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Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces

Ndambomve Patrice, Kpoumie Moussa El-Khalil, Ezzinbi Khalil

Journal of Applied Analysis

2023

Vol. 29, nr 1

127--142

In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces.We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and theMönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), no. 3, 536-546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.

Controllability of nonlinear implicit fractional integrodifferential systems

Balachandran K., Divya S.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 4

713--722

In this paper, we study the controllability of nonlinear fractional integrodifferential systems with implicit fractional derivative. Sufficient conditions for controllability results are obtained through the notion of the measure of noncompactness of a set and Darbo’s fixed point theorem. Examples are included to verify the result.

On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces

Yang Y., Wang J.

Opuscula Mathematica

2011

Vol. 31, no. 3

443-455

In this paper, we study a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. Existence results of mild solutions are proved for a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. The main results are obtained by using the Schaefer fixed point theorem and semigroup theory. Finally, an example is given for demonstration.

A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions

Wang J., Yan X., Zhang X. H., Wang T. M, Li X. Z.

Opuscula Mathematica

2011

Vol. 31, no. 1

119-135

In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: [formula]. Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.