Discrete non-linear inequalities and applications to boundary value problems. (English) Zbl 1116.26016

The authors establish explicit upper bounds for some nonlinear discrete inequalities of Gronwall-Bellman-Ou-Yang type. The results are further generalizations of inequalities of this type of W.-S. Cheung [J. Difference Equ. Appl. 10, No. 2, 213–223 (2004; Zbl 1045.26007)] and B. G. Pachpatte [J. Math. Anal. Appl. 189, No. 1, 128–144 (1995; Zbl 0824.26010) and JIPAM, J. Inequal. Pure Appl. Math. 3, No. 2, Paper No. 18 (2002; Zbl 0994.26017)]. Applications of the results obtained to study the boundedness, uniqueness, and continuous dependence of solutions of certain discrete boundary value problems for difference equations are given.


26D15 Inequalities for sums, series and integrals
39A10 Additive difference equations
Full Text: DOI


[1] Bellman, R., The stability of solutions of linear differential equations, Duke math. J., 10, 643-647, (1943) · Zbl 0061.18502
[2] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta math. acad. sci. hungar., 7, 71-94, (1956) · Zbl 0070.08201
[3] Cheung, W.S., On some new integrodifferential inequalities of the Gronwall and Wendroff type, J. math. anal. appl., 178, 438-449, (1993) · Zbl 0796.26007
[4] 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
[5] W.S. Cheung, Some retarded Gronwall-Bellman-Ou-Iang-type inequalities and applications to initial boundary value problems, preprint
[6] W.S. Cheung, Q.H. Ma, Nonlinear retarded integral inequalities for functions in two variables, J. Concrete Appl. Math., in press · Zbl 1091.26012
[7] Dafermos, C.M., The second law of thermodynamics and stability, Arch. ration. mech. anal., 70, 167-179, (1979) · Zbl 0448.73004
[8] Haraux, H., Nonlinear evolution equation: global behavior of solutions, Lecture notes in math., vol. 841, (1981), Springer-Verlag Berlin · Zbl 0461.35002
[9] Lipovan, O., A retarded Gronwall-like inequality and its applications, J. math. anal. appl., 252, 389-401, (2000) · Zbl 0974.26007
[10] Ma, Q.M.; Yang, E.H., On some new nonlinear delay integral inequalities, J. math. anal. appl., 252, 864-878, (2000) · Zbl 0974.26015
[11] Ma, Q.M.; Yang, E.H., Some new gronwall – bellman – bihari type integral inequalities with delay, Period. math. hungar., 44, 225-238, (2002) · Zbl 1006.26011
[12] Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M., Inequalities involving functions and their integrals and derivatives, (1991), Kluwer Academic Dordrecht · Zbl 0744.26011
[13] Ou-Iang, L., The boundedness of solutions of linear differential equations \(y'' + A(t) y = 0\), Shuxue jinzhan, 3, 409-415, (1957)
[14] Pachpatte, B.G., Inequalities for differential and integral equations, (1998), Academic Press New York · Zbl 1032.26008
[15] Pachpatte, B.G., On some new inequalities related to a certain inequality arising in the theory of differential equations, J. math. anal. appl., 251, 736-751, (2000) · Zbl 0987.26010
[16] Pachpatte, B.G., On some new inequalities related to certain inequalities in the theory of differential equations, J. math. anal. appl., 189, 128-144, (1995) · Zbl 0824.26010
[17] Pachpatte, B.G., On some retarded integral inequalities and applications, J. inequal. pure appl. math., 3, (2002) · Zbl 0994.26017
[18] 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
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.