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

Authors : Ahmet YANTİR, Duygu SOYOĞLU

Pages : 153-163

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Publication Date : 2015-02-01

Article Type : Research Paper

Abstract :In this article, we prove an existence theorem regarding the weak solutions to the hyperbolic-type partial dynamic equation $$z^{\Gamma\Delta}insert ignore into journalissuearticles values(x,y);=finsert ignore into journalissuearticles values(x,y,zinsert ignore into journalissuearticles values(x,y););\;\; x\in T_{1},\;\; y\in T_{2}$$ $$zinsert ignore into journalissuearticles values(x,0);=0,\;\; zinsert ignore into journalissuearticles values(0,y);=0$$ 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