Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations

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10 Scopus citations

Abstract

The concept of asymptotic proportionality and conditional asymptotic equality which is presented here aims at making global asymptotic stability statements for time-heterogeneous difference and differential equations. For such non-autonomous problems (apart from special cases) no prominent special solutions (equilibra, periodic solutions) exist which are natural candidates for the asymptotic behaviour of arbitrary solutions. One way out of this dilemma consists in looking for conditions under which any two solutions to the problem (with different initial conditions) behave in a similar or even the same way as time tends to infinity. We study a general sublinear difference equation in an ordered Banach space and, for illustration, time-heterogeneous versions of several well-known differential equations modelling the spread of gonorrhea in a heterogeneous population, the spread of a vector-borne infectious disease, and the dynamics of a logistically growing spatially diffusing population.

Original languageEnglish (US)
Pages (from-to)237-268
Number of pages32
JournalJournal of Differential Equations
Volume73
Issue number2
DOIs
StatePublished - Jun 1988
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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