Almost convergence of solutions of nonlinear Volterra equations in Banach space
The author studies the asymptotic behavior of a solution of a nonlinear Volterra integral equation in a real uniformly convex Banach space E. Let A be an m-accretive operator in \(E\times E\) and b [resp. f] be a locally absolutely continuous function from [0,\(\infty)\) into R [resp. E] such that \(b(0)>0\) [resp. f(0)\(\in \overline{D(A)}]\). Assume that u is a bounded solution of the following Volterra equation: \[ u(t)+\int^{t}_{0}b(t-s)A(u(s))ds\ni f(t),\quad t\geq 0. \] Under some additional conditions the weak convergence of (1/T)\(\int^{T}_{0}u(s)ds\) as \(T\to \infty\) is obtained. It is proved by applying directly a nonlinear ergodic theorem for contraction semigroups in Banach spaces.
scientific article; zbMATH DE number 3957728
scientific article