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On solving Lipschitz pseudocontractive operator equations
Journal of Inequalities and Applications volume 2014, Article number: 314 (2014)
Abstract
We analyze the convergence of the Manntype double sequence iteration process to the solution of a Lipschitz pseudocontractive operator equation on a bounded closed convex subset of arbitrary real Banach space into itself. Our results extend the result in (Moore in Comp. Math. Appl. 43: 15851589, 2002).
MSC:47H10, 54H25.
1 Introduction
Let E be a real Banach space and {E}^{\ast} be the dual space of E. Let J be the normalized duality mapping from E to {2}^{{E}^{\ast}} defined by
for all x\in E where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. A singlevalued duality map will be denoted by j.
An operator T:E\to E is said to be

pseudocontractive if there exists j(xy)\in J(xy) such that
\u3008TxTy,j(xy)\u3009\le {\parallel xy\parallel}^{2}
for any x,y\in E;

accretive if for any x,y\in E, there exists j(xy)\in J(xy) satisfying
\u3008TxTy,j(xy)\u3009\ge 0; 
strongly pseudocontractive if there exist j(xy)\in J(xy) and a constant \lambda \in (0,1) such that
\u3008TxTy,j(xy)\u3009\le \lambda {\parallel xy\parallel}^{2}
for any x,y\in E;

strongly accretive if for any x,y\in E, there exist j(xy)\in J(xy) and a constant t\in (0,1) satisfying
\u3008TxTy,j(xy)\u3009\ge t{\parallel xy\parallel}^{2}
for all x,y\in E.
As a consequence of a result of Kato [1], the concept of pseudocontractive operators can equivalently be defined as follows:
T is strongly pseudocontractive if there exists \lambda \in (0,1) such that the inequality
holds for all x,y\in E and r>0. If \lambda =0 in the inequality (1.1), then T is pseudocontractive.
It is easy to see that T is pseudocontractive if and only if IT is accretive where I denotes the identity mapping on E.
Let C be a compact convex subset of a real Hilbert space and let T:C\to C be a Lipschitz pseudocontraction. It remains as an open question whether the Mann iteration process always converges to a fixed point of T. In [2] it was proved that the Ishikawa iteration process converges strongly to a fixed point of T. In 2001, Mutangadura and Chidume [3] constructed the following example to demonstrate that the Mann iteration process is not guaranteed to converge to a fixed point of a Lipschitz pseudocontraction mapping a compact convex subset of a real Hilbert space H into itself.
Example [3]
Let H={\mathrm{\Re}}^{2} with the usual Euclidean inner product, and for x=(a,b)\in H define {x}^{\perp}=(b,a). Now, let C={B}_{1}(o); the closed unit ball in H and let {C}_{1}=\{x\in H:\parallel x\parallel \le \frac{1}{2}\}, {C}_{2}=\{x\in H:\frac{1}{2}\le \parallel x\parallel \le 1\}. Define the map T:C\to C by
Observe that T is pseudocontractive, Lipschitz continuous (with Lipschitz constant 5) and has the origin (0,0) as its unique fixed point; C is compact and convex. However, for any x\in {C}_{1}, we have
while for any x\in {C}_{2}, we have
and therefore no Mann sequence can converge to (0,0), the unique fixed point of T, unless the initial guess is the fixed point itself.
Moore [4] introduced the concept of a Manntype double sequence iteration process and proved that it converges strongly to a fixed point of a continuous pseudocontraction which maps a bounded closed convex nonempty subset of a real Hilbert space into itself.
Definition 1.1 [4]
Let \mathcal{N} denote the set of all nonnegative integers (the natural numbers) and let E be a normed linear space. By a double sequence in E is meant a function f:\mathcal{N}\times \mathcal{N}\to E defined by f(n,m)={x}_{n,m}\in E. A double sequence \{{x}_{n,m}\} is said to converge strongly to {x}^{\ast} if given any \u03f5>0, there exist N,M>0 such that \parallel {x}_{n,m}{x}^{\ast}\parallel <\u03f5 for all n\ge N, m\ge M. If \mathrm{\forall}n,r\ge N, \mathrm{\forall}m,t\ge M, we have \parallel {x}_{n,r}{x}_{m,t}\parallel <\u03f5, then the double sequence is said to be Cauchy. Furthermore, if for each fixed n, {x}_{n,m}\to {x}_{n}^{\ast} as m\to \mathrm{\infty} and then {x}_{n}^{\ast}\to {x}^{\ast} as n\to \mathrm{\infty}, then {x}_{n,m}\to {x}^{\ast} as n,m\to \mathrm{\infty}.
Theorem 1.1 [4]
Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H, and let T:C\to C be a continuous pseudocontractive map. Let {\{{\alpha}_{n}\}}_{n\ge 0},{\{{a}_{k}\}}_{k\ge 0}\subset (0,1) be real sequences satisfying the following conditions:

