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

2019/12/30 by Chávez, Alan, Pinto, Manuel, Zavaleta, Ulices
#45G10 #46B25 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1912.13048

Abstract

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 λ-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.

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