Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

Xingyue Jian

doi:10.4236/jamp.2019.71019

Abstract

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.

Highlights

In recent years, there are few studies on boundary value problems of conformable fractional differential equations under new definitions [1] [2] [3]
We study a class of boundary value problems for conformable fractional differential equations under a new definition
Conformable fractional derivatives have good operational properties (Four Operational Rules of Derivatives, Chain Rule and Leibniz Rule), this definition can construct fractional Newton equation and Euler-Lagrange equation from fractional variational method, this is of great significance to the study of uniform or uniformly accelerated motion of particles and to the solution of Newton’s fractional-order mechanical problems [4] [5]

Summary

Introduction

There are few studies on boundary value problems of conformable fractional differential equations under new definitions [1] [2] [3]. This method has gradually become an important method for studying nonlinear differential equations [6] [7] [8] [9]. The indefinite sign of solutions of nonlinear differential equations determines that some problems (anti-periodic boundary value problems and their generalizations) cannot be studied directly by the method of upper and lower solutions for monotone iteration. Motivated by the above work, in this paper, the existence of solutions for a class of boundary value problems of conformable fractional differential equations under a new definition is proved by using the method of coupled upper and lower solution, and the range of solutions is obtained.

Preliminaries

Conclusions

Examples