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Almost automorphic solutions of second order evolution equations. (English) Zbl 1085.34045
Summary: This article is concerned with the existence of almost automorphic mild solutions to second-order evolution equations of the form $\ddot u(t)= Au(t)+ f(t),\tag{$$*$$}$ where $$A$$ generates a strongly continuous semigroup and $$f$$ is an almost automorphic function. Using the notion of uniform spectrum of a function and the method of sums of commuting operators in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to $$(*)$$ in terms of spectrum of $$A$$ and uniform spectrum of $$f$$. Moreover, we study the nonlinear perturbation of this equation and obtain an extension of results by T. Diagana, the second author [Far East J. Math. Sci. (FJMS) 8, 313–322 (2003; Zbl 1040.34068)] and T. Diagana, the second and the third author [Proc. Am. Math. Soc. 132, 3289–3298 (2004; Zbl 1053.34050)].

##### MSC:
 34G10 Linear differential equations in abstract spaces 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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