Journal of Mathematics
Volume 2019 (2019), Article ID 8509494, 15 pages
https://doi.org/10.1155/2019/8509494

Research Article

Common Fixed Points of − Rational Contractions with Applications

Mujahid Abbas,1,2 Hira Iqbal,³ and Manuel De La Sen⁴

¹Department of Mathematics, Government College University, Lahore 54000, Pakistan
²Department of Mathematics and Applied Mathematics, University of Pretoria Hatfield 002, Pretoria, South Africa
³Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Pakistan
⁴Institute of Research and Development of Processes, University of The Basque Country, Campus of Leioa (Bizkaia), 48080 Leioa, Spain

Correspondence should be addressed to Manuel De La Sen; manuel.delasen@ehu.eus

Received 3 December 2018; Revised 6 February 2019; Accepted 17 February 2019; Published 26 March 2019

Academic Editor: Basil K. Papadopoulos

Copyright © 2019 Mujahid Abbas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present the notion of set valued rational contraction mappings and then some common fixed point results of such mappings in the setting of metric spaces are established. Some examples are presented to support the concepts introduced and the results proved in this paper. These results unify, extend, and refine various results in the literature. Some fixed point results for both single and multivalued rational contractions are also obtained in the framework of a space endowed with partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

1. Introduction and Preliminaries

Let be a metric space. A mapping is called a contraction if there exists a constant such that, for any , we have

The widely known Banach contraction theorem [1] states that a contraction mapping on a complete metric space has a unique fixed point; that is, there exists a point in such that .

In the last few decades, several authors have extended and generalized this principle in various directions.

Jleli and Samet [2] presented a new type of contractive mapping, namely -contraction mapping and established an interesting fixed point theorem for such mappings in a generalized metric space. The concept of generalized metric spaces was introduced by Branciari [3], where the triangle inequality is replaced by the inequality for all pairwise distinct points .

Jleli and Samet [2] considered the set of real valued functions which satisfy the following conditions:

is nondecreasing;for each sequence , if and only if ;there exist and such that .

Example 1. Define where by

Then .

A mapping on a metric space is called a contraction if for any and , we have

whenever and .

Theorem 2 (see [2]). Let be a complete generalized metric space and . If there exist and such that

holds for any whenever . Then has a fixed point.

Ahmad et al. [4] modified the class of mappings as follows:

where is continuous.

Example 3. Define for by

Then, .

Authors in [4] considered the following result of Jleli and Samet [2] with the function instead of :

Theorem 4. Let be a complete metric space and a -contraction, where . Then has a unique fixed point and for any , the sequence converges to .

Note that the Banach contraction theorem immediately follows from the above theorem.

Let be a nonempty set endowed with a metric . Let be the set of all nonempty subsets of , denotes the set of all nonempty compact subsets of , and denotes the set all nonempty closed and bounded subsets. For and , define distance of a point from the set by

A mapping defined by

is called the generalized Pompeiu-Hausdorff distance induced by .

Let . A point is called a fixed point of if

Nadler [5] obtained the following multivalued version of Banach contraction principle.

Theorem 5. Let be a complete metric space. If satisfies

for any and , then has a fixed point.

Afterwards, many researchers have obtained fixed point results for multivalued mappings satisfying certain generalized contractive conditions. Hançer et al. [6] introduced multivalued contraction mappings as follows.

Let be a metric space, , and . Then, is called a multivalued -contraction if, for any ,

holds whenever where .

They established the following fixed point results for multivalued -contraction mappings on complete metric spaces.

Theorem 6. Let be a complete metric space and a multivalued -contraction. Then has a fixed point.

For further results in this direction, we refer to [7–10].

Another variation of contraction mapping that can be found in literature is -contraction mapping.

Asl et al. [11] initiated the concept of -admissibility in case of multivalued mappings, whereas Mohammadi et al. [12] presented the notion of -admissibility in case of multifunctions.

Karapinar et al. [13] presented the idea of a triangular -admissible mapping.

Definition 7. Let . A pair is called triangular if for any , and imply that .

Recently, Abbas et al. [14] proposed a concept of -closed mappings for set valued mappings. We present the following generalization of the definition.

Definition 8. Let and . We say that a pair is triangular -closed if the pair is triangular and for any with we have for all and .

If then a mapping which is triangular closed is referred to simply as a triangular -closed mapping.

