On the boundedness and the asymptotic behaviour of the non-negative solutions of Volterra-Hammerstein integral equations

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Abstract

We study the Volterra-Hammerstein integral equation {Mathematical expression} t≥0, x∈D. We derive sufficient conditions for the boundedness of all non-negative solutions U. We show that, for bounded non-negative solutions U, U(t,.) is positive on D for sufficiently large t>0, if we impose appropriate positivity assumptions on f and h. If we additionally assume that, for y∈D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases, the following alternative holds for any bounded non-negative solution U: Either U(t,.) converges toward zero for t→∞, pointwise on D, or U(t,.) converges, for t→∞, toward the unique bounded positive solution of the corresponding Hammerstein integral equation, uniformly on D. We indicate conditions for the occurrence of each of the two cases.

Original languageEnglish (US)
Pages (from-to)379-412
Number of pages34
JournalManuscripta Mathematica
Volume31
Issue number4
DOIs
StatePublished - Aug 1 1980
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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