Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Qualitative and quantitative analysis for solutions to a class of Volterra-Fredholm type difference equation | Zheng Advances in Difference Equations 2011 2011 30 http content 2011 1 30 RESEARCH o Advances in Difference Equations a SpringerOpen Journal Open Access Qualitative and quantitative analysis for solutions to a class of Volterra-Fredholm type difference equation Bin Zheng Correspondence zhengbin2601@ School of Science Shandong University of Technology Zhangzhou Road 12 Zibo Shandong 255049 China Abstract In this paper we present some new discrete Volterra-Fredholm type inequalities based on which we study the qualitative and quantitative properties of solutions of a class of Volterra-Fredholm type difference equation. Some results on the boundedness uniqueness and continuous dependence on initial data of solutions are established under some suitable conditions. Mathematics Subject Classification 2010 26D15 Keywords discrete inequalities Volterra-Fredholm type difference equations qualitative analysis quantitative analysis bounded 1 Introduction In this paper we study a class of Volterra-Fredholm type difference equation with the following form TO TO TO zp m n g1 m n g2 s n zp s n s m 1 11 TOTO EE 22 Fii s t m n z s t 22 22F2i ị n m n z ị n i 1 s m 1 t n 1 12 TOTO EE E G1i s t m n z s t EE G2i ị n m n z ị n i 1 s M 1 t N 1 ị s n t TO TO ị s n t where z m n g1 m n g2 m n are R-valued functions defined on o F1i F2i i 1 2 . l2 and G1i G2i i 1 2 . l2 are R-valued functions defined on o2 X R p 1 is an odd number. Volterra-Fredholm type difference equations can be considered as the discrete analog of classical Volterra-Fredholm type integral equations which arise in the theory of parabolic boundary value problems the mathematical modeling of the spatio-temporal development of an epidemic and various physical and biological problems. For Eq. 1 if we take 11 I2 1 F21 Z h m n z Z h G21 Z h m n z Z h s 0 then Eq. 1 becomes the discrete version with infinite sum upper limit of 1 Eq. . Some concrete forms of Eq. 1 are also .
