×
Ma, Qing-Hua

Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications. (English) Zbl 1184.26024

J. Comput. Appl. Math. 233, No. 9, 2170-2180 (2010).
The author gives some explicit bounds to some power nonlinear Volterra-Fredholm type discrete inequalities involving a function of two variables. These inequalities can be used as effective tools in the study of quantitative properties of solutions of certain classes of partial finite and sum-difference equations. The author presents applications of these inequalities to the study of boundedness, uniqueness and continuous dependence of the solutions of certain Volterra-Fredholm type sum-difference equations.
Reviewer: Stamatis Koumandos (Nicosia)

Cited in 2 Reviews
Cited in 26 Documents

MSC:

26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
39A10 Additive difference equations

Keywords:

Power nonlinear discrete inequalities; Volterra-Fredholm type; Gronwall-Bellman-Ou-Iang type; explicit bounds; sum-difference equations
Cite Review PDF
Full Text: DOI

References:

[1] Gronwall, T. H., Note on the derivatives with respect to a parameter of solutions of a system of differential equations, Ann. of Math., 20, 292-296 (1919) · JFM 47.0399.02
[2] Bellman, R., The stability of solutions of linear differential equations, Duke Math. J., 10, 643-647 (1943) · Zbl 0061.18502
[3] Bainov, D.; Simeonov, P., Integral Inequalities and Applications (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0759.26012
[4] Mitrinović, D. S.; Pečarić, J. K.; Fink, A. M., Inequalities Involving Functions and their Integrals and Derivatives (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dortlrecht, Boston, London · Zbl 0744.26011
[5] Pachpatte, B. G., Inequalities for Differential and Integral Equations (1998), Academic Press: Academic Press New York · Zbl 1032.26008
[6] Ou-Iang, L., The boundedness of solutions of linear differential equations \(y'' + A(t) y^\prime = 0\), Shuxue Jinzhan, 3, 409-418 (1957)
[7] Cheung, W. S.; Ma, Q. H., Nonlinear retarded integral inequalities for functions in two variables, J. Concr. Appl. Math., 2, 119-134 (2004) · Zbl 1091.26012
[8] Cheung, W. S.; Ma, Q. H., On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications, J. Inequal. Appl., 8, 347-361 (2005) · Zbl 1112.26018
[9] Cheung, W. S., Some discrete nonlinear inequalities and applications to boundary value problems for difference equations, J. Difference Equ. Appl., 10, 213-223 (2004) · Zbl 1045.26007
[10] Cheung, W. S.; Ren, J. L., Discrete nonlinear inequalities and applications to boundary value problems, J. Math. Anal. Appl., 319, 708-724 (2006) · Zbl 1116.26016
[11] Cheung, W. S., Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Anal. TMA, 64, 2112-2128 (2006) · Zbl 1094.26011
[12] Jiang, F. C.; Meng, F. W., Explicit bounds on some new nonlinear integral inequalities with delay, J. Comput. Appl. Math., 205, 479-486 (2007) · Zbl 1135.26015
[13] Li, W. N.; Han, M. A.; Meng, F. W., Some new delay integral inequalities and their applications, J. Comput. Appl. Math., 180, 191-200 (2005) · Zbl 1067.26019
[14] Lipovan, O., A retarded integral inequality and its applications, J. Math. Anal. Appl., 285, 436-443 (2003) · Zbl 1040.26007
[15] Ma, Q. H.; Yang, E. H., Some new nonlinear delay integral inequalities, J. Math. Anal. Appl., 252, 864-878 (2000) · Zbl 0974.26015
[16] Ma, Q. H., N-independent-variable discrete inequalities of Gronwall-Ou-Iang type, Ann. Differential Equations, 16, 813-820 (2000)
[17] Ma, Q. H.; Debnath, L., A more generalized Gronwal-like integral inequality with applications, Int. J. Math. Math. Sci., 33, 927-934 (2003) · Zbl 1014.26015
[18] Ma, Q. H.; Cheung, W. S., Some new nonlinear difference inequalities and their applications, J. Comput. Appl. Math., 202, 339-351 (2007) · Zbl 1121.26019
[19] Ma, Q. H., Some new nonlinear Volterra-Fredholm-type discrete inequalities and their applications, J. Comput. Appl. Math., 216, 451-466 (2008) · Zbl 1152.26324
[20] 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
[21] Meng, F. W.; Li, W. N., On some new integral inequalities and their applications, Appl. Math. Comput., 148, 381-392 (2004) · Zbl 1045.26009
[22] Meng, F. W.; Ji, D., On some new nonlinear discrete inequalities and their applications, J. Comput. Appl. Math., 208, 425-433 (2007) · Zbl 1128.26015
[23] Pachpatte, B. G., On some new inequalities related to certain inequality arising in the theory of differential equations, J. Math. Anal. Appl., 189, 128-144 (1995) · Zbl 0824.26010
[24] Pang, P. Y.H.; Agarwal, R. P., On an integral inequality and discrete analogue, J. Math. Anal. Appl., 194, 569-577 (1995) · Zbl 0845.26009
[25] Yang, E. H., On some nonlinear integral and discrete inequalities related to Ou-Iang’s inequality, Acta Math. Sin. (Engl. Ser.), 14, 353-360 (1998) · Zbl 0913.26007
[26] Yang, E. H., A new integral inequality with power nonlinear and its discrete analogue, Acta Math. Appl. Sin., 17, 233-239 (2001) · Zbl 0987.26007
[27] McKee, S.; Tang, T., Integral inequalities and their application in numerical analysis, Fasc. Math., 23, 67-76 (1991) · Zbl 0753.65047
[28] McKee, S.; Tang, T.; Diogo, T., An Euler-type method for two-dimensional Volterra integral equations of the first kind, IMA J. Numer. Anal., 20, 423-440 (2000) · Zbl 0965.65143
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.