A survey of recent results for the generalizations of ordinary differential equations. (English) Zbl 1476.34002
Summary: This is a review paper on recent results for different types of generalized ordinary differential equations. Its scope ranges from discontinuous equations to equations on time scales. We also discuss their relation with inclusion and highlight the use of generalized integration to unify many of them under one single formulation.
MSC:
| 34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |
| 34A06 | Generalized ordinary differential equations (measure-differential equations, set-valued differential equations, etc.) |
| 34N05 | Dynamic equations on time scales or measure chains |
| 34A36 | Discontinuous ordinary differential equations |
Keywords:
review paper; generalized ordinary differential equations; discontinuous equations; time scalesReferences:
| [1] | Wu, Z-Q., The ordinay differential equations with discontinuous right members and the discontinuous solutions of the quasilinear partial differential equations, Scientia Sinica, 13, 1901-1917 (1964) · Zbl 0154.10901 |
| [2] | Wend, D. V. V., Existence and uniqueness of solutions of ordinary differential equations, Proceedings of the American Mathematical Society, 23, 27-33 (1969) · Zbl 0183.35604 · doi:10.1090/S0002-9939-1969-0245879-4 |
| [3] | Biles, D. C., Existence of solutions for discontinuous differential equations, Differential and Integral Equations, 8, 6, 1525-1532 (1995) · Zbl 0824.34003 |
| [4] | Rzymowski, W.; Walachowski, D., One-dimensional differential equations under weak assumptions, Journal of Mathematical Analysis and Applications, 198, 3, 657-670 (1996) · Zbl 0849.34003 · doi:10.1006/jmaa.1996.0106 |
| [5] | Biles, D. C.; Binding, P. A., On Carathéodory’s conditions for the initial value problem, Proceedings of the American Mathematical Society, 125, 5, 1371-1376 (1997) · Zbl 0869.34004 · doi:10.1090/S0002-9939-97-03942-7 |
| [6] | Hassan, E. R.; Rzymowski, W., Extremal solutions of a discontinuous scalar differential equation, Nonlinear Analysis—Theory, Methods & Applications, 37, 8, 997-1017 (1999) · Zbl 0949.34005 · doi:10.1016/S0362-546X(97)00687-1 |
| [7] | Biles, D. C.; Schechter, E., Solvability of a finite or infinite system of discontinuous quasimonotone differential equations, Proceedings of the American Mathematical Society, 128, 11, 3349-3360 (2000) · Zbl 0959.34012 · doi:10.1090/S0002-9939-00-05584-2 |
| [8] | Cid, J. Á.; Pouso, R. L., Ordinary differential equations and systems with time-dependent discontinuity sets, Proceedings of the Royal Society of Edinburgh, 134, 4, 617-637 (2004) · Zbl 1079.34002 · doi:10.1017/S0308210500003383 |
| [9] | Hu, S. C., Differential equations with discontinuous right-hand sides, Journal of Mathematical Analysis and Applications, 154, 2, 377-390 (1991) · Zbl 0724.34017 · doi:10.1016/0022-247X(91)90044-Z |
| [10] | Pouso, R. L., Upper and lower solutions for first-order discontinuous ordinary differential equations, Journal of Mathematical Analysis and Applications, 244, 2, 466-482 (2000) · Zbl 0962.34008 · doi:10.1006/jmaa.2000.6717 |
| [11] | Aubin, J. P.; Cellina, A., Differential Inclusions (2006), Amsterdam, The Netherlands: Elsevier, Amsterdam, The Netherlands |
| [12] | Pouso, R. L.; Biles, D. C., First-order singular and discontinuous differential equations, Boundary Value Problems, 2009 (2009) · Zbl 1190.34007 · doi:10.1155/2009/507671 |
