Asynchronous exponential growth in differential equations with homogeneous nonlinearities. (English) Zbl 0827.47035
Dore, Giovanni (ed.) et al., Differential equations in Banach spaces. Proceedings of the 2nd conference on differential equations in Banach spaces held at Bologna, Italy, 1991. New York: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 148, 225-233 (1993).
Let \(A\) be an infinitesimal generator of a strongly continuous semigroup of positive linear operators in a Banach lattice \(X\) and \(F\) be a nonlinear operator in \(X_+\) satisfying \(F(cx)= cF(x)\), \(x\in X_+\), \(c\geq 0\). The author considers the abstract differential equation
\[
Z_x' (t)= AZ_x (t)+ F(Z_x (t)), \qquad t\geq 0, \quad Z_x (0)= x,
\]
and gives some sufficient conditions under which solutions of this equation are of asynchronous exponential growth on \(U\subset X_+\), that is that there is a real constant \(\lambda\) such that \(Q_x:= \lim_{t\to \infty} e^{-\lambda t} Z_x (t)\) exists, \(Q_x\) is nonzero for all \(x\in U\), and \(R(Q_x)\) lies in a one-dimensional subspace of \(X\). Such type of behavior occurs in models of population growth. The underlying ideas of the proved results are illustrated by some examples.
For the entire collection see [Zbl 0785.00028].
For the entire collection see [Zbl 0785.00028].
Reviewer: V.Marić (Novi Sad)
MSC:
47D06 | One-parameter semigroups and linear evolution equations |
47B65 | Positive linear operators and order-bounded operators |
46B42 | Banach lattices |
47E05 | General theory of ordinary differential operators |
34G20 | Nonlinear differential equations in abstract spaces |
92D25 | Population dynamics (general) |