Example 9. Let . Let be defined by

Define the mappings by

and

It is obvious that the pair is triangular -closed.

The following lemma is crucial in our results.

Lemma 10. Let Suppose that the pairs and are a triangular -closed. Assume that there exists with where . Define sequences and , then for all with .

Proof. By assumption, there exist and such that . Since is -closed and we obtain where and . As is -closed, we have in such that . Continuing this way, we have sequences and with and for all .
Since the pair is triangular we obtain that

Thus by induction, we have

for any with .

Parvaneh et al. [15] introduced the concept of -contraction with respect to a family of functions and obtained some contraction fixed point results in metric and ordered metric spaces.

They introduced the following family of functions:

Let denote the set of functions satisfying condition :

For all with there exists such that

Following are some examples of such functions [15].

Example 11. , where and .

Example 12. , where and .

The following definition which is a generalization of continuity [16] is needed in the sequel.

Definition 13. Let be a metric space, and . A pair is -continuous at the point if, for any sequence in , and for all implies that . We say that pair is -continuous on if the pair is -continuous on each .

In this paper, we introduce multivalued - rational contraction pair of multivalued mappings and prove the existence of common fixed points of the pair in a metric space. We also obtain some fixed point results for both single and multivalued rational contraction mappings in a space endowed with a partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

2. Common Fixed Point Results

Throughout this section we assume that is a metric space and where satisfies , and . Let be a family of continuous and nondecreasing functions where for .

We now present the following definitions:

Definition 14. Let , , , and

(1)A pair is called a multivalued rational contraction pair if, for any with and , the following condition holds:

 where

 and

(2) A pair is called a multivalued rational contraction if, for any with and , the following condition holds:

 where

 and

(3) A mapping is called a multivalued rational contraction if, for any with and , the following condition holds:

 where

 and

Remark 15.

(1)If are defined as for all in Definition 14, then the pairs of mappings and are multivalued generalized rational contractions.(2)If are defined as for all in Definition 14, then is a multivalued generalized rational contraction mapping.

Theorem 16. Let . Suppose that the pairs and are multivalued rational contractions such that

(C1)the pairs and are triangular -closed;(C2)there exist and with ;(C3) and are -continuous.

Then there exists such that .

Proof. If , for some , then we have our conclusion. Assume that , for all . By assumption there exist and such that . If or then the result follows. Assume that and then

This implies

Since and is multivalued rational contraction, we obtain that

where

Since , by there exists such that

Therefore, from (29) and using the fact that is nondecreasing we obtain

Also,

since

Thus,

Replacing (31) and (34) in (28) we get

If , then

a contradiction. Hence . Therefore,

By we have

Thus, there exists such that

Then from (37) we have

Since , , , and is -closed, we have . If or then the result follows. Suppose that and . Thus

As , we have

where

Since , by there exists such that

Therefore, from (43) and using the fact that is nondecreasing we have

Further,

Since

we get

Replacing (45) and (48) in (42) we have

If , then

a contradiction. Hence . Therefore,

By we have

Thus, there exists such that

Therefore, from (51) we have

Furthermore, from (40) we have

Proceeding in the same manner, we obtain a sequence in such that , , , , and with and it satisfies

for each . As , , and , we have . Then,

Therefore,

Thus,

Since by there exists such that . Thus, from definition of and (59) we obtain

Also,

Then

Therefore, replacing (60) and (62) in (58) we obtain

If then

a contradiction. Further,

Again, using we have

Therefore, there exists such that

Thus, we have

From (56), we have

Hence, we have a sequence in and such that

for all . On taking limit as , we obtain

which implies that . Then by we obtain

Now, we show that is a Cauchy sequence. If is not Cauchy, then there exist and for all such that

Thus,

Therefore, from the above inequality and (72), we obtain

Also,

and

On taking limit as in (76) and using (75) (77), we have

Therefore,

Similarly, we obtain that

By Lemma 10, we have

If , we have

Taking limit we have

a contradiction to our assumption. Thus, assume that . Therefore, we have

where

Taking limit in (85),

Now, by there exists such that . Thus, using the continuity of and (86),

Moreover,

Taking limits as and using (72) and (80) we obtain that

Thus, using (84) and the continuity of we have

From (89), we obtain

a contradiction. Hence, is a Cauchy. Since is a complete metric space, there exists such that As the pair is -continuous, we have
Note that

On taking limit as on both sides of the above inequality, we obtain that and hence . As is -continuous, we have Also,

On taking limit as on both sides of the above inequality, we obtain that and hence . Thus there exists such that .

