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Fixed Points Results for Graphic Contraction on Closed Ball

Aftab Hussain

Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan

Abstract: In this paper, we introduce a new class of ciric fixed point theorem of (α,ψ)-contractive mappings on a closed ball in complete metric space. As an application, we have derived some new fixed point theorems for ciric ψ-graphic contractions defined on a metric space endowed with a graph in metric space. Our results provide extension as well as substantial generalizations and improvements of several well known results in the existing comparable literature.

Keywords: fixed point, α-admissible, (α,ψ)-contraction, closed ball

Citation: Aftab Hussain, Fixed Points Results for Graphic Contraction on Closed Ball. Turkish Journal of Analysis and Number Theory, 4 (4). 92-97. January 2016. doi:10.12691/tjant-4-4-2

1. Introduction

In 2012, Samet et al. [18], introduced a concept of - contractive type mappings and established various fixed point theorems for mappings in complete metric spaces. Afterwards Karapinar and Samet [6], refined the notions and obtain various fixed point results. Hussain et al. [9], enlarged the concept of -admissible mappings and obtained useful fixed point theorems. Subsequently, Abdeljawad [4] introduced pairs of admissible mappings satisfying new sufficient contractive conditions different from [9] and [18], and proved fixed point and common fixed point theorems. Lately, Salimi et al. [17], modified the concept of - contractive mappings and established fixed point results. Mohammadi et al. [7] introduced a new notion of -contractive mappings and show that this is a real generalization for some old results. Arshad et al. [2] established fixed point results of a pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space. Hussain et al. [8], introduced the concept of an -admissible map with respect to and modify the -contractive condition for a pair of mappings and established common fixed point results for two, three, and four mappings in a closed ball in complete dislocated metric spaces. Over the years, fixed point theory has been generlized in multi-directions by several mathematicians(see [1-18]).

Let be a family of nondecreasing functions such that for each .

Lemma 1. ([17]). If then for all .

Definition 2. ([18]). Let be a metric space. A mapping is an contractive mapping if there exist two functions and such that

for all .

Definition 3. ([18]). Let and . We say that is -admissible if implies that

Example 4. Let and an identity mapping on Define by

Then is admissible.

Definition 5. ([17]). Let and two functions. We say that is -admissible mapping with respect to if implies that

If then above definition reduces to definition 3. If then is called an -subadmissible mapping.

Definition 6. ([7]). Let and by

We say that is -admissible. If then and so This implies Also

2. Main Results

We prove ciric fixed point results for contraction mappings on a closed ball in complete metric space.

Theorem 7. Let be a complete metric space and is admissible mapping with respect to . For , and , assume that,

[picture] | (1)

where

and

[picture] | (2)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

Proof. Let in be such that , . Continuing this process, we construct a sequence of points in such that, By assumption and is -admissible mapping with respect to we have from which we deduce that which also implies that Continuing in this way we obtain for all First, we show that for all . Using inequality (2), we have,

It follows that,

Let for some . Using inequality (1), we obtain,

So

[picture] | (3)

the case is impossible

Which is a contradiction. Otherwise, in other case

Thus we have,

[picture] | (4)

Now,

Thus Hence for all . Now inequality can be written as

[picture] | (5)

Fix and let such that Let with Then, by the triangle inequality, we have

Hence is a Cauchy sequence in . As every closed ball in a complete metric space is complete, so there exists such that Also

[picture] | (6)

So by given assumption from (ii), we have for all Now from (1), we obtain

[picture] | (7)

where

If then for every Thus

[picture] | (8)

which on taking limit as gives

Hence The result follows.

Example 8. Let with metric on defined by Let be defined by,

Consider and

Now then

Also if then

Then the contractive condition does not hold on Also if, then

If in the Theorem 7, we have the following corollary.

Corollary 9. Let be a complete metric space and is admissible mapping. For , and , assume that,

[picture] | (9)

where

and

[picture] | (10)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

If in the Theorem 7, we have the following corollary.

Corollary 10. Let be a complete metric space and is -subadmissible mapping. For , and , assume that,

[picture] | (11)

where

and

[picture] | (12)

If following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

Corollary 11. Let be a complete metric space and is admissible mapping with respect to . For , and , assume that,

[picture] | (13)

where

and

[picture] | (14)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

If in the corollary 11, we have the following corollary.

Corollary 12. Let be a complete metric space and is admissible mapping. For , and , assume that,

[picture] | (15)

where

and

[picture] | (16)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

If in the corollary 11, we have the following corollary.

Corollary 13. Let be a complete metric space and is admissible mapping. For , and , assume that,

[picture] | (17)

and

[picture] | (18)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

If in the corollary 11, we have the following corollary.

Corollary 14. Let be a complete metric space and is admissible mapping. For , and , assume that,

[picture] | (19)

and

[picture] | (20)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

If , we obtain the following corollary.

Corollary 15. Let be a complete metric space and is admissible mapping with respect to . For , and , assume that,

[picture] | (21)

and

[picture] | (22)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

If in the corollary 11, we have the following corollary.

Corollary 16. Let be a complete metric space and is admissible mapping. For , and , assume that,

[picture] | (23)

and

[picture] | (24)

Suppose that the following assertions hold:

•

• for any sequence in such that for all and as then for all .

Then, there exists a point in such that

3. Fixed Point Results for Graphic Contractions

Consistent with Jachymski [13], let be a metric space and denotes the diagonal of the Cartesian product . Consider a directed graph such that the set of its vertices coincides with , and the set of its edges contains all loops, i.e., . We assume has no parallel edges, so we can identify with the pair . Moreover, we may treat as a weighted graph (see [13]) by assigning to each edge the distance between its vertices. If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that and for A graph is connected if there is a path between any two vertices. is weakly connected if is connected(see for details [1,5,12,13]).

Definition 17. ([13]). We say that a mapping is a Banach -contraction or simply -contraction if preserves edges of , i.e.,

and decreases weights of edges of in the following way:

Now we extend concept of -contraction as follows.

Definition 18. Let be a metric space endowed with a graph and be self-mappings. Assume that for , and , following conditions hold,

where

Then the mappings is called ciric -graphic contractive mappings. If for some , then we say is -contractive mappings.

Definition 19. Let be a metric space endowed with a graph and be self-mappings. Assume that for , and , following conditions hold,

where

Then the mappings is called ciric -graphic contractive mappings. If for some , then we say is -contractive mappings.

Theorem 20. Let be a complete metric space endowed with a graph and be ciric -graphic contractive mappings and . Suppose that the following assertions hold:

• and for all ;

• if is a sequence in such that for all and as , then for all .

Then has a fixed point.

Proof. Define, by

First we prove that the mapping is -admissible. Let with , then . As is ciric -graphic contractive mappings, we have, . That is, Thus is -admissible mapping. From (i) there exists such that . That is, If with , then . Now, since is ciric -graphic contractive mapping, so That is,

Let with as and for all . Then, for all and as . So by (ii) we have, for all . That is, Hence, all conditions of Corollary 9 are satisfied and has a fixed point.

Corollary 21. Let be a complete metric space endowed with a graph and and be a mapping. Suppose that the following assertions hold:

• is Banach -contraction on ;

• and ;

• if is a sequence in such that for all and as , then for all .

Then has a fixed point.

Corollary 22. Let be a complete metric space endowed with a graph and and be a mapping. Suppose that the following assertions hold:

• is Banach -contraction on and there is such that ;

• if is a sequence in such that for all and as , then for all .

Then has a fixed point.

Conflict of Interests

The authors declare that they have no competing interests.

References

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