zbMATH — the first resource for mathematics

About some new generalizations of bellman-bihari results for integro-functional inequalities with discontinuous functions and applications. (English) Zbl 1239.26014
Summary: We present some new nonlinear Bellman-Bihari type integral inequalities with delay for discontinuous functions (integro-sum inequalities; impulse integral inequalities). Some applications of the results are included: conditions of boundedness (uniformly), stability by Lyapunov (uniformly), practical stability by Chetaev (uniformly) for the solutions of impulsive differential and integro-differential systems of ordinary differential equations.

26D15 Inequalities for sums, series and integrals
34K45 Functional-differential equations with impulses
34K20 Stability theory of functional-differential equations
Full Text: DOI
[1] Bihari, I., A generalization of lemma of Bellman and its application to uniqueness problems of differential equations, Acta math. acad. sci. hung., 7, 1, 71-94, (1956) · Zbl 0070.08201
[2] Bainov, D.; Simeonov, P.S., Integral inequalities and applications, (1992), Kluwer Acad. Publ Dordrecht · Zbl 0759.26012
[3] S.D. Borysenko, G. Iovane, Integro-Sum Inequalities and Qualitative Analysis Dynamical Systems with Perturbations, Tipografia-Legatoria ELDA Università of Salerno, Salerno, 2006
[4] G. Iovane, S.D. Borysenko, Boundedness,stability, practical stability of motion impulsive systems, in: Proc. DE@CAS, Brest, 2005, pp. 15-21
[5] Iovane, G., Some new integral inequalities of bellman – bihari type with delay for discontinuous functions, Nonlinear anal., 66, 498-508, (2007) · Zbl 1118.26022
[6] Hu, S.C.; Lakshmikantham, V.; Leela, S., Impulsive differential systems and the pulse phenomena, J. math. anal. appl., 137, 2, 605-612, (1989) · Zbl 0684.34003
[7] Lakshmikantham, V.; Leela, S., Differential and integral inequalities, theory and applications, (1969), Academic Press New York · Zbl 0177.12403
[8] Lakshmikantham, V.; Leela, S.; Mohan Rao Rama, M., Integral and integro-differential inequalities, Appl. anal., 24, 3, 157-164, (1987) · Zbl 0589.26007
[9] Lakshmikantham, V.; Bainov, D.; Simeonov, P.S., Theory of impulsive differential equations, Ser. modern appl. math., 6, (1989) · Zbl 0719.34002
[10] Martyhyuk, A.A.; Lakshmikantham, V.; Leela, S., Stability of motion: the method of integral inequalities, (1989), Naukova Dumka Kyiv
[11] Mitropolskiy, Yu.A.; Leela, S.; Martyhyuk, A.A., About some directions of investigations of V. lakshmikantham in theory of differential equations and applications, Differ. uravn., 22, 4, 550-572, (1986)
[12] Mitropolskiy, Yu.A.; Samoilenko, A.M.; Perestyuk, N., On the problem of substantiation of overgoing method for the second equations with impulse effect, Ukraïn. mat. zh., 29, 6, 750-762, (1977)
[13] Mitropolskiy, Yu.A.; Iovane, G.; Borysenko, S.D., About a generalization of bellman – bihari type inequalities for discontinuous functions and their applications, Nonlinear anal., 66, 10, 2140-2165, (2007) · Zbl 1119.26026
[14] Samoilenko, A.M.; Borysenko, S.D.; Cattani, C.; Matarazzo, G.; Visinsky, V., Differential models: stability, inequalities and estimates, (2001), Naukova Dumka Kyiv · Zbl 1051.34001
[15] Samoilenko, A.M.; Perestyuk, N., Differential equations with impulse effect, (1987), Visha Shkola Kyiv
[16] Akinyele, O., On gronwall – bellman – bihari type integral inequalities in several variables with retardation, J. math. anal. appl., 104, 1-26, (1984) · Zbl 0604.26011
[17] Bellman, R.; Cooke, K.L., Differential – difference equations, (1963), Academic Press New York · Zbl 0118.08201
[18] Borysenko, D.S.; Gallo, A.; Toscano, R., Integral inequalities gronwall – bellman type for discontinuous functions, Visnik kyiv univ., 1, 63-67, (2005)
