On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

Yanyuan Xing,Fang Liu,Feng Jiao

doi:10.1155/2020/8679465

Yanyuan Xing, Fang Liu + Show 1 more

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https://doi.org/10.1155/2020/8679465

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Abstract

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray–Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.

Highlights

Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes
Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, polymer rheology, aerodynamics, capacitor theory, chemistry, biology, control theory, and electrodynamics of complex medium. e initial and boundary value problems for nonlinear fractional differential equations arise from the study of models of viscoelasticity, electrochemistry, porous media, and electromagnetics
Boundary value problems of fractional differential equations have been investigated for many years

Summary

Introduction

Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Mathematical Problems in Engineering where CDαt and CDβt denote the Caputo fractional derivative of order α, β; f: [0, T] × R ⟶ R is a given continuous function; and T is a fixed positive constant. In [24], the author discussed the nonlinear fractional differential equations with nonseparated type integral boundary conditions. By applying the Leray–Schauder degree theory and some standard fixed point theorems, some new existence and uniqueness results are obtained. Motivated by the abovementioned papers and many known results, in this paper, we concentrate on the existence and uniqueness of solutions for the nonlinear fractional integro-differential equations and inclusions of order α ∈ To the best of our knowledge, no paper has considered the generalization of nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions (3).

Preliminaries and Lemmas

Main Results

Examples