Mixed Integral Equations and Inequalities

Abstract

In the study of many basic models in parabolic differential equations which describe diffusion or heat transfer phenomena and epidemiology, the integral equation of the form are given functions and u is the unknown function, occur in a natural way, see [5,8,28,31,32,37-39,69,134 ]. The integral equation (3.1.1) appears to be Volterra-type in t, and of Fredholm-type with respect to x and hence it can be viewed as a mixed Volterra-Fredholm-type integral equation. In the general case solving integral equation (3.1.1) is highly nontrivial problem and handling the study of its qualitative properties need a fresh outlook. The method of integral inequalities with explicit estimates serve as an important tool which provides valuable information of various dynamic equations, without the need to know in advance the solutions explicitly. Recently in [99,103,105,107,116,119] explicit estimates on a number of new integral inequalities are considered and used in various applications. In the present chapter, we offer some fundamental mixed integral inequalities with explicit estimates established in the above noted papers and also focus our attention on some basic qualitative aspects of solutions of equations of the form (3.1.1). A particular feature of our approach here is that it is elementary and provide some basic results for future advanced studies in the field.