EXISTENCE OF SOLUTIONS FOR NONLINEAR IMPULSIVE DYNAMIC EQUATIONS ON A TIME SCALE

Abstract

Let T be a time scale such that 0,t_{i},T∈T, i=1,2,…,n, and 0<t_{i}<t_{i+1}. Assume each t_{i} is dense. Using a fixed point theorem due to Krasnoselskii-Burton, we show that the nonlinear impulsive dynamic equation

    {<K1.1/>┊

<K1.1 ilk="MATRIX" >
y^{Δ}(t)=-a(t)h(y^{σ}(t))+f(t,y(t)), t∈(0,T],
y(0)=0,
y(t_{i}⁺)=y(t_{i}⁻)+I(t_{i},y(t_{i})), i=1,2,…,n,
</K1.1>
where y(t_{i}^{±})=lim_{t→t_{i}^{±}}y(t), and y^{Δ} is the Δ-derivative on T, has a solution.