On almost automorphic type solutions of abstract integral equations, a Bohr-Neugebauer type property and some applications

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Castro Alayo, Efraín Manuelito
Cayo-Colca, Ilse Silvia
Fernández-Jeri, A.B.
Idrogo-Vásquez, G.
Medina-Mendoza, Marleni
Rodriguez Pérez, Roxana Jakeline
Rojas-Ocampo, E.
Torrejón-Valqui, Llisela
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Academic Press Inc.
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In the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions, both situations are posed on Banach spaces. Also, we develop a Bohr-Neugebauer type result for the abstract integral equations. Before that, we introduce the notion of ?-bounded functions, develop the appropriate abstract theory and discuss the almost periodic situation. As applications, we study the existence of an asymptotically almost automorphic solution to integro-differential equations modeling heat conduction in materials with memory and also the existence of the almost automorphic solution to semilinear parabolic evolution equations with finite delay. © 2020 Elsevier Inc.
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Mild solutions, Abstract integral equations, Almost automorphic functions, Asymptotically almost automorphic functions, Bohr-Neugebauer property