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Asymptotically typed solutions to a semilinear integral equation. (English) Zbl 1326.45004
Summary: We investigate the existence of $$\mu$$-pseudo almost automorphic solutions to the semilinear integral equation $$x(t)=\int_{-\infty}^{t}a(t-s)[Ax(s)+f(s,x(s))]\,ds$$, $$t\in\mathbb{R}$$ in a Banach space $$\mathbf{X}$$, where $$a\in L^{1}(\mathbb{R}_{+})$$, $$A$$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $$\mathbf{X}$$, and $$f:\mathbb{R}\times\mathbf{X}\to\mathbf{X}$$ is a $$\mu$$-pseudo almost automorphic function. The main results are proved by using integral resolvent families combined with the theory of $$\mu$$-pseudo almost automorphic functions.

##### MSC:
 45G10 Other nonlinear integral equations 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 34F05 Ordinary differential equations and systems with randomness 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
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