Journal of Function Spaces
Volume 2016 (2016), Article ID 3690421, 10 pages
http://dx.doi.org/10.1155/2016/3690421

Research Article

On the Existence of Coincidence and Common Fixed Point of Two Rational Type Contractions and an Application in Dynamical Programming

Ahmed H. Soliman¹ and Tamer Nabil²

¹Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
²Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt

Received 29 March 2016; Revised 23 May 2016; Accepted 12 June 2016

Academic Editor: Pasquale Vetro

Copyright © 2016 Ahmed H. Soliman and Tamer Nabil. 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 establish some coincidence point results for self-mappings satisfying rational type contractions in a generalized metric space. Presented coincidence point theorems weaken and extend numerous existing theorems in the literature besides furnishing some illustrative examples for our results. Finally, our results apply, in particular, to the study of solvability of functional equations arising in dynamic programming.

1. Introduction

Banach contraction principle is one of the most important aspects of fixed point theory as a source of the existence and uniqueness of solutions of many problems in various branches inside and outside mathematics (see, [1–3]). Some generalizations of this theorem replace the contraction condition by a weaker. For instance, in 1975, Dass and Gupta [4] defined the following rational type contraction which is more general than the contraction condition:

where is a mapping from a metric space into itself.

Recently, in 2015, Almeida et al. [5] introduced an extension of condition (1) of Dass and Gupta [4] as follows:

where is defined by

and is a nondecreasing upper semicontinuous function with for all .

Another offshoot of generalizations of Banach’s theorem is based on extending the axioms of metric spaces. It is worth mentioning that the use of triangle inequality in a metric space is of extreme importance, since it implies that is continuous; each open ball is an open set, a sequence may converge to unique point, and every convergent sequence is a Cauchy sequence and other things. In 2000, Branciari [6] introduced a new concept of generalized metric space by replacing the triangle inequality of a metric space by a so-called rectangular inequality. Since then, various works have dealt with fixed point results in such spaces (see, [7–16]). It was not directly noted that such generalized metric spaces (GMS, for short) may fail to satisfy the conditions which were mentioned above in metric spaces.

In this paper, we introduce coincidence point theorems for two contraction self-mappings of rational type in generalized metric spaces. Our results improve the results of Almeida et al. [5]. These theoretical theorems are applied to the study of the existence solutions to a system of functional equations in dynamic programming.

2. Preliminaries

In this section, we present some preliminaries and notations related to rational type contraction and GMS.

Definition 1 (Branciari [6]). Suppose that be a nonempty set and be a distance function such that for all and ,

(i),(ii),(iii) (quadrilateral inequality).

Then we called GMS.

The following example shows that GMS are more general than metric spaces.

Example 2. Suppose that Define on as follows:

Then is a GMS but not metric space.

Definition 3 (see [6, 17]). Suppose that be a GMS and let be a sequence in Then

(i) converges to in GMS if ,(ii) is a Cauchy in GMS if, , such that , ,(iii) is called complete GMS if every Cauchy sequence in converges to a point in

Remark 4 (Sarma et al. [18]). Definition 1 of GMS does not ensure the following properties:

(a) is continuous on its domain.(b)A GMS is Hausdorff.(c)There is a unique limit of a convergence sequence.(d)Any convergent sequence is a Cauchy sequence.

In 2009, Samet [19] and Sarma et al. [18] introduced the following example which shows Remark 4.

Example 5 (see [18, 19]). Suppose that , where and Define from into as follows:

and if and .
Then is a complete GMS. Moreover, one can see that

(1) and is not Cauchy sequence,(2)there is no such that ; hence, GMS is not Hausdorff with the respective topology, where ,(3) is not continuous.

Lemma 6 (see [16]). Any Cauchy sequence in GMS converges to a unique point.

Definition 7 (see [17]). Let and . The mapping is -admissible if, for all such that , we have . If is the identity mapping, then is called -admissible.

Definition 8 (see [17]). Let be a GMS and is -regular if, for each sequence in such that for all and , there exists a subsequence of such that

Definition 9 (see [17]). Suppose that be two mappings from a nonempty set into itself. The mappings are said to be weakly compatible if implies A point is called point of coincidence of and if there exists a point such that

3. Main Results

In this section we introduce some coincidence point results for two rational contraction self-mappings on GMS.

Theorem 10. Let be a GMS and let and be self-mappings on such that Suppose that is a complete GMS and the following condition holds:

where is defined by

and is a continuous nondecreasing function and
Then and have a unique point of coincidence in Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. Define the sequence and in defined by

If , then is a point of coincidence of and Consequently, we can suppose that for all
Now, by (6), we have

where

We consider the following cases:

(i)If , from (9), we have

(ii)If , from (9), we obtain

Hence,

that is, (11) holds.

