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Convergence results of multi-valued nonexpansive mappings in Banach spaces
Journal of Inequalities and Applications volume 2014, Article number: 483 (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.
Let E be a Banach space, K be a nonempty, closed, and convex subset of E. We denote the family of all subsets of E by , the family of nonempty, closed, and bounded subsets of E by , the family of nonempty compact subsets of E by , and the family of nonempty, compact, and convex subsets of E by . Let be Hausdorff metric on , i.e.,
A multi-valued mapping is called nonexpansive (respectively, contractive), if for any , we have
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.
In 1999, Sahu  obtained the strong convergence theorems of the nonexpansive type and non-self multi-valued mappings for the following iteration process:
where , , , and . Unfortunately, a gap exists in the proof of Theorem 1 in p.405 of ; there are the following inequalities:
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 .
Remark 1.1 To revise the gap we have found in Theorem 1 of , in this paper, we change the fixed point set of T () into . The above problem is solved easily. Indeed,
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.
Motivated by Sahu  and the above results, we propose two new iteration processes (1.6) and (1.7) and we study them in this paper. Let K be a nonempty, bounded, closed, and convex subset of E, and u and be fixed elements of K. We have
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.
Let E be a real uniformly convex Banach space and let J denote the normalized duality mapping from E to defined by
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.
Recall that the norm of Banach space E is said to be Gâteaux differentiable (or E is said to be smooth), if the limit
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.
A Banach space E is called uniformly convex, if for each there is a such that for with , and , holds. The modulus of convexity of E is defined by
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.
Let LIM be a continuous linear functional on satisfying . Then we know that LIM is a mean on N if and only if
for every . According to time and circumstances, we use instead of . A mean LIM on N is called a Banach limit if
Lemma 2.1 ([, Lemma 1])
Let C be a nonempty, closed, and convex subset of a Banach space E with uniformly Gâteaux differentiable norm. Let be a bounded sequence of E and let be a mean LIM on N and . Then
if and only if
for all .
Definition 2.1 A multi-valued mapping is said to satisfy Condition I if there is a nondecreasing function with , for such that
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.
Proof Firstly, let , , we show that exists and is bounded. Using (2.1), we obtain
If , then apparently holds.
If , from (3.1) we have
we find that is a decreasing sequence, so
exists. Hence is bounded, and so is .
Secondly, we show that .
It follows from (1.6) and from and being bounded, that there exists a real number such that
Since , we have
We define by for each , since E is uniformly convex (hence reflexive) and ϕ is continuous, convex, and as , ϕ attains its infimum over K (see ). Let
Then M is nonempty, bounded, closed, and convex (see [, Theorem 1.3.11]).
Next, we show that M is singleton.
Since M and are bounded, there exists such that M, for all . Then, by inequality (3.12) of  for we have
If , we have
This is a contradiction. Therefore, M has a unique element.
Now we show that is the fixed point of T. Since T is compact valued, we have some for such that
indeed, , since Tz is a compact set, and we take .
It follows from the above inequality that
and hence , it follows that M is singleton, so .
Therefore, and so by the assumption, and is nonempty.
On the other hand, for , we have
It follows from (1.6) that
Hence, from (3.2), (3.3), and (3.4), we obtain
Finally, we show that . Let , then by Lemma 1 of  we get
Let be arbitrary, then since J is norm-to-weak∗ uniformly continuous on bounded subsets of E, there exists such that for all , we have
since , and it is a minimizer of ϕ over K. Now, since ε is arbitrary this implies that
Combining inequalities (3.5) and (3.6) we get
Therefore, there is a subsequence of which converges strongly to z.
Since exists , we get
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.
Proof It follows from the proof of Theorem 3.1 that
Then Condition I implies that
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.
Proof Firstly, let and , then
Let , ; we show that is bounded. Using (2.1), we obtain
If , then is apparently bounded.
If , from (3.7) we have
Thus, is bounded.
Secondly, we show that .
It follows from (1.7) and since and are bounded, there exists a real number such that
Since , we have
We define by for each . Since E uniformly convex (hence reflexive) and ϕ is continuous, convex and as , ϕ attains its infimum over K (see, e.g., ). Let
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.
By Lemma 2.1, we have
for all . In particular, we have
It follows from the proof of (3.6) of Theorem 3.1 that (3.6) also holds here. Thus
where is a constant such that . Thus
Therefore, there is a subsequence of which converges strongly to z. To complete the proof, suppose there is another subsequence of which converges strongly to . Then is a fixed point of T by (3.9) and because K is closed. It then follows from (3.10) that
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.
Example 4.1 Let , , , , whichis nonexpansive, for every . Then is the sequence generated by
and as , where 0 is the fixed point of T.
Example 4.2 Let , , , , which is nonexpansive, , for every . Then is the sequence generated by
and as , where 0 is the fixed point of T.
Remark 4.1 We can prove Example 4.1 and Example 4.2 by Theorem 3.1 and Theorem 3.3, respectively, and we show two numerical experiments (Figure 1 and Figure 2) which can explain that the sequence strongly converges to 0.
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.
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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).
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.
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He, H., Liu, S. & Chen, R. Convergence results of multi-valued nonexpansive mappings in Banach spaces. J Inequal Appl 2014, 483 (2014). https://doi.org/10.1186/1029-242X-2014-483
- multi-valued nonexpansive mappings
- fixed point
- strong convergence
- implicit iteration