Journal of Mathematics
Volume 2016 (2016), Article ID 5190718, 8 pages
http://dx.doi.org/10.1155/2016/5190718

Research Article

Multivalued Fixed Point Theorems for Generalized Contractions and Their Applications

Muhammad Shoaib and Muhammad Sarwar

Department of Mathematics, University of Malakand, Chakdara, Dir (Lower), Khyber Pakhtunkhwa 18800, Pakistan

Received 31 July 2016; Accepted 14 September 2016

Academic Editor: Nan-Jing Huang

Copyright © 2016 Muhammad Shoaib and Muhammad Sarwar. 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 give common hybrid fixed point results for generalized weak contraction satisfying and properties in the framework of metric spaces. An application to functional equations is also discussed.

1. Introduction and Preliminaries

Metric fixed point theory provides one of the best important and useful techniques for the existence of fixed point, coupled fixed point, coincidence point, and common fixed point for self-map under different situations. It is applicable for the solution of fractional differential equations, matrix equations, integral equations, functional equations, linear inequalities, or integrodifferential equations (see [1, 2]). In this area Banach [3] has sorted out fruitful and well known result; such result was later on called Banach contraction principle.

Banach contraction principle generalizes in numerous spaces [4, 5]; mainly, in 1969, Nadler [6] further modified and elaborated the Banach contraction principle to set-valued mapping with the Hausdorff metric. He proved the following theorem.

Theorem 1 (see [6]). Map on complete metric space which holds the condition

Here, , are the collection of bounded closed and nonempty subset of . Then has a fixed point.

By this virtue fixed point theory has been usefully applicable to various disciplines to solve integral inclusion and functional equations and so forth.

Kutbi and Sintunavarat [7] investigated fixed point theorems for generalized -contraction multivalued mappings in -complete metric spaces. Shen and Hong [8] demonstrated common fixed point results using generalized contractive multivalued operators in complete metric space.

In Hilbert space, Alber and Guerre-Delabriere [4] presented weak contraction by generalizing contraction further and showed the presence of fixed points for a self-map. Rhoades [9] proved this results in metric space under -weak contraction. Dutta and Choudhury [10] generalized -weak contraction to the concept of weak contraction and examined results for fixed point. In [11] new fixed point results for -contractive multivalued mappings on -complete metric spaces and their consequences are studied. Zhang and Song [12] described weak contraction under two self-maps. The result proved by Zhang and Song [12] generalized by Đorić [13] for the presence of common fixed point under weak contraction.

Jungck [14] originated the clue of commutativity of mappings. The idea of commutativity of mappings generalized by the same researcher to compatible mappings [15], later to weakly compatible mappings [16]. Lately, Al-Thagafi and Shahzad considered the notion of occasionally weakly compatible (owc) self-mappings [17] which generalized nontrivial weakly compatible self-maps.

Theory of hybrid fixed point (single-valued and multivalued maps) is a new region in the field of multivalued contractive theory; see [18–20]. Particularly, Aliouche and Popa [21] demonstrated some results for two maps for occasionally weakly compatible hybrid mappings in symmetric space and discuss their application.

In 1989, Kaneko and Sessa [22] extended the notion of compatible mapping to hybrid mapping. Subsequently Jungck [15] presented the notion of weakly compatible for hybrid (single- and multivalued mapping). In 2007, Abbas and Rhoades [23] defined owc property for hybrid mapping.

The concept of -property was introduced by Aamri and El Moutawakil [24]. Sintunavarat and Kumam [25] proved that for the existence of common fixed point, -property always required the completeness of the subspace and gave the idea of common limit in the range property (CLR) to overcome the mention drawback. Abdou [26] generalized this property to one pair of hybrid (multivalued) maps and then the same author in [27] extended the CLR property to two pairs of hybrid mapping. In [28] common fixed point theorems in fuzzy metric spaces with CLRg property are studied.

In the current work we derived some common hybrid fixed point results for weak contraction in metric space. Throughout the paper , , and stand for the set of all nonnegative real numbers, the set of positive integers, and the set of nonnegative integers, respectively.

Definition 2. Suppose is nonempty set and let be a function satisfying the following conditions:

(1) implies that .(2).(3), where .(4).

Then is a metric on and the pair is called metric space.

Definition 3 (see [23]). Maps , are said to be occasionally weakly compatible (shortly, (owc)-property) if for some with .

Lemma 4 (see [29]). Let be a set and be mappings. If and are bounded, then

Definition 5 (see [15]). Maps , on metric space are said to be weakly compatible if whenever .

