Filippov Lemma for certain second order differential inclusions

• Autorzy: Grzegorz Bartuzel , Andrzej Fryszkowski
• Języki publikacji: EN

Abstrakty

EN

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y′(0) = 0, where the matrix A ∈ ℝd×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y 0 ∈ W 2,1 be an arbitrary function fulfilling the above initial conditions and such that where p 0 ∈ L 1[0, 1]. Then there exists a solution y ∈ W 2,1 to the above differential inclusions such that a.e. in [0, 1], .

Słowa kluczowe

EN

Differential inclusion; Differential operator; Lipschitz multifunction; Filippov Lemma

Kategorie tematyczne

• 39A05: General theory
• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
• 46G05: Derivatives
• 28A15: Abstract differentiation theory, differentiation of set functions
• 28A05: Classes of sets (Borel fields, σ -rings, etc.), measurable sets, Suslin sets, analytic sets

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 6
• Strony: 1944-1952

Daty

wydano

2012-12-01

online

2012-10-12

Twórcy

autor

Grzegorz Bartuzel

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662, Warsaw, Poland,

autor

Andrzej Fryszkowski

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662, Warsaw, Poland,

Bibliografia

• [1] Aubin J.P., Cellina A., Differential Inclusions, Grundlehren Math. Wiss., 264, Springer, Berlin, 1984 http://dx.doi.org/10.1007/978-3-642-69512-4[Crossref]
• [2] Aubin J.-P., Frankowska H., Set-Valued Analysis, Systems Control Found. Appl., 2, Birkhäuser, Boston, 1990
• [3] Bartuzel G., Fryszkowski A., A topological property of the solution set to the Sturm-Liouville differential inclusions, Demonstratio Math., 1995, 28(4), 903–914
• [4] Bartuzel G., Fryszkowski A., Stability of principal eigenvalue of the Schrödinger type problem for differential inclusions, Topol. Methods Nonlinear Anal., 2000, 16(1), 181–194
• [5] Bartuzel G., Fryszkowski A., A class of retracts in L p with some applications to differential inclusion, Discuss. Math. Differ. Incl. Control Optim., 2001, 22(2), 213–224 http://dx.doi.org/10.7151/dmdico.1038[Crossref]
• [6] Bressan A., Cellina A., Fryszkowski A., A class of absolute retracts in spaces of integrable functions, Proc. Amer. Math. Soc., 1991, 112(2), 413–418 http://dx.doi.org/10.1090/S0002-9939-1991-1045587-8[Crossref]
• [7] Cellina A., On the set of solutions to Lipschitzian differential inclusions, Differential Integral Equations, 1988, 1(4), 495–500
• [8] Cellina A., Ornelas A., Representation of the attainable set for Lipschitzian differential inclusions, Rocky Mountain J. Math., 1992, 22(1), 117–124 http://dx.doi.org/10.1216/rmjm/1181072798[Crossref]
• [9] Cernea A., On the existence of solutions for a higher order differential inclusion without convexity, Electron. J. Qual. Theory Differ. Equ., 2007, #8
• [10] Colombo R.M., Fryszkowski A., Rzezuchowski T., Staicu V., Continuous selections of solution sets of Lipschitzean differential inclusions, Funkcial. Ekvac., 1991, 34(2), 321–330
• [11] Filippov A.F., Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control., 1967, 5, 609–621 http://dx.doi.org/10.1137/0305040[Crossref]
• [12] Fryszkowski A., Fixed Point Theory for Decomposable Sets, Topol. Fixed Point Theory Appl., 2, Kluwer, Dordrecht, 2004
• [13] Fryszkowski A., Rzezuchowski T., Continuous version of Filippov-Wazewski relaxation theorem, J. Differential Equations, 1991, 94(2), 254–265 http://dx.doi.org/10.1016/0022-0396(91)90092-N[Crossref]
• [14] Hartman Ph., Ordinary Differential Equations, Birkhäuser, Boston, 1982
• [15] Hu Sh., Papageorgiou N.S., Handbook of Multivalued Analysis I, Math. Appl., 419, Kluwer, Dordrecht, 1997
• [16] Naselli Ricceri O., A-fixed points of multi-valued contractions, J. Math. Anal. Appl., 1988, 135(2), 406–418 http://dx.doi.org/10.1016/0022-247X(88)90164-3[Crossref]
• [17] Naselli Ricceri O., Ricceri B., Differential inclusions depending on a parameter, Bull. Polish Acad. Sci. Math., 1989, 37(7–12), 665–671
• [18] Papageorgiou N.S., Boundary value problems for evolution inclusions, Comment. Math. Univ. Carolin., 1988, 29(2), 355–363
• [19] Repovš D., Semenov P.V., Continuous Selections of Multivalued Mappings, Math. Appl., 455, Kluwer, Dordrecht, 1998
• [20] Rybinski L.E., A fixed point approach in the study of the solution sets of Lipschitzian functional-differential inclusions, J. Math. Anal. Appl., 1991, 160(1), 24–46 http://dx.doi.org/10.1016/0022-247X(91)90287-A[Crossref]
• [21] Tolstonogov A.A., On the structure of the solution set for differential inclusions in a Banach space, Mat. Sb., 1982, 118(160)(1), 3–18 (in Russian)

Identyfikatory

DOI

10.2478/s11533-012-0119-2