#### 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. **Let**be a nonempty set and**be a real number. A function**is called complex-valued**-metric, if for all**the 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]). **Let**be a complex-valued**-metric space, and then, a point**is*(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]). **Let**be a sequence in a complex-valued**-metric space**and**then*(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]). *Let**be a complex-valued**-metric space and**be a sequence in**Then,**converges to**if and only if**as*

Lemma 6 (see [28]). *Let**be a complex-valued**-metric space and**be a sequence in**Then,**is a Cauchy sequence if and only if**as*

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

*Definition 7 (see [19]). **Let**be a complex-valued**-metric space and**be a sequence in**Denote**and*for and For we denote

*Remark 8. **Let**be a complex-valued**-metric space with**If**then**is a metric space. Moreover, for* *is the Hausdorff metric induced by*

*Definition 9 (see [22]). **Let**be 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]). **Let**be a complex-valued**-metric space and**be 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. *Let**be a complete complex-valued**-metric space and let**be multivalued mappings with g.l.b. property. Let**be 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. *Let**be a complete complex-valued**-metric space and let**be multivalued mappings with g.l.b. property. Let**be 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. *Let**be a complete complex valued**-metric space and let**be multivalued mappings with g.l.b. property. Let**be mappings such that*

Then, and have a unique common fixed point.

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

Corollary 14. *Let**be a complete complex-valued**-metric space and let**be multivalued mappings with g.l.b. property. Let**be mappings such that*

Then, and have a unique common fixed point.

Corollary 15. *Let**be a complete complex-valued**-metric space and let**be multivalued mappings with g.l.b property. Let**be 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. **Let**Define**by**where**Then,**is a complete complex-valued**-metric space. Consider the mappings*