(iii)If , from (9), we get

which is impossible.
In any case, we proved that (11) holds. Since is decreasing sequence, it converges to a nonnegative number, If , then, letting in (9), we deduce

which implies that ; that is,

Suppose that for all and show that is GMS Cauchy sequence. First, we prove that the sequence is bounded. Since , there exists such that for all If , for all , from (6), we have

where

Hence,

Thus the sequence is decreasing and hence is bounded. Now, if

dose not hold, then there exists a subsequence of such that From

we obtain that

Now, by (6), with and , we have

which tends to as ; hence,
Now, if possible, let be not a Cauchy sequence. Then there exists such that, for , there exist for which we can find subsequences and of such that is the smallest index for which

Now, using (24) and the rectangular inequality, we get

Letting in the above inequality, using (16) and (20), we obtain

From

letting , we obtain

From (6), with and , we get

where

Now, using the continuity of as , we obtain

which implies that , a contradiction with Hence, is a GMS Cauchy sequence. Since is complete GMS, there exists such that Let be such that , applying (6) with :

where

We get from (32) that

which implies that ; that is, and so is a point of coincidence for and .
Now, we prove that is the unique point of coincidence of and Let and be arbitrary points of coincidence of and such that and Using condition (6), it follows that

which implies that Thus, and have a unique point of coincidence.
Next, we prove that If is a point of coincidence of and as and are weakly compatible, we obtain that and so Consequently, is unique common fixed point of and

Example 11. Suppose as in Example 2; let and , defined by , , and .
Then and satisfy all the conditions of Theorem 10. Hence, is unique coincidence and common fixed point of and

Corollary 12. Replace condition (6) in Theorem 10 with the following condition:

where and
Then and have a unique point of coincidence in Moreover, if and are weakly compatible, then and have a unique common fixed point.

Corollary 13. Put (the identity mapping) in Theorem 10. Then one can get a unique fixed point of

Remark 14 (see [5, Theorem  7]). It is spatial case of Theorem 10. Next, we introduce some coincidence point theorems for two -contractions self-mappings of rational type in complete GMS.

Theorem 15. Let be a GMS and let be two self-mappings satisfying the following conditions:

where is as in Theorem 10; , and is a complete GMS.
Consider also that the next conditions hold:

(i) such that (ii) is -admissible.(iii) is -regular and , for each , and (iv)Either or whenever and (v) is a lower semicontinuous function and

Then and have a unique point of coincidence in Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. Suppose that Define and as two sequences in such that . If , then , which implies that is a coincidence point of and Consequently, we can suppose that for all From (i), we get that Also, by (ii), we have that Continuous with this process, we obtain that Now, by using (37), we get

where

We consider the following cases:

(i)If , from (38), we have

Since is nondecreasing, we have

(ii)If , from (38), we obtain

The nondecreasing property of implies that

Hence, (41) is obtained.

(iii)If , by (38), we obtain

this is a contradiction.
In any case, we proved that (41) holds. Since is decreasing, it converges to a nonnegative number, If , then, letting in (37), we deduce

which implies that ; that is,

Suppose that for all and prove that is GMS Cauchy sequence. First, we show that the sequence is bounded. Since , there exists such that for all If , for all , from (37), we have

where

Hence,

Thus the sequence is decreasing and hence is bounded. If, for some , we have and , then, from (49), we get

which is a contradiction. Then is bounded. Now, if

dose not hold, then there exists a subsequence of such that From

we obtain that

Now, by (37), one can obtain that

which implies that
Now, if possible, let be not a Cauchy sequence. Then there exists for which we can find subsequences and of with such that

Further, corresponding to , we can choose in such a way that it is the smallest integer for which

Now, using (55) and (56) and the rectangular inequality, we get

Letting in the above inequality, using (46) and (51), we obtain

From

letting , we obtain

From (37), with and , we get

where

Now, using the continuity of as , we obtain

A contradiction is obtained with , and then , and hence is a GMS Cauchy sequence. Since is complete GMS, there exists such that Let be such that , applying (37) with :

where

We get from (64) that

which implies that ; that is, and so is a coincidence point for and
Now, we prove that is the unique coincidence point of and Let and be arbitrary coincidence points of and such that and Using condition (37), it follows that

which implies that Thus, and have a unique coincidence point.
As in the conclusion in the last paragraph of the proof of Theorem 10 and the weakly compatible property of and , we obtain that and have unique common fixed point.

Corollary 16. Put in Theorem 15. Then one can get a unique fixed point of

Remark 17 (see [5, Theorem  16]). It is spatial case of Theorem 15.

4. An Application in Dynamical Programming

The aim of this section is to use Theorem 10 to study the existence and uniqueness of solutions of the following system of functional equations:

which are used in dynamic programming (see [20–22]), where is a state space, is a decision space, and , , , and are considered operators.

We denote by the set of all bounded functionals on Also, we define by

Remark 18. We note that the space is Banach, where the distance function in is defined as follows:

Lemma 19 (see [5]). Let be bounded functionals; then

Proposition 20. Suppose that are three bounded functionals and

Also, let be an operator defined as follows:

For all and , is well defined.

Proof. It is enough to prove that is bounded for all From the boundedness of , and , we have

which give the boundedness of on Hence, is well defined.

Theorem 21. Consider the assumptions of Proposition 20 and the following property:

where

For all and all , all Also, the function is as in Theorem 10.
Then (68) has a unique common solution

Proof. First, we prove that the mappings in system (68) satisfy condition (6). Indeed, by using Lemma 19, we have that , and we consider

Then all the conditions of Theorem 10 are satisfied; hence, system (68) has a unique solution.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

The two authors contributed equally to this work. Both authors read and approved the final paper.

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