Weak Solutions of a Hyperbolic-Type Partial Dynamic Equation in Banach Spaces

Abstract

In this article, we prove an existence theorem regarding the weak solutions to the hyperbolic-type partial dynamic equation \begin{equation*}\begin{array}{l} z^{\Gamma\Delta}(x,y)=f(x, y, z(x, y)), z(x, 0)=0, \ \ \ z(0, y)=0 \end{array}, \ \ x\in\mathbb{T}_1, \ \ y\in \mathbb{T}_2\end{equation*} in Banach spaces. For this purpose, by generalizing the definitions and results of Cicho\'n \emph{et.al.} we develop weak partial derivatives, double integrability and the mean value results for double integrals on time scales. DeBlasi measure of weak noncompactness and Kubiaczyk's fixed point theorem for the weakly sequentially continuous mappings are the essential tools to prove the main result.