Common Fixed Point Results for a Pair of Multivalued Mappings in Complex-Valued -Metric Spaces

Abstract

Several fixed point results for the existence of common fixed points of multivalued contractive mappings have been established in complex-valued metric space. In this paper, we study the existence of common fixed points for a pair of multivalued contractive mappings satisfying some rational inequalities in the framework of complex-valued -metric spaces. The contractive condition used in this paper generalizes many contractive conditions used by other authors in the literature. Employing our results, we check the existence solution to the Riemann-Liouville equation.

1. Introduction

Fixed point theory is a well-researched area of mathematics; in particular, results concerning fixed points of contractive type mappings are found useful for determining the existence and uniqueness of solutions of various mathematical models. In this field, Banach [1] introduced the notion of contraction mapping in a complete metric space and gave a fixed point result for finding the fixed point of the contraction mapping. Later in 1969, Kannan [2] gave another contractive type mapping that demonstrated the fixed point theorem. However, in the Kannan contraction result, the continuity property required for the result of Banach was shown to be not necessary. Other authors have also studied several contraction mappings with differing properties (see, for instance, Chaterjea [3]). Since then, the theory of fixed points has been developed regarding results on finding fixed points of self and nonself mappings which are single-valued in a metric space.

Moreover, the study of fixed points for multivalued type contractive mappings was pioneered by Nadler [4] and further studied by Markin [5]. Since then, many researchers have generalized and extended various fixed point results from single-valued contractive mappings to multivalued contractive type mappings. For more literature concerning such extensions and generalizations, see, for instance, [6–12] and other references therein.

On the other hand, the axiomatic development of metric spaces was started by M. Fréchet, a French mathematician in the year 1906. The importance of metric spaces in the natural growth of functional analysis is huge. Several authors have drawn inspirations from the impact of this natural idea to mathematics and functional analysis in particular. Therefore, there have been several generalizations of this notion in the forms of rectangular metric spaces, semimetric spaces, quasimetric spaces, quasisemimetric spaces, -metric spaces, cone metric spaces, and more recently the graphical rectangular -metric spaces. We refer the reader to the following references for surveys on these generalizations [1, 13–18].

One of these generalizations in the last decade is that of Azam et al. [19, 20]. They introduced the notion of complex-valued metric spaces, and some fixed point theorems for mappings with some rational inequalities were established. The central and core idea is to define rational expressions which are not well posed in the cone metric spaces, and thus, such results of analysis cannot be extended to cone metric spaces but to complex-valued metric spaces. Complex-valued metric spaces find interesting applications in many branches of mathematics such as algebraic geometry and number theory as well as in field of studies such as physics, thermodynamics, and electrical engineering.

Furthermore, the idea of -metric was introduced in 1989 by Bakhtin [21]. Based on this presentation, Rao et al. [22] introduced the concept of fixed point theorems on complex-valued -metric spaces which is a natural generalization of the complex-valued metric spaces. The relationship between the complex-valued -metric and cone metric space is well known. Inspired by [22], many authors have proven the existence of fixed points of different mappings satisfying rational inequalities in the framework of complex-valued -metric spaces (see [19, 23] for more details).

In 2016, Singh et al. [24] introduced a contractive type mapping satisfying some rational inequalities. They obtained the existence of common fixed point for a pair of single-valued mappings satisfying more general contraction conditions in the framework of complex-valued metric spaces. Since then, fixed point theory has been the center of extensive research for many authors (see, e.g., [25, 26]).

Our motivation in this work is in twofolds: we extend the results of [19, 24, 27] from complex-valued metric spaces to complex-valued -metric spaces; we also generalize the contraction mappings used therein by introducing a multivalued contraction mapping satisfying a general condition in complex-valued -metric spaces. Our result also improves and strengthens the results of [20, 22, 28, 29] and many other related results in the literature.

2. Preliminaries

In this section, we give some basic definitions and results which will be useful in establishing our main result. Let be the set of complex numbers and Also, we define partial order and on as follows: (i) if and only if and (ii) if and only if and

Definition 1. Letbe a nonempty set andbe a real number. A functionis called complex-valued-metric, if for allthe following conditions hold:(i) and if and only if (ii)(iii).Then, a set satisfying such metric written in pair as is called a complex-valued -metric space.

