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On weakly singular versions of discrete nonlinear inequalities and applications. (English) Zbl 1474.26098

Summary: Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing weakly singular inequalities and can be used in the analysis of nonlinear Volterra type difference equations with weakly singular kernels. A few applications to the upper bound and the uniqueness of solutions of nonlinear difference equations are also involved.

MSC:

26D15 Inequalities for sums, series and integrals
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[1] Agarwal, R. P.; Deng, S.; Zhang, W., Generalization of a retarded Gronwall-like inequality and its applications, Applied Mathematics and Computation, 165, 3, 599-612 (2005) · Zbl 1078.26010 · doi:10.1016/j.amc.2004.04.067
[2] Deng, S., Nonlinear discrete inequalities with two variables and their applications, Applied Mathematics and Computation, 217, 5, 2217-2225 (2010) · Zbl 1205.26026 · doi:10.1016/j.amc.2010.07.022
[3] Martyniuk, A. A.; Lakshamikantham, V.; Leela, S., Motion Stability: The Method of Integral Inequalities (1977), Kiev, Russia: Naukova Dumka, Kiev, Russia
[4] 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
[5] Deng, S.; Wu, Y.; Li, X., Nonlinear delay discrete inequalities and their applications to Volterra type difference equations, Advances in Difference Equations, 2010 (2010) · Zbl 1187.39018 · doi:10.1186/1687-1847-2010-795145
[6] Beesack, P. R., More generalised discrete Gronwall inequalities, Zeitschrift für Angewandte Mathematik und Mechanik, 65, 12, 583-595 (1985) · Zbl 0578.26004 · doi:10.1002/zamm.19850651202
[7] Dixon, J.; McKee, S., Weakly singular discrete Gronwall inequalities, Zeitschrift für Angewandte Mathematik und Mechanik, 66, 11, 535-544 (1986) · Zbl 0627.65136 · doi:10.1002/zamm.19860661107
[8] Dauer, J. P.; Mahmudov, N. I., Integral inequalities and mild solutions of semilinear neutral evolution equations, Journal of Mathematical Analysis and Applications, 300, 1, 189-202 (2004) · Zbl 1074.34059 · doi:10.1016/j.jmaa.2004.06.040
[9] Henry, D., Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840 (1981), New York, NY, USA: Springer, New York, NY, USA · Zbl 0456.35001
[10] Januszewski, J., On Volterra integral equations with weakly singular kernels in Banach spaces, Demonstratio Mathematica, 26, 1, 131-136 (1993) · Zbl 0791.45006
[11] Medveď, M., A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, Journal of Mathematical Analysis and Applications, 214, 2, 349-366 (1997) · Zbl 0893.26006 · doi:10.1006/jmaa.1997.5532
[12] Medved, M., Nonlinear singular difference inequalities suitable for discretizations of parabolic equations, Demonstratio Mathematica, 33, 3, 517-525 (2000) · Zbl 0985.65110
[13] Medved, M., On singular versions of Bihari and Wendroff-Pachpatte type integral inequalities and their application, Tatra Mountains Mathematical Publications, 38, 163-174 (2007) · Zbl 1164.26021
[14] Ma, Q. H.; Yang, E. H., Estimates on solutions of some weakly singular Volterra integral inequalities, Acta Mathematicae Applicatae Sinica, 25, 3, 505-515 (2002) · Zbl 1032.26009
[15] Yang, E. H.; Ma, Q. H.; Tan, M. C., Discrete analogues of a new class of nonlinear Volterra singular integral inequalitie, Journal of Jinan University, 28, 1, 1-6 (2007)
[16] Cheung, W. S.; Ma, Q. H.; Tseng, S., Some new nonlinear weakly singular integral inequalities of wendroff type with applications, Journal of Inequalities and Applications, 2008 (2008) · Zbl 1151.26014 · doi:10.1155/2008/909156
[17] Deng, S.; Prather, C., Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay, Journal of Inequalities in Pure and Applied Mathematics, 9, 2, article 34 (2008) · Zbl 1172.26321
[18] Furati, K. M.; Tatar, N., Power-type estimates for a nonlinear fractional differential equation, Nonlinear Analysis: Theory, Methods & Applications, 62, 6, 1025-1036 (2005) · Zbl 1078.34028 · doi:10.1016/j.na.2005.04.010
[19] Lakhal, F., A new nonlinear integral inequality of Wendroff type with continuous and weakly singular kernel and its application, Journal of Mathematical Inequalities, 6, 3, 367-379 (2012) · Zbl 1254.26024 · doi:10.7153/jmi-06-36
[20] Mazouzi, S.; Tatar, N.-E., New bounds for solutions of a singular integro-differential inequality, Mathematical Inequalities & Applications, 13, 2, 427-435 (2010) · Zbl 1189.42008 · doi:10.7153/mia-13-32
[21] Szufla, S., On the Volterra integral equation with weakly singular kernel, Mathematica Bohemica, 131, 3, 225-231 (2006) · Zbl 1110.45003
[22] Szufla, S., On the Hammerstein integral equation with weakly singular kernel, Funkcialaj Ekvacioj, 34, 2, 279-285 (1991) · Zbl 0753.45011
[23] Tatar, N.-E., On an integral inequality with a kernel singular in time and space, Journal of Inequalities in Pure and Applied Mathematics, 4, 4, article 82 (2003) · Zbl 1058.26017
[24] Wang, H.; Zheng, K., Some nonlinear weakly singular integral inequalities with two variables and applications, Journal of Inequalities and Applications, 2010 (2010) · Zbl 1204.26046 · doi:10.1155/2010/345701
[25] Zheng, K., Bounds on some new weakly singular Wendroff-type integral inequalities and applications, Journal of Inequalities and Applications, 2013, article 159 (2013) · Zbl 1284.26024 · doi:10.1186/1029-242X-2013-159
[26] Pachpatte, B. G., On some new inequalities related to certain inequalities in the theory of differential equations, Journal of Mathematical Analysis and Applications, 189, 1, 128-144 (1995) · Zbl 0824.26010 · doi:10.1006/jmaa.1995.1008
[27] Ma, Q. H.; Pečarić, J., Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations, Journal of Mathematical Analysis and Applications, 341, 2, 894-905 (2008) · Zbl 1142.26015 · doi:10.1016/j.jmaa.2007.10.036
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