Abstract
In this paper, we prove strong convergence for the modified Ishikawa iteration process of a total asymptotically nonexpansive mapping satisfying condition (A) in a real uniformly convex Banach space. Our result generalizes the results due to Rhoades (J. Math. Anal. Appl. 183:118-120, 1994).
MSC:47H05, 47H10.
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1 Introduction
Let X be a real Banach space, let C be a nonempty closed convex subset of X, and let T be a mapping of C into itself. Then T is said to be asymptotically nonexpansive [2] if there exists a sequence , , with , such that
for all and . T is said to be uniformly L-Lipschitzian if there exists a constant such that
for all and . If T is asymptotically nonexpansive, then it is uniformly L-Lipschitzian. We denote by ℕ the set of all positive integers. T is said to be total asymptotically nonexpansive (in brief, TAN) [3] if there exist two nonnegative real sequences and with as , such that
for all and , where and if and only if ϕ is strictly increasing, continuous on and . It is clear that if we take for all and for all in (1.2), it is reduced to (1.1). Approximating fixed points of the modified Ishikawa iterative scheme under total asymptotically nonexpansive mappings has been investigated by several authors; see, for example, Chidume and Ofoedu [4, 5], Kim [6], Kim and Kim [7] and others. For a mapping T of C into itself in a Hilbert space, Schu [8] considered the following modified Ishikawa iteration process (cf. Ishikawa [9]) in C defined by
where and are two real sequences in . If for all , then iteration process (1.3) becomes the following modified Mann iteration process (cf. Mann [10]):
where is a real sequence in .
Rhoades [1] proved the following results which extended Theorems 1.5 and 2.3 of Schu [8] to uniformly convex Banach spaces.
Theorem 1.1 Let X be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of X, and let be a completely continuous asymptotically nonexpansive mapping with satisfying , , . Then, for any , the sequence defined by (1.4), where satisfies for all and some , converges strongly to some fixed point of T.
Theorem 1.2 Let X be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E, and let be a completely continuous asymptotically nonexpansive mapping with satisfying , , . Then, for any , the sequence defined by (1.3), where , satisfy for all and some , converges strongly to some fixed point of T.
On the other hand, Kim [11] proved the following result which generalized Theorem 1 of Senter and Dotson [12].
Theorem 1.3 Let X be a real uniformly convex Banach space, let C be a nonempty closed convex subset of X, and let T be a nonexpansive mapping of C into itself satisfying condition (A) with . Suppose that for any in C, the sequence is defined by , for all , where and are sequences in such that and . Then converges strongly to some fixed point of T.
In this paper, we prove that if T is a total asymptotically nonexpansive self-mapping satisfying condition (A), the iteration defined by (1.3) converges strongly to some fixed point of T, which generalizes the results due to Rhoades [1].
2 Preliminaries
Throughout this paper, we denote by X a real Banach space. Let C be a nonempty closed convex subset of X, and let T be a mapping from C into itself. Then we denote by the set of all fixed points of T, i.e., . We also denote by . A Banach space X is said to be uniformly convex if the modulus of convexity , , of X defined by
satisfies the inequality for every . When is a sequence in X, then will denote strong convergence of the sequence to x.
Definition 2.1 [12]
A mapping with is said to satisfy condition (A) if there exists a nondecreasing function with and for all such that
for all , where .
3 Strong convergence theorem
We first begin with the following lemma.
Lemma 3.1 [13]
Let , and be sequences of nonnegative real numbers such that , and
for all . Then exists.
Lemma 3.2 [14]
Let X be a uniformly convex Banach space. Let . If , and , then for .
Lemma 3.3 Let C be a nonempty closed convex subset of a uniformly convex Banach space X, and let be a TAN mapping with . Suppose that , and ϕ satisfy the following two conditions:
-
(I)
such that for all .
-
(II)
, .
Suppose that the sequence is defined by (1.3). Then exists for any .
Proof For any , we set
From (I) and strict increasing of ϕ, we obtain
By using (3.1), we have
where and . Since
and thus
where , , and . So, we have
where and . Hence
By Lemma 3.1, we see that exists. □
Theorem 3.4 Let X be a uniformly convex Banach space, and let C be a nonempty closed convex subset of X. Let be a uniformly continuous and TAN mapping with . Suppose that , and ϕ satisfy the following two conditions:
-
(I)
such that for all .
-
(II)
, .
Suppose that for any in C, the sequence defined by (1.3) satisfies and . Then converges strongly to some fixed point of T.
Proof For any , by Lemma 3.3, is bounded. We set
By Lemma 3.3, we see that exists. Without loss of generality, we assume . As in the proof of Lemma 3.3, we obtain
where and . By using Lemma 3.2 and Takahashi [15], we obtain
Hence we obtain
Thus
Since is strictly increasing, continuous and , we obtain
By using (3.1) in the proof of Lemma 3.3, we have
where and . Thus
and hence
where , , and . So, we have
where and . By using Lemma 3.2 and Takahashi [15], we obtain
By the same method as above, we obtain
Since is bounded and T is a TAN mapping, we obtain
where . By using , we have
Since
by (3.2) and (3.4), we obtain
By using (3.3) and (3.4), we obtain
Since
by using (3.3) and (3.4), we have
Since
by (3.4) and (3.6), we get
From
by (3.7) and (3.8), we obtain
Since
and by the uniform continuity of T, (3.4), (3.5) and (3.9), we have
By using condition (A), we obtain
for all . As in the proof of Lemma 3.3, we obtain
Thus
By using Lemma 3.1, we see that exists. We first claim that . In fact, assume that . Then we can choose such that for all . By using condition (A), (3.10) and (3.11), we obtain
as . This is a contradiction. So, we obtain . Next, we claim that is a Cauchy sequence. Since , we obtain . Let be given. Since and , there exists such that for all , we obtain
Let and . Then, by (3.12), we obtain
Taking the infimum over all on both sides and by (3.13), we obtain
for all . This implies that is a Cauchy sequence. Let . Then . Since is closed, we obtain . Hence converges strongly to some fixed point of T. □
Remark 3.5 If is completely continuous, then it satisfies demicompact and, if T is continuous and demicompact, it satisfies condition (A); see Senter and Dotson [12].
Remark 3.6 If is bounded away from both 0 and 1, i.e., for all and some , then and hold. However, the converse is not true. For example, consider .
We give an example of a mapping which satisfies all the assumptions of T in Theorem 3.4, i.e., is a uniformly continuous mapping with which is TAN on C, not Lipschitzian and hence not asymptotically nonexpansive.
Example 3.7 Let and . Define by
Note that for all and and . Clearly, T is both uniformly continuous and TAN on C. We show that T satisfies condition (A). In fact, if , then . Similarly, if , then
So, we get for all . But T is not Lipschitzian. Indeed, suppose not, i.e., there exists such that
for all . If we take and , then
This is a contradiction.
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Kim, G.E. Strong convergence to a fixed point of a total asymptotically nonexpansive mapping. Fixed Point Theory Appl 2013, 302 (2013). https://doi.org/10.1186/1687-1812-2013-302
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DOI: https://doi.org/10.1186/1687-1812-2013-302