Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

Shuqin Zhang,Lei Hu

doi:10.3390/math7030286

Abstract

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

Highlights

In this paper, we will observe and study the unique existence of approximate solution to the following initial value problem of variable order ( p(t) q(t)D0+ x (t) = f (t, x, D0+ x ), 0 < t < +∞, (1)x (0) = 0, q(t) where 0 < q(t) < p(t) < 1, f (t, x, D0+ x ) are given real functions, and D0+, D0+ denote derivatives of variable order p(t) and q(t) defined by p(t) D0+ x (t) = D0+ x (t) d dt d = dtZ t ( t − s )− p(t)
The following example illustrates that the semigroup property of the variable order fractional integral does not holds for the piecewise constant functions p(t) and q(t) defined in the same partition of finite interval [ a, b]
F (t)|t=3, which implies that the semigroup property of the variable order fractional integral does not hold for the piecewise constant functions p(t) and q(t) defined in the same partition [0, 1]

Summary

Introduction

We will observe and study the unique existence of approximate solution to the following initial value problem of variable order Let α : [ a, b] → (n − 1, n] (n is a natural number), the left Riemann–Liouville fractional derivative of order α(t) for function x (t) are defined as the following two types α(t). I0α+ D0α+ x (t) = x (t) + ctα−1 , c ∈ R These properties play a very important role in considering the existence of the solutions of differential equations for the Riemann–Liouville fractional derivative, for details, please refer to [22,23,24,25,26]. It brings us extreme difficulties, that we cannot get these properties like Lemmas 3–5 for the variable order fractional operators (integral and derivative).

Some Preliminaries on Approximate Solution

Existence of Approximate Solution

Conclusions