(i)
{lim}_{k\to \mathrm{\infty}}{a}_{k}=1,

(ii)
{lim}_{k,r\to \mathrm{\infty}}({a}_{k}{a}_{r})/(1{a}_{k})=0, \mathrm{\forall}0<r\le k,

(iii)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0,

(iv)
{\sum}_{n\ge 0}{\alpha}_{n}=\mathrm{\infty}.
For an arbitrary but fixed \omega \in C, and for each k\ge 0, define {T}_{k}:C\to C by {T}_{k}x=(1{a}_{k})\omega +{a}_{k}Tx, \mathrm{\forall}x\in C. Then the double sequence {\{{x}_{k,n}\}}_{k\ge 0,n\ge 0} generated from an arbitrary {x}_{0,0}\in C by
converges strongly to a fixed point {x}_{\mathrm{\infty}}^{\ast} of T in C.
The following lemma will be useful in the sequel.
Lemma 1.2 [5]
Let \{{\delta}_{n}\} and \{{\sigma}_{n}\} be two sequences of nonnegative real numbers satisfying the inequality
Here \gamma \in [0,1). If {lim}_{n\to \mathrm{\infty}}{\sigma}_{n}=0, then {lim}_{n\to \mathrm{\infty}}{\delta}_{n}=0.
It is our purpose in this paper to extend Theorem 1.1 from Hilbert space to an arbitrary real Banach space with no further assumptions on the real sequences {\{{\alpha}_{n}\}}_{n\ge 0}, {\{{a}_{k}\}}_{k\ge 0}.
2 Main results
Theorem 2.1 Let C be a bounded closed convex subset of a Banach space E and T:C\to C be a Lipschitz pseudocontraction with F(T)\ne \mathrm{\varnothing}. Let {\{{\alpha}_{n}\}}_{n\ge 0},{\{{a}_{k}\}}_{k\ge 0}\subset (0,1) be real sequences satisfying the following conditions:

(i)
{lim}_{k\to \mathrm{\infty}}{a}_{k}=1,

(ii)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0.
For an arbitrary but fixed \omega \in C, and for each k\ge 0, define {T}_{k}:C\to C by {T}_{k}x=(1{a}_{k})\omega +{a}_{k}Tx, \mathrm{\forall}x\in C. Then the double sequence {\{{x}_{k,n}\}}_{k\ge 0,n\ge 0} generated from an arbitrary {x}_{0,0}\in C by
converges strongly to a fixed point {x}^{\ast} of T in C.
Proof Since T is Lipschitzian, there exists L>0 such that
Since T is pseudocontractive, for each k\ge 0, we have
Hence, {T}_{k} is Lipschitz and strongly pseudocontractive. Also, C is invariant under {T}_{k} for all k\ge 0, by convexity. Thus, for each k\ge 0, {T}_{k} has a unique fixed point {x}_{k}^{\ast}, say, in C.
Now, we proceed in the following steps.

(I)
for each k\ge 0, {x}_{k,n}\to {x}_{k}^{\ast}\in C as n\to \mathrm{\infty}.

(II)
{x}_{k}^{\ast}\to {x}^{\ast}\in C as k\to \mathrm{\infty}.

