- Open Access
Existence and approximation results for SKC mappings in spaces
© Abbas et al.; licensee Springer. 2014
Received: 28 February 2014
Accepted: 7 May 2014
Published: 23 May 2014
Recently, Karapınar and Tas (Comput. Math. Appl. 61:3370-3380, 2011) extended the class of Suzuki-generalized nonexpansive mappings to the class of SKC mappings. In this paper, we investigate SKC mappings to get a criterion to guarantee a fixed point, via extending the results proved by Karapınar and Tas into the class of spaces. Further, by using Ishikawa-type iteration scheme for two mappings, we derive approximation fixed point sequence. Our results extend, improve and unify some existing results in this direction, such as (Nonlinear Anal. Hybrid Syst. 4:25-31, 2010) by Nanjaras et al. or (Comput. Math. Appl. 61:109-116, 2011) by Khan and Abbas.
MSC:47H09, 47H10, 49M05.
which lies between the class of mappings nonexpansiveness and quasi-nonexpansiveness. Later, such mappings were called Suzuki-type nonexpansive. In this interesting paper , Suzuki determine the existence of a fixed point of such mappings. In 2009 Dhompongsa et al.  improved the results of Suzuki . In this distinguished paper , the authors obtained a fixed point result for mappings with condition (C) on a Banach space under certain conditions. Afterwards Nanjaras et al.  gave some characterization of existing fixed point results for mappings with condition (C) in the framework of spaces. Recently, Khan and Abbas  derived some fixed point results via different iterative schemes for nonexpansive mappings in spaces (see also ). Very recently, Karapınar and Tas  proposed some new classes of mappings which substantially generalized the notion of Suzuki-type nonexpansive mappings. The subject of this paper is to extend the mentioned results above for the class of SKC mappings  in the framework of spaces. Furthermore, by using Ishikawa-type iteration scheme, we derive some common fixed point results via approximation fixed point sequences. The results we present in this article improve and unify some existing results in this direction, such as  and .
First of all, we recollect some fundamental definition and results from the report of Dhompongsa and Panyanak .
is called a geodesic from x to y in X. The image of c is said to be a geodesic segment joining the points x and y. A geodesic segment is denoted by , if it is unique.
Let . The subset Y of X is called convex if Y includes each geodesic segment joining for any two points in Y.
Definition 2.1 A metric space is called a geodesic space if all are joined by a geodesic.
In a geodesic metric space , the triple is said to be a geodesic triangle where the points , , in X are considered as the vertices of △ and a geodesic segment between each pair of vertices becomes the edges of △. A triangle in the Euclidean plane such that for is called a comparison triangle for the geodesic triangle . A geodesic space is called a space [8–12] if all geodesic triangles of appropriate size satisfy the following comparison axiom.
for all and all comparison points .
In fact, a geodesic space is a space if and only if it satisfies the (CN) inequality; please, see .
Lemma 2.1 ()
is uniquely geodesic.
- (ii)Let be points of X, let , and let and denote, respectively, the points of and which satisfy and . Then(2.1)
Let , and such that . Then .
- (iv)Let . For each , there exists a unique point such that(2.2)
Throughout the paper, we will use the notation for the unique point z satisfying (2.2).
In a space, asymptotic center consists of exactly one point .
We set .
Lemma 2.2 ()
Every bounded sequence in X has a △-convergent subsequence.
If C is a closed and convex subset of X, and if is a bounded sequence in C, then the asymptotic center of is in C.
- (3)The following inequality:
- (4)The following inequality:
holds, for all and .
Nanjaras et al.  proved that a self mapping satisfying condition (C), and defined on a nonempty bounded and closed subset of a complete space has a fixed point.
The following definitions are basically due to Karapınar and Tas  but here we state them in the framework of spaces.
- (1)a Suzuki-Ćirić conditioned mapping (SCC) if
- (2)a Suzuki-Karapınar conditioned mapping (SKC) if
- (3)a Kannan-Suzuki conditioned mapping (KSC) if
- (4)a Chatterjea-Suzuki conditioned mapping (CSC) if
For further details on these mappings and their implications, we refer to  and references therein.
The following are some basic properties of SKC mappings whose proofs in the setup of spaces follow the same lines as those of Propositions 11, 14, and 19 in , and therefore we omit them.
Proposition 2.1 Let K be a nonempty subset of a space X. An SKC mapping is quasi-nonexpansive provided that the set of fixed point of T is nonempty.
Proposition 2.2 Let K be a nonempty closed subset of a space X and an SKC mapping then the set of fixed point of T is closed.
holds, for all x, y in K.
