Annali di Matematica Pura ed Applicata

, Volume 94, Issue 1, pp 63–74 | Cite as

Some point-wise estimates for solutions of a class of nonlinear functional-integral inequalities

  • Jagdish Chandra
  • V. Lakshmikantham


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.


Differential Equation General Result Functional Differential Equation Comparison Theorem Integral Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    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.CrossRefMathSciNetGoogle Scholar
  3. [3]
    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.CrossRefMathSciNetGoogle Scholar
  4. [4]
    C. Corduneanu,On a class of functional-integral equations, Bull. Math. Soc. Sci. Math. R. S. Roumanie,12 (1968), pp. 43–53.zbMATHMathSciNetGoogle Scholar
  5. [5]
    S. G. Deo - M. G. Murdeshwar,A note on Gronwall’s inequality, to appear.Google Scholar
  6. [6]
    H. E. Gollwitzer,A note on a functional inequality, Proc. Amer. Math. Soc.,23 (1969), pp. 642–647.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    L. J. Grimm -K. Schmitt,Boundary value problem for differential equations with deviating arguments, Aequa. Math.,4 (1970), pp. 176–190.CrossRefMathSciNetGoogle Scholar
  8. [8]
    M. A. Kransoselskii,Positive Solutions of Operator Equations, Noordhoff, Groningen, 1961.Google Scholar
  9. [9]
    V. Lakshmikantham -S. Leela,Differential and Integral Inequalities, vol. I, Academic Press, New York, 1969.Google Scholar
  10. [10]
    W. Walter,Differential and Integral Inequalities, Springer-Verlag, Berlin, 1970.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1972

Authors and Affiliations

  • Jagdish Chandra
    • 1
  • V. Lakshmikantham
    • 2
  1. 1.DurhamUSA
  2. 2.KingstonUSA

Personalised recommendations