Alka Chadha

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.

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.

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.

Faedo–Galerkin approximate solutions of a neutral stochastic fractional differential equation with finite delay

Abstract In this work, we study a class of neutral stochastic fractional differential equation in an arbitrary separable Hilbert space H . We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by projection of considered associated integral equation onto finite dimensional space. The existence and uniqueness of solutions to every approximate integral equation are obtained by using Banach fixed point theorems and analytic semigroup theory. We show the convergence of the solutions by using Faedo–Galerkin approximations and give an example to show the effectiveness of the main theory. Finally, we provide the conclusion at the end.

Existence of the Mild Solutions for Nonlocal Fractional Differential Equations of Sobolev Type with Iterated Deviating Arguments

This paper investigates a nonlocal differential equation of Sobolev type of fractional order with iterated deviating arguments in Banach space. The sufficient condition for providing the existence of mild solution to the nonlocal Sobolev-type fractional differential equation with iterated deviating arguments is obtained via technique of fixed-point theorems and analytic semigroup method. Finally, an example is given to explain the applicability of the abstract results developed.

Existence of a Mild Solution for Impulsive Neutral Fractional Differential Equations with Nonlocal Conditions

In the present work, we study the existence of a mild solution of a fractional-order differential equation with impulsive conditions in a Banach space X. We establish the existence and uniqueness of the mild solution by using some fixed-point theorems and resolvent semigroup theory.