Abstract
The asymptotically nonexpansive mappings have been introduced by Goebel and Kirkin 1972. Since then, a large number of authors have studied the weak and strongconvergence problems of the iterative algorithms for such a class of mappings.It is well known that the asymptotically nonexpansive mappings is a propersubclass of the class of asymptotically pseudocontractive mappings. In thepresent paper, we devote our study to the iterative algorithms for finding thefixed points of asymptotically pseudocontractive mappings in Hilbert spaces. Wesuggest an iterative algorithm and prove that it converges strongly to the fixedpoints of asymptotically pseudocontractive mappings.
MSC: 47J25, 47H09, 65J15.
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1 Introduction
Let H be a real Hilbert space with inner product and norm , respectively. Let C be a nonempty, closed,and convex subset of H. Let be a nonlinear mapping. We use to denote the fixed point set of T.
Recall that T is said to be L-Lipschitzian if there exists such that
for all . In this case, if , then we call T anL-contraction. If , we call T nonexpansive. T is said to be asymptotically nonexpansiveif there exists a sequence with such that
for all and all .
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that, if C is a nonempty bounded, closed,and convex subset of a uniformly convex Banach space E, then everyasymptotically nonexpansive self-mapping T of C has a fixed point.Further, the set of fixed points of T is closed and convex.
Since then, a large number of authors have studied the following algorithms for theiterative approximation of fixed points of asymptotically nonexpansive mappings(see, e.g., [2–29] and the references therein).
-
(A)
The modified Mann iterative algorithm. For arbitrary , the modified Mann iteration generates a sequence by
(1.2) -
(B)
The modified Ishikawa iterative algorithm. For arbitrary , the modified Ishikawa iteration generates a sequence by
(1.3) -
(C)
The CQ algorithm. For arbitrary , the CQ algorithm generates a sequence by
(1.4)
An important class of asymptotically pseudocontractive mappings generalizing theclass of asymptotically nonexpansive mapping has been introduced and studied by Schuin 1991; see [19].
Recall that is called an asymptotically pseudocontractivemapping if there exists a sequence with for which the following inequality holds:
for all and all . It is clear that (1.5) is equivalent to
for all and all .
Recall also that T is called uniformly L-Lipschitzian if there exists such that
for all and all .
Now, we know that the class of asymptotically nonexpansive mappings is a propersubclass of the class of asymptotically pseudocontractive mappings. If we define amapping by the formula , then we can verify that T is asymptoticallypseudocontractive but it is not asymptotically nonexpansive.
In order to approximate the fixed point of asymptotically pseudocontractive mappings,the following two results are interesting.
One is due to Schu [19], who proved the following convergence theorem.
Theorem 1.1 Let H be a Hilbert space, C be a nonempty closed bounded and convex subset of H. Let T be a completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractiveself-mapping of C with and , where for all . Let and be two sequences satisfying for all . Then the sequence generated by the modified Ishikawa iteration (1.3) converges stronglyto some fixed point of T.
Another one is due to Chidume and Zegeye [30] who introduced the following algorithm in 2003.
Let a sequence be generated from by
where the sequences and satisfy
-
(i)
and ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
.
They gave the strong convergence analysis for the above algorithm (1.7) with somefurther assumptions on the mapping T in Banach spaces.
Remark 1.2 Note that there are some additional assumptions imposed on theunderlying space C and the mapping T in the above two results. In(1.7), the parameter control is also restricted.
Inspired by the results above, the main purpose of this article is to construct aniterative method for finding the fixed points of asymptotically pseudocontractivemappings. We construct an algorithm which is based on the algorithms (1.2) and(1.7). Under some mild conditions, we prove that the suggested algorithm convergesstrongly to the fixed point of asymptotically pseudocontractive mappingT.
2 Preliminaries
It is well known that in a real Hilbert space H, the following inequalityand equality hold:
and
for all and .
Lemma 2.1 ([31])
Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let be a uniformly L-Lipschtzian and asymptotically pseudocontraction. Then is demiclosed at zero.
Lemma 2.2 ([32])
Let be a sequence of real numbers. Assume does not decrease at infinity, that is, there exists at leasta subsequence of such that for all . For every , define an integer sequence as
Then as , and for all
Lemma 2.3 ([33])
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(i)
;
-
(ii)
or .
Then .
3 Main results
Now we introduce the following iterative algorithm for asymptoticallypseudocontractive mappings.
