Well-posedness questions and approximation schemes for a general class of functional differential equations

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1986

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Virginia Polytechnic Institute and State University

Abstract

In this paper we consider approximation schemes and questions of well-posedness for a general class of functional differential equations of neutral-type (NFDE) where the difference operator does not have an atom at zero. Equations of this type occur in the modeling of certain aeroelastic control problems and include many singular integro-differential equations.

We obtain general necessary and sufficient conditions for the well-posedness of functional differential equations of neutral-type on the Banach-spaces Rn</sup?xLp. As an example of the well-posedness of the non-atomic NFDE-system that arises in the study of aeroelasticity is established on Rn</sup?xLp, 1≤p<2.

Employing the equivalence between generalized solutions of NFDEs and mild solutions of the “corresponding” abstract Cauchy-problems, we make use of general approximation results for well-posed Cauchy-problems to establish and analyze the convergence of the “averaging projection” scheme on the Banach spaces Rn</sup?xLp, 1<p<∞, for a class of problems with atomic difference operators.

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