Some Studies on the Existence Theorems for Differential and Integral Equations involving Caratheodory Nonlinearities

Abstract

It is well known that most of the natural and physical phenomena in the universe are not straight forward, since there is nonlinear nature of phenomenon in the general area of Biological sciences, Physical sciences and Engineering. The mathematical description of such phenomena results, many processes in Biological sciences, Physical sciences and Engineering. These are not continuous and involve jumps or discontinuous terms. Such effect of jumps of the environment on the productivity of animals, body growth, wool growth, milk production, semen production, female reproduction etc. The effect of jumps, heat stress on buffaloes as the increased water loss, increased basal metabolite etc. (see [56,71]). The effects of discontinuity or jumps on temperature, water, humidity, air, and soil nutrients as the plant characters (see [7, 70]). Therefore some of those types of problems may be formulated as nonlinear differential and integral equations involving discontinuous terms. Therefore it is of interest to study the nonlinear differential and integral equations and its applications to day to day problems of life. Integer order ordinary nonlinear differential and integral equations have attracted the attention of several mathematicians of the world since long time and much have been done concerning the various aspects of the solutions for nonlinear differential and integral equations (see [10,11,12,14,16,17]). newlineThe origin of nonlinear integral equation in Banach Algebras lies in the works of famous physicist Chandrasekhar (1980) in his studies on radiative heat transfer in the subject of thermodynamics which gave birth to the well-known Chandrasekhar H-equation in thermodynamics. newlineThe method developed for proving existence the solution to the nonlinear differential and integral equations is very much cumbersome and involves several technicalities. Therefore there was a need to establish a general tool for solving nonlinear differential and integral equations, since the various differential and integral equations of integer