Volterra integro-differential equations with accretive nonlinearity
The topic of this paper is the initial value problem \[ (1)\quad d/dt(\gamma (u(t)-x)+\int^{t}_{0}k(t-s)(u(s)-x)ds)+A(u(t))\ni G(u)(t),\quad t\in R^+,\quad u(0)=x, \] which is set in a real Banach space X. Here \(\gamma\geq 0\), \(k: R^+\to R\) is a locally integrable, nonnegative, and nonincreasing, A is an m-accretive operator in X with domain D(A), \(x\in \overline{D(A)}\), and G is a mapping of \(C(R^+;X)\) when \(\gamma >0\), or of \(L^ 1_{loc}(R^+;X)\) when \(\gamma =0\) into \(L^ 1_{loc}(R^+;X)\) satisfying a Lipschitz condition. The author discusses the existence, uniqueness, regularity and continuous dependence upon the data of solutions of (1). The basic idea of the approach is to replace the operator L: D(L)\(\subset L^ 1_{loc}(R^+;X)\to L^ 1_{loc}(R^+;X)\), \(L(v)(t)=d/dt(v(t)+\int^{t}_{0}k(t-s)v(s)ds)\), \(D(L)=\{v\in L^ 1_{loc}(R^+;X)/t\to \gamma v(t)+\int^{t}_{0}k(t-s)v(s)ds\) is locally absolutely continuous and a.e. differentiable on \(R^+\), and takes the value 0 at \(t=0\}\) by its Yosida approximation \(L_{\lambda}\), \(\lambda >0\), and then to pass to the limit as \(\lambda \to 0^+\).
scientific article; zbMATH DE number 3919058
scientific article