About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications

Nonlinear Analysis, Theory, Methods and Applications

Volumn 66, Issue 10, 2007, Pages 2140-2165

  Mitropolskiy, Yu A ^a   Iovane, G ^b   Borysenko, S D ^c  

^a INSTITUTE OF HYDROBIOLOGY   (Ukraine)

^b UNIVERSITY OF SALERNO   (Italy)

^c NATIONAL TECHNICAL UNIVERSITY OF UKRAINE   (Ukraine)

Author keywords

Boundedness; Estimates; Impulsive integro differential equations; Integral inequalities; Integro sum inequalities

Indexed keywords

INTEGRODIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

BOUNDEDNESS; IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS; INTEGRAL INEQUALITIES; INTEGRO SUM INEQUALITIES;

FUNCTION EVALUATION;

EID: 33847207835     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2006.03.006     Document Type: Article

References (64)

1. R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, New York, 1992
2. Akinyele O. On Gronwall-Bellman-Bihari-type integral inequalities in several variables with retardation. J. Math. Anal. Appl. 104 1 (1984) 1-26
3. Azbelev N.V., and Tsalyuk Z.B. On Chaplygin's problem. Ukraïn. Mat. Zh. 10 1 (1958) 3-12
4. Bellman R. Theory of Solution Stability of Differential Equations (1954), Isdatelstvo Inostz. Lit, Moscow
5. Bihari J. A generalization of lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Scient. Hung. 7 1 (1956) 71-94
6. Blaquiere A. Differential games with piecewise continuous trajectories. Lecture Notes in Control and Inform. Sci. 3 (1977) 34-69
7. Borysenko S.D. Estimates technical stability on nonlinear approximation. Mat. Fiz. 29 (1981) 12-18
8. Borysenko S.D. About asymptotical stability on linear approximation of the systems with impulse influence. Ukraïn. Mat. Zh. 35 2 (1983) 144-150
9. Borysenko S.D. On solution stability on linear approximation of the systems with impulse. Differential Equations 22 5 (1986) 884-886
10. Borysenko S.D. Integro-sum inequalities for functions of many independent variables. Differential Equations 25 9 (1989) 1638-1641
11. S.D. Borysenko, Integro-sum inequalities and their use in the study of impulse systems, in: Proceeding of VI Internat. M. Kravchuk Conf., 1997, pp. 41-53
12. Borysenko S.D. On integro-sum functional inequalities. Differential Equations 34 6 (1998) 850
13. Borysenko S.D. Multidimensional integro-sum inequalities. Ukraïn. Mat. Zh. 50 2 (1998) 172-177
14. D.S. Borysenko, About one integral inequality for piece-wise continuous functions, in: Proc. X Int. Kravchuk Conf., Kyiv, 2004, p. 323
15. S.D. Borysenko, Construction of Mathematical Models, Vipol, Kiev, 1995
16. Borysenko S.D., Iovane G., and Giordano P. Investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces. Nonlinear Anal. 62 (2005) 345-363
17. Borysenko S.D., Ciarletta M., and Iovane G. Integro-sum inequalities and motion stability of systems with impulse perturbations. Nonlinear Anal. 62 (2005) 417-428
18. S.D. Borysenko, G. Iovane, P. Giordano, About some hyperbolic impulsive equation and estimates solutions, in: Proc. DE&CAS, Brest, 2005, pp. 9-14
19. M. Ciarletta, S.D. Borysenko, G. Iovane, About stability systems with impulse influence on nonlinear approximation, in: Proc. X Int. Kravchuk Conf., Kiev, 2004, pp. 275-276
20. Demidovich B.P. Lectures on Mathematical Stability Theory of Motion (1967), Nauka, Moscow
21. G. Iovane, S.D. Borysenko, Boundedness, stability, practical stability of motion impulsive systems, in: Proc. DE&CAS, Brest, 2005, pp. 15-21
22. Kaul S., Lakshmikantham V., and Leela S. Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times. Nonlinear Anal. 22 10 (1994) 1263-1270
23. Krylov N.N., and Bogolyubov N.N. Introduction in nonlinear mechanics (1937), Izolateestvo AS USSR, Kiev
24. Heikkilä S., Lakshmikantham V., and Leela S. Applications of monotone techniques to differential equations with discontinuous right-hand side. Differential Integral Equations 1 3 (1988) 287-297
25. Hu S.C., Lakshmikantham V., and Leela S. Impulsive differential systems and the pulse phenomena. J. Math. Anal. Appl. 137 2 (1989) 605-612
26. Lakshmikantham V., and Leela S. Differential and Integral Inequalities, Theory and Applications (1969), Academic Press, New York
27. V. Lakshmikantham, Trends in the theory of impulsive differential equations, in: Proc. Internat. Conf. on the Theory and Applications of Differential Equations, Ohio University, 1988, pp. 76-87
28. Lakshmikantham V., Leela S., and Mohan Rao Rama M. Integral and integro-differential inequalities. Appl. Anal. 24 3 (1987) 157-164
29. Lakshmikantham V., and Liu X. Impulsive hybrid systems and stability theory. Dynam. Systems Appl. 7 1 (1998) 1-9
30. Lakshmikantham V., Leela S., and Martynyuk A. Stability analysis of nonlinear systems. Monogr. Textbooks Pure Appl. Math. 125 (1989)
31. Lakshmikantham V., and Liu X. Stability for impulsive differential systems in terms of two measures. Appl. Math. Comput. 29 (1989) 89-98
