Multi-valued version of , , , and conditions in Ptolemy metric spaces
© Hosseini Ghoncheh and Razani; licensee Springer. 2014
Received: 16 February 2014
Accepted: 13 November 2014
Published: 26 November 2014
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.
Definition 1.1 
is called a Ptolemy inequality, where .
Now, the definition of Ptolemy metric space is as follows.
Definition 1.2 
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  and ). Moreover, Burckley et al.  proved that spaces are Ptolemy metric spaces. They presented an example to show the converse is not true. Espinola and Nicolae in  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  introduced the C condition.
for all .
Karapınar and Taş  presented some new definitions which are modifications of Suzuki’s C condition as follows.
- (i)T is said to satisfy the condition ifwhere
- (ii)T is said to satisfy the condition ifwhere
- (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 [, 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 
Karapınar and Taş  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  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.
Definition 1.8 
A sequence in a space X is said to be Δ-convergent to , if x is the unique asymptotic center of every subsequence of .
Every bounded sequence in X has a Δ-convergent subsequence [, p.3690].
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 [, Proposition 2.1].
If C is a closed convex subset of X and if is a nonexpansive mapping, then the conditions, -converges to x and , imply and [, Proposition 3.7].
Lemma 1.10 
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 
Let and be bounded sequences in metric space K and . Suppose and for all . Then .
Theorem 1.12 
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
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 
Espinola and Nicolae  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.
- (i), where
- (ii), where
Remark 2.4 Notice that any or map is a map.
either or ,
either or ,
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 .
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 .
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 [, p.6] we construct a function , which is and has a fixed point.
X is a geodesic Ptolemy space, but it is not a space (see ).
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  the multi-valued version of the and conditions.
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.
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 .
Thus by the Opial property, . □
Example 2.14 
this shows T satisfies the condition. Since , .
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