Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations

Abstract

This paper is aimed at proving some common fixed point theorems for mappings involving generalized rational-type fuzzy cone-contraction conditions in fuzzy cone metric spaces. Some illustrative examples are presented to support our work. Moreover, as an application, we ensure the existence of a common solution of the Fredholm integral equations: $\mu \left(\tau \right)={\int }_0^{\tau }\Gamma \left(\tau ,v,\mu \left(v\right)\right)dv$ and $\nu \left(\tau \right)={\int }_0^{\tau }\Gamma \left(\tau ,v,\nu \left(v\right)\right)dv$ , for all $\mu \in U$ , $v\in \left[0,\eta \right]$ , and $0<\eta \in ℝ$ , where $U=C\left(\left[0,\eta \right],ℝ\right)$ is the space of all $ℝ$ -valued continuous functions on the interval $\left[0,\eta \right]$ and $\Gamma :\left[0,\eta \right]\times \left[0,\eta \right]\times ℝ⟶ℝ$ .