Interpolative fuzzy Z-contraction with its application to Fredholm non-linear integral equation

Abstract

The aim of this paper is to introduce the notion of interpolative fuzzy Z-contractive condition for a pair of self maps in a fuzzy metric space, which enlarges, unifies and generalizes the existing fuzzy contractions (by Gregori and Sapena [6]), ψ contraction (by Mihet [11]), fuzzy Z-contractions (by Shukla et al. [21]) and Tirado contraction (by Tirado [22]), which are for only one self map. Using this, we establish a unique common fixed point theorem for two self maps satisfying condition (S), which was introduced by Shukla et al. in [21] through weak compatibility. As an applications, we discuss fixed point theorem for Fredholm non-linear integral equation. The article also includes an example, which shows the validity of our results.