Weak Solutions of a Hyperbolic-Type Partial Dynamic Equation in Banach Spaces
Analysis of PDEs
2014-12-24 v1
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.
Cite
@article{arxiv.1412.7351,
title = {Weak Solutions of a Hyperbolic-Type Partial Dynamic Equation in Banach Spaces},
author = {Ahmet Yantir and Duygu Soyoglu},
journal= {arXiv preprint arXiv:1412.7351},
year = {2014}
}