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International Journal of Analysis
Volume 2013 (2013), Article ID 539723, 7 pages
http://dx.doi.org/10.1155/2013/539723
Research Article

Some Existence and Convergence Theorems for Nonexpansive Type Mappings

1Department of Mathematics, Walter Sisulu University, Mthatha 5117, South Africa
2Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India
3Department of Mathematics, Rhodes University, Grahamstown 6140, South Africa

Received 27 August 2012; Accepted 4 November 2012

Academic Editor: Zhijun Qiao

Copyright © 2013 S. N. Mishra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some existence and convergence theorems for a class of nonexpansive type mappings are obtained in a normed space. The results obtained herein generalize certain known results.

1. Introduction

Let be a nonempty subset of a normed space . A mapping is said to be nonexpansive if for all . Suppose is another mapping on . Then the mapping is said to be -nonexpansive if for all . The class of -nonexpansive mappings is more general than nonexpansive mappings [13].

We extend the above notion of -nonexpansive mappings to a more general class of nonexpansive mappings.

Definition 1. Let be a normed space, a nonempty subset of and . We say that is a generalized -nonexpansive type mapping if for all , where Further will be called a generalized nonexpansive type mappings if for all , where

In this paper we obtain some common fixed point theorems for generalized nonexpansive type mappings in normed spaces. Specifically, in Section 3, we obtain some coincidence and common fixed point theorems while in Section 4, the weak convergence of Mann iterations (cf. Mann [4]) to a common fixed point for the above class of mappings is discussed. The results obtained herein generalize certain results of Kim et al. [1], Rhoades and Temir [3], and Shazad [5] among others.

2. Preliminaries

Let be a nonempty subset of a normed space and . A point is called a coincidence point of and if and a common fixed point if .

Now onwards will denote the set of naturals while and the set of fixed points of and , respectively.

Definition 2. Let be a normed space, a nonempty subset of , and . The pair of mappings is called(i)commuting if for all .(ii)weakly commuting if for all , (see [6]).(iii)-weakly commuting if for all , there exists such that (see [7]).
The following example illustrates that weakly commuting mappings are -weakly commuting but the converse is not true in general.

Example 3. Let be endowed with usual norm . Let be mappings defined by Then Therefore and the pair is -weakly commuting with but not weakly commuting.
In general, commuting weakly commuting -weakly commuting.

Definition 4 (cf. [1]). Let be a normed space and a nonempty subset of . The set is called -starshaped with if for all , the segment joining to , is contained in ; where .
Further if is a nonempty -starshaped subset of a normed space , then the mapping is said to be -affine if for all and .

Definition 5 (cf. [1]). Let be a normed space, a nonempty subset of , and such that . Suppose and is -starshaped. Then the pair of mappings is called -subweakly commuting on if for all , there exists a real number such that where .
We note that -subweakly commuting mappings are -weakly commuting but the converse is not true in general.

Example 6. Let (set of reals) with norm and . Define by Then Therefore holds for and and are -weakly commuting on .
On the other hand, , and for all , So, there does not exist any such that for all , holds. Thus and are not -subweakly commuting on .

Definition 7. Let be a nonempty subset of a normed space and . Let be a sequence in . We denote the weak and strong convergence of to by and , respectively. The mapping is said to be demicontinuous if is a sequence in such that , then .

Definition 8. A Banach space is said to satisfy the Opial's condition (see [8]), if whenever a sequence in , converges weakly to ( ), then for all , .
We note that the spaces, do not satisfy Opial' condition while all spaces do (see for details Goebel and Kirk [9]).

3. Existence Results

The following common fixed point theorem is due to Shahzad [5, Theorem 2.1]. For related results we refer to [2, 1012].

Theorem 9. Let be a metric space and a nonempty subset of . Let be a pair of mappings such that(i); (ii), , ;(iii) the pair is -weakly commuting on .
If (closure of ), is complete and is continuous, then is a singleton.

Now we obtain a more general version of the above theorem, where the continuity condition on has been dispensed with and the completeness of has been replaced by completeness of .

Theorem 10. Let be a metric space and a nonempty subset of . Let be a pair of mappings such that(i);(ii), ;(iii) the pair is -weakly commuting on .
Then we have the following:(a) is a singleton if is complete,(b) is a singleton if is complete.

Proof . Pick . Since , we can construct a sequence in such that for all (see [12]). By (ii), we have Now if then Therefore Since , is a Cauchy sequence in (see [1, 5]).
(a) Suppose that is complete. Then there exists a point such that . Thus, . Since , there exists such that . Again by (ii), we have Making , yields a contradiction. Therefore and . Since the pair is -weakly commuting on , it follows that Therefore and . Again by (ii), we have Making , yields which implies . Using (ii), uniqueness of can be proved. Since , we conclude that .
(b) Suppose is complete. Then for some and there exist such that . As in part (a), we can show that . Thus .

Recently Kim et al. [1] obtained the following result for -nonexpansive type mappings in a normed space.

