A NEW EXISTENCE THEOREM FOR STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS ON A MEASURE CHAIN

Abstract

In this paper we consider the following differential equation on a measurechain $T$\[u^{\Delta \Delta }(t)+f(u(\sigma (t)))=0,t\in [a,b]\cap T,\]satisfying Sturm-Liouville boundary value conditions\begin{eqnarray*}\alpha u(a)-\beta u^\Delta (a) &=&0, \\[12pt]\gamma u(\sigma (b))+\delta u^\Delta (\sigma (b)) &=&0.\end{eqnarray*}An existence result is obtained by using a Fixed Point Theorem due toKrasnoselskii and Zabreiko. Our conditions imposed on $f$ are very easy toverify and our result is even new for the special cases of differentialequations and difference equations, as well as in the general time scalesetting.