Semilinear functional differential equations in Banach space

In this paper we prove existence and study the asymptotic behavior of mild solutions to a class of semi-linear abstract functional differentia1 equations which involve a nonlinear delay term. This class is characterized by the fact that the associated homogeneous linear differential equation generates a strongly continuous linear evolution system of compact operators. We also prove a regularity result by placing additional restrictions on our nonlinear delay term. Our approach is closely patterned on the recent treatment of abstract semilinear ordinary differential equations by Pazy [lo] and is similar to recent work on abstract functional differential equations by Travis and Webb [12]. More precisely we consider the nonlinear Banach space Volterra integral equation:

[1]  C. C. Travis,et al.  Existence, stability, and compactness in the -norm for partial functional differential equations , 1978 .

[2]  V. Lakshmikantham,et al.  Differential equations in abstract spaces , 1972 .

[3]  Avner Friedman,et al.  Partial differential equations , 1969 .

[4]  D. Brewer.  A nonlinear semigroup for a functional differential equation , 1978 .

[5]  L. E. El'sgolt's.  Introduction to the Theory of Differential Equations with Deviating Arguments , 1966 .

[6]  Felix E. Browder,et al.  Non-Linear Equations of Evolution , 1964 .

[7]  C. Travis,et al.  Existence and stability for partial functional differential equations , 1974 .

[8]  E. S. Noussair,et al.  Differential equations in Banach spaces , 1973, Bulletin of the Australian Mathematical Society.

[9]  A. Pazy,et al.  A class of semi-linear equations of evolution , 1975 .

[10]  B. Calvert.  Nonlinear equations of evolution , 1971 .

[11]  W. Fitzgibbon.  Stability for Abstract Nonlinear Volterra Equations involving Finite Delay , 1977 .

[12]  L. Ė. Ėlʹsgolʹt︠s︡.  Introduction to the theory of differential equations with deviating arguments , 1966 .

[13]  Glenn F. Webb,et al.  Partial differential equations with deviating arguments in the time variable , 1976 .

[14]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

[15]  J. Dyson,et al.  20.—Functional Differential Equations and Non-linear Evolution Operators , 1976, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  W. Fitzgibbon.  Nonlinear volterra equations with infinite delay , 1977 .

[17]  J. Hale.  Functional Differential Equations , 1971 .

[18]  G. Webb.  Asymptotic stability for abstract nonlinear functional differential equations , 1976 .

[19]  S. Krein,et al.  Linear Differential Equations in Banach Space , 1972 .

[20]  G. Webb.  Autonomous nonlinear functional differential equations and nonlinear semigroups , 1974 .