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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.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
35K55 Nonlinear parabolic equations
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[1] E. Ait Dads, Contribution a l’existence de solutions pseudo presque periodiques d’ une classe d’equations fonctionnelles, Doctorat d’Etat, Universite Cadi Ayyad, Marrakesch, 1994.
[2] E. Ait Dads, K. Ezzinbi, Positive pseudo almost periodic solution for some non linear delay integral equation, J. Cybernet. (1994) 134-145. · Zbl 0834.45006
[3] Ait Dads, E.; Ezzinbi, K., Pseudo almost periodic solutions of some delay differential, J. math. anal. appl., 201, 840-850, (1996) · Zbl 0858.34055
[4] B. Amir, L. Maniar, Composition of pseudo almost periodic functions and semilinear Cauchy problems with non dense domain, Tübinger Berichte Zur Funktionalanalysis, Helf 6, Jahrgang (1996/97), pp. 6-15.
[5] B. Amir, L. Maniar, Existence and asymptotic behavior of solutions to the inhomogeneous Cauchy problems with non dense domain via extrapolation spaces, Preprint. · Zbl 1019.34058
[6] S. Castillo, M. Pinto, An asymptotic theory of functional differential equations, Submitted for publication. · Zbl 1070.34101
[7] Da Prato, G.; Griovard, E., On extrapolation spaces, Rend. accad. naz. lincei, 72, 330-332, (1982) · Zbl 0527.46055
[8] Da Prato, G.; Sinestrari, E., Differential operators with non dense domain, Ann. scuola normale superiore Pisa, 14, 295-344, (1989) · Zbl 0652.34069
[9] A.M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer, New York, Berlin, 1974. · Zbl 0325.34039
[10] J. Hale, V. Lunel, Introduction to Functional Differential Equations, Springer, New York, Berlin, 1993. · Zbl 0787.34002
[11] E. Hille, R.S. Philips, Functional Analysis and Semigroups, American Mathematical Society, Providence, RI, 1975.
[12] Nagel, R., Sobolev spaces and semigroup, Semesterberiche funktional-analysis, band, 4, 1-20, (1983)
[13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, Berlin, 1983. · Zbl 0516.47023
[14] S. Zaidman, Almost periodic functions in abstract spaces, Research Notes in Mathematics, Pitman, London, 1985. · Zbl 0648.42006
[15] C. Zhang, Pseudo almost periodic functions and their applications, Thesis, the University of Western Ontario, 1992.
[16] Zhang, C., Pseudo almost periodic solutions of some differential equations, J. math. anal. appl., 181, 62-76, (1994) · Zbl 0796.34029
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