American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape
  • PROC Home
  • This Issue
  • Next Article
    Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition
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, (1990).   Google Scholar

[2]

P. Billingsley, Weak convergence of measures: Applications in probability,, in, (1971).   Google Scholar

[3]

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

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions",, in Lecture Notes in Math. 580, (1977).   Google Scholar

[5]

M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete,, Adv. Diff. Equations 2014, (2014).   Google Scholar

[6]

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

[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), 252 (2012), 3816.   Google Scholar

[8]

D. Fraňková, Regulated functions,, Math. Bohem. 116 (1991), 116 (1991), 20.   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures,, in, 4416 (2007), 385.   Google Scholar

[13]

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

[14]

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

[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), (2013), 283.   Google Scholar

[16]

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

[17]

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

[18]

S. Saks, "Theory of the Integral",, Monografie Matematyczne, (1937).   Google Scholar

[19]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints",, Dordrecht, (1979).   Google Scholar

[20]

A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems",, Dordrecht, (1997).   Google Scholar

[21]

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

[22]

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

[23]

M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions",, Habil. Thesis, (2001).   Google Scholar

show all references

References:
[1]

J.P. Aubin and H. Frankowska, "Set-Valued Analysis",, Birkhäuser, (1990).   Google Scholar

[2]

P. Billingsley, Weak convergence of measures: Applications in probability,, in, (1971).   Google Scholar

[3]

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

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions",, in Lecture Notes in Math. 580, (1977).   Google Scholar

[5]

M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete,, Adv. Diff. Equations 2014, (2014).   Google Scholar

[6]

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

[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), 252 (2012), 3816.   Google Scholar

[8]

D. Fraňková, Regulated functions,, Math. Bohem. 116 (1991), 116 (1991), 20.   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures,, in, 4416 (2007), 385.   Google Scholar

[13]

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

[14]

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

[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), (2013), 283.   Google Scholar

[16]

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

[17]

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

[18]

S. Saks, "Theory of the Integral",, Monografie Matematyczne, (1937).   Google Scholar

[19]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints",, Dordrecht, (1979).   Google Scholar

[20]

A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems",, Dordrecht, (1997).   Google Scholar

[21]

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

[22]

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

[23]

M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions",, Habil. Thesis, (2001).   Google Scholar

[1]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[2]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[3]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[4]

Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054
[5]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
[6]

Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359
[7]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437
[8]

Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779
[9]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[10]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
[11]

Jagannathan Gomatam, Isobel McFarlane. Generalisation of the Mandelbrot set to integral functions of quaternions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107
[12]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[13]

Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020
[14]

Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229
[15]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039
[16]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315
[17]

Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1399-1406. doi: 10.3934/mbe.2017072
[18]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555
[19]

Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467
[20]

Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753

 Impact Factor: 

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences