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2015, 2015(special): 287-296. doi: 10.3934/proc.2015.0287

On the properties of solutions set for measure driven differential inclusions

Mieczysław Cichoń 1, and  Bianca Satco 2,

1. 

Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
2. 

Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University, Universitatii 13, 720229 Suceava, Romania

Received  July 2014 Revised  November 2014 Published  November 2015

The aim of the paper is to present properties of solutions set for differential inclusions driven by a positive finite Borel measure. We provide for the most natural type of solution results concerning the continuity of the solution set with respect to the data similar to some already known results, available for different types of solutions. As consequence, the solution set is shown to be compact as a subset of the space of regulated functions. The results allow one (by taking the measure $\mu$ of a particular form) to obtain information on the solution set for continuous or discrete problems, as well as impulsive or retarded set-valued problems.
Keywords: regulated function, Measure differential inclusion, solution set., Lebesgue-Stieltjes integral.
Mathematics Subject Classification: Primary: 34A60, 93C30; Secondary: 28A33, 26A42, 34A3.
Citation: Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287
References:
[1]

J.P. Aubin and H. Frankowska, "Set-Valued Analysis", Birkhäuser, Boston, 1990.

[2]

P. Billingsley, Weak convergence of measures: Applications in probability, in "DCBMS-NSF Regional Conference Series in Applied Mathematics", 1971.

[3]

A.M. Bruckner, J.B. Bruckner and B.S. Thomson, "Real Analysis", Prentice-Hall, 1997.

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions", in Lecture Notes in Math. 580, Springer, Berlin, 1977.

[5]

M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete, Adv. Diff. Equations 2014, 2014:56, 18 pp.

[6]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739-765.

[7]

M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations 252 (2012), 3816-3847.

[8]

D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), 20-59.

[9]

Fremlin, D.H., Measure Theory. Vol. 2, Torres Fremlin, Colchester (2003).

[10]

Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ. 16 (2009), 299-313.

[11]

R. Lucchetti, G. Salinetti and R. J-B. Wets, Uniform convergence of probability measures: topological criteria, Jour. Multivariate Anal. 51 (1994), 252-264.

[12]

J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures, in "Hybrid Systems: Computation and Control", Lecture Notes in Computer Science 4416 (2007), 385-398.

[13]

B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estimation and Control 4 (1994), 1-21.

[14]

B. Miller and E.Y. Rubinovitch, "Impulsive Control in Continuous and Discrete-Continuous Systems", Kluwer Academic Publishers, Dordrecht, 2003.

[15]

G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), 283-303.

[16]

W.R. Pestman, Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. 2 (1995), 381-388.

[17]

R.R. Rao, Relations between weak and uniform convergence of measures with applications The Annals of Mathematical Statistics 33 (1962), 659-680.

[18]

S. Saks, "Theory of the Integral", Monografie Matematyczne, Warszawa, 1937.

[19]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints", Dordrecht, Praha, 1979.

[20]

A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems", Dordrecht, Kluwer Academic, 1997.

[21]

G.N. Silva and R.B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl. 202 (1996), 727-746.

[22]

A. Slavik, Well-posedness results for abstract generalized differential equations and measure functional differential equations, Journal of Differential Equations 259 (2015), 666-707.

[23]

M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions", Habil. Thesis, Praha, 2001.

show all references

References:
[1]

J.P. Aubin and H. Frankowska, "Set-Valued Analysis", Birkhäuser, Boston, 1990.

[2]

P. Billingsley, Weak convergence of measures: Applications in probability, in "DCBMS-NSF Regional Conference Series in Applied Mathematics", 1971.

[3]

A.M. Bruckner, J.B. Bruckner and B.S. Thomson, "Real Analysis", Prentice-Hall, 1997.

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions", in Lecture Notes in Math. 580, Springer, Berlin, 1977.

[5]

M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete, Adv. Diff. Equations 2014, 2014:56, 18 pp.

[6]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739-765.

[7]

M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations 252 (2012), 3816-3847.

[8]

D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), 20-59.

[9]

Fremlin, D.H., Measure Theory. Vol. 2, Torres Fremlin, Colchester (2003).

[10]

Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ. 16 (2009), 299-313.

[11]

R. Lucchetti, G. Salinetti and R. J-B. Wets, Uniform convergence of probability measures: topological criteria, Jour. Multivariate Anal. 51 (1994), 252-264.

[12]

J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures, in "Hybrid Systems: Computation and Control", Lecture Notes in Computer Science 4416 (2007), 385-398.

[13]

B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estimation and Control 4 (1994), 1-21.

[14]

B. Miller and E.Y. Rubinovitch, "Impulsive Control in Continuous and Discrete-Continuous Systems", Kluwer Academic Publishers, Dordrecht, 2003.

[15]

G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), 283-303.

[16]

W.R. Pestman, Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. 2 (1995), 381-388.

[17]

R.R. Rao, Relations between weak and uniform convergence of measures with applications The Annals of Mathematical Statistics 33 (1962), 659-680.

[18]

S. Saks, "Theory of the Integral", Monografie Matematyczne, Warszawa, 1937.

[19]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints", Dordrecht, Praha, 1979.

[20]

A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems", Dordrecht, Kluwer Academic, 1997.

[21]

G.N. Silva and R.B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl. 202 (1996), 727-746.

[22]

A. Slavik, Well-posedness results for abstract generalized differential equations and measure functional differential equations, Journal of Differential Equations 259 (2015), 666-707.

[23]

M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions", Habil. Thesis, Praha, 2001.

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