| [13] | Pouso, R. L., Necessary conditions for solving initial value problems with infima of superfunctions, Mathematical Inequalities & Applications, 8, 4, 633-641 (2005) · Zbl 1085.34003 · doi:10.7153/mia-08-59 |
| [14] | Biles, D. C., A necessary and sufficient condition for existence of solutions for differential inclusions, Nonlinear Analysis—Theory, Methods & Applications, 31, 3-4, 311-315 (1998) · Zbl 0910.34029 · doi:10.1016/S0362-546X(96)00312-4 |
| [15] | Binding, P., The differential equation \(\dot{x} = f \circ x\), Journal of Differential Equations, 31, 2, 183-199 (1979) · Zbl 0363.34001 · doi:10.1016/0022-0396(79)90143-8 |
| [16] | Biles, D. C.; Pouso, R. L., Extremal solutions to first-order differential inclusions under weak assumptions, Dynamics of Continuous, Discrete & Impulsive Systems, 18, 3, 341-352 (2011) · Zbl 1231.34025 |
| [17] | Biles, D. C., Existence of solutions for singular differential inclusions, Dynamics of Continuous, Discrete & Impulsive Systems, 14, 4, 557-564 (2007) · Zbl 1141.34006 |
| [18] | Biles, D. C.; Cid, J. Á.; Pouso, R. L., Initial value problems for singular and nonsmooth second order differential inclusions, Mathematische Nachrichten, 280, 12, 1335-1343 (2007) · Zbl 1136.34012 · doi:10.1002/mana.200510549 |
| [19] | Pouso, R. L., Necessary and sufficient conditions for existence and uniqueness of solutions of second-order autonomous differential equations, Journal of the London Mathematical Society, 71, 2, 397-414 (2005) · Zbl 1070.34014 · doi:10.1112/S0024610705006319 |
| [20] | Hilger, S., Ein masskettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten [Ph.D. thesis] (1988), Würzburg, Germany: Universität Würzburg, Würzburg, Germany · Zbl 0695.34001 |
| [21] | Bohner, M.; Peterson, A., Dynamic Equations on Time Scales (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1021.34005 · doi:10.1007/978-1-4612-0201-1 |
| [22] | Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1025.34001 · doi:10.1007/978-0-8176-8230-9 |
| [23] | Atici, F. M.; Guseinov, G. Sh., On Green’s functions and positive solutions for boundary value problems on time scales, Journal of Computational and Applied Mathematics, 141, 1-2, 75-99 (2002) · Zbl 1007.34025 · doi:10.1016/S0377-0427(01)00437-X |
| [24] | Atici, F. M.; Biles, D. C., First order dynamic inclusions on time scales, Journal of Mathematical Analysis and Applications, 292, 1, 222-237 (2004) · Zbl 1064.34009 · doi:10.1016/j.jmaa.2003.11.053 |
| [25] | Sheng, Q.; Fadag, M.; Henderson, J.; Davis, J. M., An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Analysis—Real World Applications, 7, 3, 395-413 (2006) · Zbl 1114.26004 · doi:10.1016/j.nonrwa.2005.03.008 |
| [26] | Rogers, J. W.; Sheng, Q., Notes on the diamond-\( \alpha\) dynamic derivative on time scales, Journal of Mathematical Analysis and Applications, 326, 1, 228-241 (2007) · Zbl 1114.26003 · doi:10.1016/j.jmaa.2006.03.004 |
| [27] | Sheng, Q., Hybrid approximations via second-order crossed dynamic derivatives with the diamond-\( \alpha\) derivative, Nonlinear Analysis—Real World Applications, 9, 2, 628-640 (2008) · Zbl 1134.39302 · doi:10.1016/j.nonrwa.2006.12.006 |
