Some results on fixed points of asymptotically strict quasi-ϕ-pseudocontractions in the intermediate sense
© Qin et al.; licensee Springer 2012
Received: 16 April 2012
Accepted: 14 August 2012
Published: 7 September 2012
In this paper, a new nonlinear mapping, asymptotically strict quasi-ϕ-pseudocontraction in the intermediate sense, is introduced. Projection algorithms are considered for the mapping. Strong convergence theorems for fixed points of the mapping are established based on projection algorithms in a real Banach space.
MSC:47H09, 47J05, 47J25.
Fixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. The theory itself is a beautiful mixture of analysis, topology, and geometry. Lots of problems arising in economics, engineering, and physics can be studied by fixed point techniques. The study of fixed point approximation algorithms for computing fixed points is now a topic of intensive research efforts. Many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection , where is some positive integer, is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such point. The well-known convex feasibility problem, which captures applications in various disciplines such as image restoration and radiation therapy treatment planning, is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings, see, for example, [1–6].
For iterative algorithms, the oldest and simplest one is the Picard iterative algorithm. It is known that T, where T stands for a contractive mapping, enjoys a unique fixed point, and the sequence generated by the Picard iterative algorithm can converge to the unique fixed point. However, for more general nonexpansive mappings, the Picard iterative algorithm fails to converge to fixed points of nonexpansive mappings even when they enjoy fixed points. The Krasnoselskii-Mann iterative algorithm (one-step iterative algorithm) and the Ishikawa iterative algorithms (two-step iterative algorithm) have been studied for approximating fixed points of nonexpansive mappings and their extensions. However, both the Krasnoselskii-Mann iterative algorithm and the Ishikawa iterative algorithms are weak convergence for nonexpansive mappings only; see  and  for the classic weak convergence theorems. In many disciplines, including economics , image recovery , quantum physics , and control theory , problems arise in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property so that the energy of the error between the iterate and the solution x eventually becomes arbitrarily small. The importance of strong convergence is also underlined in , where a convex function f is minimized via the proximal-point algorithm: it is shown that the rate of convergence of the value sequence is better when converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space. Projection methods, which were first introduced by Haugazeau , have been considered for the approximation of fixed points of nonexpansive mappings and their extensions. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.
In this paper, a new class of new nonlinear mappings is introduced and studied. Based on a simple hybrid projection algorithm, a theorem of strong convergence for common fixed points is obtained. The results presented in this paper mainly improve the known corresponding results announced in the literature sources listed in this work.
The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, the hybrid projection algorithm is proposed and analyzed. With the help of the generalized projections, theorems of strong convergence are established. Some subresults of the main results are discussed.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk . Since 1972, a host of authors have studied the convergence of iterative algorithms for such a class of mappings.
The class of mappings which are asymptotically nonexpansive in the intermediate sense was considered by Bruck, Kuczumow, and Reich . It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous; see [16–18].
The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn . It is easy to see that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings as a special case. In 2007, Marino and Xu  proved that the fixed point set of strict pseudocontractions is closed and convex. They also obtained a strong convergence theorem for fixed points of the class of mappings based on hybrid projection algorithms; see  for more details.
The class of asymptotically strict pseudocontractions was introduced by Qihou  in 1996. Kim and Xu  proved that the fixed point set of asymptotically strict pseudocontractions is closed and convex. They also obtained a strong convergence theorem for fixed points of the class of asymptotically strict pseudocontractions based on projection algorithms; see  for more details.
The class of mappings was introduced by Sahu, Xu, and Yao . They proved that the fixed point set of asymptotically strict pseudocontractions in the intermediate sense is closed and convex. They also obtained a strong convergence theorem for fixed points of the class of mappings based on projection algorithms; see  for more details.
where denotes the generalized duality pairing of elements between E and . It is well known that if is strictly convex, then J is single valued; if is uniformly convex, then J is uniformly continuous on bounded subsets of E; if is reflexive and smooth, then J is single valued and demicontinuous.
It is also well known that if C is a nonempty closed convex subset of a Hilbert space H and is the metric projection of H onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber  introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Recall that a Banach space E is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in E such that and . Let be the unit sphere of E. Then the Banach space E is said to be smooth provided exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all .
Recall that a Banach space E has the Kadec-Klee property if for any sequence and with and , then as . For more details on the Kadec-Klee property, the readers can refer to [25–27] and the references therein. It is well known that if E is a uniformly convex Banach space, then E satisfies the Kadec-Klee property.
