Existence of positive bounded solutions of system of three dynamic equations with neutral term on time scales

Urszula Ostaszewska, Ewa Schmeidel, Małgorzata Zdanowicz

Abstract


In this paper the system of three dynamic equations with neutral term
in the following form

$$

\left\{
\begin{array}{l}
( x (t) + p(t)  x (u_1 (t) )  )^\delta
= a(t)  f (y(u_2(t)))

\\
y^\delta (t) 

= b(t) g (z( u_3(t) )  )

\\
z^\delta(t)

= c(t) h( x( u_4(t) )  )

\right.

\end{array}

$$
on time scales is considered. The aim of this paper is to present sucient
conditions for the existence of positive bounded positive solutions of the
considered system for 0 < p(t) const < 1. The main tool of the proof
of presented here result is Krasnoselskii's xed point theorem. Also, the
useful generalization of the Arzela-Ascoli theorem on times scales to the
three dimensional case is proved.


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