We may omit the -continuity condition in the above theorem by condition (H).

If is a sequence in with for all and for some , then for all .

Theorem 17. Let . Suppose that the pairs and are multivalued rational contractions such that

(C1) and are triangular -closed mapping;(C2)there exist and with ;(C3)the pair satisfies condition (H).

Then there exists such that .

Proof. As in Theorem 16, we obtain a Cauchy sequence in the complete metric space with where and . As the pair satisfies condition (H), for all . We need to show that is the common fixed point. Suppose on the contrary that .
From condition (H), we obtain as and . If then

Taking limit in the above equation we obtain

a contradiction to our assumption. Thus, we assume . Then,

Further,

where

Taking in the above inequality we have

Thus, as previously shown, by and continuity of we have

Also,

Hence

Moreover, from (97) we have

On taking limit as in the above inequality and using , we obtain that

It follows from that

That is,

a contradiction. Hence, . Similarly, we can show . Hence,

Example 18. Let and

Define the mappings by

and

Define and by

and

It is evident that both the pairs and are triangular -closed. Now, we show that the pair is a rational contraction for . That is, we need to show that

for all .
Let Without any loss of generality, we may assume that . Thus,

Thus, the pair is a . Similarly, is rational for as defined above. If , then . For any sequence for all , we have and converges to some . Thus . All the conditions of Theorem 17 are satisfied and is the common fixed point of and .

Corollary 19. Let . Suppose that the pairs and are continuous multivalued rational contractions, then there exists such that .

Proof. Define for all . Then the result follows from Theorem 16.

Theorem 20. Let be a multivalued rational contraction mapping such that

(C1) is a triangular -closed mapping;(C2)there exist and with ;(C3) is -continuous.

Then there exists such that .

Proof. By choosing in Theorem 16 the result follows.

Theorem 21. Let be a multivalued rational contraction mapping such that

(C1) is a triangular -closed mapping;(C2)there exist and with ;(C3)the pair satisfies condition (H).

Then there exists such that .

Proof. By choosing in Theorem 17 the result follows.

Theorem 22. Let be a multivalued mapping. Suppose that satisfies the following conditions:

(C1)For any such that and , we have

 where

(C2) is a triangular -closed mapping;(C3)there exist and with ;(C4) is -continuous or satisfies condition (H).

Then there exists such that .

Proof. Taking and where in Theorems 20 and 21, the result follows.

Example 23. Let and

Define the mapping by

Define as

and

Clearly, is a triangular -closed mapping. If and , then . Note that (C(3)) is also satisfied. Let , then if . Assume that , then

Therefore, is a multivalued rational contraction with and . Thus, all the conditions of Theorem 21 are satisfied and is a fixed point of .

Corollary 24. Let . Suppose that, for any such that , and , the pairs and satisfy

whenever and

whenever . If the following conditions also hold:

(C1) and are triangular -closed mappings;(C2)there exist and with ;(C3) and are -continuous.

Then there exists such that .

Now, we apply the results for the existence of common fixed points of single valued mappings on a complete metric space.

Definition 25. Let be two mappings on a nonempty set and . A pair is called -admissible if for any , with , we have .

Denote the set of fixed points of and by and , respectively.

Theorem 26. Let . Suppose that the pairs and are rational contractions such that

(C1)the pairs and are triangular -admissible mappings;(C2)there exists such that ;(C3) and are -continuous.

Then .

Proof. Define as and . Then Theorem 16 implies the result.

Theorem 27. Let . Suppose that the pairs and are rational contractions such that

(C1) and are triangular -admissible mapping;(C2)there exists such that ;(C3)the pair satisfies condition (H).

Then

Proof. Define as and . Then Theorem 17 implies the result.

Example 28. Let and

Define by

Define by

and

Clearly, and are triangular -admissible mappings.
Define as

Note that pairs and are -continuous. Also, and are rational contractions for . If and , then . Thus all the conditions of Theorem 26 are satisfied. Thus, and have a common fixed point in .

Corollary 29. Let be an -continuous rational contraction. Then has a fixed point in if there exists such that and is triangular -admissible. Furthermore, the fixed point is unique if .