Remark 2.1 If E is a reflexive, strictly convex, and smooth Banach space, then for all , if and only if . It is sufficient to show that if , then . From (2.6), we have . This implies that . From the definition of J, we see that . It follows that ; see [25, 27] for more details.
A point p in C is said to be an asymptotic fixed point of T  if C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by .
- (2)T is said to be relatively nonexpansive if
- (3)T is said to be relatively asymptotically nonexpansive if
where is a sequence such that as .
- (4)T is said to be quasi-ϕ-nonexpansive if
- (5)T is said to be asymptotically quasi-ϕ-nonexpansive if there exists a sequence with as such that
Remark 2.3 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings were considered in Qin, Cho, and Kang , and Zhou, Gao, and Tan ; see also [37–42].
Remark 2.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require .
- (6)T is said to be a strict quasi-ϕ-pseudocontraction if and there exists a constant such that
- (7)T is said to be an asymptotically strict quasi-ϕ-pseudocontraction if and there exists a sequence with as and a constant such that
Remark 2.6 It is clear that strict quasi-ϕ-pseudocontractions are asymptotically strict quasi-ϕ-pseudocontractions with the sequence . The class of asymptotically strict quasi-ϕ-pseudocontractions was first considered in Qin et al.; see  for more details on asymptotically strict quasi-ϕ-pseudocontractions and see  for more details on quasi-strict pseudocontractions and the references therein.
Remark 2.7 If , then the class of asymptotically strict quasi-ϕ-pseudocontractions is reduced to asymptotically quasi-ϕ-nonexpansive mappings.
- (8)The mapping T is said to be asymptotically regular on C if, for any bounded subset K of C,
- (9)T is said to be an asymptotically strict quasi-ϕ-pseudocontraction in the intermediate sense if and there exists a sequence with as and a constant such that(2.8)
Remark 2.9 The class of asymptotically strict quasi-ϕ-pseudocontractions in the intermediate sense is a generalization of the class of asymptotically strict quasi-pseudocontractions in the intermediate sense in the framework of Banach spaces. For examples of the mapping in , we refer the readers to Sahu, Xu, and Yao .
Remark 2.10 If and , then we call T an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense.
Remark 2.11 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense in the framework of Banach spaces.
In order to prove our main results, we also need the following lemmas:
Let E be a uniformly convex and smooth Banach space. Letandbe two sequences in E. Ifand eitheroris bounded, thenas.
3 Main results
Now, we are in a position to give the main results in this paper.
Then the sequenceconverges strongly to.
In view of Lemma 2.12, we see that as for each . This implies as for each . From the closedness of , we obtain . This proves that is convex. This completes the proof that is closed and convex.
where . It follows that is closed, and convex. This, in turn, implies that is closed and convex.
which implies that . This proves, for each , that . This implies that .
That is, . From the closedness of , we find for each . This proves .
This shows . This completes the proof. □
Based on Theorem 3.1, we have the following.
Then the sequenceconverges strongly to.
Proof Putting and , we can conclude from Theorem 3.1 the desired conclusion immediately. □
In the framework of Hilbert spaces, we have the following results for an uncountable family of asymptotically strict quasi-pseudocontractions in the intermediate sense and an uncountable family of asymptotically quasi-nonexpansive mappings in the intermediate sense.
Then the sequenceconverges strongly to.
Then the sequenceconverges strongly to.
In this section, we consider minimizers of proper, lower semicontinuous, and convex functionals, and solutions of variational inequalities.
Rockafellar  proved that ∂f is a maximal monotone operator. It is easy to verify that if and only if .
where, . Thenconverges strongly to, wherestands for the generalized projection from E onto.
This shows that . In view of Example 2.3 in Qin, Cho, and Kang , we find that is closed quasi-ϕ-nonexpansive with . Notice that every quasi-ϕ-nonexpansive mapping is an asymptotically strict quasi-ϕ-pseudocontraction in the intermediate sense. Following the proof of Theorem 3.1, we can immediately conclude the desired conclusion. This completes the proof. □
where, . Thenconverges strongly to, wherestands for the generalized projection from E onto.
This implies that . In view of Example 2.3 in Qin, Cho, and Kang , we find that is closed quasi-ϕ-nonexpansive with . Notice that every quasi-ϕ-nonexpansive mapping is an asymptotically strict quasi-ϕ-pseudocontraction in the intermediate sense. Following the proof of Theorem 3.1, we can immediately conclude the desired conclusion. □
This research is partially supported by Natural Science Foundation of Zhejiang Province (Q12A010097) and National Natural Science Foundation of China (11126334).
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