Pacific Journal of Mathematics

Discrete generalized Gronwall inequalities in three independent variables.

B. G. Pachpatte and S. M. Singare

Article information

Source
Pacific J. Math., Volume 82, Number 1 (1979), 197-210.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102785072

Mathematical Reviews number (MathSciNet)
MR549844

Zentralblatt MATH identifier
0413.26009

Subjects
Primary: 39A12: Discrete version of topics in analysis
Secondary: 26D15: Inequalities for sums, series and integrals

Citation

Pachpatte, B. G.; Singare, S. M. Discrete generalized Gronwall inequalities in three independent variables. Pacific J. Math. 82 (1979), no. 1, 197--210. https://projecteuclid.org/euclid.pjm/1102785072


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References

  • [x-ly-1z] , u(x, !/f2) + Q(k, I, n)u(k, I, n) which in view of the definition of m(x, y, z) implies *mxy{x, y, z + 1) - 2mxy(x,y, z) (16)ar-l y-1 z~l~] ^ p(x, V, z)\ m{x, i/,2) + Q(k, I, ri)m(k, I, n). If we put 35 -- 1 V -- lZ -- l (17)v(x, y, z) = m(x, 1/,^) + ^(^, I,n)m{k, I,n) , so that v(0, y, z) = (0) + b(y) + c() , v(x, 0, ^ - ( ) + 6(0) + c(z) ,
  • [1] E. F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin, 1961.
  • [2] P. R. Beesack, Gronwall Inequalities, Carleton Mathematical Lecture Notes No.11, May 1975.
  • [3] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20 (1919), 292-296.
  • [4] B. G. Pachpatte, On the discrete generalizations of Gronwals inequality, J. Indian Math. Soc., 37 (1973), 147-156.
  • [5] B. G. Pachpatte, Finite difference inequalities and their applications,Proc. Nat. Acad. Sci. India, 43 (A) (1973), 348-356.
  • [6] B. G. Pachpatte,On discrete inequalitiesrelated to Gronwals inequality,Proc. Indian Acad. Sci., 85 (A) (1977), 26-40.
  • [7] B. G. Pachpatte,A note on some fundamental discrete inequalities of the Gronwall-Bellman type, Bull. Inst. Math. Acad. Sinica, 5 (1977), 121-128.
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  • [10] B. G. Pachpatte, An integral inequality similar to Bellman-Bihari inequality, Bull. Soc. Math. Grece, 15 (1974), 7-15.