Asynchronous exponential growth of semigroups of nonlinear operators

Abstract

AbstractThe property of asynchronous exponential growth is analyzed for the abstract nonlinear differential equation z′(t) = Az(t) + F(z(t)), t ⩾ 0, z(0) = x ϵ X, where A is the infinitesimal generator of a semigroup of linear operators in the Banach space X and F is a nonlinear operator in X. Asynchronous exponential growth means that the nonlinear semigroup S(t), t ⩾ 0 associated with this problem has the property that there exists λ > 0 and a nonlinear operator Q in X such that the range of Q lies in a one-dimensional subspace of X and limt → ∞ e−λtS(t)x = Qx for all x ϵ X. It is proved that if the linear semigroup generated by A has asynchronous exponential growth and F satisfies ∥F(x)∥ ⩽ f(∥x∥) ∥x∥, where f: R+ → R+ and ∝∞(f(r)r) dr < ∞, then the nonlinear semigroup S(t), t ⩾ 0 has asynchronous exponential growth. The method of proof is a linearization about infinity. Examples from structured population dynamics are given to illustrate the results