Study On Existence And Regularity Results Of Solutions For Partial Functional Integrodifferential Equations In Banach Spaces

In this paper, by using a new method, we consider the following Cauchy problems: where (E,|·|) is a real Banach space,f∈C([0,∞)×C, E), A:D(A) C E→E is the infinitesimal generator of a c0-semigroup on E, B(t) for t≥0is a closed linear operator with domain D(B) D D(A), and for each t≥0, ut(θ)=u(t+θ) for-r≤θ≤0, and where A(t) for t≥0is closed linear operator with dense domain D(A) which is independent of t, for0≤s≤t, B(t, s) is a closed linear operator with domain D(B) D D(A).We obtain some new existence and regularity results.By comparing with the related known results and using a new method to this equation firstly, we prove that our obtained results are interesting(see chapterl).This paper is divided into6chapters:In chapter1, we introduce the re-search background, the most recent results for properties of solutions of the par-tial functional integrodifferential equations, the new method, our main results, the proving method, the structure of this paper; In chapter2, we introduce some marks which are used in the paper, some definitions and some lemmas; In chap-ter3, we consider the existence and regularity of solutions of the first Cauchy problem; In chapter4, we consider the existence and uniqueness of mild solutions of the nonautonomous case; In chapter5, we give two examples to apply our conclusions; In chapter6, we summarize the paper.