-Hausdorff Functions and Common Fixed Points of Multivalued Operators in a -Metric Space and Their Applications

Abstract

-Hausdorff functions for are introduced, and common fixed-point theorems for a pair of multivalued operators satisfying generalized contraction conditions are proven in a -metric space. Our results are proper extensions and new variants of many contraction conditions existing in literature. In order to demonstrate applications of our result, we have proven an existence theorem for a unique common multivalued fractal of a pair of iterated multifunction systems and also an existence theorem for a common solution of a pair of Volterra-type integral equations.

1. Introduction

In the last few decades, a wide range of extensions, generalizations, and applications of the infamous Banach contraction principle came into existence. In the sequel, Bakhtin [1] initiated the idea of a -metric space followed by Czerwik [2], in which the author by weakening the triangular inequality formally defined a -metric space and proved the Banach contraction principle in a -metric space. Some examples and other details of a -metric space can be found in Kirk and Shahzad [3] whereas a wide range of generalized fixed-point theorems in a -metric space can be found in [4–7]. On the other hand, the study of a metric function on the set of closed and bounded subsets of a metric space was initiated by Pompeiu in [8] and then continued by Hausdorff [9]. Such a metric function is referred to as the Hausdorff-Pompeiu metric. Banach’s contraction principle was extended to a multivalued function in a metric space by Nadler [10] and in a -metric space by Czerwik [2] using the Hausdorff-Pompeiu metric . Further generalized results of multivalued contractions can be found in ([11–14]). Czerwik’s contraction was also generalized in many directions to name a few: -quasi-contraction [15], Hardy-Rogers contraction [16], weak quasi-contraction [17], Ciric contraction [18], etc. More results on multivalued contraction mappings in a -metric space can be found in [19–23]. Very recently, Debnath [24] proved the set-valued Meir–Keeler-type as well as Geraghty- and Edelstein-type fixed-point theorems in a -metric space whereas Altun et al. [25] and Kumar and Luambano [26] proved fixed-point results for multivalued -contraction mappings in complete metric space and partial metric space, respectively. In [27], the authors introduced the concept of -Hausdorff-Pompeiu -metric for some and proved fixed-point theorems for multivalued mappings belonging to various classes of multivalued -contractions in a -metric space. Applications of fixed-point results in dealing with solutions of nonlinear problems arising in engineering and science are an important area in present-day research. Fruitful applications of fixed-point problems in solution of various types of integral equations, fractional differential equations, and optimization problems can be found in [28–32]. Barnsley [33] introduced the idea of data interpolation using the fractal methodology of iterated function systems. Nowadays, fractal functions constitute a method of approximation of nondifferentiable mappings, providing suitable tools for the description of irregular signals (see [34–39]). The aim of this work is to prove common fixed-point theorems for a pair of multivalued mappings in a -metric space using -Hausdorff-Pompeiu -metric and thereby extend and introduce new variants of various fixed-point results for multivalued mappings existing in literature. We have provided two applications of our main results: one to prove the existence of a unique common multivalued fractal of a pair of iterated multifunction system defined on a -metric space and the second to prove the existence of a common solution of a pair of Volterra-type nonlinear integral equations.

2. Preliminaries

In this section, we provide some preliminary definitions, lemmas, and propositions required in our main results.

Definition 1 (see [1]). Let be a nonempty set and satisfy the following: (1) if and only if for all (2) for all (3)There exists a real number such that for all Then, is a -metric on and is a -metric space with coefficient .

Let be the collection of all nonempty closed and bounded subsets of a -metric space . For , define , , and . Czerwik [2] has shown that is a -metric in the set and is called the Hausdorff-Pompeiu -metric induced by . In [27], the authors introduced the function for some and showed that is a -metric for the set . They called this function the -Hausdorff-Pompeiu -metric induced by the -metric . Note that for or , is the Hausdorff-Pompeiu metric .

Proposition 2 (see [27]). For any , .

Definition 3 (see [18]). The -metric is -continuous if and only if for any and sequence in with , we have .

Proposition 4 (see [19]). For any ,

Lemma 5 (see [18]). Let be a sequence in (). If there exists such that for all , then is a Cauchy sequence.

The following lemma follows immediately from the above lemma.

Lemma 6. If for some , with , for all , then is a Cauchy sequence.

3. Main Results

We introduce pairwise -Hausdorff functions as follows:

Definition 7. Let . For any , and any , if there exist such that then we say that and are pairwise -Hausdorff functions.

For , we get the following.

Definition 8. For any and any if there exist such that then we say that is a -Hausdorff function.

Remark 9. (i)For , is always a -Hausdorff function(ii)If for any , the function is a -Hausdorff function, then for any , the function is a -Hausdorff function

Example 10. Let , and be as follows:

We will show that the functions and satisfy (2). We will consider the values of in as follows: (i). In this case, and are singleton sets and so (2) is obviously true(ii). . If , then we have and , , , and . Thus, (2) is true for all . If , then inequality (2) holds with (iii). , and the result follows in the same way as in (ii) above.(iv). . If , then we have and , , , and . Thus, (2) is true for all . If , then we take and then , , , and . Thus, (2) is true for all . If , inequality (2) holds with

Thus, and are pairwise -Hausdorff functions for . However, and are not pairwise -Hausdorff functions for , as we see that inequality (2) is not satisfied for , , and . In fact, and are not pairwise -Hausdorff functions for .

We now present our main result as follows:

Theorem 11. Let be a complete -metric space with constant , be -continuous, and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and some , , with , , and . Then, and have a common fixed point.

Proof. Let , , and . By (2), there exist , such that . By (2) again, there exist , such that
Continuing these ways, we construct the sequence such that Now, Therefore, Again, Thus, we have where .
By Lemma 6, the sequence is a Cauchy sequence. Since is complete, there exists such that the Cauchy sequence is convergent to . We will show that . By the definition of , we have

It follows that

Since we have

This implies

Again, we have

It follows that

Since we have

This implies

Now

Using (17) and (22) in the above two inequalities, we get

This gives and . Since and are closed, we have and .

Our next result provides an extension and new variants of Ciric’s quasi-contraction [15] and multivalued weak quasi-contraction [17], for a pair of multivalued mappings in a -metric space.

Theorem 12. Let be a complete -metric space with constant , be -continuous, and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all , some with and . Then, and have a common fixed point.

Proof. Proceeding as in the proof of Theorem 11, for some , , and , we construct the sequence satisfying (7). Then, we have Therefore, Again, and we get Thus, we have where

By Lemma 6, the sequence is a Cauchy sequence. Since is complete, there exists such that the Cauchy sequence is convergent to . We will show that . By the definition of , we have

It follows that

Since we have

This implies

Again, we have

It follows that

Since we have

This implies

Now,

Using (35) and (40) in the above two inequalities, we get

Since and , we get and . As and are closed, we have and .

Applying the same technique as in the proof of Theorem 12, we can prove the following extension and new variant of Ciric’s contraction for a pair of multivalued mappings in a -metric space.

Theorem 13. Let be a complete -metric space with constant , be -continuous, and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and some with . Then, and have a common fixed point.

For in Theorem 11, we get the following result:

Corollary 14. Let be a complete -metric space with constant , be -continuous, and be a multivalued -Hausdorff function for some and satisfying the following condition: for all and some ,