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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.

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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