Common Fixed Points of − Rational Contractions with Applications

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 thatwhereSince , by there exists such that Therefore, from (29) and using the fact that is nondecreasing we obtainAlso, 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 haveSince , , , and is -closed, we have . If or then the result follows. Suppose that and . Thus As , we have whereSince , by there exists such that Therefore, from (43) and using the fact that is nondecreasing we haveFurther, Since we getReplacing (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 satisfiesfor each . As , , and , we have . Then, Therefore, Thus, Since by there exists such that . Thus, from definition of and (59) we obtainAlso, 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 obtainNow, 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 obtainAlso,andOn taking limit as in (76) and using (75) (77), we have Therefore,Similarly, we obtain thatBy Lemma 10, we have If , we have Taking limit we have a contradiction to our assumption. Thus, assume that . Therefore, we havewhere 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 thatThus, 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 byfor 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 byClearly, 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 byDefine 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 bywhere and is a continuous function. The problem in (156) can be written asProblem (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.