Abstract
In this paper, we present some weakly compatible and quasi-contraction results for self-mappings in fuzzy cone metric spaces and prove some coincidence point and common fixed point theorems in the said space. Moreover, we use two Urysohn type integral equations to get the existence theorem for common solution to support our results. The two Urysohn type integral equations are as follows:x(l)=∫01K1(l,v,x(v))dv+g(l),y(l)=∫01K2(l,v,y(v))dv+g(l),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} &x(l)= \\int _{0}^{1}K_{1}\\bigl(l,v,x(v) \\bigr)\\,dv+g(l), \\\\ &y(l)= \\int _{0}^{1}K_{2}\\bigl(l,v,y(v) \\bigr)\\,dv+g(l), \\end{aligned}$$ \\end{document} where lin [0,1] and x,y,gin mathbf{E}, where E is a real Banach space and K_{1},K_{2}:[0,1]times [0,1]times mathbb{R}to mathbb{R}.
Highlights
1 Introduction In 2007, Huang et al [1] introduced the concept of cone metric space and proved some fixed point theorems for the underlying cone
The initial version of fuzzy set theory was given by Zadeh [12], while Kramosil et al in [13] introduced the fuzzy metric space or
In this paper we use the concept of complete FCM-spaces given by Rehman and Li [25] and prove some coincidence point and common fixed point theorems for weakly compatible three self-mappings and some quasi-contraction results in FCM-spaces
Summary
In 2007, Huang et al [1] introduced the concept of cone metric space and proved some fixed point theorems for the underlying cone. Definition 2.9 ([20]) Let (X, M, ∗) be an FCM-space, and a mapping F1 : X → X is said to be fuzzy cone contractive if ∃β ∈ (0, 1) such that (2.1). “A self-mapping F1 in a complete FCM-space in which the contractive sequences are Cauchy and hold (2.1), F1 has a unique fixed point in X” is a Banach contraction principle, which has been obtained in [20]. We note that fuzzy cone contractive sequences can be proved to be Cauchy sequences for weakly compatible self-mappings in FCM-spaces (see the proof of Theorem 3.1). In this paper we use the concept of complete FCM-spaces given by Rehman and Li [25] and prove some coincidence point and common fixed point theorems for weakly compatible three self-mappings and some quasi-contraction results in FCM-spaces.
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