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Ezzinbi, Khalil; Kyelem, Bila Adolphe; Ouaro, Stanislas

Periodic solutions in the \(\alpha\)-norm for some neutral partial functional differential equations with finite delay. (English) Zbl 1306.34105

Afr. Mat. 24, No. 4, 625-645 (2013).
Summary: The aim of this work is to study the existence of periodic solutions in the \(\alpha\)-norm for some partial differential equations of neutral type with finite delay. We assume that the linear part is densely defined and is the generator of an analytic semigroup. The delayed part is assumed to be periodic with respect to the first argument. In the nonhomogeneous linear case, we show that the existence of a bounded solution in \(\mathbb R^+\) implies the existence of periodic solution. In nonlinear case, we use two approaches, the first one is based on the ultimate boundedness of the solutions and the second one is based on the multivalued fixed point theory.

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MSC:

34K13 Periodic solutions to functional-differential equations
34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
47N20 Applications of operator theory to differential and integral equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations

Keywords:

analytic semigroup; partial functional differential equations of neutral type; \(\alpha\)-norm; multivalued maps; condensing maps; periodic solutions; fixed point theorem
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