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Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain. (English) Zbl 0985.34052
Existence and uniqueness of so-called pseudo-almost periodic solutions to the abstract semilinear evolution equation \[ x'(t)= Ax(t)+ f(t, x(t)),\qquad t\in\mathbb{R}, \] is proved, where \(A\) is a linear unbounded (not necessarily dense defined) operator in some Banach space that generates a continuous semigroup. Moreover, \(f\) is some continuous mapping that is pseudo-almost periodic in \(t\) and Lipschitzian with respect to the second argument.
The results can be applied to delay and partial differential equations as it is done by the authors in two examples for a semilinear first-order partial differential equation.

34G20 Nonlinear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
35K55 Nonlinear parabolic equations
Full Text: DOI
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