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Nonlinear integral inequalities of the Volterra type

Published online by Cambridge University Press:  24 October 2008

Ryszard Szwarc
Ryszard Szwarc
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, Wi 53706, U.S.A. Institute of Mathematics, Wroclaw University, pl. Grunwaldzki 2/4, 50–384 Wroclaw, Poland
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Extract

We are studying the integral inequality

where all functions appearing are defined and increasing on the right half-axis and take the value zero at zero. We are interested in determining when the inequality admits solutions u(x) which are non-vanishing in a neighbourhood of zero. It is well-known that if ψ(x) is the identity function then no such solution exists. This due to the fact that the operator defined by the integral on the right-hand side of the equation is linear and compact. So if we are interested in non-trivial solutions it is natural to require that ψ(x) > 0 at least for all non-zero points in some neighbourhood of zero. One of the typical examples is the power function ψ(x) = xα, where α < 1. This situation was explored in [2]. The functions a(x) that admit non-zero solutions were characterized by Bushell in [1]. For a general approach to the problem we refer to [2], [3] and [4].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 599 - 608
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Bushell, P. J.. On a class of Volterra and Fredholm non-linear integral equations. Math. Proc. Cambridge Philos. Soc. 79 (1979), 329–335.Google Scholar
[2]Bushell, P. J. and Okrasiński, W.. Uniqueness of solutions for a class of non-linear Volterra integral equations with convolution kernel. Math. Proc. Cambridge Philos. Soc. 106 (1989), 547–552.Google Scholar
[3]Okrasiński, W.. On a nonlinear Volterra equation. Math. Methods Appl. Sci. 8 (1986), 345–350.Google Scholar
[4]Szwarc, R.. Attraction principle for nonlinear integral operators of the Volterra type. J. Math. Anal. Appl., to appear.Google Scholar