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

$({\mathrm R},{\mathcal B})$-Multidimensional-almost automorphic functions with applications to integral and partial differential equations

“…• We also mention that, in [11] the authors have used the notion of bi-almost automorphy and the one of λ-boundedness in order to study the following nonlinear abstract integral equations of advanced and delayed type: Observing the previous works (and references cited therein), we emphasize that the notion of bi-almost automorphic function is crucial in the study of the almost automorphic dynamics in differential, integrodifferential and difference equations.…”

“…The concept of bi-almost automorphy was introduced in literature in [11,Definition 2.7] to study the asymptotic behavior of nonautonomous evolution equations in Banach spaces, see also [5,9] and references therein. The research study [9] by A. Chávez, S. Castillo and M. Pinto where the authors have used the notion of bi-almost automorphy of the Green functions in their investigation of almost automorphic solutions of abstract differential equations with piecewise constant arguments.…”

“…Results about existence and uniqueness of the asymptotically almost automorphic solution to the integrodifferential equation (3.14)-(3.15) have been studied in [11,20]. This subsection is an adaptation of the results on asymptotically almost automorphic solutions to integrodifferential equations in [11] to the case of asymptotically R-almost automorphic ones, and we give details for the reader convenience. Let us start with the following definition Definition 3.6.…”

“…It is well known that, (usually mild) solutions to nonautonomous evolution differential equations satisfy a concrete integral equation in which the integral kernels are expressed using two parameter evolution families {U (t, s)} t≥s≥0 , see [36]. Thanks to the bi-almost automorhy of the evolution operator {U (t, s)} t≥s≥0 we may prove the existence of almost automorphic solutions, see [9,10,11,17,46] for more details.…”

“…to mention a recent research study [11] by A. Chávez, M. Pinto and U. Zavaleta, where the authors systematically analyzed the notion of bi-almost automorphy in the study of abstract nonlinear integral equations that are simultaneously of advanced and delayed type, as well as the research studies [14] by Y.-K. Chang, S. Zheng, [27] by Z. Hu, Z. Jin, [44] by Z. Xia and [45] by Z. Xia, D. Wang.…”