Borysenko, Sergiy and Gallo, Angela and Piccirillo, Anna Maria (2010) Multidimensional Impulse Inequalities and General Bihari-Type Inequalities for Discontinuous Functions with Retardation. (Submitted)

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Item Type: Article
Title: Multidimensional Impulse Inequalities and General Bihari-Type Inequalities for Discontinuous Functions with Retardation
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Borysenko, SergiyUNSPECIFIED
Gallo, AngelaUNSPECIFIED
Piccirillo, Anna MariaUNSPECIFIED
Date: 2010
Department: Matematica e applicazioni "Renato Caccioppoli"
Section of the department: INGEGNERIA
Date: 2010
References: [ 1 ] R.P.Agarwal,On an integral inequality in n-independent variables. J.Math. Anal. Appl. 85 (1982), pp.192-196. [ 2 ] R.P.Agarwal,Difference Equations and Inequalities: Theory, Methods and Applications, New York.1992. [ 3 ] O. Akinyele, On Gronwall-Bellman-Bihari type integral inequalities in several variables with retardation, J.Math.Anal.Appl. 104 (1) (1984) ,pp.1-26 .  4  N.V. Azbelev, Z.B.Tsalyuk,On Chaplygin’s problem, Ukrain. Mat. Zh. 10 (1) (1958), pp.3-12 .  5  D.Bainov,P.S.Simeonov, Integral Inequalities and Applications, Kluwer Acad. Pubbl., Dordreht,1992.  6 R.Bellman,K.L.Cooke, Differential – Difference equations, Academic Press, New York,1963. [ 7 ] I.Bihari, 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. [8 ] D.S.Borysenko, About the integral inequality for piecewise continuous functions,in:Proc.X Int.Kravchuk Conf.,Kyiv 2004,p.323 [ 9 ] D.S.Borysenko, A.Gallo, R.Toscano, Integral inequalities Gronwall-Bellman type for discontinuous functions, Visnik Kyiv University (1) (2005) 63-67. [10] 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. [11] S.D.Borysenko, Integro-sum inequalities for functions of many independent variables, Differential equations, 25 (9) (1989) ,pp.1638-1641. [12] S.D.Borysenko, G.Iovane, P.Giordano, Investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces, Nonlinear Anal. 62(2005), pp. 345-363. [13] S.D.Borysenko,M.Ciarletta, G.Iovane, Integro-sum inequalities and motion stability of systems with impulse perturbations, Nonlinear Anal. 62(2005) 417-428. [14] S.D.Borysenko, G.Iovane, P.Giordano,About some hyperbolic impulsive equations and estimate solutions,in:Proc.DE@CAS,Brest,2005, pp.9-14. [15] S.D.Borysenko,G.Iovane,Integro-Sum Inequalities and Qualitative Analysis Dinamical Systems with Perturbations, Tipografia-Legatoria “ELDA” Università of Salerno, Salerno, 2006. [16] S.D.Borysenko,G.Iovane,Integro-Sum Inequalities and Qualitative Analysis Dinamical Systems withPerturbations, Tipografia-Legatoria “ELDA” Università of Salerno, Salerno, 2006. [17 ] S.D.Borysenko, G.Iovane, About some integral inequalities of Wendroff type for discontinuous functiolns. Nonlinear Anal. 66(10) 2007 ,pp.2190-2203 . [ 18 ] A.Gallo, A.M. Piccirillo,About new analogies Gronwall – Bellman – Bihari type inequalities for discontinuous functions and estimates solutions impulsive differential systems, Nonlinear Anal.67(2007) 5, pp.1550-1559. [19] G.Iovane,S.D.Borysenko,Boundedness,stability, practical stability of motion impulsive systems,in:Proc.DE@CAS,Brest,2005,pp.15-21. [20] G.Iovane, Some new integral inequalities of Bellman Bihari type with delay for discontinuous functions, Nonl.Anal. 66 (2007), pp. 498-508. [21] S.C. Hu, V.Lakshmikantham, S.Leela, Impulsive differential systems and the pulse phenomena, J.Math. Anal. Appl. 137 (2) (1989) 605-612. [22] V.Lakshmikantham, S.Leela, Differential and Integral Inequalities, Theory and Applications, Academic Press, New York, 1969. [23]V.Lakshmikantham, S.Leela, M.Mohan Rao Rama, Integral and integro-differential inequalities, Appl. Anal. 24(3)(1987) 157-164. [24] V.Lakshmikantham, D.Bainov, P.S.Simeonov, Theory of Impulsive Differential Equation, Ser.Modern Appl. Math. 6, 1989. [25] A.A. Martyhyuk, V.Lakshmikantham, S.Leela, Stability of Motion: the Method of Integral Inequalikties, Naukova Dumka, Kyiv, 1989. [26] Yu.A.Mitropolskiy, S.Leela, A.A. Martyhyuk,About some derections of investigations of V.Lakshmikantham in theory of differential equations and applications, Differ.Uravn. 