Convergence results of multi-valued nonexpansive mappings in Banach spaces
© He et al.; licensee Springer. 2014
Received: 30 April 2014
Accepted: 19 November 2014
Published: 4 December 2014
The purpose of this paper is to establish two new iteration schemes as follows: , , , , , , for a multi-valued nonexpansive mapping T in a uniformly convex Banach space and prove that and converge strongly to a fixed point of T under some suitable conditions, respectively. Moreover, a gap in Sahu (Nonlinear Anal. 37:401-407, 1999) is found and revised.
MSC:47H04, 47H10, 47J25.
A point x is called a fixed point of T if . In this paper, stands for the fixed point set of the mapping T.
In 1968, Markin  firstly established the nonexpansive multi-valued convergence results in Hilbert space. Banach’s contraction principle was extended to a multi-valued contraction by Nadler  in 1969.
In 1974, a breakthrough was achieved by TC Lim using Edelstein’s method of asymptotic centers .
Theorem 1.1 (Lim )
Let K be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space E and a multi-valued nonexpansive mapping. Then T has a fixed point.
In 1990, Kirk and Massa  obtained another important result for the multi-valued nonexpansive mappings.
Theorem 1.2 (Kirk and Massa )
Let K be a nonempty, bounded, closed, and convex subset of a Banach space E and a multi-valued nonexpansive mapping. Suppose that the asymptotic center in K of each bounded sequence of E is nonempty and compact. Then T has a fixed point.
Clearly, if the above inequality holds, then the inequality must be assumed, for all , but this inequality does not hold generally, based on the definition of the Hausdorff metric on , for all .
In 2001, Xu  extended Theorem 1.2 to a multi-valued nonexpansive nonself-mapping and obtained the fixed theorems. Some recent fixed point results for multi-valued nonexpansive mappings can be found in [8–13] and the references therein.
where satisfy certain conditions, and we prove some strongly convergence theorems for the multi-valued nonexpansive mappings in Banach spaces. The results presented in this paper mainly extend and improve the corresponding results of Sahu  on the iteration algorithms.
where denotes the dual space of E and denotes the generalized duality pair. It is well known that if E is smooth or if is strictly convex, then J is single-valued.
exists for each x, y on the unit sphere of E. Moreover, if for each y in the limit defined by (2.2) is uniformly attained for x in , we say that the norm of E is uniformly Gâteaux differentiable. It is also well known that if E has a uniformly Gâteaux differentiable norm, then the duality mapping J is norm-to-weak star uniformly continuous on each bounded subset of E.
for all . E is said to be uniformly convex if , and for all .
Throughout this paper, we write (respectively, ) to indicate that the sequence weakly (respectively, weak*) converges to x, and as usual will symbolize strong convergence. In order to show our main results, the following definitions and lemmas are needed.
Lemma 2.1 ([, Lemma 1])
for all .
An example of mappings that satisfy Condition I can be found in reference .
3 Strong convergence theorems
Theorem 3.1 Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty closed convex subset of E, be a multi-valued nonexpansive mapping. Assume that and for each . Let be an implicit Mann type iteration defined by (1.6), where and , then the sequence converges strongly to a fixed point of T.
If , then apparently holds.
exists. Hence is bounded, and so is .
Secondly, we show that .
Then M is nonempty, bounded, closed, and convex (see [, Theorem 1.3.11]).
Next, we show that M is singleton.
This is a contradiction. Therefore, M has a unique element.
indeed, , since Tz is a compact set, and we take .
and hence , it follows that M is singleton, so .
Therefore, and so by the assumption, and is nonempty.
Therefore, there is a subsequence of which converges strongly to z.
The proof is completed. □
Theorem 3.2 Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K a nonempty, closed, and convex subset of E, be a multi-valued nonexpansive mapping that satisfies Condition I, assume and for each . Let be an implicit Mann type iteration defined by (1.6), where and , then the sequence converges strongly to a fixed point of T.
The remainder of the proof is the same as Theorem 2.4 of . □
Theorem 3.3 Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty, closed, and convex subset of E, be a multi-valued nonexpansive mapping. Assume that and for each . Let be the modified implicit Mann type iteration defined by (1.7), where and . Then the sequence converges strongly to a fixed point of T.
If , then is apparently bounded.
Thus, is bounded.
Secondly, we show that .
Then M is nonempty, and it is also bounded, closed, and convex (see [, Theorem 1.3.11]).
In the same way as the proof of Theorem 3.1, we see that M is also singleton, and it follows from (3.9) that z is the fixed point of T.
This proves the strong convergence of to .
The proof is completed. □
4 Numerical examples
Now, we give two real numerical examples in which the conditions satisfy the ones of Theorem 3.1 and Theorem 3.3.
and as , where 0 is the fixed point of T.
and as , where 0 is the fixed point of T.
Remark 4.2 From the above numerical examples, we can see that the convergence results in this paper are important. The main reason is that the convergence of the two iteration schemes in this paper can easily be implemented by the software of Matlab 7.0, so they can be applied for numerical calculations in practical problems.
This work is supported by the Fundamental Research Funds for the Central Universities (No. K5051370004), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JQ1020), the National Science Foundation of China (No. 61202178 and No. 61373174), the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11426167).
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