Abstract
The 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 lim t → ∞ e − λt S( 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.
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