Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales

Abstract

In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation*} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation*} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.

Keywords

• cauchy dynamic problem
• banach space
• measure of weak noncompactness
• weak solutions
• time scales
• fixed point