×

Integral inequalities for retarded Volterra equations. (English) Zbl 1103.26018

Summary: Integral inequalities are very useful in the qualitative analysis of differential and integral equations. Starting with [the author, J. Math. Anal. Appl. 252, No. 1, 389–401 (2000; Zbl 0974.26007)], several recent investigations [see the author, ibid. 285, No. 2, 436–443 (2003; Zbl 1040.26007); B. G. Pachpatte, ibid. 267, No. 1, 48–61 (2002; Zbl 0996.26008); JIPAM, J. Inequal. Pure Appl. Math. 3, No. 2, Paper No. 18 (2002; Zbl 0994.26017); ibid. 5, No. 1, Paper No. 19 (2004; Zbl 1068.26011) and ibid. 5, No. 3, Paper No. 80 (2004; Zbl 1068.26020)] were devoted to retarded integral inequalities.
In this paper we consider the case of retarded Volterra integral equations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the asymptotic behaviour of the solutions.

MSC:

26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7, 81-94 (1965) · Zbl 0070.08201
[2] Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math., 20, 292-296 (1919)
[3] Lipovan, O., A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl., 252, 389-401 (2000) · Zbl 0974.26007
[4] Morro, A., A Gronwall-like inequality and its application to continuum thermodynamics, Boll. Un. Mat. Ital. B, 6, 553-562 (1982) · Zbl 0498.34036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.