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Some difference inequalities for iterated sums with applications. (English) Zbl 1470.26041

Summary: The main objective of this paper is to establish two new nonlinear sum-difference inequalities with multiple iterated sums. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. These new inequalities can be used as handy tools in the study of the estimation of solutions of difference equations.

MSC:

26D15 Inequalities for sums, series and integrals
39B72 Systems of functional equations and inequalities
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[1] Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20, 4, 292-296 (1919) · JFM 47.0399.02 · doi:10.2307/1967124
[2] Bellman, R., The stability of solutions of linear differential equations, Duke Mathematical Journal, 10, 643-647 (1943) · Zbl 0061.18502 · doi:10.1215/S0012-7094-43-01059-2
[3] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Mathematica Academiae Scientiarum Hungaricae, 7, 81-94 (1956) · Zbl 0070.08201 · doi:10.1007/BF02022967
[4] Baĭnov, D.; Simeonov, P., Integral Inequalities and Applications. Integral Inequalities and Applications, Mathematics and Its Applications, 57, xii+245 (1992), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0759.26012
[5] Pachpatte, B. G., Inequalities for Differential and Integral Equations. Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering, 197, x+611 (1998), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 1032.26008
[6] Lipovan, O., A retarded Gronwall-like inequality and its applications, Journal of Mathematical Analysis and Applications, 252, 1, 389-401 (2000) · Zbl 0974.26007 · doi:10.1006/jmaa.2000.7085
[7] Zhang, W.; Deng, S., Projected Gronwall-Bellman’s inequality for integrable functions, Mathematical and Computer Modelling, 34, 3-4, 393-402 (2001) · Zbl 0992.26013 · doi:10.1016/S0895-7177(01)00070-X
[8] Kim, B.-I., On some Gronwall type inequalities for a system integral equation, Bulletin of the Korean Mathematical Society, 42, 4, 789-805 (2005) · Zbl 1090.26017 · doi:10.4134/BKMS.2005.42.4.789
[9] Pachpatte, B. G., On certain nonlinear integral inequalities involving iterated integrals, Tamkang Journal of Mathematics, 37, 3, 261-271 (2006) · Zbl 1129.26017
[10] Cheung, W.-S., Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 64, 9, 2112-2128 (2006) · Zbl 1094.26011 · doi:10.1016/j.na.2005.08.009
[11] Wang, W.-S., A generalized retarded Gronwall-like inequality in two variables and applications to BVP, Applied Mathematics and Computation, 191, 1, 144-154 (2007) · Zbl 1193.26014 · doi:10.1016/j.amc.2007.02.099
[12] Abdeldaim, A.; Yakout, M., On some new integral inequalities of Gronwall-Bellman-Pachpatte type, Applied Mathematics and Computation, 217, 20, 7887-7899 (2011) · Zbl 1220.26012 · doi:10.1016/j.amc.2011.02.093
[13] Guo, Z.; Zhou, X.; Wang, W.-S., Interval oscillation criteria of second order mixed nonlinear impulsive differential equations with delay, Abstract and Applied Analysis, 2012 (2012) · Zbl 1245.34070 · doi:10.1155/2012/351709
[14] Wang, W. S.; Huang, D.; Li, X., Generalized retarded nonlinear integral inequalities involving iterated integrals and an application, Journal of Inequalities and Applications, 2013, article 376 (2013) · Zbl 1285.26046 · doi:10.1186/1029-242X-2013-376
[15] Hull, T. E.; Luxemburg, W. A. J., Numerical methods and existence theorems for ordinary differential equations, Numerische Mathematik, 2, 1, 30-41 (1960) · Zbl 0089.29003 · doi:10.1007/BF01386206
[16] Willett, D.; Wong, J. S. W., On the discrete analogues of some generalizations of Gronwall’s inequality, Monatshefte für Mathematik, 69, 362-367 (1965) · Zbl 0145.06003
