Hatvani László honlapja

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Cím:On the existence of periodic solutions for linear inhomogeneous and quasilinear functional differential equations.
Szerző:Hatvani, L.; Krisztin, T.
Forrás:J. Differ. Equations 97, No.1, 1-15 (1992).
Absztrakt:The equation $x'(t)=\int\sp{+\infty}\sb{-\infty}(dE(s))x(t+s)+f(t)$ is considered with $T$-periodic $f$. Associate the characteristic equation $\text{det} \Delta(\mu)=0$ $\Delta(\mu)=\mu I-\int\sp{+\infty}\sb{- \infty}e\sp{\mu s}dE(s)$. Define $\mu\sb R={2k\pi i\over T}$ and let $\hat f(k)$ be the $k$th Fourier coefficient of $f$. An elementary proof is given for the following: The equation has a $T$-periodic solution iff $a\hat f(k)=0$ for all row vectors $a$ for which $a\Delta(\mu\sb k)=0$. For the ``if'' part a proof using Leray-Schauder arguments allows generalizations to nonlinear situations.} \RV{A.Halanay (Bucure\c{s}ti) 
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