Nonlinear Analysis: Theory, Methods & Applications
Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces☆
Introduction
Let be a real Banach space and be the dual space of . Let denote the normalized duality mapping from into defined by where denotes the generalized duality pairing between and . It is well known that if is smooth then is single-valued. In the sequel, we shall denote the single-valued normalized duality mapping by .
Recall that a mapping with domain and range in is called pseudocontractive if the inequality holds for each and for all . From a result of Kato [1], we know that (1) is equivalent to (2) below; there exists such that for all .
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 with domain and range in is called accretive if the inequality holds for every and for all .
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 be a Banach space. Suppose that is a nonempty closed convex subset of and is a continuous pseudocontractive mapping satisfying the weakly inward condition: is the closure of for each , where . Then for each , there exists a unique continuous path , satisfying the following equation:
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 where is a convex subset of a Banach space , a mapping and 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 be a nonempty subset of a Banach space and let be a nonexpansive mapping. Let be a real sequence satisfying the following control conditions: ; .
Let be defined by(3)such that for all . If is bounded then as .
The interest and importance of Theorem I1 lie in the fact that strong convergence of the sequence is achieved under certain mild compactness assumptions either on 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 is a compact convex subset of a Hilbert space , is a Lipschitzian pseudocontractive mapping and is any point in , then the sequence converges strongly to a fixed point of , where is defined iteratively for each positive integer bywhere are sequences of positive numbers satisfying the following conditions: ; ; .
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 be a compact convex subset of a real Hilbert space , a hemicontractive mapping. Let be a real sequence in [0,1] satisfying for some . For arbitrary , the sequence is defined byThen converges strongly to a fixed point of .
All of above motivate us to construct an iterative algorithm for approximating fixed points of pseudocontractive mappings without compactness assumptions on operator 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 be a closed convex subset of a real Banach space and be a mapping. Define in the following way: where is an anchor and and are three real sequences in satisfying some appropriate conditions.
Section snippets
Preliminaries
Let be a real Banach space with dual . Let denote the unit sphere of . The norm on is said to be Gâteaux differentiable if the limit exists for each and in this case is said to be smooth. is said to have a uniformly Frechet differentiable norm if the limit (6) is attained uniformly for and in this case is said to be uniformly smooth. It is well known that if 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 be a nonempty closed convex subset of a real uniformly smooth Banach space . Let be a continuous pseudocontractive mapping. Let and be three real sequences in satisfying the following conditions: ; and ; .
For arbitrary initial value and a fixed anchor , the sequence is defined by(5). Then converges strongly to a fixed point of .
Proof
First, we observe that
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Cited by (29)
Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space
2011, Applied Mathematics LettersCitation Excerpt :Then, a strong convergence theorem is established under some suitable conditions. The results presented in this paper improve and extend the corresponding results announced in [2] and many others. We need the following lemmas for the proof of our main results.
Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces
2009, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :Thus, we have the following corollary. (2) Theorem 3.1 generalizes Rafiq [13, Theorem 3.1] and Yao, Liou and Chen [14, Theorem 3.1] to multimaps. It improves, complements and develops Xu and Yin [2, Theorem 1 and Corollary 2], Jung [18, Theorem 1], Kim and Jung [7, Theorem 4.1], Sahu [20, Theorem 1], Ceng and Yao [17, Theorems 2.1 and 2.2], and Shahzad and Zegeye [10, Theorem 3.1].
A hybrid algorithm for pseudo-contractive mappings
2009, Nonlinear Analysis, Theory, Methods and ApplicationsOn Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings
2009, Nonlinear Analysis, Theory, Methods and ApplicationsStrong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces
2009, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :Within the past 40 years or so, mathematicians have been devoting their study to the existence and iterative construction of fixed points for pseudo-contractions and of zeros for accretive mappings (see, e.g., [1–20]).
On Mann implicit iterations for strongly accretive and strongly pseudo-contractive mappings
2008, Applied Mathematics and Computation
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This research was partially supported by grant NSC 95-2221-E-230-017.