Singular Impulsive Two-point Boundary Value Problems For Differential Equations On A Half Line

This paper discusses the two-point boundary value problems for impulsive differential equations on a half linewhere Δu|_t=t1 = u(t₁+) - u(t₁-), Δu'|_t=t1 = u'(t₁+) - u'(t₁-), u(t) is left continuous at t₁. In this paper, f(t, x, y) ∈ C((0, +∞)³, (0, +∞)) may be singular t = 0, x = 0, y = 0, and I_0,1(x,y), I_1,1(x, y) ∈ C((0, +∞), (0, +∞)) may be singular x = 0, y = 0. This paper presents the existence and multiplicity of positive solutions of (1).There are two chapters in this paper.In Chapter 1, we present the existence of positive solutions of (1).(1) is the case in [11] when n = 2, k=1.The difference from the conditions in [11] is that f(t,x,y) may be singular at t = 0, x = 0, y = 0,and I_0,1(x,y), I_1,1(x,y) may be singular at x = 0, y = 0. Moreover, even if there is no impulse, our works improve the results in [35-36] also. The method what we used is Leray-Schauder theorem by which we first obtain the approximate solutions. Then, using Arzela-Ascoli theorem, we consider the set of the approximate solutions and we obtain a convergent subsequence. The limit is a positive solution for equation (1).In Chapter 2, we establish a special cone in a special Banach space and present the existence of multiple positive solutions for (1). The difference from the conditions in [11],[27],[32-34] is that f(t,x,y) depends on y and may be suplinear in x at x = +∞. Moreover, under the influence of the impulse, (1) may have multiple positive solutions also even if f(t, x, y) may be sublinear at x = +∞. We first transfer the equation (1) into an equivalent equation. By the theory of fixed point in a cone, we get the existence of at...