Abstract

The aim of this paper is to establish common fixed point results for multivalued mappings satisfying generalized F-contractive conditions of Hardy Rogers type with respect to generalized dynamic process in b-metric space. Our results improve and generalize several well known results in the existing literature.

1. Introduction and Preliminaries

Let be a non empty set, be a mapping. A point is called a fixed point of if Fixed points results of mappings, which satisfies some specific contractive conditions on some space have been very useful in research activity (see ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}).

Recently, Wardowski ³⁰ introduced a new concept of contraction named F-contraction and proved a fixed point theorem which generalizes Banach contraction principle. Klim et al. ²² further established fixed point result for F-contractive mapping in dynamic process. Cosentino et al. ¹⁶ further generalized this concept as F-Contractive Mappings of Hardy-Rogers-Type. Arshad et al. ⁴ proved fixed point result in contraction of Hardy-Rogers-type. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for multivalued mappings in b-metric space with generalized dynamic process. This paper contain common fixed point results for two mappings. Throughout our paper and represent set of real numbers, set of natural numbers and family of non-empty closed bounded subsets respectively.

Definition 1 ¹⁰ Let be a non-empty set and let be a given real number. A function is called a b-metric provided that, for all

1) iff

2)

3) The pair is called a b-metric space.

Definition 2 ¹⁵ Let be a b-metric space. Then a sequence in is called a Cauchy sequence if and only if for all there exist such that for each , we have

Definition 3 ³⁰ Let be a mapping satisfying:

(F1) F is strictly increasing.

(F2) for each sequence of positive numbers if and only if

(F3) there exists such that

We denote with the family of all functions that satisfy the conditions (F1)-(F3).

Let be a metric space. A self-mapping on is called an F-contraction if there exist such that

Theorem 4 ³⁰ Let be a complete metric space and let be an F-contraction. Then has a unique fixed point and for every a sequence is convergent to

Definition 5 ¹⁶ Let be a metric space. A self-mapping on is called a generalized F-contraction of Hardy-Rogers-type if there exist and such that

for all with where and

Theorem 6 ¹⁶ Let be a complete metric space and let be a self-mapping on . Assume that there exist and such that is an F-contraction of Hardy-Rogers-type, that is,

for all where , and Then has a fixed point. Moreover, if , then the fixed point of is unique.

Definition 7 ²⁰ Let be a metric space. For the Hausdorff metric on induced by metric is given as:

where and is the Hausdorff metric induced by

Definition 8 ²² Let and let satisfies (F1)-(F3). The mapping is called a set-valued F-contraction with respect to a dynamic process if there exists a function such that

In the above discussion denotes the collection of all non empty closed subsets of and is called a dynamic process of starting at The dynamic process will be written simply as where

Definition 9 ³ Let , and be an arbitrary but fixed element in The set for all is called a generalized dynamic process of and starting at The generalized dynamic process will simply be written as The sequence for which is a generalized dynamic process is called iterative sequence of starting at

Lemma 10 ²³ Suppose be a b-metric space and be a sequence in such that

where Then the sequence is a Cauchy sequence in provided that

2. Main Result

Definition 11 Let be a mapping satisfying:

(F1) F is strictly increasing.

(F2) for each sequence of positive numbers if and only if

We denote with the family of all functions that satisfy the conditions (F1) and (F2).

Let be a b-metric space. A self-mapping on is called an F₁ -contraction if there exist such that

(1)

Definition 12 Let be a b-metric space, is continuous and be an arbitrary point in . A mapping is called a generalized multivalued -contraction of Hardy-Rogers-type with respect to a generalized dynamic process if there exist and is non decreasing such that

where

for all with where and

Now we state and prove our main result.

Theorem 13 Let be a complete b-metric space and If is a generalized multivalued -contraction of Hardy-Rogers-type with respect to a generalized dynamic process Then and have a common fixed point.

Proof: Let be an arbitrary point, by definition of generalized multivalued -contraction of Hardy-Rogers-type with respect to a generalized dynamic process we have

As is strictly increasing, therefore

As and we deduce that and so

This implies that

Continuing this process, we can easily say that

Now, to show that is a Cauchy sequence in . Let with

Using Lemma 10, and taking limit we get

which proves that is a Cauchy, so there exist some such that Now we prove that For this, we have

By (F2)

Therefore

From above we can write

Again by (F2) Therefore So . Moreover

Hence, , so , that is is the common fixed point of and .

Putting , we obtain a generalize form of Kannan's result in dynamic process.

Corollary 14 Let be a complete b-metric space and Assume that there exist and such that is continuous satisfying:

for all with , where , and . Then and have a common fixed point.

Choosing , we obtain a generalize version of Reich's result.

Corollary 15 Let be a complete b-metric space and Assume that there exist and such that is continuous satisfying:

for all with , where , and . Then and have a common fixed point.

Theorem 16 Let be a complete b-metric space, is continuous and Assume that there exist and such that is continuous from the right. Now if is a generalized dynamic process such that

(2)

where

for all with where and . Then and have a common fixed point.

Proof: Let be an arbitrary point of By definition of generalized dynamic process . Since is continuous from the right, there exists a real number such that

Now, from we deduce that there exists such that Consequently, we get

which implies

As is strictly increasing we deduce

As and hence we deduce that and so consequently,

Continuing this way we get

and hence

Proceeding this as in Theorem 13, we obtain that is a Cauchy sequence. Since is a complete metric space, there exists some such that . Now we prove that . For this, since

By (F2)

Therefore, Hence

By using (F2) which implies Also

Therefore hence that is

Replacing b-metric space by metric space in the above result we get the following corollary:

Corollary 17 Assume that is a complete metric space, is continuous and Assume that there exist and such that is continuous from the right. Now if is a generalized dynamic process in such a way that

where

for each with where and Then and possess a common fixed point.

Example 18 Assume that and is defined by

where Define and by and Define a sequence by for all with and Then

and so on. Here

Fix and clearly . We can check that is a b-metric. Now checking for all we can find some that satisfy the inequality (2) in such a way that for each Moreover is found to be the common fixed point of and

Putting and in the above result we get the following corollary:

Corollary 19 Assume that is a complete b-metric space, is continuous, there exist a function and let satisfy (F1)-(F2). Now if is a generalized dynamic process in such a way that

then there is a common fixed point of and i.e.

Putting as an identity function in Corollary 19 we get:

Corollary 20 Assume that is a complete metric space, is continuous there exist a function and suppose satisfy (F1)-(F2). Now if is a dynamic process in such a way that:

then there exists a fixed point of

Interests

The authors declare that they have no competing interests.

References