On a New Nonlinear Sum-difference Inequality with Application

A new nonlinear sum-difference inequality in two variables which generalize some existing results and can be used as handy tools in the analysis of certain partial difference equation is discussed. An example to show boundedness of solutions of a difference value problem is also given.

Authors:

• Kelong Zheng
• Shouming Zhong

Keywords:

• Sum-Difference inequality
• Nonlinear
• Boundedness.

References:

[1] W. S. Cheung, Q. H. Ma, J. Peˇcari'c, Some discrete nonlinear inequalities
and applications to difference equations, Acta Mahtematica Scientia
2008, 28B(2) 417-430.
[2] W. S. Cheung, J. Ren, Discrete non-linear inequalities and applications
to boundary value problems, J. Math. Anal. Appl. 319 (2006) 708-724.
[3] S. Deng, Nonlinear discrete inequalities with two variables and their applications,
Appl. Math. Comput. (2010), doi:10.1016/j.amc.2010.07.022.
[4] F. C. Jiang, F. W. Meng, Explicit bounds on some new nonlinear integral
inequalities with delay, J. Comput. Appl. Math. 205(2007), 479-486.
[5] Y. H. Kim, On some new integral inequalities for functions in one and
two variables, Acta Math. Sinica, 2(2)(2005), 423-434.
[6] O. Lipovan, A retarded integral inequality and its applications, J. Math.
Anal. Appl. 285(2003), 436-443.
[7] Q. H. Ma, W. S. Cheung, Some new nonlinear difference inequaities and
their applications, Journal of Computational and Applied Mathematics
202(2007), 339-351.
[8] B. G. Pachpatte, Inequalities for Finite Difference Equations, Marcel
Dekker, New York, 2002.
[9] Y. Wu, X. Li, S. Deng, Nonlinear delay discerte inequalities and
their applications to Voliterra type difference equations, Advances in
Difference Equations, Volumn 2010, Aritcle ID 795145, 14 pages.
[10] K. Zheng, Some retarded nonlinear integral inequalities in two variables
and applications, JIPAM. J. Inequal. Pure Appl. Math. 9(2)(2008),
Article 57, 11 pp.