On the asymptotic behavior of solutions of nonlinear Volterra equations (Q1086782)

The author investigates the asymptotic behavior of the solutions of the nonlinear Volterra equation (V) \(u(t)+b^*Au(t)\ni g(t)\), \(t\in R^+\), considered in a real, uniformly convex Banach space X under the following assumptions: A is an odd m-accretive operator in X, \(b\in AC_{loc}(R^+;R)\), \(b(0)>0\), \(b'\in BV_{loc}(R^+;R)\), \(g\in W^{1,1}_{loc}(R^+;X)\), \(g'\in L^ 1(R^+;X)\), g(0)\(\in \overline{D(A)}\), and b is completely positive on \(R^+\) with \(b(\infty)>0\). The main result is: Let u be a generalized solution of (V) and such that for each \(h>0\lim_{t\to \infty}\| u(t+h)-u(t)\| =0\). Then \(\lim_{t\to \infty}u(t)\) exists and belongs to \(A^{-1}(0)\). Some consequences of this result are pointed out, and relations to earlier work by \textit{J.-B. Baillon} and \textit{P. Clement} [Nonlinear Anal., Theory Methods Appl. 5, 789-801 (1981; Zbl 0541.45009)], by \textit{I. Miyadera} and \textit{K. Kobayasi} [ibid. 6, 349-365 (1982; Zbl 0507.47041)], and by the reviewer [SIAM J. Math. Anal. 8, 950-970 (1977; Zbl 0379.45011)], are discussed.