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http://dx.doi.org/10.14403/jcms.2013.26.1.071

ON SOME NONLINEAR INTEGRAL INEQUALITIES ON TIME SCALES  

Choi, Sung Kyu (Department of Mathematics Chungnam National University)
Koo, Namjip (Department of Mathematics Chungnam National University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.26, no.1, 2013 , pp. 71-84 More about this Journal
Abstract
In this paper we study some nonlinear Pachpatte type integral inequalities on time scales by using a Bihari type inequality. Our results unify some continuous inequalities and their corresponding discrete analogues, and extend these inequalities to dynamic inequalities on time scales. Furthermore, we give some examples concerning our results.
Keywords
time scales; dynamic inequality; Gronwall-Bellman inequality; integral inequality;
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1 W. N. Li, Some Pachpatte type inequalitis on time scales, Comput. Math. Appl. 57 (2009), 275-282.   DOI   ScienceOn
2 W. N. Li, Some new dynamic inequalitis on time scales, J. Math. Anal. Appl. 319 (2006), 802-814.   DOI   ScienceOn
3 R. P. Agarwal, Difference Equations and Inequalities, 2nd ed., Marcel Dekker, Inc., New York, 2000.
4 R. P. Agarwal, M. Bohner, and A. Peterson, Inequalities on time scales: A survey, J. Math. Inequal. Appl. 4 (2001), 535-557.
5 D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequlities involving functions and their integrals and derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
6 J. A. Oguntuase, On an inequality of Gronwall, J. Inequal. Pure and App. Math. 2 (2001), Issue 1, Article 9, 6 pages.
7 B. G. Pachpatte, A note on Gronwall-Bellman inequality, J. Math. Anal. Appl. 44 (1973), 758-762.   DOI
8 B. G. Pachpatte, On some retarded integral inequalities and applications, J. Inequal. Pure and App. Math. 3 (2002), Issue 2, Article 18, 7 pages.
9 B. G. Pachpatte, Inequalities for Finite Difference Equations, Marcel Dekker, Inc., New York, 2002.
10 D. B. Pachpatte, Explicit estimates on integral inequalities with time scales, J. Inequal. Pure Appl. Math. 7 (2006), Issue 4, Article 143.
11 D. Willett and J. S. W.Wong, On the discrete analogues of some generalizations of Gronwall's inequality, Monatsh Math. 69 (1965), 362-367.   DOI
12 F. Wong. C. C. Yeh, and C. H. Hong, Gronwall inequalities on time scales, Math. Inequal. Appl. 9 (2006), 75-86.
13 E. Akin-Bohner, M. Bohner, and F. Akin, Pachpatte inequality on time scales, J. Inequal. Pure and App. Math. 6 (2005), Issue 1, Article 6, 23 pages.
14 J. Baoguo, L. Erbe and A. Peterson, Oscillation of sublinear Emden-Fowler dynamic equations on time scales, J. Difference Equ. Appl. 16 (2010), 217-226.   DOI   ScienceOn
15 D. Bainov and P. Simeonov, Integral Inequalites and Applications, Academic Publishers, Dordrecht, 1992.
16 M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
17 R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943), 643-647.   DOI
18 I. Bihari, A generalization of a lemma of Bellman and its application to unique- ness of differential equations, Acta Math. Acad. Sci. Hungar 7 (1956), 71-94.   DOI   ScienceOn
19 M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, 2001.
20 S. K. Choi, B. Kang, and N. Koo, On inequalities of Gronwall type, J. Chungcheong Math. Soc. 20 (2007), 561-568.
21 S. K. Choi and N. Koo, On a Gronwall-type inequality on time scales, J. Chungcheong Math. Soc. 23 (2010), 137-147.
22 S. K. Choi and N. Koo, Volterra discrete inequalities of Bernoulli type, J. Inequal. Appl. 2010 (2010), Article ID 546423, p. 11.
23 S. K. Choi, S. Deng, N. J. Koo, and W. Zhang, Nonlinear integral inequalities of Bihari-tpye without class H, Math. Inequal. Appl. 8 (2005), no. 4, 643-654.
24 R. A. C. Ferreira and D. F. M. Torres, Generalizations of Gronwall-Bihari inequalities on time scales, J. Difference Equ. Appl. 15 (2009), no. 6, 529-539.
25 T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 20 (1919), 292- 296.   DOI   ScienceOn
26 S. Hilger, Analysis on measure chains- a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56.   DOI
27 V. Lakshmikantham and D. Trigiante, Theory of Difference Equations with Applications to Numerical Analysis, Academic Press, San Diego, 1988.
28 V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York, 1989.