Theorem 11. Let be a nonempty -star shaped subset of a normed space and two mappings satisfying the following conditions:(i)the mapping is -nonexpansive and is -affine with ;(ii);(iii) the pair is -subweakly commuting.

Suppose is compact. Then we have the following:(a) and have a coincidence point .(b)If or is demicontinuous, then .

We extend the above theorem to a more general class of nonexpansive type mappings. In the sequel we will need the following Lemma 12 and Proposition 13.

Lemma 12. Let be a metric space and a nonempty subset of . Let be a pair of mappings such that (i);(ii), ;(iii) the pair is -weakly commuting on .
Then we have the following: (a) is a singleton if is complete;(b) is a singleton if is complete.

Proof. The proof can be completed on the lines of the proof of Theorem 10, where is replaced by

Proposition 13. Let be a nonempty -star shaped subset of a normed space and two mappings such that(i) is a generalized -nonexpansive type mapping and is -affine with ;(ii); (iii)the pair is -subweakly commuting;(iv) is complete.
Then there exist exactly one point such that for all .

Proof. Define by , for all and for each .
Since is -subweakly commuting and is -affine, we have for all . Thus, the pair is -weakly commuting on .
Also for all . For , we have , that is, there exist a point such that .
Observe that It follows that for all . Now for each , we conclude that (i)*,(ii)*,(iii)* is complete, (iv)* is -weakly commuting on .
Therefore by Lemma 12, there exist exactly one point such that which implies that .

Now we obtain a common fixed point theorem for generalized -nonexpansive type mappings.

Theorem 14. Let be a nonempty subset of a normed space . Let be two mappings satisfying conditions of Proposition 13. Suppose is compact. Then we have the following:(a) and have a coincidence point ,(b)If or is demicontinuous, then .

Proof. Let be a sequence in such that . By Proposition 13, there exists exactly one point such that for all .
Set . Since is compact, there exist a subsequence of such that Thus for some .
The assumption implies that is bounded. It follows that Thus . By the condition (3), we have Making , we get a contradiction. Therefore and .(a)Since the pair is -subweakly commuting, we have Thus .(b)Suppose is demicontinuous. Since , it follows from the demicontinuity of that . But . Thus we conclude that . Similarly we can prove that when is demicontinuous.

The following example shows the generality of Theorem 14 over Theorem 11.

Example 15. Let (set of reals) with norm and . Define by For and , we have and the condition (2) of Theorem 11 is not satisfied. Further, it can be easily verified that and satisfy all the hypotheses of Theorem 14 and , is a common fixed point of and .

4. Convergence Results

Recently, Rhoades and Temir [3] obtained the following theorem.

Theorem 16. Let be a Banach space and a closed convex subset of which satisfies Opial's condition. Let be mappings such that(i) is -nonexpansive;(ii) is nonexpansive. Suppose is a real sequence in . Then the sequence of Mann iterates defined for an arbitrary by converges weakly to a common fixed point of and .

The following theorem extends Theorem 16 to generalized -nonexpansive type mappings.

Theorem 17. Let be a Banach space and a closed convex subset of which satisfies Opial's condition. Let be such that(i) is generalized -nonexpansive type;(ii) is nonexpansive.
Suppose is a real sequence in . Then the sequence of Mann iterates defined for an arbitrary by converges weakly to a common fixed point of and .

Proof. If is nonempty and singleton, then the proof is complete. Assume that is nonempty and that is not a singleton. Let . Then Since is generalized -nonexpansive type, we have Now the following cases arise.
Case 1. . Then Since is nonexpansive on , the above inequality reduces to
Case 2. . Then
Nonexpansiveness of on implies
Case 3. . Then
Again, since is nonexpansive on , it follows that Therefore in all the cases, we get
By (44) and (51), we get Thus, for , is a nonincreasing sequence. Hence, exists. Now we show that converges weakly to a common fixed point of and . Let and be two subsequences of which converge weakly to and , respectively. We will show that . Suppose the contrary. Since satisfies Opial's condition and exists for any , we get a contradiction. Hence .

Corollary 18. Theorem 16.

Proof. It comes from Theorem 17, when .

Corollary 19. Let be a Banach space and a closed convex subset of which satisfies Opial's condition. Let be generalized nonexpansive type mapping.
Suppose is a real sequence in . Then the sequence of Mann iterates defined for an arbitrary by converges weakly to a common fixed point of and .

Proof. It comes from Theorem 17, when is an identity mapping on .

Corollary 20. Let be a Banach space and a closed convex subset of which satisfies Opial's condition. Let be a nonexpansive mapping.
Suppose is a real sequence in . Then the sequence of Mann iterates defined for arbitrary defined converges weakly to a common fixed point of and .

Proof. It comes from Corollary 19 when .

Acknowledgments

The authors would like to thank Professor M. Abbas for his useful comments. The authors would like to thank the referees for their constructive comments and useful suggestions.

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