Weakly compatible and quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations

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: $$\begin{aligned} &x(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &y(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned}$$ x(l)=∫01K1(l,v,x(v))dv+g(l),y(l)=∫01K2(l,v,y(v))dv+g(l), where $l\in [0,1]$l∈[0,1] and $x,y,g\in \mathbf{E}$x,y,g∈E, where E is a real Banach space and $K_{1},K_{2}:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$K1,K2:[0,1]×[0,1]×R→R.

Rational Type Fuzzy-Contraction Results in Fuzzy Metric Spaces with an Application

“…is new theory will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an integral type application in the sense of Jabeen et al [42] to prove a result for a unique solution to support our work.…”

“…Notice that ℓ is well defined and (40) has a unique solution if and only if ℓ has a unique fixed point in U. Now we have to show that eorem 1 applies to the integral operator ℓ. en, ∀μ 1 , μ * ∈ U, we have the following two cases: (44), then, from (42) and ( 43), we have…”

“…e integral operator ℓ has a unique fixed point, i.e., (40) has a solution in U. 44), then, from (42) and ( 43), we have…”

Some Novel Generalized Strong Coupled Fixed Point Findings in Cone Metric Spaces with Application to Integral Equations

“…In this manuscript, we study the more general cyclic coupled cone contraction results in complete CMS ðU, d c Þ and prove that a cyclic mapping Γ : U × U ⟶ U has a strong coupled FP theorem in ðU, d c Þ. Moreover, we present an integral-type application by using the concept of Chen et al [24] and Jabeen et al [25] to support our work. We shall present the illustrative examples and the two Urysohn integral equations (UIEs) for finding the solution of problems to hold up our consequences.…”

Coupled fixed point analysis in fuzzy cone metric spaces with an application to nonlinear integral equations

“…Oner et al [31] introduced the fuzzy cone metric space or shortly (FCM-space) and proved a fuzzy cone Banach contraction theorem for a fixed point in FCM-spaces with the assumption of Cauchy sequences. Some more topological properties, fixed point and common fixed point results can be found in, e.g., [32][33][34][35][36].…”