Impulsive dynamic equations on a time scale

Abstract

Let $mathbb{T}$ be a time scale such that $0, t_i, T in mathbb{T}$, $i = 1, 2, dots, n$, and $0 < t_i < t_{i+1}$. Assume each $t_i$ is dense. Using a fixed point theorem due to Krasnosel'skii, we show that the impulsive dynamic equation $$displaylines{ y^{Delta}(t) = -a(t)y^{sigma}(t)+ f ( t, y(t) ),quad t in (0, T],cr y(0) = 0,cr y(t_i^+) = y(t_i^-) + I (t_i, y(t_i) ), quad i = 1, 2, dots, n, }$$ where $y(t_i^pm) = lim_{t o t_i^pm} y(t)$, and $y^Delta$ is the $Delta$-derivative on $mathbb{T}$, has a solution. Under a slightly more stringent inequality we show that the solution is unique using the contraction mapping principle. Finally, with the aid of the contraction mapping principle we study the stability of the zero solution on an unbounded time scale.

Keywords

• Fixed point theory
• nonlinear dynamic equation
• stability
• impulses