[17] Sugiyama, S., On the stability problems of difference equations, Bulletin of Science and Engineering Research Laboratory. Waseda University, 45, 140-144 (1969)
[18] Pachpatte, B. G.; Deo, S. G., Stability of discrete-time systems with retarded argument, Utilitas Mathematica, 4, 15-33 (1973) · Zbl 0274.93041
[19] Pachpatte, B. G., On discrete inequalities related to Gronwall’s inequality, Proceedings of the Indian Academy of Sciences A, 85, 1, 26-40 (1977) · Zbl 0349.39002
[20] Pachpatte, B. G., Finite difference inequalities and discrete time control systems, Indian Journal of Pure and Applied Mathematics, 9, 12, 1282-1290 (1978) · Zbl 0393.93018
[21] Yang, E., A new nonlinear discrete inequality and its application, Annals of Differential Equations, 17, 3, 261-267 (2001) · Zbl 0997.26009
[22] Pachpatte, B. G., On some fundamental integral inequalities and their discrete analogues, Journal of Inequalities in Pure and Applied Mathematics, 2, 2, article 15 (2001) · Zbl 0989.26011
[23] Meng, F. W.; Li, W. N., On some new nonlinear discrete inequalities and their applications, Journal of Computational and Applied Mathematics, 158, 2, 407-417 (2003) · Zbl 1032.26019 · doi:10.1016/S0377-0427(03)00475-8
[24] Cheung, W.-S.; Ren, J., Discrete non-linear inequalities and applications to boundary value problems, Journal of Mathematical Analysis and Applications, 319, 2, 708-724 (2006) · Zbl 1116.26016 · doi:10.1016/j.jmaa.2005.06.064
[25] Pachpatte, B. G., Integral and Finite Difference Inequalities and Applications. Integral and Finite Difference Inequalities and Applications, North-Holland Mathematics Studies, 205, x+309 (2006), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1104.26015
[26] Sheng, W.; Li, W. N., Bounds on certain nonlinear discrete inequalities, Journal of Mathematical Inequalities, 2, 2, 279-286 (2008) · Zbl 1152.26327 · doi:10.7153/jmi-02-25
[27] Ma, Q.-H.; Cheung, W.-S., Some new nonlinear difference inequalities and their applications, Journal of Computational and Applied Mathematics, 202, 2, 339-351 (2007) · Zbl 1121.26019 · doi:10.1016/j.cam.2006.02.036
[28] Cho, Y. J.; Dragomir, S. S.; Kim, Y.-H., On some integral inequalities with iterated integrals, Journal of the Korean Mathematical Society, 43, 3, 563-578 (2006) · Zbl 1102.26015 · doi:10.4134/JKMS.2006.43.3.563
[29] Wang, W.-S., A generalized sum-difference inequality and applications to partial difference equations, Advances in Difference Equations, 2008 (2008) · Zbl 1149.39010 · doi:10.1155/2008/695495
[30] Wang, W.-S., Estimation on certain nonlinear discrete inequality and applications to boundary value problem, Advances in Difference Equations, 2009 (2009) · Zbl 1168.26318 · doi:10.1155/2009/708587
[31] Zheng, K.-L.; Zhong, S.-M.; Ye, M., Discrete nonlinear inequalities in time control systems, Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA ’09) · doi:10.1109/ICACIA.2009.5361069
[32] Wang, W.-S.; Li, Z.; Cheung, W.-S., Some new nonlinear retarded sum-difference inequalities with applications, Advances in Difference Equations, 2011, 1, article 41 (2011) · Zbl 1272.26014 · doi:10.1186/1687-1847-2011-41
[33] Zhou, H.; Huang, D.; Wang, W.-S.; Xu, J.-X., Some new difference inequalities and an application to discrete-time control systems, Journal of Applied Mathematics, 2012 (2012) · Zbl 1263.93143 · doi:10.1155/2012/214609
[34] He, Q.; Sun, T.; Xi, H., Dynamics of a family of nonlinear delay difference equations, Abstract and Applied Analysis, 2013 (2013) · Zbl 1272.39008 · doi:10.1155/2013/456530
[35] Chen, Z.-X.; Shon, K. H., Fixed points of meromorphic solutions for some difference equations, Abstract and Applied Analysis, 2013 (2013) · Zbl 1272.30043 · doi:10.1155/2013/496096
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