Abstract
We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.
Highlights
Introduction and preliminariesMany evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly
It is known that many biological phenomena involving threshold, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems do exhibit impulse effects
We mainly study the following second-order impulsive differential equations with antiperiodic boundary value conditions: u f t, u, u, t ∈ 0, T \ Ω, u tk u tk Ik u tk, u tk u tk Jk u tk, k 1, 2, . . . , m, 1.1 u 0 −u T, u 0 −u T, Boundary Value Problems where Ω : m i1 ti and f
Summary
Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. We mainly study the following second-order impulsive differential equations with antiperiodic boundary value conditions:. We should mention the work by Cabada et al in 17 which is concerned with a certain nth order linear differential equation with constant impulses at fixed times and nonhomogeneous periodic boundary conditions. To the best of our knowledge, this is the first work to deal with the antiperiodic solutions to second-order differential equations with nonconstant impulses. Our method to prove the existence of antiperiodic solutions is based on the works in 13, 18, 19. We introduce and denote limt → t−k f t, x, y exist the Banach space P C and 0, T f t−k, x, y , Rn by. The following fixed point theorem is our main tool to prove the existence of at least one solution to 1.1. Either i the operator equation x λAx has a solution for λ 1, or ii the set S : {x ∈ X, x λAx, λ ∈ 0, 1 } is unbounded
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