Variational Methods for an Impulsive Fractional Differential Equations with Derivative Term

Yulin Zhao,Jiafa Xu,Haibo Chen

doi:10.3390/math7100880

Abstract

This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main results.

Highlights

In this paper we are interested in the solvability of solutions for the following impulsive fractional differential equations with derivative dependence
Α −1 c α tl +1 = T, the operator ∆ is defined as ∆(t DTα−1 (0c Dtα u)(t j )) = t DTα−1 (0c Dtα u)(t+
We studied a class of impulsive fractional boundary value problems with nonlinear derivative dependence

Summary

Introduction and Main Results

In this paper we are interested in the solvability of solutions for the following impulsive fractional differential equations with derivative dependence αcαcα t DT ( a(t)0 Dt u(t)) + b(t)u(t) = f (t, u(t), 0 Dt u(t)), t 6= t j , a.e t ∈ [0, T ]. Zhang [21] by establishing a variational structure and applying Mountain Pass theorem and iterative technique, investigated the solvability of solutions to the following nonlinear fractional differential equations In case α ∈ ( 12 , 1], Galewski and Molica Bisci in [22] by using variational methods, proved that the following fractional boundary problems α −1 c α d (0 Dt u(t)) − t DTα−1 (ct DTα u(t)). The existence of at least one nontrivial solution to the problem (1)

Preliminaries

Proof of Theorems 1–3

Conclusions