Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

Naveed Iqbal,Shamsullah Zaland,Azmat Ullah Khan Niazi,Ramsha Shafqat,Muhammad Gulzar

doi:10.1155/2021/5795065

Abstract

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form 0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.

Highlights

A significant range of physical processes in real-world events may be modeled using dynamical equations with fractionalorder derivatives
Fractional calculus authorizes the operations of differentiation and integration of fractional order
Agarwal et al [10] proposed the notion of the fractional differential equation in 2010

Summary

Introduction

A significant range of physical processes in real-world events may be modeled using dynamical equations with fractionalorder derivatives. In 2019, Melliani et al [20] worked on controlled fuzzy evolution equation which is given below: u′ðIÞ = αuðIÞ + PðI, uðIÞ, uðρðssÞÞÞ + AðIÞcðIÞÞ, I ∈ I, uðI0Þ = u0, ð4Þ where the intervalI = 1⁄2I0, I1is on real line and fuzzy metric space isE1. By the inspiration of the above works, we adopted the Caputo derivative to prove the uniqueness and existence for the below controlled fuzzy differential equation of fractional order c0DγIxðIÞ = αxðIÞ + PðI, xðIÞÞ + AðIÞWðIÞ, I ∈ 1⁄20, T, ð5Þ xðI0Þ = x0, ð6Þ where E1 is a fuzzy metric space and I = 1⁄20, T is an interval of real line. The purpose of this paper consists of the study of the existence of mild solution which depends on fuzzy differential equation of fractional order [23, 24].

Preliminaries

Main Results

Λγ α wðΛÞðΛγI

Examples

Conclusion