32. Lakshmikantham V., and McRae Farzana A. Basic results for fuzzy impulsive differential equations. Math. Inequal. Appl. 4 2 (2001) 239-246
33. Lakshmikantham V., Murty K.N., and Silvasundaram S. Impulsive differential systems for two-point boundary value problems. Appl. Math. Comput. 50 2-3 (1992) 157-166
34. V. Lakshmikantham, Recent trends in the theory of impulsive differential equations. in: Conference of Differential Equations and Applications, Ruse, 1989, 1991, pp. 12-25
35. Lakshmikantham V., Leela S., and Kaul S. Comparison principle for impulsive differential equations with variable times and stability theory. Nonlinear Anal. 22 4 (1994) 499-503
36. Lakshmikantham V., Leela S., and Martynyuk A.A. Practical Stability of Nonlinear Systems (1990), World Scientific Publishing Co.
37. Lakshmikantham V., Bainov D.D., and Simeonov P.S. Theory of impulsive differential equations. Ser. Modern Appl. Math. 6 (1989)
38. Lakshmikantham V., Liu X., and Sathananthan S. Impulsive integro- differential equations and extension of Lyapunov's method. Appl. Anal. 32 3-4 (1989) 203-214
39. Lakshmikantham V. Trends in the theory of impulsive differential equations. Differential Equations Appl. 1 (1989) 76-87
40. Lakshmikantham V., and Liu X. On quasistability for impulsive differential systems. Nonlinear Anal. 13 7 (1989) 819-828
41. Lakshmikantham V., and Liu X. Stability criteria for impulsive differential equations in terms of two measures. J. Math. Anal. Appl. 137 2 (1989) 591-604
42. V. Lakshmikantham, Monotone iterative technique for discontinuous differential systems, in: Proceedings of Dynamic Systems and Applications, vol. 1, 1993, pp. 199-204
43. Lakshmikantham V. Comparison principle for impulsive differential equations with variable times. Comp. Methods Stab. Theory (1993) 227-236
44. A.M. Lyapunov, General problem on motion stability, Collect. Works, Moscow, 1956
45. A.A. Martynyuk, V. Lakshmikantham, S. Leela, Stability of motion: the method of integral inequalities, 1989, p. 272
46. Milman V.D., and Myshkis A.D. About stability of motion with pusches. Sibirsk. Mat. Zh. 1 2 (1960) 233-237
47. Mitropolskiy Yu.A., Leela S., and Martynyuk A. About some directions of investigations of V. Lakshmikantham in the theory of differential equations and applications. Differ. Uravn. 22 4 (1986) 55-572
48. Mitropolskiy Yu.A., Samoilenko A.M., and Perestyuk N. On the problem of substantiation of averaging method for the second order equations with impulse effect. Ukraïn. Mat. Zh. 29 6 (1977) 750-762
49. Mitropolskiy Yu.A., Perestyuk N., and Chernikova O.S. Convergence of the systems of differential equations with impulse excitation. Proc. AS USSR. Ser. A 11 (1983) 11-15
50. Myshkis A.D., and Samoilenko A.M. Systems with impulses in the present instants. Mat. Sb. 74 2 (1967) 202-208
51. Moiseev N.D. On some methods of the theory of technical stability. Proc. N. Zhukovskiy Military-Air. Acad, Vyp. 135 (1945) 47-60
52. Pandit S.G. Differential systems with impulse perturbations. Pacific J. Math. 86 2 (1980) 553-560
53. Pandit S.G. On the stability of impulsively perturbed differential systems. Bull. Austral. Math. Soc. 17 3 (1977) 423-432
54. Pavlidis T., and Jury E.I. Analysis of new class of pulse-frequency modulated feed-back systems. IEEE Trans. Automat. Control. Nol. AC 10 1 (1965) 35-43
55. Pavlidis T. Stability of a class of discontinuous dynamical systems. Inf. Control 9 6 (1966) 298-322
56. Rakhmatullina L.F. On application of solvability conditions of Chaplygin problems of boundedness and solution stability of differential equations. Jzv. Vuzov 2 (1959) 198-201
57. Rama Mohana Rao M., and Sree Har Rao V. Stability of impulse by perturbed systems. Bull. Austral. Math. Soc. 16 1 (1976) 99-110
58. Sree Har Rao V. Asymptotically of measure differential equations. Nonlinear Anal. 27 2 (1978) 483-489
59. A.M. Samoilenko, S.D. Borysenko, Integro-sum inequalities and stability of processes with discrete disturbances, in: Proc. of Third Int. Conf. "Differential Equations and Applications" Russe, Bulgaria, 1987, pp. 377-380
60. Samoilenko A.M., and Borysenko S.D. On functional inequalities of Bihari type for discontinuous functions. Uspekhi Mat. Nauk 53 4 (1998) 147-148
61. Samoilenko A.M., Borysenko S.D., Laserra E., and Matarazzo G. Wendroff type integro-sum inequalities and applications. Math. Notes 3 2 (2002) 123-132
62. Samoilenko A.M., Borysenko S.D., Cattani C., and Yasinsky V. Differential Models: Construction, Representations and Applications (2001), Naukova Dumka, Kyiv
63. Samoilenko A.M., Borysenko S.D., Cattani C., Matarazzo G., and Yasinsky V. Differential Models: Stability, Inequalities and Estimates (2001), Naukova Dumka, Kyiv
64. Samoilenko A.M., and Perestyuk N. Differential Equations with Impulse Effect (1987), Visha Shkola, Kyiv