Corollary 30. Let be a triangular -admissible rational contraction. Then has a fixed point in provided that there exists such that and satisfies condition (H). Furthermore, the fixed point is unique if .

Example 31. Let . Note that

defines the metric on . Define by

Define and

and

Clearly, is -continuous. Define as

Let . We only consider the case where ; all other cases are trivial. Note that

Thus, is an rational contraction. Also, is triangular -admissible. Let , then . All the conditions of Corollary 29 are satisfied and is a fixed point of .

3. Application to Nonlinear Elastic Beam Equations

We study the existence of solutions of fourth-order two-point boundary value problem given by

which represents the bending of an elastic beam clamped at both ends. The boundary value problem in (135) can be written as [17]

where the Green function associated with the given boundary value problem is given by

where (see [18]).

Let be the space of all continuous functions defined on . The metric on is given by

for all . Note that the space is complete metric space.

Theorem 32. Suppose that the following hypotheses are satisfied:

(1) is continuous;(2) is nondecreasing for each ;(3)for for , we have

 for any , where

(4)there exists such that, for all ,

Then problem (135) has a solution in .

Proof. Let . Define the operator by

Clearly, is continuous. Define as

and

Clearly, is triangular. Also, since is nondecreasing, then for any such that for all we obtain

Hence, . Since , is -admissible. Now, for all such that for all we have

where . Using the fact that we have

Now,

Now, passing through exponential we obtain

Thus, satisfies

with and . Since all the conditions of Corollary 29 are satisfied, then problem (135) has a solution in .

4. An Application to First-Order Periodic Problem

In this section, we establish the necessary conditions for existence of a fixed point of a mapping in the setting of a partially ordered metric space. Throughout this section, we assume that is a partially ordered metric space.

Definition 33. A sequence is called -preserving if for all .

Definition 34. A mapping is called -closed if, for any with , for all and .

Definition 35. A mapping is called -continuous at a point if, for any sequence in , and implies that for all . We say that the mapping is -continuous on if -continuous at every .

Corollary 36. Let be rational contraction such that is a -closed mapping and is -continuous. If there exists such that , then there exists such that .

Proof. Define by whenever and whenever and whenever . Thus, the result follows from Theorem 21.

Example 37. Let . Define a metric on by

Define by

Note that is a partially ordered metric space. Define the mapping by and . It can easily be shown that is -closed, -continuous, and rational contraction for and . If and , then we have . Note that all the conditions of Corollary 36 are satisfied and is the set of fixed points of .

Corollary 38. Suppose is a rational contraction. If is -closed and -continuous and there exists such that , then has a fixed point in .

Corollary 39. Suppose is a -closed and -continuous mapping that satisfies

for any with . Then has a fixed point in .

We now apply the Corollary 38 in proving the existence of solution of the first-order periodic problem.

Let be the space of all continuous functions defined on . The metric on is given by

Define the partial order on by

for all . Note that the space is partially ordered complete metric space. The following first-order periodic problem is given by

where and is a continuous function. The problem in (156) can be written as

Problem (157) is equivalent to

where is defined by

Note that

The following definition will be used in our theorem.

Definition 40. A lower solution for (156) is a function differentiable on such that

Theorem 41. Suppose that following conditions hold:

(1) is a continuous function;(2) is a nondecreasing function for each ;(3)there exists where such that

 where

 holds for all and ;(4)there exists a lower solution of problem (156).

Then problem (156) has a solution in .

Proof. Let . Define the integral operator as

Clearly, is continuous. For any such that we obtain

Since we obtain

Passing through exponential, we have

Setting , we obtain that

Thus,

for , where and . Since is nondecreasing then for any such that for all , we have

Therefore, is -closed. If is a lower solution of (156), then by simple calculations we have

Hence, by the Corollary 38, problem (156) has a solution in .

Remark 42. Our results generalize, extend, and refine several results in the literature.

(1)Our results dealing with single valued mappings can be viewed as an extension and generalization of Banach fixed point theorem [1]. It is worth mentioning that the results in [15] are not a generalization of the Banach fixed point theorem.(2)Theorems 20 and 21 extend Nadler’s theorem [5], Bianchini’s Theorem [19], and Hancer’s theorem [6].(3)Corollaries 29 and 30 generalize Theorems 2.3 and 2.4 in [4] and refine Theorems 2.5 and 2.7 in [15].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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