the authors show that the modified Ishikawa iteration sequence with errors converges strongly to the unique solution of the equation Tx=f if T is a Lipschitzian strongly accretive mapping. The results of this paper improve and extend some recent results.
2 Preliminaries
Throughout this paper, we assume that E is a real Banach space with dual {E}^{\ast}. Suppose that \u3008\cdot ,\cdot \u3009 is the dual pair between E and {E}^{\ast}, and J:E\to {2}^{{E}^{\ast}} is the normalized duality mapping defined by
First, we recall some concepts. A mapping T:E\to E is said to be:

(i)
uniformly Lipschizian if there exists a constant L>0 such that
\parallel {T}^{n}x{T}^{n}y\parallel \le L\parallel xy\parallel \phantom{\rule{1em}{0ex}}\text{for all}x,y\in E,n\ge 1; 
(ii)
strongly successively pseudocontractive if for every x,y\in E there exist t>1 and j(xy)\in J(xy) such that
\u3008{T}^{n}x{T}^{n}y,j(xy)\u3009\le \frac{1}{t}{\parallel xy\parallel}^{2}\phantom{\rule{1em}{0ex}}\text{for all}n\ge 1;(2.1) 
(iii)
strongly pseudocontractive if for every x,y\in E there exist t>1 and j(xy)\in J(xy) such that
\u3008TxTy,j(xy)\u3009\le \frac{1}{t}{\parallel xy\parallel}^{2}.(2.2)
Example 2.1 Let E=R=(\mathrm{\infty},\mathrm{\infty}) with the usual norm. Take K=[0,1] and define T:K\to K by
for all n\ge 0. Then F(T)=\{0\} and T is not continuous at x=1. We can verify that
Thus {T}^{2} is continuous in K and {T}^{2}K\subset [0,{3}^{n}] for all n\ge 1. Then for any x\in K, there exists j(x0)\in J(x0) satisfying
for all n\ge 1. That is, T is a strongly successively pseudocontractive mapping.
Lemma 2.2 (see [[14], Lemma 1.1])
Let E be a Banach space and x,y\in E. Then \parallel x\parallel \le \parallel x+\gamma y\parallel for all \gamma >0 if and only if there exists j(x)\in J(x) such that Re\u3008y,j(x)\u3009\ge 0.
In the sequel, let k=\frac{t1}{t}, where t is the constant appearing in (2.1). It follows from (2.1) that
Therefore, it follows from Lemma 2.2 and (2.3) that the definition of a strongly successively pseudocontractive mapping is equivalent to the following definition.
Definition 2.3 T is strongly successively pseudocontractive if there exists t>1 such that
for all x,y\in E, s>0 and n\ge 1.
Definition 2.4 Let T:E\to E be a mapping. For a given {x}_{0}\in E. \{{a}_{n}\}, \{{b}_{n}\} are sequences in [0,1], and \{{u}_{n}\}, \{{v}_{n}\} are sequences in E. The sequence \{{x}_{n}\}\subset E defined by
is said to be a modified Ishikawa iteration sequence with errors.
The following lemmas will be needed in proving our main results.
Lemma 2.5 (see [15])
Let \{{\lambda}_{n}\}, \{{\mu}_{n}\}, \{{c}_{n}\} be nonnegative real sequences satisfying the inequality
If {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}, then (i) {lim}_{n\to \mathrm{\infty}}{\lambda}_{n} exists, and (ii) in particular, if \{{\lambda}_{n}\} has a subsequence \{{\lambda}_{{n}_{k}}\} converging to 0, then {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0.
From Lemma 2.5 we have the following.
Lemma 2.6 Let \{{\lambda}_{n}\}, \{{\mu}_{n}\}, \{{d}_{n}\} be nonnegative real sequences satisfying
where \{{t}_{n}\} is a sequence in [0,1] such that {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}=\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty} and {d}_{n}=o({t}_{n}). Then {\lambda}_{n}\to 0 as n\to \mathrm{\infty}.
Proof Since {d}_{n}=o({t}_{n}), there exists a natural number {n}_{0} such that {d}_{n}\le \frac{{t}_{n}{\lambda}_{n}}{2} for n\ge {n}_{0}. It follows from (2.6) that
By Lemma 2.5, we see that {lim}_{n\to \mathrm{\infty}}{\lambda}_{n} exists. Therefore, there exists D>0 such that {\lambda}_{n}\le D.
Let {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=\delta, then \delta =0. Indeed, if \delta >0, there exists a natural number {N}_{0} such that {\lambda}_{n}\ge (\delta /2), {d}_{n}\le ({t}_{n}{\lambda}_{n})/2 for n\ge {N}_{0}. It follows from (2.6) that
This implies \frac{\delta {t}_{n}}{4}\le \frac{{\lambda}_{n}{t}_{n}}{2}\le {\lambda}_{n}{\lambda}_{n+1}+D{\mu}_{n}+{c}_{n}, \mathrm{\forall}n\ge {N}_{0}. Therefore,
Note that {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}, and we have {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}<\mathrm{\infty}, a contradiction with {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}=\mathrm{\infty}. Then {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0. This completes the proof of Lemma 2.6. □
Let E be a Banach space and T a selfmap of E. Suppose {x}_{0}\in E and {x}_{n+1}=f({x}_{n},T) defines an iteration procedure which yields a sequence of points \{{x}_{n}\}\subset E. Let F(T)=\{p\in E:Tp=p\}\ne \mathrm{\varnothing} denote the fixed point of T and let \{{x}_{n}\} converge to a fixed point p of T. Let \{{y}_{n}\}\subset E and let {\epsilon}_{n}=\parallel {y}_{n+1}f({y}_{n},T)\parallel be a sequence in [0,\mathrm{\infty}).
Definition 2.7 (see [13, 16, 17])
If {lim}_{n\to \mathrm{\infty}}{\epsilon}_{n}=0 implies that {lim}_{n\to \mathrm{\infty}}{y}_{n}=p, then the iteration procedure defined by {x}_{n+1}=f({x}_{n},T) is said to be Tstable. If {\sum}_{n=0}^{\mathrm{\infty}}{\epsilon}_{n}<\mathrm{\infty} implies {lim}_{n\to \mathrm{\infty}}{y}_{n}=p, then the iteration procedure defined by {x}_{n+1}=f({x}_{n},T) is said to be almost Tstable.
3 Main results
Theorem 3.1 Let T:E\to E be a uniformly Lipschitzian and strongly successively pseudocontractive mapping with F(T)\ne \mathrm{\varnothing}. Let \{{x}_{n}\} be defined by (2.5) and \{{a}_{n}\},\{{b}_{n}\}\subset [0,1] and \{{u}_{n}\},\{{v}_{n}\}\subset E satisfying the conditions:

