Abstract
Discrete Bihari-type inequalities with n nonlinear terms are discussed, which generalize some known results and may be used in the analysis of certain problems in the theory of difference equations. Examples to illustrate the boundedness of solutions of a difference equation are also given.
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References
Agarwal, R.P. On an integral inequality in n independent variables. J. Math. Anal. Appl., 85: 192–196 (1982)
Agarwal, R.P., Deng, S., Zhang, W. Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput., 165: 599–612 (2005)
Bihari, I. A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hungar., 7: 81–94 (1956)
Borysenko, S., Matarazzo, G., Pecoraro, M. A generalization of Bihari’s lemma for discontinuous functions and its application to the stability problem of differential equations with impulse disturbance. Georgian Math. J., 13: 229–238 (2006)
Cheung, W. Gronwall-Bellman-Bihari integral inequalities and application to nonlinear Volterra integral equation. Southeast Asian Bull. Math., 29: 447–454 (2005)
Choi, S.K., Deng, S., Koo, N.J., Zhang, W. Nonlinear integral inequalities of Bihari-type without class H. Math. Inequal. Appl., 8: 643–654 (2005)
Deng, S., Prather, C. Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay. J. Inequal. Pure Appl. Math., 9: Article 34 (2008)
Horváth, L. Generalizations of special Bihari type integral inequalities. Math. Inequal. Appl., 8: 441–449 (2005)
Lungu, N. On some Gronwall-Bihari-Wendorff-type inequalities. Semin. Fixed Point Theory Cluj-Napoca, 3: 249–254 (2002)
Pachpatte, B.G. On generalizations of Bihari’s inequality. Soochow J. Math., 31: 261–271 (2005)
Pachpatte, B.G. Integral inequalities of the Bihari type. Math. Inequal. Appl., 5: 649–657 (2002)
Pachpatte, B.G. On Bihari like integral and discrete inequalities. Soochow J. Math., 17: 213–232 (1991)
Phat, V.N., Park, J.Y. On the Gronwall inequality and asymptotic stability of nonlinear discrete systems with multiple delays. Dynam. Systems Appl., 10: 577–588 (2001)
Pinto, M. Integral inequalities of Bihari-type and applications. Funkcial. Ekvac., 33: 387–403 (1990)
Popenda, J., Agarwal, R.P. Discrete Gronwall inequalities in many variables. Comput. Math. Appl., 38: 63–70 (1999)
Lü, T., Huang, Y. A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind. J. Math. Anal. Appl., 282: 56–62 (2003)
Wang, W., Zhou, X. An extension to nonlinear sum-difference inequality and applications. Advances in Difference Equations, Article ID 486895 (2009)
Wong, F., Yeh, C., Hong, C. Gronwall inequalities on time scales. Math. Inequal. Appl., 9: 75–86 (2006)
Zhang, W., Deng, S. Projected Gronwall-Bellman’s inequality for integrable functions. Math. Comput. Modelling, 34: 393–402 (2001)
Zheng, K., Wu, Y., Deng, S. Nonlinear integral inequalities in two independent variables and their applications. J. Inequal. Appl., Article ID 32949 (2007)
Zheng, K., Wu, Y., Zhong, S. Discrete nonlinear integral inequalities in two variables and their applications. Appl. Math. Comput., 207: 140–147 (2009)
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Supported by the Program of Education Department of Sichuan Province (No. 10ZA173).
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Wu, Y. Nonlinear discrete inequalities of Bihari-type and applications. Acta Math. Appl. Sin. Engl. Ser. 29, 603–614 (2013). https://doi.org/10.1007/s10255-013-0235-1
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DOI: https://doi.org/10.1007/s10255-013-0235-1