Monotone iterative techniques for time-dependent problems with applications (Q1306846)

The authors use the theory of positive semigroups to develop a monotone iterative technique in the spirit of the paper of \textit{J. J. Nieto} [Nonlinear Anal., Theory Methods Appl. 28, No. 12, 1923-1933 (1997; Zbl 0883.47058)] for the time-dependent nonlinear abstract problem \[ u'(t)=Au(t)+N(t)u(t), t\in J;\quad u(0)=\phi_{0}, \] with \(J=[0, T_{0})\), \(0<T_{0}\leq\infty\), \(A:D(A)\subset X\to X\) is an infinitesimal generator of a strongly continuous linear semigroup, \(X\) being an ordered Banach space; \(\phi_{0}\in X\), and \(N(t)\) is a nonlinear operator satisfying a one-sided Lipschitz condition. The existence of a couple of ordered lower and upper solutions to this problem is assumed. The main results are applied to investigate the existence of solutions to certain impulsive reaction-diffusion systems and to a system of elliptic equations arising in population dynamics. In the last case, some stability results are achieved.