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Comportamento asintotico di una equazione integrale non lineare

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Summary

In the present note is considered the asymptotic behaviour of the following non-linear integral equations:

$$\begin{gathered} (1)\sigma (t) = f(t) + \mathop \smallint \limits_0^t k(t - \tau )\varphi (\sigma (\tau ),{\text{ }}\tau )d\tau . \hfill \\ (2)\sigma (t) = f(t) + \mathop \smallint \limits_0^t k(t - \tau )\varphi (\sigma (\tau ),{\text{ }}\tau )d\tau \hfill \\ \end{gathered} $$

with the kernel not Fourier-transformable.

In the case where the Laplace-transform of\(k(t),\tilde k{\text{(}}s{\text{)}}\), has a first order pole for s=0, it is proved that, if the condition:

$$(i) - \frac{1}{\beta } + Re(1 + \gamma s)\tilde k(s) \leqslant 0$$

for equation (2), or

$$(ii) - \frac{1}{\beta } + Re \tilde k(s) \leqslant 0$$

for equation (1), is fulfilled, then any bounded solution σ(t) has the property:

$$\mathop {\lim }\limits_{t \to \infty } \sigma (t) = 0$$

Bibliografia

  1. [1]

    C. Corduneanu, Anal. St. Univ. Cuza, 9, 1963, p. 369–375.

  2. [2]

    I. W. Sandberg, Bell Syst. Techn. J., 43, 1964, p. 1581–1599.

  3. [3]

    S. Albertoni eG. P. Szegö,Comptes rendus, 261, 1965, p. 29–32.

  4. [4]

    V. M. Popov,Avtomatika i telemechanika, 22, 1961, p. 961–979.

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Gli autori ringraziano il Prof Corduneanu per i preziosi consigli.

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Albertoni, S., Cellina, A. & Szegö, G.P. Comportamento asintotico di una equazione integrale non lineare. Annali di Matematica 72, 133–140 (1966) doi:10.1007/BF02414331

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