STUDY SOME IMPULSIVE SECOND-ORDER BOUNDARY PROBLEMS ON BOUNDED AND UNBOUNDED DOMAINS

Abstract

In this memory, we study two second-order impulsive boundary value problems : one on a boun ded domain [0, T] and the other on an unbounded domain [0, +∞[, using variational methods and critical point theorems, such as the Mountain Pass Lemma, the Saddle Point Theorem, and the Lax-Milgram Theorem. This study is applied in Chapter 2 to establish the existence and multiplicity of solutions for the following second-order impulsive boundary value problem :    −u ′′(t) + g(t)u(t) = f(t, u(t)), a.e.t ∈ [0, T]; u(0) = u(T) = 0, ∆u ′ (tj ) = u ′ (t + j ) − u ′ (t − j ) = Ij (u(tj )), j = 1, 2, . . . , p, In the final chapter, we apply this study to obtain at least one solution for the following second order impulsive boundary value problem :    −u ′′(t) + λu(t) = f(t, u(t)), a.e.t ∈ [0, +∞), t ̸= tj , ∆u ′ (tj ) = g(tj )Ij (u(t − j )), j ∈ N∗ , u(0) = u(+∞) = 0