22 (4) (1986) 550-572. [27]Yu.A.Mitropolskiy, G. Iovane, S.D. Borysenko, About a generalization of Bellman- Bihari type inequalities for discontinuous functions and their applications. Nonlinear Anal. 66(2007) (10), pp.2140-2165. 28 J.J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J.Math.Anal.Appl., 1997 (205) , pp. 423-433. 29 J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Letters, 2002 /15), pp. 489- 493. [30] B.G. Pachpatte, On some integral inequalities similar to Bellman-Bihari inequality, J. Math. Anal. Appl. 49 (1975) 794-802. [31] B.G. Pachpatte, On some new integral and integrodifferential inequalities in two independent variables and their applications. J. Differential Equations 33 (1979), pp.249- 272. [32] B.G. Pachpatte, On some new integro-differential inequalities of the Wendroff type. J.Math. Anal. Appl. 73 (1980), pp.491-500. 33A.M. Piccirillo, Estimates of the solutions hyperbolic equations with impulse perturbations, in: Proc. XI Int. Kravchuk Conf. Kyiv, 2006,pp.77-80. 34A.M. Piccirillo, I. Verigina, About some generalization Bihari result for integro-sum inequalities, in: Proc. XI Int. Kravchuk Conf. Kyiv, 2006,p. 187. 35 L.F. Rakhmatullina, On some application of stability conditions of Chaplyging problems for the problems of boundedness and solution stability of differential equations. Izv. Vuzov, 2(1959), pp. 198-201. 36 A.M.Samoilenko, S.D.Borysenko, C.Cattani, G.Matarazzo, V.Visinsky, Differential Models: Stability, Inequalities and Estimates, Naukova Dumka, Kyiv, 2001. [37] A.M.Samoilenko,N. Perestyuk, Differential Equations with Impulse Effect, Visha Shkola, Kyiv, 1987. (1977) 423-432 . [38] E.Thahdapani, R.D. Agarwal, On some new inequalities in n-indipendent variables, J. Math. Anal. Appl. 86 (1982), pp.542-561. [39] C.G. Yeh, Wendroff inequalitity in n-indipendent variables, J. Math. Anal. Appl.78 (1980), pp. 78-87. [40] C.G. Yeh, M.H. Shih, The Gronwall- Bellman inequality in several variables, J.Math.Anal. Appl. 86 (1982), pp.157-167. [41] W. Walter, Differential and Integral Inequalities, Springer- Verlag, New York, 1970. [42] Z.B. Tsalyuk, Multidimensional integral inequalities, Differents Uravneniya ,(1983) (10), pp. 1828-1839. •••••••••• [ 1 ] R.P.Agarwal,On an integral inequality in n-independent variables. J.Math. Anal. Appl. 85 (1982), pp.192-196. [ 2 ] R.P.Agarwal,Difference Equations and Inequalities: Theory, Methods and Applications, New York.1992. [ 3 ] O. Akinyele, On Gronwall-Bellman-Bihari type integral inequalities in several variables with retardation, J.Math.Anal.Appl. 104 (1) (1984) ,pp.1-26 .  4  N.V. Azbelev, Z.B.Tsalyuk,On Chaplygin’s problem, Ukrain. Mat. Zh. 10 (1) (1958), pp.3-12 .  5  D.Bainov,P.S.Simeonov, Integral Inequalities and Applications, Kluwer Acad. Pubbl., Dordreht,1992.  6 R.Bellman,K.L.Cooke, Differential – Difference equations, Academic Press, New York,1963. [ 7 ] I.Bihari, 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. [8 ] D.S.Borysenko, About the integral inequality for piecewise continuous functions,in:Proc.X Int.Kravchuk Conf.,Kyiv 2004,p.323 [ 9 ] D.S.Borysenko, A.Gallo, R.Toscano, Integral inequalities Gronwall-Bellman type for discontinuous functions, Visnik Kyiv University (1) (2005) 63-67. [10] 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. [11] S.D.Borysenko, Integro-sum inequalities for functions of many independent variables, Differential equations, 25 (9) (1989) ,pp.1638-1641. [12] S.D.Borysenko, G.Iovane, P.Giordano, Investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces, Nonlinear Anal. 62(2005), pp. 345-363. [13] S.D.Borysenko,M.Ciarletta, G.Iovane, Integro-sum inequalities and motion stability of systems with impulse perturbations, Nonlinear Anal. 62(2005) 417-428. [14] S.D.Borysenko, G.Iovane, P.Giordano,About some hyperbolic impulsive equations and estimate solutions,in:Proc.DE@CAS,Brest,2005, pp.9-14. [15] S.D.Borysenko,G.Iovane,Integro-Sum Inequalities and Qualitative Analysis Dinamical Systems with Perturbations, Tipografia-Legatoria “ELDA” Università of Salerno, Salerno, 2006. [16] S.D.Borysenko,G.Iovane,Integro-Sum Inequalities and Qualitative Analysis Dinamical Systems withPerturbations, Tipografia-Legatoria “ELDA” Università of Salerno, Salerno, 2006. [17 ] S.D.Borysenko, G.Iovane, About some integral inequalities of Wendroff type for discontinuous functiolns. Nonlinear