On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

Mikhail Kamenskii,Jen-Chih Yao,Paul Raynaud De Fitte,Garik Petrosyan

doi:10.3390/math10020219

Abstract

In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

Highlights

Fractional calculus and the theory of fractional differential equations have gained significant popularity and importance over the past three decades, mainly due to demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering
If the differential equation is linear with respect to the desired function and the boundary conditions are linear and homogeneous, a unified approach based on the introduction and study of the properties of the so-called operator of the boundary value problem can be applied to such problems
We studied the periodic boundary value problem for fractional quasilinear differential equations in a Hilbert space

Summary

Introduction

Fractional calculus and the theory of fractional differential equations have gained significant popularity and importance over the past three decades, mainly due to demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering (see monographs [1,2], papers [3,4,5,6,7,8,9]). This branch of mathematics really provides many useful tools for various problems related to special functions of mathematical physics, as well as their extensions and generalizations for one or more variables (see e.g., paper [10]). For the quasilinear case with a fractional derivative of order q ∈ (1, 2), this kind of problems were not study until now

Preliminaries

Results

Example

Conclusions