Asynchronous Exponential Growth of Semigroups of Nonlinear Operators
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 167, 443467 (1992) Asynchronous Exponential Growth of Semigroups of Nonlinear Operators M GYLLENBERG Lulea University of Technology, Department of Applied Mathematics, S-95187, Lulea, Sweden AND G. F. WEBB* Vanderbilt University, Department of Mathematics, Nashville, Tennessee 37235 Submitted by Ky Fan Received August 9, 1990 The property of asynchronous exponential growth is analyzed for the abstract nonlinear differential equation i’(f) = AZ(~) + F(z(t)), t 2 0, z(0) =x E 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), I B 0 associated with this problem has the property that there exists i, > 0 and a nonlinear operator Q in X such that the range of Q lies in a one-dimensional subspace of X and lim,,, em”‘S(r)x= Qx for all XE X. It is proved that if the linear semigroup generated by A has asynchronous exponential growth and F satisfies \/F(x)11 < f( llxll) ilxll, where f: [w, --t Iw + and 1” (f(rYr) dr < ~0, then the nonlinear semigroup S(t), t >O has asynchronous exponential growth. The method of proof is a linearization about infinity. Examples from structured population dynamics are given to illustrate the results. 0 1992 Academic Press. Inc. INTRODUCTION The concept of asynchronous exponential growth arises from linear models of cell population dynamics.
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