On Continuous and Discrete Inequalities

Abstract

The inequalities furnish a very general comparison principle in studying many qualitative as well as quantitative properties of solutions of related equations. The Gronwall inequality, an example of an inequality for monotone operator k in which the exact solution w = a+kw, provides an upper bound on all solutions of u < or equal to a+ku . This inequality has been extended on various motivations and used in various contexts. The various types of these inequalities are studied in this thesis, the known results are deduced or compared as remark following the main results. These results are used to study asymptotic behaviour and oscillation of solutions of functional differential equations. The discrete analogue of the results is discussed; Several known results are improved and some applications to discrete stochastic models are described. The inequalities involving higher order derivatives are directly dealt with and estimates are obtained in terms of known functions. Some basic inequalities have been derived which are used to study the approximate solutions; Several existence, uniqueness results for hyperbolic delay differential equations has been established. An iterative scheme is provided which converges to the maximal solution of the related system which is the basis to study several properties of the solutions of the original system.