Definition 6 (see [25]). Maps are said to satisfy the common limit in the range of with respect to (shortly, the -property with respect to ) if there exists a sequence in such that, for some , .

The following definitions can be found in [27].

Definition 7. Mappings , on metric space are said to satisfy the common limit in the range of with respect to (shortly, the -property with respect to ) if there exists a sequence in and such that, for some , .

Definition 8. Mappings and on metric space are said to satisfy the common limit in the range of with respect to (shortly, the -property with respect to ) if there exist sequences and in and such that, for some ,   , , and .

Now, we give some definitions for multivalued mappings defined in a metric space . Define the function for by

where

where

The following can be deduced from the definition of :

Lemma 9. Let be a metric space. For any and , we have the following:

(1), .(2).(3), .(4).(5).(6).(7)

Lemma 10 (see [6]). Assume a metric space Furthermore, ; then for every and for each there exist such that .

In [6] it is shown that the above lemma is also true for .

Lemma 11. Assume a metric space . Furthermore, ; then for every and for each there exist such that .

2. Main Results

In our main results we used the following two classes:

Theorem 12. Let on metric space . Furthermore assume that and have -property and

where

Here, , and , there exists unique common fixed point of , , , and .

Proof. Since, , pairs hold -property, therefore, there exist such that

which implies that and .
Now, we prove that . Suppose ; then, from (8), we have

where

and we have

If

then, (11) becomes

But is nondecreasing, so Since and , therefore, we have

which gives us contradiction. Hence
If

then, (11) becomes

Since, is nondecreasing and and , therefore, one can get

which is again not possible. Hence Similarly it is easy to show in the case if

then .
Next, we have to show that is a fixed point of . Assume that . Then, by using (8), we have

Here,

If

then, (21) implies that

If

then, (21) implies that

Similarly it gives us contradiction if

which gives us contradiction in all three cases. Hence . Similarly, we can prove . Thus we have

Therefore, is a common fixed point of , , , and . Furthermore, by using condition (8), we have

where

which implies that .
Next, to show that fixed is unique, suppose there exists another fixed other than ; then, from condition (8), we have

where

By simple calculation we conclude that the common fixed point is unique.

Theorem 13. Let on metric space . Furthermore, assume and satisfy the (CLR f)-property and

where

, , and . Further assume the following conditions also hold:

(i), have a coincidence point.(ii), have a coincidence point.(iii)If , are weakly compatible at and, moreover, , then, , have a common fixed point.(iv)If , are weakly compatible at and, moreover, , then, , have a common fixed point.(v)If both conditions (iii) and (iv) hold, then, there exist common fixed point of , , , and .

Proof. Since and satisfy the -property, there exists two sequences and such that

and, by closeness of and , we have

Now, we claim that . To Justify the said claim, using condition (33) with and , we have

where

By taking limit, we get

where

Equation (39) becomes

which implies that

Since, , by definition of Hausdorff metric

Hence we have .
In the other side using condition (33) with and , we have

here,

By taking limit, we get

where

condition (33) becomes

which implies

from this, we have

By definition of Hausdorff metric

Thus, maps , have a coincidence point and the maps , have a coincidence point . Furthermore, by virtue of condition we obtain and . So, ; this proved . A similar argument proves ; that is, . From , we deduced .

3. Applications to System of Functional Equations

In this section, we discuss common solution for four functional equations given below with the help of Theorems 12 and 13. Throughout this unit signifies inf or sup, and stand for Banach spaces, the state space is , the decision space is , and the space of all bounded real-valued functions on is which is Banach space. Define , by

Here,

and , , .

Consider the system

Here and denote the state vectors and decision vectors, respectively, , , , and signify the transformations of the process, and , , , and symbolized the optimal return functions under the initial state .

Let be defined by

where .

Theorem 14. Assume are maps given by (55) which hold the following conditions:

(1) , and , for , are bounded.(2) and pairs share (CLR) property.(3) and pairs are weakly compatible; further, and .(4) ,

 here

Then, (54) system equations have a common solution in .

Proof. Since , , and , for , are bounded, there exists such that

Thus by Lemma 4, (55), and (58) , , and are self-mappings in .
Let . Suppose that . Then using (55) we can find such that

where .
Next, with help of (59) and (62), we have

Similarly, using (60) and (61), we have

So we can write

where , , , and .
Similarly, one can obtain (66), if . Now, with the help of (66), we have

where
Since is arbitrary one can get

By taking , and in Theorem 13 we get (68). Hence, we deduce that the mappings , , , and have a common fixed point in ; that is, system (54) has a solution.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors read and approved the final version.

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