×

zbMATH — the first resource for mathematics

Nonlinear discrete inequalities with two variables and their applications. (English) Zbl 1205.26026
Summary: An error in the proof of Theorem 1 by K. Zheng, Y. Wu and S. Zhong [Appl. Math. Comput. 207, No. 1, 140–147 (2009; Zbl 1178.26032)] is reported. This paper gives the right proof under some additional condition. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.P., On an integral inequality in n independent variables, J. math. anal. appl., 85, 192-196, (1982) · Zbl 0487.26008
[2] Agarwal, R.P.; Deng, S.; Zhang, W., Generalization of a retarded Gronwall-like inequality and its applications, Appl. math. comput., 165, 599-612, (2005) · Zbl 1078.26010
[3] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta math. acad. sci. hung., 7, 81-94, (1956) · Zbl 0070.08201
[4] Borysenko, S.; Matarazzo, G.; Pecoraro, M., A generalization of bihari’s lemma for discontinuous functions and its application to the stability problem of differential equations with impulse disturbance, Georgian math. J., 13, 229-238, (2006) · Zbl 1124.45005
[5] Cheung, W., Some new nonlinear inequalities and applications to boundary value problems, Nonlinear anal., 64, 2112-2128, (2006) · Zbl 1094.26011
[6] Cheung, W., Some discrete nonlinear inequalities and applications to boundary value problems for difference equations, J. difference equ. appl., 10, 213-223, (2004) · Zbl 1045.26007
[7] Cheung, W.; Ren, J., Discrete non-linear inequalities and applications to boundary value problems, J. math. anal. appl., 319, 708-724, (2006) · Zbl 1116.26016
[8] Choi, S.K.; Deng, S.; Koo, N.J.; Zhang, W., Nonlinear integral inequalities of bihari-type without class H, Math. inequal. appl., 8, 643-654, (2005) · Zbl 1090.26015
[9] S. Deng, Nonlinear delay discrete inequalities and their applications to Volterra type difference equations, submitted for publication.
[10] S. Deng, C. Prather, Nonlinear discrete inequalities of Bihari-type, submitted for publication.
[11] Horväth, L., Generalizations of special bihari type integral inequalities, Math. inequal. appl., 8, 441-449, (2005) · Zbl 1078.26015
[12] Lungu, N., On some gronwall – bihari – wendorff-type inequalities, Semin. fixed point theory cluj-napoca, 3, 249-254, (2002) · Zbl 1136.26304
[13] Pachpatte, B.G., On generalizations of bihari’s inequality, Soochow J. math., 31, 261-271, (2005) · Zbl 1076.26014
[14] Pachpatte, B.G., Integral inequalities of the bihari type, Math. inequal. appl., 5, 649-657, (2002) · Zbl 1019.26009
[15] Pachpatte, B.G., On bihari like integral and discrete inequalities, Soochow J. math., 17, 213-232, (1991) · Zbl 0745.26013
[16] Phat, V.N.; Park, J.Y., On the Gronwall inequality and asymptotic stability of nonlinear discrete systems with multiple delays, Dyn. syst. appl., 10, 577-588, (2001) · Zbl 1017.34077
[17] Pinto, M., Integral inequalities of bihari-type and applications, Funkcial. ekvac., 33, 387-403, (1990) · Zbl 0717.45004
[18] Popenda, J.; Agarwal, R.P., Discrete Gronwall inequalities in many variables, Comput. math. appl., 38, 63-70, (1999) · Zbl 0941.39011
[19] Salem, Sh.; Raslan, K.R., Some new discrete inequalities and their applications, JIPAM J. inequal. pure appl. math., 5, (2004), (article 2, 9 pp.) · Zbl 1059.26022
[20] Wong, F.; Yeh, C.; Hong, C., Gronwall inequalities on time scales, Math. inequal. appl., 9, 75-86, (2006) · Zbl 1091.26020
[21] Zheng, K.; Wu, Y.; Zhong, S., Discrete nonlinear integral inequalities in two variables and their applications, Appl. math. comput., 207, 140-147, (2009) · Zbl 1178.26032
[22] Zhang, W.; Deng, S., Projected gronwall – bellman’s inequality for integrable functions, Math. comput. modell., 34, 393-402, (2001) · Zbl 0992.26013
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.