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


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


Zbl 0519.34045
Full Text: DOI


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