Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces

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Abstract

Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, T:CC a pseudocontractive mapping. Let {αn}, {βn} and {γn} be three real sequences in (0,1) satisfying appropriate conditions; then for fixed uC and arbitrary x0C, the sequence {xn} generated by xn=αnu+βnxn1+γnTxn,n1, converges strongly to a fixed point of T.

Introduction

Let X be a real Banach space and X be the dual space of X. Let J denote the normalized duality mapping from X into 2X defined by J(x)={fX:x,f=xf,f=x},xX, where , denotes the generalized duality pairing between X and X. It is well known that if X is smooth then J is single-valued. In the sequel, we shall denote the single-valued normalized duality mapping by j.

Recall that a mapping T with domain D(T) and range R(T) in X is called pseudocontractive if the inequality xyxy+r((IT)x(IT)y), holds for each x,yD(T) and for all r>0. From a result of Kato [1], we know that (1) is equivalent to (2) below; there exists j(xy)J(xy) such that TxTy,j(xy)xy2, for all x,yD(T).

The class of pseudocontractive mappings is one of the most important classes of mappings in nonlinear analysis and it has been attracting mathematicians’ interest. Apart from them being a generalization of nonexpansive mappings, interest in pseudocontractive mappings stems mainly from their firm connection with the class of accretive mappings, where a mapping A with domain D(A) and range R(A) in X is called accretive if the inequality xyxy+s(AxAy), holds for every x,yD(A) and for all s>0.

Within the past 30 years or so, many authors have been devoting their study to the existence of zeros of accretive mappings or fixed points of pseudocontractive mappings and iterative construction of zeros of accretive mappings, and of fixed points of pseudocontractive mappings (see [2], [3], [4], [5], [6]). In particular, in 2000, Morales and Jung [7] studied the existence of paths for pseudocontractive mappings in Banach spaces. They proved the following result.

Theorem MJ

Let X be a Banach space. Suppose that C is a nonempty closed convex subset of X and T:CX is a continuous pseudocontractive mapping satisfying the weakly inward condition: T(x)IC(x)¯(IC(x)¯ is the closure of IC(x)) for each xC , where IC(x)=x+{c(ux):uX and c1} . Then for each zC , there exists a unique continuous path tytC,t[0,1) , satisfying the following equation:yt=tTyt+(1t)z.

On the other hand, several algorithms have been introduced and studied by various authors for approximating fixed points (zeros) of nonexpansive and pseudocontractive mappings (accretive mappings) in Hilbert spaces and Banach spaces; you may wish to consult [8], [9], [10], [11], [12], [13], [14], [15].

In 1953, Mann [16] introduced an iterative algorithm which is now referred to as the Mann iterative algorithm. Most of the literature deals with the special case of the general Mann iterative algorithm which is defined by x0C,xn+1=αnxn+(1αn)Txn,n0, where C is a convex subset of a Banach space X, T:CC a mapping and {αn} a sequence of positive numbers satisfying certain control conditions.

It is well known that the Mann iterative algorithm can be employed to approximate fixed points of nonexpansive mappings and zeros of strongly accretive mappings in Hilbert spaces and Banach spaces. Many convergence theorems have been announced and published by a good numbers of authors.

A typical convergence result in connection with the Mann iterative algorithm is the following theorem proved by Ishikawa [17].

Theorem I1

Let C be a nonempty subset of a Banach space X and let T:CX be a nonexpansive mapping. Let {cn} be a real sequence satisfying the following control conditions:

  • (a)

    n=0cn= ;

  • (b)

    0cnc<1 .

Let {xn} be defined by(3)such that xnC for all n0 . If {xn} is bounded then xnTxn0 as n .

The interest and importance of Theorem I1 lie in the fact that strong convergence of the sequence {xn} is achieved under certain mild compactness assumptions either on T or on its domain. It is much more interesting and important for one to establish strong convergence theorems without compactness assumptions on the mapping considered or on its domain.

In connection with the iterative approximation of fixed points of pseudocontractions, the following question is still open: Does the Mann iterative algorithm always converge for continuous pseudocontractions or even Lipschitz pseudocontractions? However, in 2001, Chidume and Mutangadura [18] provided an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iterative algorithm failed to converge and they stated there “This resolves a long standing open problem”. Of this nature, the following strong convergence theorem is basic and important.

Theorem I2 [19]

If K is a compact convex subset of a Hilbert space H , T:KK is a Lipschitzian pseudocontractive mapping and x0 is any point in K , then the sequence {xn} converges strongly to a fixed point of T , where {xn} is defined iteratively for each positive integer n0 by{xn+1=(1αn)xn+αnTyn,yn=(1βn)xn+βnTxn,where {αn},{βn} are sequences of positive numbers satisfying the following conditions:

  • (i)

    0αnβn1 ;

  • (ii)

    limnβn=0 ;

  • (iii)

    n=0αnβn= .

Since its publication in 1974, Theorem I2, as far as we know, has never been extended to more general Banach spaces.

Very recently, Rafiq [20] introduced a Mann type implicit iterative algorithm, (5) below, and proved the following theorem.

Theorem R

Let K be a compact convex subset of a real Hilbert space H , T:KK a hemicontractive mapping. Let {αn} be a real sequence in [0,1] satisfying {αn}[δ,1δ] for some δ(0,1) . For arbitrary x0K , the sequence {xn} is defined byx0K,xn=αnxn1+(1αn)Txn,n1.Then {xn} converges strongly to a fixed point of T .

All of above motivate us to construct an iterative algorithm for approximating fixed points of pseudocontractive mappings without compactness assumptions on operator T or its domain in more general Banach spaces. It is our purpose in this paper to introduce the following iterative algorithm associated with a pseudocontractive mapping, to have a strong convergence in the setting of Banach spaces.

Let C be a closed convex subset of a real Banach space X and T:CC be a mapping. Define {xn} in the following way: x0C,xn=αnu+βnxn1+γnTxn,n1, where u is an anchor and {αn},{βn} and {γn} are three real sequences in (0,1) satisfying some appropriate conditions.

Section snippets

Preliminaries

Let X be a real Banach space with dual X. Let U={xX:x=1} denote the unit sphere of X. The norm on X is said to be Gâteaux differentiable if the limit limt0x+tyxt exists for each x,yU and in this case X is said to be smooth. X is said to have a uniformly Frechet differentiable norm if the limit (6) is attained uniformly for x,yU and in this case X is said to be uniformly smooth. It is well known that if X is uniformly smooth then the duality mapping is norm-to-norm uniformly

Main results

Now we state and prove our main result.

Theorem 3.1

Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X . Let T:CC be a continuous pseudocontractive mapping. Let {αn},{βn} and {γn} be three real sequences in (0,1) satisfying the following conditions:

  • (i)

    αn+βn+γn=1 ;

  • (ii)

    limnβn=0 and limnαnβn=0 ;

  • (iii)

    n=0αnαn+βn= .

For arbitrary initial value x0C and a fixed anchor uC , the sequence {xn} is defined by(5). Then {xn} converges strongly to a fixed point of T .

Proof

First, we observe that {xn}

References (21)

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This research was partially supported by grant NSC 95-2221-E-230-017.

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