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Asymptotic behavior of solutions of nonlinear Volterra equations and mean points. (English) Zbl 0985.45007
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)].

45M05Asymptotic theory of integral equations
45G10Nonsingular nonlinear integral equations
Full Text: DOI
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