Dwijendra N. Pandey

Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach space . The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.

Approximate controllability of semilinear system with state delay using sequence method

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear system with state delay using sequence method. Instead of a C0-semigroup associated with the mild solution of the system, we use the so-called fundamental solution. Controllability results are obtained by sequential approach and operator semigroup theory. At the end, an example is given to illustrate the theory.

Existence and approximation of solution to neutral fractional differential equation with nonlocal conditions

This paper is concerned with the approximation of the solution for neutral fractional differential equation with nonlocal conditions in an arbitrary separable Hilbert space H . We study an associated integral equation and then, consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The sufficient condition for the existence and uniqueness of solutions to every approximate integral equation is derived by using analytic semigroup and Banach fixed point theorem. We demonstrate convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Moreover, we consider the Faedo-Galerkin approximations of the solution and demonstrate some convergence results. An example is also provided to illustrate the discussed abstract theory.

Approximate Controllability of Second-Order Semilinear Control System

This paper deals with the approximate controllability of second-order semilinear control system in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The control function for this system is suitably constructed, and the sufficient conditions for the approximate controllability of the proposed problem in Hilbert spaces are established. The results are obtained when the nonlinearity satisfying the monotone condition and integral contractor condition. Finally, an example is provided to illustrate the application of the obtained results.

Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay

Abstract This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanams novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.

Exact Controllability of an Impulsive Semilinear System with Deviated Argument in a Banach Space

A functional differential equation with deviated argument coupled with impulsive conditions is studied for the existence and uniqueness of the mild solution and exact controllability of the system. The results are obtained by using Banach contraction principle and semigroup theory without imposing additional assumptions such as analyticity and compactness conditions on the generated semigroup and the nonlinear term. An example is provided to illustrate the presented theory.

Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.

Periodic BVP for a class of nonlinear differential equation with a deviated argument and integrable impulses

AlkaChadhaand Dwijendra N PandeyDepartment of Mathematics,Indian Institute of Technology Roorkee,Roorkee-247667alkachaddha03@gmail.com, dwij.iitk@gmail.comABSTRACTThis paper deals with periodic BVP for integer/fractional order differential equationswith a deviated argument and integrable impulses in arbitrary Banach space X forwhich the impulses are not instantaneous. By utilizing fixed point theorems, we firstlyestablish the existence and uniqueness of the mild solution for the integer order dif-ferential system and secondly obtain the existence results for the mild solution to thefractional order differential system. Also at the end, we present some examples to showthe effectiveness of the discussed abstract theory.RESUMENEste art´iculo estudia las ecuaciones diferenciales de orden entero/fraccional con condi-ciones de frontera peri´odicas con un argumento desviado e impulsos integrables enespacios de Banach arbitrarios X donde los pulsos no son instanta´neos. Utilizando teo-remas de punto fijo, establecemos la existencia y unicidad de soluciones temperadaspara los sistemas diferenciales de orden entero, y luego obtenemos resultados de exis-tencia para soluciones temperadas del sistema diferencial de orden fraccional. Adem´as,presentamos un ejemplo para mostrar la efectividad de la teor´ia abstracta discutida.Keywords and Phrases: Deviating arguments, Fixed point theorem, Impulsive differential equa-tion, Periodic BVP, Fractional calculus.2010 AMS Mathematics Subject Classification: 34G20, 34K37, 34K45, 35R12, 45J05.

Approximations of solutions for a nonlinear differential equation with a deviating argument

In this paper, we prove the existence and convergence of approximate solution for a class of nonlinear differential equations with a deviated argument in a Hilbert space. We establish the existence and uniqueness of a solution to every approximate integral equation using the fixed point argument. Then, we prove the convergence of a solution of the approximate integral equation to the solution of the associated integral equation. We also consider the Faedo-Galerkin approximation of a solution and prove some convergence results.

Approximations of Solutions of a Class of Neutral Differential Equations with a Deviated Argument

In this paper, we study the approximations of solutions to a class of nonlinear neutral differential equations with a deviated argument in a Hilbert space. We consider an associated integral equation corresponding to the given problem and a sequence of approximate integral equations. We establish the existence and uniqueness of solutions to every approximate integral equation using the fixed point theory. Then, we prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of solutions and prove some convergence results.