[19] D.S. Borysenko, A. Gallo, R. Toscano, Integral inequalities Gronwall-Bellman type for discontinuous functions and estimates of solutions impulsive systems, in: Proc. DE@CAS, Brest, 2005, pp. 5-9 · Zbl 1098.34510
[20] Borysenko, S.D., Integro-sum inequalities for functions of many independent variables, Differ. equ., 25, 9, 1638-1641, (1989) · Zbl 0678.26007
[21] Borysenko, S.D., Multidimensional integro-sum inequalities, Ukraïn. mat. zh., 50, 2, 172-177, (1998)
[22] Borysenko, S.D.; Iovane, G.; Giordano, P., Investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces, Nonlinear anal., 62, 345-363, (2005) · Zbl 1087.34502
[23] Borysenko, S.D.; Ciarletta, M.; Iovane, G., Integro-sum inequalities and motion stability of systems with impulse perturbations, Nonlinear anal., 62, 417-428, (2005) · Zbl 1087.34003
[24] S.D. Borysenko, G. Iovane, P. Giordano, About some hyperbolic impulsive equations and estimate solutions, in: Proc. DE@CAS, Brest, 2005, pp. 9-14
[25] Borysenko, S.D.; Ciarletta, M.; Iovane, G., About stability in nonlinear approximation of systems with impulse influence, Report acad. sci ukr., 1, 8-12, (2006) · Zbl 1100.34503
[26] 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, 2, 229-238, (2006) · Zbl 1124.45005
[27] S.D. Borysenko, G. Iovane, About estimates of solutions for nonlinear hyperbolic equations with impulsive perturbations on some hypersurfaces, Preprint DIIMA n. 7, March 2006, University of Salerno
[28] Borysenko, S.; Iovane, G., About some new integral inequalities of Wendroff type for discontinuous functions, Nonlinear anal., 66, 10, 2190-2203, (2007) · Zbl 1135.26012
[29] Gallo, A.; Piccirillo, A.M., About new analogies gronwall – bellman – bihari type inequalities for discontinuous functions and estimates solutions impulsive differential systems, Nonlinear anal., 67, 5, 1550-1559, (2007) · Zbl 1124.26013
[30] A. Gallo, A.M. Piccirillo, On some generalizations Bellman-Bihari result for integro-functional inequalities for discontinuous functions and their applications, Boundary Value Problems (in press) · Zbl 1177.26035
[31] Nieto, I.I., Impulsive resonance periodic problems of first order, Appl. math. lett., 15, 489-493, (2002) · Zbl 1022.34025
[32] Nieto, I.I., Basic theory for nonresonance impulsive periodic problems of first order, J. math. anal. appl., 205, 423-433, (1997) · Zbl 0870.34009
[33] Nieto, I.I., Periodic boundary value problems for first order impulsive ordinary differential equations, Nonlinear anal., 51, 1223-1239, (2002) · Zbl 1015.34010
[34] Pachpatte, B.G., On some integral inequalities similar to bellman – bihari inequality, J. math. anal. appl., 49, 794-802, (1975) · Zbl 0305.26009
[35] Pandit, S.G., On the stability of impulsively perturbed differential systems, Bull. austral. math. soc., 17, 3, 423-432, (1977) · Zbl 0367.34038
[36] Pavlidis, J., Stability of a class of discontinuous dynamical systems, Inf. control, 9, 6, 298-322, (1966) · Zbl 0143.11904
[37] A.M. Piccirillo, Estimates of the solutions hyperbolic equations with impulse perturbations, in: Proc. XI Int. Kravchuk Conf. Kyiv, 2006, pp. 77-80
[38] A.M. Piccirillo, I. Verigina, About some generalization Bihari result for integro-sum inequalities, in: Proc. XI Int. Kravchuk Conf. Kyiv, 2006, p. 187
[39] Tsalyuk, Z.B., Multidimensional integral inequalities, Differ. uravn., 10, 1828-1839, (1983) · Zbl 0556.26008
[40] Walter, W., Differential and integral inequalities, (1970), Springer-Verlag New York
[41] Yeh, C.G.; Shih, M.H., The gronwall – bellman inequality in several variables, J. math. anal. appl., 86, 157-167, (1982) · Zbl 0507.26007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.