Multivariable Sum-Difference Inequalities and Equations

Abstract

Multivariable difference equations occur in numerous settings and forms in various applications. By computing the value of a unknown function recursively for a set of equidistant points from a given set of values, leads to the equation which may be viewed as sumdifference equation on the fixed region of summation, see [2,3,14,16,46,47,51,67,68,85]. The problems of existence of solutions for these equations can be dealt with the use of the well known fixed point theorems, see [51,54]. Besides the existence problems, there are many basic questions which are significant with respect to the theory itself or to applications to it. In practice it is often difficult to obtain explicitly the solutions, and thus need a new insight to handle the qualitative properties of their solutions. The method of finite difference inequalities with explicit estimates provides the most powerful and widely used analytic tool in the study of various discrete dynamic equations. It enable us to obtain valuable information about solutions without the need to know in advance the solution explicitly. In this chapter, we focus our attention to present some fundamental sum-difference inequalities with explicit estimates recently established in [108,104,98,95,102,111,116,114,106], which can be used as tools for handling the qualitative properties of solutions of various types of multivariable sum-difference equations . Furthermore, some important basic qualitative aspects related to the solutions of certain sum-difference equations investigated in [100,103,116,101,99,107,109] are also given.