Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain

Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain ~ In recent years, the theory of almost automorphic functions has been developed extensively see, e.g., Bugajewski and N’guérékata 1 , Cuevas and Lizama 2 , and N’guérékata 3 and the references therein. However, literature concerning pseudo-almost automorphic functions is very new cf. 4 . It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. Recently an interesting article has appeared by Liang et al. 5 concerning the composition of pseudo- almost automorphic functions. The same authors in 6 have applied the results to obtain pseudo-almost automorphic solutions to semilinear differentail equations see also 7 . On the other hand, in article by Blot et al. 8 , the authors have obtained existence and uniqueness of pseudo almost automorphic solutions to some classes of partial evolutions equations.

In this work, we study the existence and uniqueness of almost automorphic and pseudo-almost automorphic solutions for a class of abstract differential equations described in the form

x t Ax t f t, x t , t ∈ R

where A is an unbounded linear operator, assumed to be Hille-Yosida see Definition 2.5 of negative type, having the domain D A , not necessarily dense, on some Banach space X; f : R×X0 → X is a continuous function, where X0 D A . The regularity of solutions for 1.1 in the space of pseudo-almost periodic solutions was considered in Cuevas and Pinto 9 see 10–12 . We note that pseudo-almost automorphic functions are more general and complicated than pseudo-almost periodic functions cf. 5 .

The existence of almost automorphic and pseudo-almost automorphic solutions for evolution equations with linear part dominated by a Hille-Yosida type operator constitutes an untreated topic and this fact is the main motivation of this paper.

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