Abstract

In this paper, we establish some new generalized rational type common fixed point results for compatible three self-mappings in complex-valued b-metric space, in which a one self-map is continuous. In support of our results, we present some illustrative examples to verify the validity of our main work. Moreover, we present the application of two Urysohn integral type equations (UITEs) for the existence of a common solution to support our work. The UITEs are and where , where is the set of all real-valued continuous functions defined on and .

1. Introduction

The theory of fixed point is one of the most interesting area of research in Mathematics. Initially, the concept of this theory was given by Banach [1] and he proved “a Banach contraction theorem for fixed point” which is stated as “a single-valued contractive type mapping in a complete metric space has a unique fixed point.” After the publication of this research article, many authors have contributed their ideas to the theory of fixed point in the context of metric spaces and proved different contractive type fixed point results for single-valued and multivalued mappings with different types of applications. Some fixed soft points and a-fixed soft points results can be found in [2, 3].

The concept of b-metric space was first introduced by Bakhtin [4], while Czerwik [5] proved some fixed point results for nonlinear set-valued contractive type mappings in b-metric spaces. Later on, Akkouchi [6] established some common fixed point theorems (CFP-theorems) for single-valued mappings under an implicit relation in b-metric spaces. In [7], Aghajani et al. proved CFP-results under the generalized weak contraction in partially ordered b-metric spaces. Further, Aydi et al. [8, 9] presented some FP-theorems and CFP-theorems for set-valued quasi-contraction and weak -contraction, respectively, in b-metric spaces. In 2013, Roshan et al. [10] established some generalized contractive type CFP-theorems in b-metric spaces and they proved that the b-metric function used in the theorems and results are not necessarily continuous. Some more FP-results in b-metric space can be found in [11–19]; the references are therein.

In 2011, Azam et al. [20] introduced the notion of complex-valued metric space and proved some CFP-theorems for a pair of self-mappings. Though complex-valued metric space forms a special class of cone metric space, so far this concept is proposed to define rational type expressions that are not significant in cone metric spaces, and therefore, some results of the analysis cannot be generalized to cone metric spaces. Properly the notion of complex-valued metric space was introduced by Rouzkard and Imdad [21] which generalized the expression of Azam et al. [20] and proved some CFP-theorems. Some more FP-results in the context of complex-valued metric spaces can be found in [22–24].

In 2013, Rao et al. [25] introduced the notion of complex-valued b-metric space which generalized the notion of complex-valued metric spaces given by Azam et al. [20] in 2011. They presented some CFP-results for generalized contraction conditions in complex-valued b-metric space. Later on, Mukheimer [26] extended and improved the results of [20, 25] and established some unique CFP-theorems in complex-valued metric spaces with illustrative examples.

In this paper, we establish some new generalized rational type CFP-theorems for compatible three self-mappings on complex-valued b-metric spaces in which one is a continuous self-map. Our results extend and modify many results given in the literature. This paper is organized as follows: Section 2 consists of preliminary concepts. In Section 3, we present some generalized unique CFP-theorems for compatible three self-mappings in complex-valued b-metric spaces with some illustrative examples to verify the validity of our work. In Section 4, we present an application of the two UITEs for the existence of a common solution to support our main result, while in Section 5, we discuss the conclusion.

2. Preliminaries

Consider represents a set of complex numbers and . Define as , iff and , where denotes the real part and denotes the imaginary part of a complex number. Accordingly, , if any one of the following conditions holds:

(C₁) and

(C₂) and

(C₃) and

(C₄) and

In particular, we can write if and one of (C₂), (C₃), and (C₄) is satisfied.

Remark 1 (see [26]). The properties given below hold and can be verified:
(R₁) if and
(R₂)
(R₃) and

Definition 2 (see [5]). Let be a nonempty set and let be a given real number. A function is said to be a b-metric on if it holds the following conditions:
(b_m1)
(b_m2)
(b_m3)
for all . The pair is called a b-metric space, where is a coefficient of .

Definition 3 (see [25]). Let be a nonempty set and let be a given real number. A function is said to be a complex-valued b-metric on if it holds the following conditions:
(Cb_m1) and if and only if
(Cb_m2)
(Cb_m3)
for all . The pair is called a complex-valued b-metric space, where is a coefficient of .

Example 1. Let . The mapping is defined by Then is a complex-valued b-metric space with .

Definition 4 (see [25, 26]). Let is a complex-valued b-metric space and be a sequence in and . Consider the following: (1)If there is for every and such that for all , , then is called convergent, converges to , and is a limit point of . Mathematically, it can be written as or as (2)If there is for every and such that for all , , where , then is said to be Cauchy sequence.(3)If every Cauchy sequence is convergent, then is said to be complete complex-valued b-metric space.

Lemma 5 (see [25, 26]). Let be a complex-valued b-metric space and let be a sequence in . Then, converges to as .

Lemma 6 (see [25, 26]). Let be a complex-valued b-metric space and let be a sequence in . Then, is a cauchy sequence as .
To prove the main result, we will use following lemma.

Lemma 7 (see [10]). Let be a complex-valued b-metric space. Consider and be two sequences such that , whenever is a sequence in such that for some ; then, .

Proof. Given that By triangular property of , Now by applying and using (2), we have Hence, we proved that .

Definition 8 (see [27]). Let be a complex-valued b-metric space. A pair is said to be compatible , whenever is a sequence in such that

3. Main Result

Theorem 9. Let be a complete complex-valued b-metric space and let be three self-mappings satisfying the following: for all , and with , where . If is continuous and , are compatible, then and have a unique common fixed point in .

Proof. Fix , and we define some sequences in such that Now by using (6), This implies that After simplification, we get that Now there are three possibilities: (i)If is a maximum term in , then after simplification, (10) can be written as follows:(ii)If is a maximum term in , then after simplification, (10) can be written as follows:(iii)If is a maximum term in , and by using the triangular property of complex-valued b-metric space, then after simplification, (10) can be written as follows:Let ; then, from (11), (12), and (13), for all , we have Similarly, Now from (15) and (14), and by induction, we have that, Next, we show that is a Cauchy sequence. Let and , then we have Hence, is a Cauchy sequence. Since is a complete complex-valued b-metric space, there exists such that , as or , and from (7), we have Since is a continuous self-map on , therefore As a pair is compatible, so for some sequence in and by the definition of compatibility, we have that Now from (19), (20), and by using Lemma 7, we have Next, we have to show that , so by putting and , in (6), This implies that Now applying on both sides and from (18), (19), and (21), we get that After simplification, we get that Since , hence we get that Next, we have to show that , and by the view of (6), This implies that Now again applying on both sides and by using (18) and (26), we have that This implies that . Since , hence Now, we have to show that , and by using (6), This implies that Applying on both sides and by using (18) and (26), we have that This implies that . Since , hence Now from (26), (30), and (34), we get that is a common fixed point of and , i.e., Uniqueness: assume that is an other common fixed point of and along with , i.e., Then, from (6), we have that This implies that , since . Hence, we proved that and have a unique common fixed point in .