DOI QR코드

DOI QR Code

ON SOME GRONWALL TYPE INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

  • Kim, Byung-Il (Department of Mathematics and Statistics, College of Natural Sciences, Chung-Ang University)
  • Published : 2004.08.01

Abstract

The aim of the present paper is to establish some nonlinear integral inequalities in two independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential equations.

Keywords

References

  1. D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992
  2. E. F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, New York, 1961
  3. R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943), 643–647
  4. I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 71–94
  5. S. S. Dragomir and N. M. Ionescu, On nonlinear integral inequalities in two independent variables, Stud. Univ. Babe¸s-Bolyai Math. 34 (1989), 11–17
  6. S. S. Dragomir and Y. -H. Kim, On certain new integral inequalities and their applications, J. Inequal. Pure Appl. Math. 3 (2002), no. 4, 1–8
  7. S. S. Dragomir and Y. -H. Kim, Some integral inequalities for function of two variables, Electron. J. Differential Equations 2003 (2003), no. 10, 1–13
  8. T. H. Gronwall, Note on the derivatives with respect to a parameter of solutions of a system of differential equations, Ann. of Math. 20 (1919), 292–296
  9. L. Guiliano, Generalazzioni di un lemma di Gronwall, Rend. Accad., Lincei, 1946, pp. 1264–1271
  10. C. E. Langenhop, Bounds on the norm of a solution of a general differential equation, Proc. Amer. Math. Soc. 11 (1960), 795–799
  11. A. Mate and P. Neval, Sublinear perturbations of the differential equation $Y^{(n)}$ = 0 and of the analogous difference equation, J. Differential Equations 52 (1984), 234–257
  12. D. S. Mitrinovic, J. E. Peˇcari´c and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, Boston, London, 1991
  13. V. V. Nemyckii and V. V. Stepanov, Qualitative theory of differential equations (Russian), OGIZ, Moscow, 1947
  14. B. G. Pachpatte, On some fundamental integral inequalities and their discrete analogues, J. Inequal. Pure Appl. Math. 2 (2001), no. 2, 1–13

Cited by

  1. On some new integral inequalities of Gronwall–Bellman–Pachpatte type vol.217, pp.20, 2011, https://doi.org/10.1016/j.amc.2011.02.093