Abstract
We obtain a unique common triple fixed point theorem for hybrid pair of mappings in metric spaces. Our result extends the recent results of B. Samet and C. Vetro (2011). We also introduced a suitable example supporting our result.
1. Introduction
The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler [1].
Let be a metric space. We denote the family of all nonempty closed and bounded subsets of and the set of all nonempty closed subsets of . For and , we denote . Let be the Hausdorff metric induced by the metric on , that is, for every .
It is clear that for and , we have .
Definition 1.1. An element is said to be a fixed point of a set-valued mapping if and only if .
In 1969, Nadler [1] extended the famous Banach contraction principle [2] from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.
Theorem 1.2 (see, Nadler [1]). Let be a complete metric space and let be a mapping from into . Assume that there exists such that for all . Then, has a fixed point.
Lemma 1.3 (see, Nadler [1]). Let and . Then for every , there exists such that .
Lemma 1.4 (see, Nadler [1]). Let . If with , then for each , there exists such that .
Lemma 1.5 (see, Nadler [1]). Let be a sequence in with , for . If and , then .
The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer the reader to [1, 3–11] and the references therein.
The concept of coupled fixed point for multivalued mapping was introduced by Samet and Vetro [12], and later several authors, namely, Hussain and Alotaibi [13], Aydi et al. [14], and Abbas et al. [15], proved coupled coincidence point theorems in partially ordered metric spaces.
Definition 1.6 (see, Samet and Vetro [12]). Let be a given mapping. We say that is a coupled fixed point of if and only if
Definition 1.7 (see, Hussain and Alotaibi [13]). Let the mappings and be given. An element is called(1)a coupled coincidence point of a pair if and ;(2)a coupled common fixed point of a pair if and .
Berinde and Borcut [16] introduced the concept of triple fixed points and obtained a tripled fixed point theorem for single valued map.
Now we give the following.
Definition 1.8. Let be a nonempty set, (collection of all nonempty subsets of ). . (i)The point is called a tripled fixed point of if (ii)The point is called a tripled coincident point of and if (iii)The point is called a tripled common fixed point of and if
Definition 1.9. Let be a multivalued map and be a self map on . The Hybrid pair is called -compatible if whenever is a tripled coincidence point of and .
2. Main Results
Theorem 2.1. Let be a metric space and let and mappings satisfying (2.1.1), for all and with , where is a fixed number, (2.1.2) and is a complete subspace of .
Then the maps and have a tripled coincidence point.
Further, and have a tripled common fixed point if one of the following conditions holds.(2.1.3) (a) is -compatible, there exist such that , and , whenever is a tripled coincidence point of and is continuous at . (b) There exist such that , and whenever is a tripled coincidence point of and is continuous at , and .
Proof. Let . From , there exist sequences , , and in such that , and , .
For simplification, denote
From (2.1.1), we obtain
Let denoted by .
Clearly, .
Then,
It is clear that .
Now we prove by induction that
where
Equation (2.3) is true for .
Assume that (2.3) is true for some . Consider
We have
Similarly, we have
Thus (2.3) is true for all integer values of .
Now from (i)–(vi) and continuing this process, we get
for all . That is,
For , we have
Hence is a Cauchy. Similarly, we can show that and are Cauchy.
Suppose is complete, the sequences , , and are convergent to some in , respectively. There exist such that , , and .
Now, we have
Letting , we get so that . That is, . Similarly, we can show that and . Thus is a tripled coincidence point of and . Suppose (2.1.3) (a) holds.
Since is a tripled coincidence point of and , there exist such that , and .
Since is continuous at and , we have , and .
Since , we have .
Since , we have .
Since , we have .
Then is tripled coincidence point of and .
Similarly, we can show that is a tripled coincidence point of and .
Also it is clear that
From (2.1.1), we have
Letting , we obtain
which implies that
Thus . Similarly, we can show that and . Thus is a tripled common fixed point of and . Suppose (2.1.3) (b) holds.
Since is a tripled coincidence point of , there exist such that , and .
Since is continuous at and , we have , and . Thus , and . Hence is a tripled common fixed point of .
The following example illustrates Theorem 2.1.
Example 2.2. Let and defined as and . Then
It is clear that all conditions of Theorem 2.1 are satisfied and is the tripled common fixed point of and .
The following example shows that and have no tripled common fixed point if (a) or (b) is not satisfied.
Example 2.3. Let , and . Then is a tripled coincidence point of and . Clearly and have no tripled common fixed point.