Propositions similar to above can be stated for the class of KSC and CSC mappings in the framework of spaces.
for all , where .
for all , where .
for all , where .
To extend existence results given in  to the class of SKC mappings in spaces. Consequently, corresponding results for KSC and CSC mappings are also extended to spaces.
To prove some strong and △-convergence results for two SKC mappings using (2.3) in spaces.
3 Main results
In the sequel, denotes the set of fixed points of T and F the set of common fixed points of T and S. The next two theorems give the existence of fixed points of SKC mappings under different conditions on C.
Theorem 3.1 Let us consider the nonempty set C be closed, bounded and convex subset of a space X, and an SKC mapping. Define a sequence as in (2.5). Then T has a fixed point in C provided that is an approximate fixed point sequence, that is, .
Uniqueness of asymptotic centers now implies that . □
Theorem 3.2 Let C be a nonempty compact convex subset of a space X, a SKC mapping, then and given by (2.5) converge strongly to a fixed point of T provided that is an approximate fixed point sequence.
Thus exists and hence converges strongly to p. □
To prepare for our approximation results, we start with the following useful lemma.
Lemma 3.1 (See )
exists, for all .
We now give our △-convergence results.
Theorem 3.3 Let X, C, T, S and be as in Lemma 3.1. If , then △-converges to a common fixed point of T and S.
First, we show that .
Since is △-convergent to v, thus v is unique asymptotic center for every subsequence of . Hence uniqueness of asymptotic centers implies that . That is, .
A similar argument shows that and hence .
We now claim that .
a contradiction. Thus, and hence .
To show that is △-convergent to a common fixed point of T and S, it suffices to show that consists of exactly one point.
a contradiction and hence . Therefore, . □
Remark 3.1 The above theorem extends Theorem 4 of Khan and Abbas  to SKC mappings.
Although the following is a corollary to our above theorem, yet it is new in itself.
Corollary 3.1 Let C be a nonempty, closed and convex subset of a space X, an SKC mapping. Let be as in (2.4). If , then the sequence is △-convergent to a fixed point of T.
Proof Take in Theorem 3.3. □
The following corollary extends Theorem 30 of Karapınar and Tas  to the setting of a space.
Corollary 3.2 Let C be a nonempty, closed and convex subset of a space X, an SKC mapping. If , then the sequence defined in (2.5) △-converges to a fixed point of T.
Proof Take , the identity mapping, in Theorem 3.3. □
Two mappings are said to satisfy the condition if there exists a nondecreasing function with , for all such that either or for all .
This condition becomes condition of Senter and Dotson  whenever .
Nanjaras et al.  obtained a strong convergence result for a Suzuki-generalized nonexpansive mappings employing condition .
In the following, we will use condition to study the strong convergence of sequence defined in Lemma 3.1.
Theorem 3.4 Let C be a nonempty closed and convex subset of a space X, be two SKC mappings satisfying condition . If , then the sequence given in (2.3) converges strongly to a common fixed point of S and T.
Proof By Lemma 3.1, it follows that exists for all . Let this limit be c, where .
If , there is nothing to prove.
which means that and so exists.
Since f is a nondecreasing function and , it follows that .
Next, we show that is a Cauchy sequence in C.
Hence is a Cauchy sequence in a closed subset C of a complete space and so it must converge to a point p in C.
Now, gives and closedness of F forces p to be in F. □
Remark 3.2 The above theorem extends Theorem 6 of Khan and Abbas  to SKC mappings.
Although the following is a corollary to Theorem 3.4, yet it is new in itself.
Corollary 3.3 Let C be a nonempty, closed and convex subset of a space X, an SKC mapping satisfying condition . Let be as in (2.4). If , then converge strongly to a fixed point of T.
Proof Take in Theorem 3.4. □
The following corollary extends Theorem 5.5 of Nanjaras et al.  to SKC mappings and, in turn, the results involving KSC and CSC mappings.
Corollary 3.4 Let C be a nonempty, closed and convex subset of a space X, an SKC mapping satisfying condition . Let be as in (2.5). If , then converge strongly to a fixed point of T.
Proof Take , the identity mapping, in Theorem 3.4. □
Theorem 5 of Khan and Abbas  can also be extended to SKC mappings.
Theorem 25 and Theorem 32 of Karapınar and Tas  and their corollaries can now be extended to the setting of a space.
Results for KSC and CSC mappings or for mappings given in  satisfying the so-called conditions and in the setup of spaces can also be obtained from corresponding results proved in this paper. As a matter of fact, these results are special cases of our results presented here.
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