Abstract
By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.
Highlights
Let T be a time scale, that is, T is a nonempty closed subset of R
We are concerned with the existence of positive solutions for the following nonlinear first-order periodic boundary value problem on time scales: xΔ tptxσt λf t, x σ t, t ∈ J : 0, T T, t / tk, k 1, 2, . . . , m, x tk − x t−k Ik x t−k, k 1, 2, . . . , m, 1.1
The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects
Summary
Let T be a time scale, that is, T is a nonempty closed subset of R. We are concerned with the existence of positive solutions for the following nonlinear first-order periodic boundary value problem on time scales: xΔ tptxσt λf t, x σ t , t ∈ J : 0, T T, t / tk, k 1, 2, . Some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales 20–27. YΔ tk − yΔ t−k Ik y t−k , y 0 y 1 0 They proved the existence of one solution to the problem 1.3 by applying Schaefer’s fixed point theorem and the nonlinear alternative of Leray-Schauder type. The existence of positive solutions to the problem 1.1 was obtained by means of the well-known Guo-Krasnoselskii fixed point theorem. A has at least three fixed points x1, x2, x3 in Kc satisfying x1 < d, a < α x2 , x3 > d with α x3 < a
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