2011
DOI: 10.7494/opmath.2011.31.3.457
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A new composition theorem for $S^p$-weighted pseudo almost periodic functions and applications to semilinear differential equations

Abstract: Abstract. In this paper, we establish a new composition theorem for S p -weighted pseudo almost periodic functions under weaker conditions than the Lipschitz ones currently encountered in the literatures. We apply this new composition theorem along with the Schauder's fixed point theorem to obtain new existence theorems for weighted pseudo almost periodic mild solutions to a semilinear differential equation in a Banach space.

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Cited by 2 publications
(1 citation statement)
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“…[35,48,47,70]), is devoted to the problem of existence and uniqueness of a bounded (µ-pseudo) almost periodic mild solution to (52) in a Banach space X. The adopted approach is based on superposition theorems in the Banach space S p AP(R, X) (or S p PAP(R, X, µ)) combined with the Banach's fixed-point principle, applied to the nonlinear operator (Γu)(t) = To our knowledge, all existing results use the fact that Γ maps AP(R, X) into itself, but Γ does not map S p AP(R, X) into AP(R, X) nor into S p AP(R, X) (see in particular [35,48,47,70]). The proposed proofs may be summarized as follows: if u ∈ AP(R, X), then u satisfies the compactness condition (Com) of Subsection 2.4, and u ∈ S p AP(R, X).…”
Section: Comments and Concluding Remarksmentioning
confidence: 99%
“…[35,48,47,70]), is devoted to the problem of existence and uniqueness of a bounded (µ-pseudo) almost periodic mild solution to (52) in a Banach space X. The adopted approach is based on superposition theorems in the Banach space S p AP(R, X) (or S p PAP(R, X, µ)) combined with the Banach's fixed-point principle, applied to the nonlinear operator (Γu)(t) = To our knowledge, all existing results use the fact that Γ maps AP(R, X) into itself, but Γ does not map S p AP(R, X) into AP(R, X) nor into S p AP(R, X) (see in particular [35,48,47,70]). The proposed proofs may be summarized as follows: if u ∈ AP(R, X), then u satisfies the compactness condition (Com) of Subsection 2.4, and u ∈ S p AP(R, X).…”
Section: Comments and Concluding Remarksmentioning
confidence: 99%