Let C be a nonempty, closed, and convex subset of a real Hilbert spaceH. Let be a uniformly L-Lipschitzian asymptoticallypseudocontractive mapping satisfying . Let be a ρ-contractive mapping. Let, , and be three real number sequences in.
Algorithm 3.1 For , define the sequence by
Next, we prove our main result as follows.
Theorem 3.2 Suppose that . Assume the sequences , , and satisfy the following conditions:
-
(i)
and ;
-
(ii)
and ;
-
(iii)
for all .
Then the sequence defined by (3.1) converges strongly to , which is the unique solution of the variationalinequality for all .
Proof From (3.1), we have
Using the equality (2.2), we get
Picking up in (1.6) we deduce
for all .
From (3.1), (3.4), and (2.2), we obtain
By (3.1), we have
Noting that T is uniformly L-Lipschitzian, from (3.5) and (3.6),we deduce
By condition (iii), we know that for all n. Then we deduce that for all .
Therefore, from (3.7), we derive
Note that and substituting (3.8) to (3.3), we obtain
Thus,
Since , without loss of generality, we assume that for all . It follows from (3.2) and (3.9) that
An induction induces
This implies that the sequence is bounded by the condition .
From (2.1) and (3.1), we have
From (3.7), we deduce
By condition (ii), we have for all . Hence, by (3.10) and (3.11), we get
It follows that
Since and are bounded, there exists such that . So,
Next, we consider two possible cases.
Case 1. Assume there exists some integer such that is decreasing for all .
In this case, we know that exists. From (3.13), we deduce
By conditions (ii) and (iii), we have . Thus, from (3.14), we get
It follows from (3.6) and (3.15) that
Since T is uniformly L-Lipschitzian, we have. This together with (3.16) implies that
Note that
Combining (3.15), (3.17), and (3.18), we have
From (3.1), we deduce
Therefore,
Since T is uniformly L-Lipschitzian, we derive
By (3.15), (3.20), and (3.21), we have immediately
Since is bounded, there exists a subsequence of satisfying
and
Thus, we use the demiclosed principle of T (Lemma 2.1) and (3.22) todeduce
So,
Returning to (3.12) to obtain
It follows that
In Lemma 2.3, we take , , and . We can easily check that and . Thus, we deduce that .
Case 2. Assume there exists an integer such that . At this case, we set . Then we have . Define an integer sequence for all as follows:
It is clear that is a non-decreasing sequence satisfying
and
for all . From (3.22), we get
This implies that . Thus, we obtain
Since , we have from (3.23) that
It follows that
Combining (3.24) and (3.25), we have
and hence
From (3.23), we obtain
It follows that
This together with (3.26) imply that
Applying Lemma 2.2 to get
Therefore, . That is, . The proof is completed. □
Since the class of asymptotically nonexpansive mappings is a proper subclass of theclass of asymptotically pseudocontractive mappings and asymptotically nonexpansivemapping T is L-Lipschitzian with . Thus, from Theorem 3.2, we get the followingcorollary.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an asymptotically nonexpansive mapping satisfying . Suppose that . Let be a ρ-contractive mapping. Let , , and be three real number sequences in . Assume the sequences , , and satisfy the following conditions:
-
(i)
and ;
-
(ii)
and ;
-
(iii)
for all , where .
Then the sequence defined by (3.1) converges strongly to , which is the unique solution of the variationalinequality for all .
Remark 3.4 Our Theorem 3.2 does not impose any boundedness or compactnessassumption on the space C or the mapping T. The parameter controlconditions (i)-(iii) are mild.
Remark 3.5 Our Corollary 3.3 is also a new result.
4 Conclusion
This work contains our dedicated study to develop and improve iterative algorithmsfor finding the fixed points of asymptotically pseudocontractive mappings in Hilbertspaces. We introduced our iterative algorithm for this class of problems, and wehave proven its strong convergence. This study is motivated by relevant applicationsfor solving classes of real-world problems, which give rise to mathematical modelsin the sphere of nonlinear analysis.
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Yao, Y., Postolache, M. & Kang, S.M. Strong convergence of approximated iterations for asymptoticallypseudocontractive mappings. Fixed Point Theory Appl 2014, 100 (2014). https://doi.org/10.1186/1687-1812-2014-100
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DOI: https://doi.org/10.1186/1687-1812-2014-100