Summary
The celebrated Gronwall-Bellman Lemma provides explicit bounds on solutions of a class of linear integral inequalities. The aim of this paper is to prove sufficiently general results analogous to this Lemma for functional-integral inequalities. These results are useful for obtaining point-wise estimates or comparison theorems for solutions of functional differential equations and functional-integral equations of Volterra type.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Bihari,A generalization of lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hung.,7 (1956), pp. 81–94.
G. Butler -T. Rogers,A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations, Jour. Math. Anal. Appl.,33, (1971), pp. 77–81.
J. Chandra -B. A. Fleishman,On a generalization of the Gronwall-Bellman Lemma in partially ordered Banach spaces, Jour. Math. Anal. Appl.,31 (1970), pp. 668–681.
C. Corduneanu,On a class of functional-integral equations, Bull. Math. Soc. Sci. Math. R. S. Roumanie,12 (1968), pp. 43–53.
S. G. Deo - M. G. Murdeshwar,A note on Gronwall’s inequality, to appear.
H. E. Gollwitzer,A note on a functional inequality, Proc. Amer. Math. Soc.,23 (1969), pp. 642–647.
L. J. Grimm -K. Schmitt,Boundary value problem for differential equations with deviating arguments, Aequa. Math.,4 (1970), pp. 176–190.
M. A. Kransoselskii,Positive Solutions of Operator Equations, Noordhoff, Groningen, 1961.
V. Lakshmikantham -S. Leela,Differential and Integral Inequalities, vol. I, Academic Press, New York, 1969.
W. Walter,Differential and Integral Inequalities, Springer-Verlag, Berlin, 1970.
Author information
Authors and Affiliations
Additional information
Entrata in Redazione il 1o settembre 1971.
Rights and permissions
About this article
Cite this article
Chandra, J., Lakshmikantham, V. Some point-wise estimates for solutions of a class of nonlinear functional-integral inequalities. Annali di Matematica 94, 63–74 (1972). https://doi.org/10.1007/BF02413602
Issue Date:
DOI: https://doi.org/10.1007/BF02413602