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

MSC:
26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
39A10 Additive difference equations
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[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 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 Dortlrecht, Boston, London · Zbl 0744.26011
[5] Pachpatte, B.G., Inequalities for differential and integral equations, (1998), 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
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