Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations via Common Fixed-Point Techniques in Complex Valued Fuzzy Metric Spaces

Humaira Humaira,Muhammad Sarwar,Thabet Abdeljawad,Ivan Giorgio

doi:10.1155/2020/7042715

Abstract

The main purpose of this paper is to study the existence theorem for a common solution to a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations. In this respect, we study the fuzzy version of some essential common fixed-point results from metric spaces in the newly introduced notion of complex valued fuzzy metric spaces. Also, we provide an illustrative example to demonstrate the validity of our derived results.

Highlights

Introduction and PreliminariesZadeh initiated the concept of fuzzy sets in 1965 [1], which introduced a deep research activity leading to the improvement of attractive theory of fuzzy system
We study the existence of solution for a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations (1). is problem is equivalent to the integral equation g
I1 I2, we find that integral equation (67) has a solution in Θ, and the nonlinear implicit boundary value problems of impulsive fractional differential equations (1) have a solution

Summary

Introduction and Preliminaries

Zadeh initiated the concept of fuzzy sets in 1965 [1], which introduced a deep research activity leading to the improvement of attractive theory of fuzzy system. Us, many results cannot be extended to cone metric spaces To overcome this problem, very recently, in 2011, Azam et al [10] initiated a new setting of metric fixed-point theory which is known as complex valued metric spaces. Shukla et al [13] introduced an innovative concept of complex valued fuzzy metric spaces where they defined several associated topological features for complex valued fuzzy metric spaces They established the fuzzy version of the well-celebrated Banach contraction principle in different directions and discussed its applications. We extend fixed-point results under the general contractive condition in [25] to the setting of complex valued fuzzy metric spaces. (b) If xb ≺ w, ∀b ∈ N and limb⟶∞xb x, x ≺ w. (c) If w ≺ xb, ∀b ∈ N and limb⟶∞xb x, w ≺ x

Common Fixed-Point Results

Existence Theorem for Nonlinear Impulsive Fractional Differential Equations

Conclusion