Anal. 66(10) 2007 ,pp.2190-2203 . [ 18 ] A.Gallo, A.M. Piccirillo,About new analogies Gronwall – Bellman – Bihari type inequalities for discontinuous functions and estimates solutions impulsive differential systems, Nonlinear Anal.67(2007) 5, pp.1550-1559. [19] G.Iovane,S.D.Borysenko,Boundedness,stability, practical stability of motion impulsive systems,in:Proc.DE@CAS,Brest,2005,pp.15-21. [20] G.Iovane, Some new integral inequalities of Bellman Bihari type with delay for discontinuous functions, Nonl.Anal. 66 (2007), pp. 498-508. [21] S.C. Hu, V.Lakshmikantham, S.Leela, Impulsive differential systems and the pulse phenomena, J.Math. Anal. Appl. 137 (2) (1989) 605-612. [22] V.Lakshmikantham, S.Leela, Differential and Integral Inequalities, Theory and Applications, Academic Press, New York, 1969. [23]V.Lakshmikantham, S.Leela, M.Mohan Rao Rama, Integral and integro-differential inequalities, Appl. Anal. 24(3)(1987) 157-164. [24] V.Lakshmikantham, D.Bainov, P.S.Simeonov, Theory of Impulsive Differential Equation, Ser.Modern Appl. Math. 6, 1989. [25] A.A. Martyhyuk, V.Lakshmikantham, S.Leela, Stability of Motion: the Method of Integral Inequalikties, Naukova Dumka, Kyiv, 1989. [26] Yu.A.Mitropolskiy, S.Leela, A.A. Martyhyuk,About some derections of investigations of V.Lakshmikantham in theory of differential equations and applications, Differ.Uravn. 22 (4) (1986) 550-572. [27]Yu.A.Mitropolskiy, G. Iovane, S.D. Borysenko, About a generalization of Bellman- Bihari type inequalities for discontinuous functions and their applications. Nonlinear Anal. 66(2007) (10), pp.2140-2165. 28 J.J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J.Math.Anal.Appl., 1997 (205) , pp. 423-433. 29 J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Letters, 2002 /15), pp. 489- 493. [30] B.G. Pachpatte, On some integral inequalities similar to Bellman-Bihari inequality, J. Math. Anal. Appl. 49 (1975) 794-802. [31] B.G. Pachpatte, On some new integral and integrodifferential inequalities in two independent variables and their applications. J. Differential Equations 33 (1979), pp.249- 272. [32] B.G. Pachpatte, On some new integro-differential inequalities of the Wendroff type. J.Math. Anal. Appl. 73 (1980), pp.491-500. 33A.M. Piccirillo, Estimates of the solutions hyperbolic equations with impulse perturbations, in: Proc. XI Int. Kravchuk Conf. Kyiv, 2006,pp.77-80. 34A.M. Piccirillo, I. Verigina, About some generalization Bihari result for integro-sum inequalities, in: Proc. XI Int. Kravchuk Conf. Kyiv, 2006,p. 187. 35 L.F. Rakhmatullina, On some application of stability conditions of Chaplyging problems for the problems of boundedness and solution stability of differential equations. Izv. Vuzov, 2(1959), pp. 198-201. 36 A.M.Samoilenko, S.D.Borysenko, C.Cattani, G.Matarazzo, V.Visinsky, Differential Models: Stability, Inequalities and Estimates, Naukova Dumka, Kyiv, 2001. [37] A.M.Samoilenko,N. Perestyuk, Differential Equations with Impulse Effect, Visha Shkola, Kyiv, 1987. (1977) 423-432 . [38] E.Thahdapani, R.D. Agarwal, On some new inequalities in n-indipendent variables, J. Math. Anal. Appl. 86 (1982), pp.542-561. [39] C.G. Yeh, Wendroff inequalitity in n-indipendent variables, J. Math. Anal. Appl.78 (1980), pp. 78-87. [40] C.G. Yeh, M.H. Shih, The Gronwall- Bellman inequality in several variables, J.Math.Anal. Appl. 86 (1982), pp.157-167. [41] W. Walter, Differential and Integral Inequalities, Springer- Verlag, New York, 1970. [42] Z.B. Tsalyuk, Multidimensional integral inequalities, Differents Uravneniya ,(1983) (10), pp. 1828-1839. ••••••••••
MIUR S.S.D.: Area 01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica
Date Deposited: 29 Jun 2011 09:15
Last Modified: 30 Apr 2014 19:46
URI: http://www.fedoa.unina.it/id/eprint/8458

Abstract

In this article we investigate some impulsive integro-functional inequalities for functions of n independent variables. The problem of reducing multidimensional integro-sum functional inequalities to one-dimensional inequalities is considered too (by using conditions of solvability Chapligin’ problem for impulsive integral inequalities). Some new analogies Wendroff-type inequalities for discontinuous functions with finite jumps are founded.

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