Three Solutions for Fourth-Order Impulsive Differential Inclusions via Nonsmooth Critical Point Theory

Dongdong Gao,Jianli Li

doi:10.1155/2018/1871453

Abstract

An existence of at least three solutions for a fourth-order impulsive differential inclusion will be obtained by applying a nonsmooth version of a three-critical-point theorem. Our results generalize and improve some known results.

Highlights

− Δ (u󸀠󸀠󸀠) = I2j (u), j = 1, 2, . . . , m, u (0) = u (1) = u󸀠󸀠 (0) = u󸀠󸀠 (1) = 0; they obtained that the boundary value problem possesses at least one solution and infinitely many solutions via some existing critical point theorems
To the best of our knowledge, there are few papers which have concerned the existence of three solutions for fourth-order differential inclusion with impulsive effects so far
By Theorem 9, we can obviously deduce that (76) possesses at least three solutions

Summary

Introduction

We will consider the following fourth-order differential inclusion with impulsive effects: u(iV) (t) + Pu󸀠󸀠 (t) + Qu (t) ∈ λ∂F (u (t)) + μ∂Gu (t, u (t)) , t ∈ [0, T] , t ≠ tj, j = 1, 2, . In [10], the authors considered the fourth-order boundary value problem with impulsive effects as follows: u(iV) (t) + Au󸀠󸀠 (t) + Bu (t) = f (t, u (t)) , a.e.t ∈ [0, T] ,. M, u (0) = u (1) = u󸀠󸀠 (0) = u󸀠󸀠 (1) = 0; they obtained that the boundary value problem possesses at least one solution and infinitely many solutions via some existing critical point theorems. To the best of our knowledge, there are few papers which have concerned the existence of three solutions for fourth-order differential inclusion with impulsive effects so far.

Preliminaries

Proof of Theorem 9

An Example