Asymptotic behavior of solutions of nonlinear Volterra equations and mean points (Q5944913)

The author studies an asymptotic behavior at infinity of the solutions of the nonlinear Volterra equation NEWLINE\[NEWLINE(V_{b,g,f}) u(t)+\int_0^t b(t-s)(Au(s)+g(s)u(s)) ds \ni f(t), \quad t\geq 0NEWLINE\]NEWLINE 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 \textit{W. Takahashi} [J. Math. Anal. Appl. 109, 130-139 (1985; Zbl 0593.47057)].