×

On the asymptotic behavior of a semilinear functional differential equation in Banach space. (English) Zbl 0598.34053

Let X be a Banach space, \(\{\) A(t):t\(\geq 0\}^ a \)family of linear operators on X that generates a linear evolution system, r a positive real number, and \(\phi: [-r,0]\to X\) a continuous function. This paper is concerned with the asymptotic behavior of the solutions to the semilinear functional differential equation \[ u'(t)+A(t)u(t)=f(t,u_ t),\quad t\geq 0,\quad u_ 0=\phi, \] where f is a sufficiently small nonlinear perturbation and the ”history” \(u_ t: [-r,0]\to X\) is defined by \(u_ t(s)=u(t+s)\) for -r\(\leq s\leq 0\). It is shown that if the linear equation \(u'(t)+A(t)u(t)=0\) exhibits asymptotic stability, so does the nonlinear one. this result improves upon that of S. M. Rankin [ibid. 88, 531- 542 (1982; Zbl 0519.34045)] because the assumed bound on f does not depend on the delay r.
Reviewer: Simeon Reich

MSC:

34G20 Nonlinear differential equations in abstract spaces
34K30 Functional-differential equations in abstract spaces
34D05 Asymptotic properties of solutions to ordinary differential equations

Citations:

Zbl 0519.34045
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Friedman, A., Partial differential equations, (1969), Holt, Rinehart & Winston New York
[2] Lightbourne, J.H.; Rankin, S.M., A partial functional differential equation of Sobolev type, J. math. anal. appl., 93, 328-337, (1983) · Zbl 0519.35074
[3] Rankin, S.M., Existence and asymptotic behavior of a functional differential equation in Banach space, J. math. anal. appl., 88, 531-542, (1982) · Zbl 0519.34045
[4] Travis, C.; Webb, G., Existence, stability and compactness in the α-norm for partial functional differential equations, Trans. amer. math. soc., 240, 129-143, (1978) · Zbl 0414.34080
[5] Walter, W., Differential and integral inequalities, (1970), Springer-Verlag Berlin/Heidelberg/New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.