Monotone iterative techniques for time-dependent problems with applications. (English) Zbl 0935.34052
The authors use the theory of positive semigroups to develop a monotone iterative technique in the spirit of the paper of 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.
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.
Reviewer: Eduardo Liz (Vigo)
MSC:
| 34G10 | Linear differential equations in abstract spaces |
| 47D03 | Groups and semigroups of linear operators |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 35K57 | Reaction-diffusion equations |
| 92D25 | Population dynamics (general) |
Citations:
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