Abstract
We study the Volterra-Hammerstein integral equation
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.
Similar content being viewed by others
References
BRAUER, F.: On a nonlinear integral equation for population growth problems. SIAM J. math. Anal.6, 312–317 (1975)
DIEKMANN, O.: Thresholds and travelling waves for the geographical spread of infection. J. math. Biology6, 109–130 (1978)
DUNFORD, N.; SCHWARTZ, J.T.: Linear Operators. I. New York: Interscience 1958
HEWITT, E; STROMBERG, K.: Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer 1969
KRASNOSEL'SKII, M.A.: Positive Solutions of Operator Equations. Groningen: Noordhoff 1964
SATTINGER, D.H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J.21, 979–1000 (1972)
THIEME, H.R.: A model for the spatial spread of an epidemic. J. math. Biology4, 337–551 (1977)
THIEME, H.R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. math. Biology8, 173–187 (1979)
THIEME, H.R.: On a class of Hammerstein integral equations. manuscripta math. 29, 49–84 (1979)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Thieme, H.R. On the boundedness and the asymptotic behaviour of the non-negative solutions of Volterra-Hammerstein integral equations. Manuscripta Math 31, 379–412 (1980). https://doi.org/10.1007/BF02320701
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02320701