Order Nonlinear Differential Equations Existence

Impulsive differential equations arise naturally in the description of physical and biological phenomena that are subjected to instantaneous changes at some time instants called moments. For a good account on this theory.which has seen a significant development over the past decades noninear boundary value problemsfor second order impulsive differential equations have received a great deal of attention. However , very few papers have been devoted to the study of higher order impulsive differential equations. In fact, very little is known the case of boundary value problems for third order impulsive differential equations. It is our aim in this paper to present a self-contained contribution to this important area. We shall introduce some auxiliary functions that will play an fundamental role in our analysis. We provide sufficient conditions on that nonlinearity and the impulse functions that guarantee the existence of at least one solution. Our approach is based on a priori estimates, the method of upper and lower solutions combined with an iterative technique and fixed point theorem, which is not necessarilymonotone. This paper is organized as follows. The thesis is divided into three chapters according to contents.In chapter 1, we consider the following third-order boundary value problem with impulse effectswhere J = [0, 1], f∈C (J×R⁺,R⁺), I_k∈C(R⁺,R⁺), R⁺ =[0, +∞), t_k(k = 1, 2, ... , n) are fixed points with 0＜t₁＜t₂＜...＜t_k＜...＜t_n＜1,Δu'|(?)= u^''(t_k⁺) - u^''(t_k^-),where u^''(t_k⁺) and u^''(t_k^-) represent the right-hand limit and lefthandlimit of u^''(t) at t = t_k respectively.In chapter 2, we investigate the problem of existence of positive solutions for the nonlinear third order boundary value problem: By using krasnoselskii's fixed-point theorem of cone, we establish various results on the existence of positive solutions the boundary value problem. Under various assumptions on f(t, u(t), u^'(t), u^''(t)), we give the existence of the positive solutions.In chapter 3, we study mainly the periodic boundary value problems for nonlinear third order differential equations subjected to impulsive effects:where f:[0, 2Ï€]×R³â†'R is a L¹-caratheodory function, g_i, h_i and k_i are given real valued functions, and t_i∈(0,2Ï€)(i = 1,2,...,m), such that 0 = t₀＜t₁＜t₂＜...＜t_k＜...＜t_m＜t_m+1 = 2Ï€,η₁∈(t₁, t₂]. We provide sufficient conditions on the nonlinearity and the impulse functions that guarantee the existence of at least one periodic solution. Our approach is based on a priori estimates, the method of upper and lower solutions combined with an iterative technique.