## 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

Zbl 0519.34045
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### 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
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