Some new nonlinear difference inequalities and their applications. (English) Zbl 1121.26019

The aim of the paper is to establish some new difference inequalities in two independent variables. A suitable choice of parameters involved gives discrete Gronwall-Ou-Iang type inequalities, cf. B.G.Pachpatte [J. Math. Anal. Appl. 222, No. 2, 438–459 (1998; Zbl 0913.39001)], and Haraux-Engler type inequalities, cf. A.Haraux [“Nonlinear evolution equations – global behavior of solutions” (1981; Zbl 0461.35002)] and H.Engler [Math. Z. 202, No. 2, 251–259 (1989; Zbl 0697.73033)] in two variables. Results also cover some inequalities proved by B.G.Pachpatte [“Inequalities for finite difference equations” (2002; Zbl 0987.39001)]. Applications to partial difference equations are given.


26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
Full Text: DOI


[1] Bainov, D.; Simeonov, P., Integral Inequalities and Applications (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0759.26012
[2] Cheung, W. S., Some discrete nonlinear inequalities and applications to boundary value problems for difference equations, J. Difference Equations Appl., 10, 2, 213-223 (2004) · Zbl 1045.26007
[3] Engler, H., Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Math. Z., 202, 251-259 (1989) · Zbl 0697.73033
[4] Haraux, A., Nonlinear Evolution Equations, (Global Behavior of Solutions, Lecture Note in Mathematics, vol. 847 (1981), Springer: Springer Berlin, New York) · Zbl 0583.35007
[5] Meng, F. W.; Li, W. N., On some new nonlinear discrete inequalities and their applications, J. Comput. Appl. Math., 158, 407-417 (2003) · Zbl 1032.26019
[6] Mitrinović, D. S.; Pečarić, J. K.; Fink, A. M., Inequalities Involving Functions and their Integrals and Derivatives (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht/Boston/London · Zbl 0744.26011
[7] Pachpatte, B. G., Some new finite difference inequalities, Comput. Math. Appl., 28, 227 (1994) · Zbl 0809.26009
[8] Pachpatte, B. G., Inequalities applicable in the theory of finite difference equations, J. Math. Anal. Appl., 222, 438-459 (1998) · Zbl 0913.39001
[9] Pachpatte, B. G., On nonlinear finite difference inequalities in two independent variables, Tamkang J. Math., 33, 67-75 (2002) · Zbl 1011.26012
[10] Pachpatte, B. G., Inequalities for Finite Difference Equations (2002), Marcel Dekker: Marcel Dekker New York · Zbl 0987.39001
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.