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Zbl 0985.45007
Jung, Jong Soo
Asymptotic behavior of solutions of nonlinear Volterra equations and mean points. (English)
[J] J. Math. Anal. Appl. 260, No.1, 147-158 (2001). ISSN 0022-247X

The author studies an asymptotic behavior at infinity of the solutions of the nonlinear Volterra equation $$(V_{b,g,f}) u(t)+\int_0^t b(t-s)(Au(s)+g(s)u(s)) ds \ni f(t), \quad t\ge 0$$ where $b\in AC_{\text{loc}}(R^+;R)$, $b(0)=1$; $b^\prime\in BV_{\text{loc}}(R^+;R)$; $g\in C(R^+;R^+)$; $f\in W^{1,1}_{\text{loc}}(R^+;X)$, $f(0)\in \overline{D(A)}$ and $R^+=[0,\infty).$ Here $A$ is an accretive operator in real reflexive Banach space $X$. Basing on the mean point, the weak and strong convergences for the ``unbounded behavior'' of solutions are given. The case $V_{1,0,0}$ was earlier considered from this point of view by {\it W. Takahashi} [J. Math. Anal. Appl. 109, 130-139 (1985; Zbl 0593.47057)].
[Nikolai K.Karapetyants (Rostov-na-Donu)]

MSC 2000:: *45M05 Asymptotic theory of integral equations
45G10 Nonsingular nonlinear integral equations

Keywords: unbounded behavior; nonlinear Volterra equation; mean point; invariant mean; completely positive; asymptotic behavior; accretive operator; Banach space; convergence

Citations: Zbl 0593.47057

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