Abstract
We deal with the existence of positive solutions to impulsive second-order differential equations subject to some boundary conditions on the semi-infinity interval.
Highlights
In recent years, impulsive differential equations have become a very active area of research and we refer the reader to the monographs [8] and the articles [6, 9, 10, 14, 15], where properties of their solutions are studied and extensive bibliographies are given
In this paper we study the existence of positive solutions for the following boundary value problem (BVP) with impulses: y + g(t, y, y ) = 0, 0 < t < ∞, t = tk, Δy tk = bk y tk, Δy tk = ak y tk, k = 1, 2, . . . , y(0) = 0, y bounded on [0,∞), (1.1)
As far as we know the publication on solvability of infinity interval problems with impulses is fewer [15]
Summary
Impulsive differential equations have become a very active area of research and we refer the reader to the monographs [8] and the articles [6, 9, 10, 14, 15], where properties of their solutions are studied and extensive bibliographies are given. In this paper we study the existence of positive solutions for the following boundary value problem (BVP) with impulses:. By a positive solution of BVP (1.1), one means a function y(t) satisfying the following conditions:. Let g : [0, ∞) × [0, ∈ b = 0, L−1 exist and is continuous. Schauder’s fixed point theorem guarantees the existence of at least a fixed point since L−1Ni is continuous and compact. Let n be a positive integer and consider the boundary value problem y + g(t, y, y ) = 0, 0 < t < n, t = tk, Δy tk = bk y tk , Δy tk = ak y tk , y(0) = 0, y (n) = 0. We apply the Leray-Schauder continuation theorem to obtain the existence of a solution for y = (Ln)−1F y.
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