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Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications. (English) Zbl 1244.26038
From the abstract: We establish some new Gronwall-Bellman-type delay dynamic inequalities on time scales, which can be used as a handy tool in the investigation of qualitative as well as quantitative analysis of solutions of delay dynamic equations on time scales.

MSC:
26D15 Inequalities for sums, series and integrals
26E70 Real analysis on time scales or measure chains
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[1] Ou-Iang, L., The boundedness of solutions of linear differential equations y″+A(t)y=0, Shuxue jinzhan, 3, 409-415, (1957)
[2] Jiang, F.C.; Meng, F.W., Explicit bounds on some new nonlinear integral inequality with delay, J. comput. appl. math., 205, 479-486, (2007) · Zbl 1135.26015
[3] Yuan, Z.L.; Yuan, X.W.; Meng, F.W., Some new delay integral inequalities and their applications, Appl. math. comput., 208, 231-237, (2009) · Zbl 1178.26031
[4] Gronwall, T.H., Note on the derivatives with respect to a parameter of solutions of a system of differential equations, Ann. math., 20, 292-296, (1919) · JFM 47.0399.02
[5] Bellman, R., The stability of solutions of linear differential equations, Duke math. J., 10, 643-647, (1943) · Zbl 0061.18502
[6] Cheung, W.S.; Ren, J.L., Discrete non-linear inequalities and applications to boundary value problems, J. math. anal. appl., 319, 708-724, (2006) · Zbl 1116.26016
[7] 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
[8] Lipovan, O., A retarded integral inequality and its applications, J. math. anal. appl., 285, 436-443, (2003) · Zbl 1040.26007
[9] Ma, Q.H.; Yang, E.H., Some new gronwall – bellman – bihari type integral inequalities with delay, Period. math. hungar., 44, 2, 225-238, (2002) · Zbl 1006.26011
[10] 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
[11] Ma, Q.H., Estimates on some power nonlinear volterra – fredholm type discrete inequalities and their applications, J. comput. appl. math., 233, 2170-2180, (2010) · Zbl 1184.26024
[12] Li, L.Z.; Meng, F.W.; He, L.L., Some generalized integral inequalities and their applications, J. math. anal. appl., 372, 339-349, (2010) · Zbl 1217.26049
[13] Lipovan, O., Integral inequalities for retarded Volterra equations, J. math. anal. appl., 322, 349-358, (2006) · Zbl 1103.26018
[14] Pachpatte, B.G., Explicit bounds on certain integral inequalities, J. math. anal. appl., 267, 48-61, (2002) · Zbl 0996.26008
[15] Erdélyi, T., Remez-type inequalities and their applications, J. comput. appl. math., 42, 2, 167-209, (1993) · Zbl 0781.30002
[16] Wang, M.J., Some inequalities via probabilistic method, Comput. math. appl., 59, 3481-3488, (2010) · Zbl 1197.60020
[17] Marinković, S.; Rajković, P.; Stanković, M., The inequalities for some types of q-integrals, Comput. math. appl., 56, 2490-2498, (2008) · Zbl 1165.33308
[18] Pachpatte, B.G., On some new nonlinear retarded integral inequalities, J. inequal. pure appl. math., 5, (2004), (Article 80) · Zbl 1068.26020
[19] Sun, Y.G., On retarded integral inequalities and their applications, J. math. anal. appl., 301, 265-275, (2005) · Zbl 1057.26022
[20] Ferreira, Rui A.C.; Torres, Delfim F.M., Generalized retarded integral inequalities, Appl. math. lett., 22, 876-881, (2009) · Zbl 1171.26328
[21] Xu, R.; Sun, Y.G., On retarded integral inequalities in two independent variables and their applications, Appl. math. comput., 182, 1260-1266, (2006) · Zbl 1118.26025
[22] Hilger, S., Analysis on measure chains-a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[23] Bohner, M.; Erbe, L.; Peterson, A., Oscillation for nonlinear second order dynamic equations on a time scale, J. math. anal. appl., 301, 2, 491-507, (2005) · Zbl 1061.34018
[24] Xing, Y.; Han, M.; Zheng, G., Initial value problem for first-order integro-differential equation of Volterra type on time scales, Nonlinear anal., 60, 3, 429-442, (2005) · Zbl 1065.45005
[25] Agarwal, R.P.; Bohner, M.; O’Regan, D.; Peterson, A., Dynamic equations on time scales: a survey, J. comput. appl. math., 141, 1-2, 1-26, (2006)
[26] Wong, F.H.; Yeh, C.C.; Yu, S.L.; Hong, C.H., Young’s inequality and related results on time scales, Appl. math. lett., 18, 983-988, (2005) · Zbl 1080.26025
[27] Sarikaya, M.Z., On weighted iyengar type inequalities on time scales, Appl. math. lett., 22, 1340-1344, (2009) · Zbl 1173.26325
[28] Liu, W.J.; Li, C.C.; Hao, Y.M., Further generalization of some double integral inequalities and applications, Acta. math. univ. Comenian., 77, 1, 147-154, (2008) · Zbl 1164.26017
[29] Cheng, X.L., Improvement of some ostrowski – grüss type inequalities, Comput. math. appl., 42, 109-114, (2001) · Zbl 0980.26011
[30] Bohner, M.; Matthews, T., The grüss inequality on time scales, Commun. math. anal., 3, 1, 1-8, (2007) · Zbl 1167.26317
[31] Ngô, Q.A., Some Mean value theorems for integrals on time scales, Appl. math. comput., 213, 322-328, (2009) · Zbl 1168.26315
[32] Agarwal, R.; Bohner, M.; Peterson, A., Inequalities on time scales: a survey, Math. inequal. appl., 4, 4, 535-557, (2001) · Zbl 1021.34005
[33] Saker, S.H., Some nonlinear dynamic inequalities on time scales and applications, J. math. inequal., 4, 561-579, (2010) · Zbl 1207.26034
[34] Saker, S.H., Some nonlinear dynamic inequalities on time scales, Math. inequal. appl., 14, 3, 633-645, (2011) · Zbl 1222.26032
[35] Li, W.N., Some delay integral inequalities on time scales, Comput. math. appl., 59, 1929-1936, (2010) · Zbl 1189.26046
[36] Ma, Q.H.; Pečarić, J., The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales, Comput. math. appl., 61, 2158-2163, (2011) · Zbl 1219.34118
[37] Saker, S.H., Nonlinear dynamic inequalities of gronwall – bellman type on time scales, E. J. qual. theory differ. equat., 86, 1-26, (2011) · Zbl 1340.26053
[38] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001
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