Behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay |
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Authors: | Mostafa Adimy Khalil Ezzinbi Aziz Ouhinou |
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Affiliation: | 1. Université de Pau, Laboratoire de Mathématiques Appliquées, CNRS UMR 5142, Avenue de l’Université, 64000 Pau, France;2. Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. 2390, Marrakesh, Morocco |
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Abstract: | The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic stationary solutions for partial functional differential equations with infinite delay. We suppose that the linear part satisfies the Hille–Yosida condition on a Banach space and it is not necessarily densely defined. Firstly, we establish a new variation of constants formula for the nonhomogeneous linear equations. Secondly, we use this formula and the spectral decomposition of the phase space to show the existence of stable and unstable manifolds. The estimations of solutions on these manifolds are obtained. For illustration, we propose to study the stability of stationary solutions for the Lotka–Volterra model with diffusion. |
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Keywords: | Semigroup Hille&ndash Yosida condition Integral solution Variation of constants formula Hyperbolic stationary solution Stable and unstable manifolds |
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