Abstract
In this paper, multi-valued version of , , , and conditions in Ptolemy metric space are presented. Then the existence of a fixed point for these mappings in a Ptolemy metric space are proved. Finally, some examples are presented.
MSC:47H10.
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1 Introduction
The definition of a Ptolemy metric space is introduced by Schoenberg [1, 2]. In order to define it, we need to recall the definition of a Ptolemy inequality as follows.
Definition 1.1 [1]
Let be a metric space, the inequality
is called a Ptolemy inequality, where .
Now, the definition of Ptolemy metric space is as follows.
Definition 1.2 [1]
A Ptolemy metric space is a metric space where the Ptolemy inequality holds.
Schoenberg proved that every pre-Hilbert space is Ptolemaic and each linear quasinormed Ptolemaic space is a pre-Hilbert space (see [1] and [3]). Moreover, Burckley et al. [4] proved that spaces are Ptolemy metric spaces. They presented an example to show the converse is not true. Espinola and Nicolae in [5] proved a geodesic Ptolemy space with a uniformly continuous midpoint map is reflexive. With respect to this, they proved some fixed point theorems.
In 2008, Suzuki [6] introduced the C condition.
Definition 1.3 Let T be a mapping on a subset K of a metric space X, then T is said to satisfy C condition if
for all .
Karapınar and Taş [7] presented some new definitions which are modifications of Suzuki’s C condition as follows.
Definition 1.4 Let T be a mapping on a subset K of a metric space X.
-
(i)
T is said to satisfy the condition if
where
-
(ii)
T is said to satisfy the condition if
where
-
(iii)
T is said to satisfy the condition if
-
(iv)
T is said to satisfy the condition if
It is clear that every nonexpansive mapping satisfies the condition [[7], Proposition 9]. There exist mappings which do not satisfy the C condition, but they satisfy the condition as the following example shows.
Example 1.5 [8]
Define a mapping T on with by
Karapınar and Taş [7] proved some fixed point theorems as follows.
Theorem 1.6 Let T be a mapping on a closed subset K of a metric space X. Assume T satisfies the , , or condition, then is closed. Moreover, if X is strictly convex and K is convex, then is also convex.
Theorem 1.7 Let T be a mapping on a closed subset K of a metric space X which satisfying the , , or condition, then holds for .
Hosseini Ghoncheh and Razani [8] proved some fixed point theorems for the , , , and conditions in a single-valued version in Ptolemy metric space. In this paper, the notation of , , , and conditions are generalized for multi-valued mappings and some new fixed point theorems are obtained in Ptolemy metric spaces.
Let X be a metric space and be a bounded sequence in X. For , let
The asymptotic radius of in K is given by
and the asymptotic center of in K is the set
Definition 1.8 [9]
A sequence in a space X is said to be Δ-convergent to , if x is the unique asymptotic center of every subsequence of .
Lemma 1.9
-
(i)
Every bounded sequence in X has a Δ-convergent subsequence [[10], p.3690].
-
(ii)
If C is a closed convex subset of X and if is a bounded sequence in C, then the asymptotic center of is in C [[11], Proposition 2.1].
-
(iii)
If C is a closed convex subset of X and if is a nonexpansive mapping, then the conditions, -converges to x and , imply and [[10], Proposition 3.7].
Lemma 1.10 [12]
If is a bounded sequence in X with and is a subsequence of with and the sequence converges, then .
The next lemma and theorem play main roles for obtaining a fixed point in the Ptolemy metric spaces.
Lemma 1.11 [13]
Let and be bounded sequences in metric space K and . Suppose and for all . Then .
Theorem 1.12 [5]
Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, a bounded sequence and nonempty closed and convex. Then has a unique asymptotic center in K.
2 Main results
Let X be complete geodesic Ptolemy space and denote the class of all subsets of X. Denote
Thus , , , , , denote the classes of bounded, closed, convex, compact, closed bounded, and compact convex subsets of X, respectively. Also is called a multi-valued mapping on X. A point is called a fixed point of T if .
Definition 2.1 [14]
Let K be a subset of a space X. A map is said to satisfy the C condition if for each , , and
there exists a such that
Espinola and Nicolae [5] used the C condition as follows.
Theorem 2.2 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K a nonempty bounded, closed, and convex subset of X. Suppose is a multi-valued mapping satisfying the C condition, then .
