Sobolev-Type Volterra-Fredholm Functional Integrodifferential Equations in Banach Spaces

Abstract

This paper deals with the problems such as the existence, uniqueness, continuous dependence and boundedness of mild solution of Volterra-Fredholm functional integrodifferential equations of sobolev type in Banach spaces. Our analysis is based on Banach fixed point principle, the integral inequality established by B. G. Pachpatte, Gronwall-Bellman inequality and the semigroup theory.

A Study on Stochastic Differential Equation Using Fractional Power of Operator in the Semigroup Theory

“…for 0 1 β α < < ≤ and the imbedding is compact whenever the resolvent operator of A is compact and this can be seen in [28].…”

On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay

“…A way to deal with determination this issue is to use integro-differential equations. Detailed subtleties on theoretical results related to integro-differential systems, one can view [1,2,6,9,10,18,25,28,37,39,40].…”

“…Very useful discussion about neutral systems involving in differential equations, one can refer [10,13,14,17,27,29,35,38]. Differential systems of Sobolev-type appear commonly in mathematical forms of much physical development, for example, in 272 V. VIJAYAKUMAR, R. UDHAYAKUMAR AND K. KAVITHA the fluid flow through fissured rocks, thermodynamics, shear in second-order fluids, etc., one can check [1,2,6,18,23,27,29,41].…”

Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces

“…Sobolev type differential equations occur naturally in the mathematical modeling of many physical phenomena such as thermodynamics, fluid flow through fissured rocks, shear in second order fluids, Kelvin-Voigt model for the non-Newtonian fluid flows, etc (see for example, [1,6,13,32,35,39,41,50] etc). Recently, some works are reported on the approximate controllability of Sobolev type differential equations via fixed point methods.…”