Weak solutions of a hyperbolic-type partial dynamic equation in Banach spaces

Year 2015, Volume: 44 Issue: 1, 153 - 163, 01.02.2015

Ahmet Yantir , Duygu Soyoğlu

Abstract

In this article, we prove an existence theorem regarding the weak solutions to the hyperbolic-type partial dynamic equation $$z^{\Gamma\Delta}(x,y)=f(x,y,z(x,y))\;\; x\in T_{1},\;\; y\in T_{2}$$ $$z(x,0)=0,\;\; z(0,y)=0$$ in Banach spaces. For this purpose, by generalizing the definitions and
results of Cichoń 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.

Keywords

Hyperbolic partial dynamic equation, Banach space, measure of weak noncompactness, time scale

References

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