A Variational Principle for Three-Point Boundary Value Problems with Impulse

Dianwu Yang

doi:10.1155/2014/840408

Abstract

We construct a variational functional of a class of three-point boundary value problems with impulse. Using the critical points theory, we study the existence of solutions to second-order three-point boundary value problems with impulse.

Highlights

In this paper, we study the following three-point boundary value problems with impulse: x󸀠󸀠 = f (t, x), t ≠ t1, t ∈ [0, 1], x󸀠 (0) = a11x (0) + a12x (t1) + a13x (1), (1)Δx󸀠 (t1) = a12x (0) + a22x (t1) + a23x (1), x󸀠 (1) = − a13x (0) − a23x (t1) − a33x (1), where 0 < t1 < 1, f : [0, 1] × R → R, Δx󸀠(t1) = x󸀠(t1+) − x󸀠(t1−), and x󸀠(t1+) (respectively, x󸀠(t1−)) denote the right limit of x󸀠(t) at t1.The existence of solutions for three-point boundary value problems has been investigated by many authors
In order to study problem (1), we define the functional φ on W by φ (x)
Let x be a critical point of the functional φ defined by (9). We prove this theorem in three steps

Summary

Introduction

In order to study problem (1), we define the functional φ on W by φ (x) Under the conditions (A1) and (A2), φ is continuously differentiable, weakly lower semicontinuous on W and (φ󸀠 (x) , y) = ((I − P) x, (I − P) y) The following theorem is the main conclusion of this paper. Assume that f satisfies the conditions (A1) and (A2). If x is a critical point of the functional φ defined by (9), x(t) is a solution of problem (1).

Results

Conclusion