(III)
{x}^{\ast}\in F(T).
Proof of (I). In fact, it follows from (2.1) that
Thus, if {x}_{k}^{\ast} is a fixed point of {T}_{k}, k\ge 0, then
Using inequality (1.1), it follows that
On the other hand, by (2.1) we obtain
Therefore,
Substituting (2.3) into (2.2), we arrive at
which implies that
and so
Since C is bounded, there exists M>0 such that
Hence, it follows from (2.4) that
Since \lambda \in (0,1) and {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, it follows from Lemma 1.2 that
i.e., {x}_{k,n}\to {x}_{k}^{\ast} as n\to \mathrm{\infty}.
Proof of (II). We prove that {\{{x}_{k}^{\ast}\}}_{k=0}^{\mathrm{\infty}}={\{{T}_{k}{x}_{k}^{\ast}\}}_{k=0}^{\mathrm{\infty}} converges to some {x}^{\ast}\in C. For this purpose, we need only to prove that {\{{x}_{k}^{\ast}\}}_{0}^{\mathrm{\infty}} is a Cauchy sequence.
In fact, we have
that is,
hence
where d=diamC. If follows from condition (i) that
This completes step (II) of the proof.
Proof of (III). In order to accomplish step (III), we first have to prove that {\{{x}_{k}^{\ast}\}}_{k=0}^{\mathrm{\infty}} is an approximate fixed point sequence for T. In fact, from {T}_{k}{x}_{k}^{\ast}=(1{a}_{k})\omega +{a}_{k}T{x}_{k}^{\ast}, we have
where d=diamC. Hence {lim}_{k\to \mathrm{\infty}}\parallel {x}_{k}^{\ast}T{x}_{k}^{\ast}\parallel =0. Since {x}_{k}^{\ast}\to {x}^{\ast} as k\to \mathrm{\infty}, T is continuous and using continuity of the norm, we get {lim}_{k\to \mathrm{\infty}}\parallel {x}^{\ast}T{x}^{\ast}\parallel =0, i.e., {x}^{\ast}=T{x}^{\ast}. This completes the proof. □
Corollary 2.2 Let C be a bounded closed convex subset of a Banach space E and T:C\to C be a nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. Let {\{{\alpha}_{n}\}}_{n\ge 0},{\{{a}_{k}\}}_{k\ge 0}\subset (0,1) be real sequences satisfying conditions (i)(ii) in Theorem 2.1. For an arbitrary but fixed \omega \in C, and for each k\ge 0, define {T}_{k}:C\to C by {T}_{k}x=(1{a}_{k})\omega +{a}_{k}Tx, \mathrm{\forall}x\in C. Then the double sequence {\{{x}_{k,n}\}}_{k\ge 0,n\ge 0} generated from an arbitrary {x}_{0,0}\in C by
converges strongly to a fixed point of T in C.
Proof Obvious, observing the fact that every nonexpansive mapping is Lipschitz and pseudocontractive. □
The following corollary follows from Theorem 2.1 on setting \omega =0\in C.
Corollary 2.3 Let C, E, T, {\{{\alpha}_{n}\}}_{n=0}^{\mathrm{\infty}}, {\{{a}_{k}\}}_{k=0}^{\mathrm{\infty}} be as in Theorem 2.1. For an arbitrary but fixed \omega \in C, and for each k\ge 0, define {T}_{k}:C\to C by {T}_{k}x={a}_{k}Tx for all x\in C. Then the double sequence {\{{x}_{k,n}\}}_{k\ge 0,n\ge 0} generated from an arbitrary {x}_{0,0}\in C by
converges strongly to a fixed point of T in C.
Remark 2.1 Theorem 2.1 improves and extends Theorem 3.1 of Moore [3] in three respects:

(1)
It abolishes the condition that {lim}_{r,k\to \mathrm{\infty}}\frac{{a}_{k}{a}_{r}}{1{a}_{k}}=0.

(2)
It abolishes the condition that {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}.

(3)
The ambient space is no longer required to be a Hilbert space and is taken to be the more general Banach space instead.
Remark 2.2

(1)
Whereas the Ishikawa iteration process was proved to converge to a fixed point of a Lipschitz pseudocontractive mapping in compact convex subsets of a Hilbert space, we imposed no compactness conditions to obtain the strong convergence of the double sequence iteration process to a fixed point of a Lipschitz pseudocontraction.

(2)
Our results may easily be extended to the slightly more general classes of Lipschitz hemicontractive and Lipschitz quasinonexpansive mappings.

(3)
Prototypes of the sequences {\{{a}_{k}\}}_{k=0}^{\mathrm{\infty}} and {\{{\alpha}_{n}\}}_{n=0}^{\mathrm{\infty}} are
{a}_{k}=\frac{k}{1+k}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\alpha}_{n}=\frac{1}{{(n+1)}^{2}}.
References
Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508520. 10.2969/jmsj/01940508
Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147150. 10.1090/S00029939197403364695
Mutangadura SA, Chidume CE: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc. Am. Math. Soc. 2001, 129: 23592363. 10.1090/S0002993901060099
Moore C: A double sequence iteration process for fixed points of continuous pseudocontractions. Comput. Math. Appl. 2002, 43: 15851589. 10.1016/S08981221(02)001219
Liu QH: A convergence theorem of the sequence of Ishikawa iterates for quasicontractive mappings. J. Math. Anal. Appl. 1990, 146: 302305.
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Abdelhakim, A.A., Gu, F. On solving Lipschitz pseudocontractive operator equations. J Inequal Appl 2014, 314 (2014). https://doi.org/10.1186/1029242X2014314
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DOI: https://doi.org/10.1186/1029242X2014314
Keywords
 Lipschitz pseudocontractions
 Manntype double sequence iteration
 strong convergence