Now, we extend the , , , and conditions to multi-valued versions.
Definition 2.3 Let K be a subset of a geodesic Ptolemy space X. A map is said to satisfy conditions (i) , (ii) , (iii) , (iv) if for each , , and
there exists a such that
-
(i)
, where
-
(ii)
, where
-
(iii)
,
-
(iv)
.
Remark 2.4 Notice that any or map is a map.
Lemma 2.5 Let X be a complete geodesic Ptolemy space, and K a nonempty closed subset of X. Suppose is a multi-valued mapping satisfying the condition, then for every , and the following hold:
-
(i)
,
-
(ii)
either or ,
-
(iii)
either or ,
where
Proof The first statement follows from the condition. Indeed, we always have
which yields
where
If we are done. If then (2.1) turns into
By simplifying (2.2), one can get (i). For the case (2.1) turns into
which implies (i). It is clear that (iii) is a consequence of (ii). To prove (ii), assume the contrary, that is,
hold for all . Thus by triangle inequality and (i), we have
□
Theorem 2.6 Let X be a complete geodesic Ptolemy space, K a nonempty closed subset of X. Suppose is a multi-valued mapping satisfying condition, then for all , , and .
Proof The proof is based on Lemma 2.5; it is proved that
holds, where
Consider the first case. If , then we have
For one can observe
Thus,
For one can obtain
Thus
Take the second case into account. For
If then
then
For the last case, and we have
Thus
Hence, the result follows from all the above cases. □
Corollary 2.7 Let X be a complete geodesic Ptolemy space, K a nonempty closed subset of X. Suppose is a multi-valued mapping satisfying condition, then for all , , and .
Theorem 2.8 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K a nonempty, bounded, closed, and convex subset of X. Suppose is a multi-valued mapping satisfying the condition and is a sequence in K with , where , then .
Proof By Theorem 1.12, has unique asymptotic center denoted by x. Let . Applying Theorem 2.6 for , x, and , respectively, it follows that there exists such that .
Let be a subsequence of that converges to some , then
taking the superior limit as and knowing that the asymptotic center of is precisely x. Thus we obtain . Hence the proof is complete. □
By the same idea of [[4], p.6] we construct a function , which is and has a fixed point.
Example 2.9 Consider the space
with metric,
X is a geodesic Ptolemy space, but it is not a space (see [4]).
Define a mapping T on X by
T satisfies the condition. Suppose and , thus , then , so
and we can choose ,
thus
One can check the condition holds for the other points of the space X.
Note that ; thus .
Corollary 2.10 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K a nonempty bounded, closed, and convex subset of X. Suppose is a multi-valued mapping satisfying the condition and is a sequence in K with , then .
One can find in [15] the multi-valued version of the and conditions.
Definition 2.11 Let K be a subset of a metric space . A map is said to satisfy the condition provided that
we say that T satisfies the condition whenever T satisfies for some .
One can replace the metric space with a Ptolemy space in the following definition.
Definition 2.12 Let K be a subset of a metric space and . A map is said to satisfy the condition if for each ,
implies
where stands for the Hausdorff distance.
Theorem 2.13 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K be a nonempty bounded, closed, and convex subset of X. Suppose is a multi-valued mapping satisfying and conditions, then .
Proof We find an approximate fixed point for T. Take , since we can choose . Define
Since K is convex, . Let be chosen such that
Similarly, set
Again we choose such that
By the same argument, we get . In this way we find a sequence such that
where and
For every
for which it follows that
since T satisfies the condition,
this implies
Now, we apply Lemma 1.11 to conclude , where . The bounded sequence is Δ-convergent, hence by passing to a subsequence . We choose such that
Since Tv is compact, the sequence has a convergent subsequence with . Moreover, , and K is closed; then . By the condition
Note that
this implies
Thus by the Opial property, . □
Example 2.14 [15]
Let and . Define a mapping T on D with by
First we show T satisfies the condition. Let , then
Let and , then
Let and , then , thus
and
Thus T satisfies the condition with . Let , then
this shows T satisfies the condition. Since , .
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Hosseini Ghoncheh, S., Razani, A. Multi-valued version of , , , and conditions in Ptolemy metric spaces. J Inequal Appl 2014, 471 (2014). https://doi.org/10.1186/1029-242X-2014-471
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DOI: https://doi.org/10.1186/1029-242X-2014-471