Some Compatible and Weakly-Compatible Four Self-Mapping Results Approach to Nonlinear Integral Equations in Fuzzy Cone Metric Spaces

This paper is aimed at proving some unique common fixed point theorems by using the compatible and weakly-compatible four self-mappings in fuzzy cone metric (FCM) space. We prove the results under the generalized rational contraction conditions in FCM spaces with the help of one self-map are continuous. Furthermore, we prove some rational contraction results with the weaker condition of the self-mapping continuity. Ultimately, our theoretical work has been utilized to prove the existence solution of the two nonlinear integral equations. This is an illustrative application of how FCM spaces can be used in other integral type operators.


Introduction
The theory of fixed-point theory was introduced by Banach [1]. He proved a "Banach contraction principle," which is stated as follows: "A self-mapping on a complete metric space verifying the contraction condition has a unique fixed point (FP)." Later on, many researchers have been generalized this principle in many directions and proved different contractive type FP and common fixed point (CFP) for single-valued and multivalued mappings in the context of metric spaces. Chatterjea [2], Chatterjea [3], and Kannan [4] proved some single-valued contractive type FP theorems. While Ali et al. [5], Covitz and Nadler [6], Daker and Kaneko [7,8], Khan [9], and Patle et al. [10] proved multivalued contractive type FP and CFP results by using different types of spaces.
Zadeh [11], in 1965, introduced the concept of fuzzy sets. Later on, this concept was used in topology and functional analysis by many researchers. Kramosil and Michalek [12] introduced the notion of fuzzy metric FM space, and they established some basic properties. After that, George and Veeramani [13] presented the stronger form of the FM. Grabiec [14] proved two FP theorems by using the concept of complete and compact FM spaces. Gregori and Sapena [15] established some FP contraction results in the sense of [13,15]. Hadzic and Pap [16] proved a FP theorem for multivalued mappings in probabilistic metric spaces and presented applications in FM spaces. Imdad and Ali [17] and Rehman et al. [18] proved some FP theorems in complete FM spaces. Pant and Chauhan [19] established some CFP theorems by using weakly-compatible mappings in menger spaces and FM spaces. Kiyani et al. [20] and Sadeghi et al. [21] proved some results for set-valued contractive type mappings in FM spaces.
The concept of cone metric space (CMS) was proposed by many researchers but it became popular after being redis-covered by Huang and Zhang [22]. They proved the convergence properties and FP theorems for nonlinear contractive type mappings. By using the concept of Huang and Zhang [22], many authors have contributed their work to the problems on CMSs. Some of such works can be found in ( [23][24][25][26][27][28]).
In 2015, the notion of fuzzy cone metric space (FCM space) was introduced by Oner et al. [29]. They proved the key attributes of FCM space and a "fuzzy cone Banach contraction theorem for FP" in FCM space. In [30], Rehman and Li extended and improved a "fuzzy cone Banach contraction theorem" and established some generalizedcontraction results for FP in FCM spaces. Rehman et al. [31,32] proved different contractive type CFP-theorems in FCM spaces. Recently, the concept of weakly compatible self-mappings in FCM spaces was given by Jabeen et al. [33]. This paper is aimed at proving some unique CFPtheorems under the generalized rational contraction conditions in FCM spaces by using compatibility and weakcompatibility of four self-mappings. We prove our results by using the one self-map are continuous. Furthermore, we prove some results without the continuity of self-mappings with supportive examples. In addition, we present an application of two nonlinear integral equations (NIEs) for the existence of a common solution to support our main work. This paper is managed as follows: in Section 2, we present the basic preliminary concept. While in Section 3, we prove our main results for unique CFP-theorems under the generalized rational contraction conditions in FCM spaces by using compatibility and weakly-compatibility of four selfmappings. In Section 4, we present NIEs as an application to support our main work.

Preliminaries
In this section, we recall some basic definitions and lemmas.
Definition 2 (see [29]). A 3-tuple ðU, M o , * Þ is called a FCM space if C is a cone of E, U is an arbitrary set, * is a contin- Definition 3 (see [29]). Let ðU, M o , * Þ be a FCM space, ∃λ 1 ∈ U and fλ j g be any sequence in U.
If we choose the self-mappings B = A and g = h in Theorem 10, we obtain the following corollary.
Next result, we shall prove without the continuity of selfmapping, i.e., h and we replaced the completeness of U by the completeness of BðUÞ or hðUÞ.