Fixed point theorems on multi valued mappings in b-metric spaces.

In this paper, we prove a fixed point theorem and a common fixed point theorem for multi valued mappings in complete b-metric spaces.

Introduction and preliminaries

Fixed point theory plays one of the important roles in nonlinear analysis. It has been applied in physical sciences, Computing sciences and Engineering. In 1922, Stefan Banach proved a famous fixed point theorem for contractive mappings in complete metric spaces. Later, Czerwik (1993, 1998) has come up with b-metrics which generalized usual metric spaces. After his contribution, many results were presented in -generalized weak contractive multifunctions and b-metric spaces (Alikhani et al. 2013; Boriceanu 2009; Mehemet and Kiziltunc 2013). The following definitions will be needed in the sequel:

Definition 1

Nadler (1969) Let X and Y be nonempty sets. T is said to be multi-valued mapping from X to Y if T is a function for X to the power set of Y. we denote a multi-valued map by:

Definition 2

Nadler (1969) A point of is said to be a fixed point of the multi-valued mapping T if .

Example 3

Joseph (2013) Every single valued mapping can be viewed as a multi-valued mapping. Let be a single valued mapping. Define by . Note that T is a multi-valued mapping iff for each . Unless otherwise stated we always assume Tx is non-empty for each .

Definition 4

Banach (1922) Led (X, d) be a metric space. A map is called contraction if there exists such that , for all .

Definition 5

Nadler (1969) Let (X, d) be a metric space. We define the Hausdorff metric on CB(X) induced by d. That isfor all , where CB(X) denotes the family of all nonempty closed and bounded subsets of X and , for all .

Definition 6

Nadler (1969) Let (X, d) be a metric space. A map is said to be multi valued contraction if there exists such that , for all

Lemma 7

Nadler (1969) Ifand, then for each, there existssuch that.

Definition 8

Aydi et al. (2012) Let X be a nonempty set and let be a given real number. A function is called a b-metric provide that, for all ,A pair(X, d) is called a b-metric space.

if and only if

.

Example 9

Boriceanu (2009) The space , , together with the function .

Example 10

Boriceanu (2009) The space for all real function such that , is b-metric space if we take .

Example 11

Aydi et al. (2012) Let and , and . Then for all . If ,the ordinary triangle inequality does not hold.

Definition 12

Boriceanu (2009) Let (X, d) be a b-metric space. Then a sequence in X is called Cauchy sequence if and only if for all there exists such that for each we have .

Definition 13

Boriceanu (2009) Let be a (X, d) b-metric space. Then a sequence in X is called convergent sequence if and only if there exists such that for all there exists such that for all we have . In this case we write

Our first result is the following theorem.

Main results

Definition 14

Let (X, d) be a b-metric space with constant . A map is said to be multi valued generalized contraction iffor all and with .

Theorem 15

Let (X, d) be a completeb-metric space with constant. Letbe a multi valued generalized contraction mapping. Then T has a unique fixed point.

Proof

Fix any . Define and let . By Lemma 7, we may choose such that .

Now,By Lemma 7, there exist such that .

Now,Continuing this process, we obtain by induction a sequence such that such thatfor all and let Since and have same radius of convergence. Then, is a Cauchy sequence. But (X, d) is a complete b-metric space, it follows that is convergent.Now,Using (1), we obtain, The above inequality is true unless . Thus, .

Now we show that u is the unique fixed point of T. Assume that v is another fixed point of T. Then we have andwe obtain, . This implies that . This completes the proof.

Theorem 16

Letbe a complete b-metric space with constant. Letbe a multi valued mapping satisfies the condition:for all x,yX andwithThen T and S have a unique common fixed point.

Fix any Define and let such that By Lemma 7, we may choose such that On the other hand and by symmetry,we haveAdding inequalities (2) and (3) , we obtainSimilarly, it can be shown that, there exists such thatContinuing this process,we obtain by induction a sequence such that such thatAlso,From (4) and (5)Therefore,Since and have same radius of convergence. Then, is a Cauchy sequence. Since is complete,there exists such that .

We shall prove that z is a common fixed point of T and S.Using (7) in (6) and letting as , we obtain,

and is closed. Thus, .

Similarly, .

We show that z is the unique fixed point of S and T. Now,Since Hence, S and T have a unique common fixed point.

Example 17

Let . We define by , for all . Then (X, d) is a complete metric space.

Define by , for all Then,

Therefore, is the unique fixed point of T.

Conclusion

Many authors have contributed some fixed point results for a self mappings in b-metric spaces. In this paper, we have proved the existence and uniqueness of fixed point results for a multivalued mappings in b-metric spaces. Our contraction mappings also generalize various known contractions like Hardy Roger contraction in the current literature.