(i)
\parallel {u}_{n}\parallel =o({a}_{n}), \parallel {v}_{n}\parallel \to 0 as n\to \mathrm{\infty}; {\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty};

(ii)
{lim}_{n\to \mathrm{\infty}}sup{a}_{n}<k/({(L+1)}^{3}+2) and {\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}{b}_{n}<\mathrm{\infty}.
Suppose \{{y}_{n}\}\subset E and define \{{\epsilon}_{n}\} by
Then the following assertions hold:

(1)
\{{x}_{n}\} converges strongly to a unique fixed point of T in E;

(2)
\{{x}_{n}\} is almost Tstable;

(3)
if {lim}_{n\to \mathrm{\infty}}{\epsilon}_{n}/{a}_{n}=0 implies {lim}_{n\to \mathrm{\infty}}{y}_{n}=p, then the iteration procedure defined by {x}_{n+1}=f({x}_{n},T) is said to be a weakly Tstable. Thus \{{x}_{n}\} is also weakly Tstable.
Proof We will show that F(T) is a singleton. Indeed, if {p}_{1},{p}_{2}\in F(T), by the definition of strongly successively pseudocontractive of T, there exists a j({p}_{1}{p}_{2})\in J({p}_{1}{p}_{2}) such that
and, since t>1, this implies that {p}_{1}={p}_{2}.
It follows from (2.5) that
It follows from (2.5), (3.1), and (3.2) that
It follows from (2.5) that
For p\in F(T), we have p=(1+{a}_{n})p+{a}_{n}(I{T}^{n}kI)p(1k){a}_{n}p. Therefore, we get
It follows from (3.4) and (2.4) that
Since {(1+{a}_{n})}^{1}\le 1, {(1+{a}_{n})}^{1}\le 1{a}_{n}+{a}_{n}^{2}, and (1+(1k){a}_{n})(1{a}_{n}+{a}_{n}^{2})=1k{a}_{n}+k{a}_{n}^{2}+(1k){a}_{n}^{3}\le 1k{a}_{n}+k{a}_{n}^{2}+(1k){a}_{n}^{2}=1k{a}_{n}+{a}_{n}^{2}, we have
Note that {lim}_{n\to \mathrm{\infty}}sup{a}_{n}<k/({(L+1)}^{3}+2), then there exists a natural number {N}_{1} such that \gamma ={sup}_{n\ge {N}_{1}}{a}_{n}<k/({(L+1)}^{3}+2). It follows from (3.5) that
holds for n\ge {N}_{1}. Let {\lambda}_{n}:=\parallel {x}_{n}p\parallel, {\mu}_{n}:=L(L+1){a}_{n}{b}_{n}, {c}_{n}=0, {d}_{n}:=(L+3)\parallel {u}_{n}\parallel +L(L+3){a}_{n}\parallel {v}_{n}\parallel, {t}_{n}:=(k\gamma ({(L+1)}^{3}+2)){a}_{n}. Thus, (3.6) becomes
Since {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, {d}_{n}=o({t}_{n}), it follows from Lemma 2.6 that we have {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0. That is, \{{x}_{n}\} converges strongly to p.
Next, we prove the conclusion (2). Let
For p\in F(T), we have
It follows from (3.7) that
By using a similar method to proving (3.6), we can prove that
Substituting (3.9) into (3.8) for n\ge {N}_{0} we get
If {\sum}_{n=1}^{\mathrm{\infty}}{\epsilon}_{n}<\mathrm{\infty}, setting {\lambda}_{n}:=\parallel {y}_{n}p\parallel, {\mu}_{n}:=L(L+1){a}_{n}{b}_{n}, {c}_{n}={\epsilon}_{n}, {d}_{n}:=(L+3)\parallel {u}_{n}\parallel +L(L+3){a}_{n}\parallel {v}_{n}\parallel, {t}_{n}:=(k\gamma ({(L+1)}^{3}+2)){a}_{n} in Lemma 2.6, we have {y}_{n}\to p as n\to \mathrm{\infty}, i.e., \{{x}_{n}\} is almost Tstable.
If {lim}_{n\to \mathrm{\infty}}\frac{{\epsilon}_{n}}{{a}_{n}}=0, setting {\lambda}_{n}:=\parallel {y}_{n}p\parallel, {\mu}_{n}:=L(L+1){a}_{n}{b}_{n}, {c}_{n}=0, {d}_{n}:=(L+3)\parallel {u}_{n}\parallel +L(L+3){a}_{n}\parallel {v}_{n}\parallel +{\epsilon}_{n}, {t}_{n}:=(k\gamma ({(L+1)}^{3}+2)){a}_{n} in Lemma 2.6, we have {y}_{n}\to p as n\to \mathrm{\infty}, i.e., \{{x}_{n}\} is weakly Tstable. This completes the proof. □
Similar to the proof of Theorem 3.1, we have the following.
Theorem 3.2 Let T:E\to E be a uniformly Lipschitzian and strictly asymptotically pseudocontractive mapping with F(T)\ne \mathrm{\varnothing}. Let \{{x}_{n}\} be defined by (2.5). Assume that \{{a}_{n}\},\{{b}_{n}\}\subset [0,1] and \{{u}_{n}\},\{{v}_{n}\}\subset E satisfy the conditions:

(i)
\parallel {u}_{n}\parallel =o({a}_{n}), \parallel {v}_{n}\parallel \to 0 as n\to \mathrm{\infty};

(ii)
there exists \delta \in (0,1k) such that {lim}_{n\to \mathrm{\infty}}sup{a}_{n}<\delta /({(L+1)}^{3}+2);

(iii)
{\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}{b}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}.
Suppose \{{y}_{n}\}\subset E and define \{{\epsilon}_{n}\} by
Then the following assertions hold:

(1)
\{{x}_{n}\} converges strongly to a unique fixed point of T in E;

(2)
\{{x}_{n}\} is both almost Tstable and weakly Tstable.
Theorem 3.3 Let T:E\to E be a Lipschitzian and strongly accretive mapping. Let \{{a}_{n}\},\{{b}_{n}\}\subset [0,1], and \{{u}_{n}\},\{{v}_{n}\}\subset E satisfy the conditions:

(i)
\parallel {u}_{n}\parallel =o({a}_{n}), \parallel {v}_{n}\parallel \to 0 as n\to \mathrm{\infty};

(ii)
{lim}_{n\to \mathrm{\infty}}sup{a}_{n}<k/({(L+1)}^{3}+2), where k is the constant of strongly accretive mapping T, and L is the Lipschitzian constant of mapping IT;

(iii)
{\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}{b}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}.
For arbitrary {x}_{0}\in E, the sequence \{{x}_{n}\} defined by
converges strongly to a solution p of Tx=f.
Proof From the result of [5], we obtain the existence of a solution for Tx=f. Since T is strongly accretive with a constant k\in (0,1), we can prove that the solution of Tx=f is unique. Define Sx=f+(IT)x, then S is a strongly pseudocontractive mapping and has a fixed point p, and it is also a Lipschitzian mapping with a constant L. For all x,y\in E, there exists s>0 such that
The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □
Remark 3.4 (1) Theorem 3.3 extends the main result of [6] from a uniformly smooth Banach space to a real Banach space and without the boundedness assumption of D(T)=R(T) and {lim}_{n\to \mathrm{\infty}}{a}_{n}={lim}_{n\to \mathrm{\infty}}{b}_{n}=0; (2) Theorem 3.3 extends and improves the corresponding results of [7] by removing the assumptions {b}_{n}\le {a}_{n} and {\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}^{2}<\mathrm{\infty}.
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Acknowledgements
The authors thank the referees and the editor for their careful reading of the manuscript and their many valuable comments and suggestions for the improvement of this article. The first author was supported in part by the Humanity and Social Sciences Research Planning Foundation of Ministry of Education of China (Grant No. 14YJAZH095). The second author was supported by National Natural Science Foundation of China (Grant No. 61374081) and the Natural Science Foundation of Guangdong Province (Grant No. S2013010013034).
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Yang, L., Peng, S. Convergence and stability of modified Ishikawa iteration sequence with errors. Fixed Point Theory Appl 2014, 224 (2014). https://doi.org/10.1186/168718122014224
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DOI: https://doi.org/10.1186/168718122014224
Keywords
 modified Ishikawa iteration sequence with errors
 strongly successively pseudocontractive mapping
 uniformly Lipschitzian mapping
 almost Tstability
 weak Tstability