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

TL;DR

This paper establishes the existence of weak solutions for a hyperbolic-type partial dynamic equation in Banach spaces by extending concepts of weak derivatives and integrability on time scales, using fixed point theorems.

Contribution

It generalizes weak derivatives and integrability on time scales to prove an existence theorem for hyperbolic partial dynamic equations in Banach spaces.

Findings

Proved existence of weak solutions in Banach spaces.

Extended definitions of weak derivatives and double integrability on time scales.

Applied fixed point theorem to establish main results.

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.