Example 2 (see [23]). Let define a mapping by for all Then, is a complex-valued -metric space with

Definition 3 (see [19]). Letbe a complex-valued-metric space, and then, a pointis(i)an interior point of a set if there exists such that (ii)the limit of a set if for every (iii) which is called an open set if every element is an interior point

Definition 4 (see [22]). Letbe a sequence in a complex-valued-metric spaceandthen(i) is the limit point of if for every with there exits such that for all we write (ii)if for every with there exists such that for all and then is a cauchy sequence in (iii) is complete if every cauchy sequence is convergent in

Lemma 5 (see [28]). Letbe a complex-valued-metric space andbe a sequence inThen,converges toif and only ifas

Lemma 6 (see [28]). Letbe a complex-valued-metric space andbe a sequence inThen,is a Cauchy sequence if and only ifas

We denote by the family of closed and bounded subsets of the set

Definition 7 (see [19]). Letbe a complex-valued-metric space andbe a sequence inDenoteandfor and For we denote

Remark 8. Letbe a complex-valued-metric space withIfthenis a metric space. Moreover, for is the Hausdorff metric induced by

Definition 9 (see [22]). Letbe a complex-valued-metric space.(i)Let be a multivalued mapping. For and defineThus, for , (ii)A mapping is said to be bounded from below if for each there exists such that for all (iii)For a multivalued mapping we say that has a lower bound property on , if for any the mapping defined by is bounded below. This implies that for , there exists an element such that for all where is the lower bound of associated with (iv)The multivalued mapping is said to have the greatest lower bound property (g.l.b. property) on if a greatest lower bound exists in for every We denote by the g.l.b. of That is,

Definition 10 (see [23]). Letbe a complex-valued-metric space andbe multivalued mappings.(i)A point is called a fixed point if (ii)A point is called a common fixed point of and if and

3. Main Result

In this section, we state and prove our main findings in the sequel.

Theorem 11. Letbe a complete complex-valued-metric space and letbe multivalued mappings with g.l.b. property. Letbe mappings such that

Then, and have a unique common fixed point.

Proof. Let be an arbitrary point in then Pick set and in (6). Then, we get This implies Since we get Thus, there exists some such that By using the g.l.b. property of and we obtain that which implies that That is, from which we get Let Clearly, then, we can inductively define a sequence such that , for , and Now, for and the fact that is complex-valued -metric space, we get That is, Letting in (21), we get This implies by Lemma 6 that is a cauchy sequence in Since is complete, there exists such that as
Next, we show that and Again, from (6), we have This implies Now, since we have From which we obtain that This implies that there exists some such that That is, By using the g.l.b. property of and we get From we have Therefore, we get

By letting in the above inequality, we get as By Lemma 5, we get that Since is closed, we have Proceeding in a similar fashion, it follows easily that Therefore, and have a common fixed point.

Corollary 12. Letbe a complete complex-valued-metric space and letbe multivalued mappings with g.l.b. property. Letbe mappings such that:

Then, and have a unique common fixed point.

Proof. The proof follows by setting , and in the proof of Theorem 11.

By setting in Theorem 11, we have the following corollary.

Corollary 13. Letbe a complete complex valued-metric space and letbe multivalued mappings with g.l.b. property. Letbe mappings such that

Then, and have a unique common fixed point.

Also, by setting in Theorem 11, we have the following consequence.

Corollary 14. Letbe a complete complex-valued-metric space and letbe multivalued mappings with g.l.b. property. Letbe mappings such that

Then, and have a unique common fixed point.

Corollary 15. Letbe a complete complex-valued-metric space and letbe multivalued mappings with g.l.b property. Letbe mappings such that

Then, and have a unique common fixed point.

By putting in our main theorem, we obtain the following corollary.

Corollary 16. Letbe a complete complex-valued-metric space and letbe multivalued mappings with g.l.b. property. Letbe mappings such that(i)(ii)(iii)

Then, has a unique fixed point in

Corollary 17. Letbe a complete complex-valued-metric space andLetbe mappings such that(i)(ii)(iii)

Then, has a unique fixed point in

Example 18. LetDefinebywhereThen,is a complete complex-valued-metric space. Consider the mappingsbe defined byObserve that by setting in Theorem 11, the contractive condition becomes trivial. Now, consider the case for which and are nonzeros, we obtain by using the definition of that Furthermore, for all and fixed Define the functions by , and Choose it is clear that for each and fixed Now, define then, Also, consider Similarly, we can show that To conclude the example, we only need to show that Indeed, for and , with and Thus, for all and for

Therefore, all conditions of Theorem 11 are satisfied and is a common fixed point of and

4. Application

4.1. Application to Riemann-Liouville Equation

In this section, we establish the existence of a solution of a Riemann-Liouville of the form where and which is the fractional integral. Let be the space of continuous function and be defined as with It is well known that is a complete complex-valued -metric space. Define by

Theorem 19. Letfor allandletbe given byand suppose thatthen Equation (43) has a solution.

Proof. It is well-known that is a fixed point of if and only if is a solution of problem (43). Note that at some point. Now, observe that for all we have that Let we get that is, Thus, we have that Clearly, all conditions of Corollary 17 are satisfied and guarantee the existence of the fixed point Thus, is the solution of the integral Equation (43).

Data Availability

No data was required for the research work.

Disclosure

Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

Conflicts of Interest

The authors declare that there is no competing interest on the paper.

Authors’ Contributions

All authors worked equally on the results and approved the final manuscript.

Acknowledgments

O.K. Oyewole acknowledges with thanks the bursary and financial support from the Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. L.O. Jolaoso is supported by the postdoctoral fellowship at the Sefako Makgatho Health Sciences University. This research is funded by the postdoctoral research fund at the Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, South Africa.