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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.

Existence and uniqueness of solutions of the dirichlet nonlocal problem with nonlocal initial condition

Byszewski L., Winiarska T.

Czasopismo Techniczne. Nauki Podstawowe

2014

R. 111, z. 2-NP

3--7

The aim of this paper is to prove the existence and uniqueness of solutions of the Dirichlet nonlocal problem with nonlocal initial condition. The considerations are extensions of results by E. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero.

W artykule udowodniono istnienie i jednoznaczność rozwiązań nielokalnego zagadnienia Dirichleta z nielokalnym warunkiem początkowym. Rozważania są rozszerzeniami rezultatów otrzymanych przez E. Andreu-Vaillo, J. M. Mazóna, J. D. Rossi i J. J. Toledo-Melero.

On a nonlinear integrodifferential evolution inclusion with nonlocal initial conditions in Banach spaces

Yan Z.

Opuscula Mathematica

2012

Vol. 32, no. 2

377-394

In this paper, we discuss the existence results for a class of nonlinear integrodifferential evolution inclusions with nonlocal initial conditions in Banach spaces. Our results are based on a fixed point theorem for condensing maps due to Martelli and the resolvent operators combined with approximation techniques.

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.