| [28] | Özkan, U. M.; Kaymakçalan, B., Basics of diamond-\( \alpha\) partial dynamic calculus on time scales, Mathematical and Computer Modelling, 50, 9-10, 1253-1261 (2009) · Zbl 1185.26066 · doi:10.1016/j.mcm.2009.01.007 |
| [29] | Dai, Q.; Tisdell, C. C., Existence of solutions to first-order dynamic boundary value problems, International Journal of Difference Equations, 1, 1, 1-17 (2006) · Zbl 1116.39009 |
| [30] | Atici, F. M.; Cabada, A.; Chyan, C. J.; Kaymakçalan, B., Nagumo type existence results for second-order nonlinear dynamic BVPS, Nonlinear Analysis—Theory, Methods & Applications, 60, 2, 209-220 (2005) · Zbl 1072.34015 · doi:10.1016/j.na.2004.07.036 |
| [31] | Fan, J.; Li, L., Existence of positive solutions for \(p\)-Laplacian dynamic equations with derivative on time scales, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.35108 · doi:10.1155/2013/736583 |
| [32] | Erbe, L.; Peterson, A.; Weiss, J., Positive solutions to a singular second order boundary value problem, Advances in Dynamical Systems and Applications, 3, 1, 89-105 (2008) |
| [33] | Liu, Z., Existence and uniqueness of solutions for singular boundary value problem on time scales, International Journal of Mathematics, 23, 7 (2012) · Zbl 1257.34075 · doi:10.1142/S0129167X1250070X |
| [34] | Eloe, P. W., Sign properties of Green’s functions for disconjugate dynamic equations on time scales, Journal of Mathematical Analysis and Applications, 287, 2, 444-454 (2003) · Zbl 1054.34023 · doi:10.1016/S0022-247X(02)00579-6 |
| [35] | Guseinov, G. Sh., Eigenfunction expansions for a Sturm-Liouville problem on time scales, International Journal of Difference Equations, 2, 1, 93-104 (2007) · Zbl 1145.39005 |
| [36] | Wang, S-P., Existence of periodic solutions for higher order dynamic equations on time scales, Taiwanese Journal of Mathematics, 16, 6, 2259-2273 (2012) · Zbl 1270.34205 |
| [37] | Benchohra, M.; Hamani, S.; Henderson, J., Oscillatory and nonoscillatory solutions for first order impulsive dynamic inclusions on time scales, Archivum Mathematicum, 43, 4, 237-250 (2007) · Zbl 1164.34003 |
| [38] | Benchohra, M.; Hamani, S.; Henderson, J., Oscillation and nonoscillation for impulsive dynamic equations on certain time scales, Advances in Difference Equations, 2006 (2006) · Zbl 1139.39008 · doi:10.1155/ADE/2006/60860 |
| [39] | Grace, S. R.; Bohner, M.; Sun, S., Oscillation of fourth-order dynamic equations, Hacettepe Journal of Mathematics and Statistics, 39, 4, 545-553 (2010) · Zbl 1228.34145 |
| [40] | Grace, S. R.; Graef, J. R.; Panigrahi, S.; Tunc, E., On the oscillatory behavior of even order neutral delay dynamic equations on time-scales, Electronic Journal of Qualitative Theory of Differential Equations, 96, 1-12 (2012) · Zbl 1340.34243 |
| [41] | Saker, S. H.; Grace, S. R., Oscillation criteria for quasi-linear functional dynamic equations on time scales, Mathematica Slovaca, 62, 3, 501-524 (2012) · Zbl 1324.34192 · doi:10.2478/s12175-012-0026-9 |
| [42] | Grace, S. R.; Pinelas, S.; Agarwal, R. P., Oscillatory behaviour of odd order neutral delay dynamic equations on time-scales, International Journal of Dynamical Systems and Differential Equations, 4, 3, 187-197 (2012) · Zbl 1258.34181 · doi:10.1504/IJDSDE.2012.049885 |
| [43] | Saker, S. H.; O’Regan, D., New oscillation criteria for second-order neutral dynamic equations on time scales via Riccati substitution, Hiroshima Mathematical Journal, 42, 1, 77-98 (2012) · Zbl 1252.34104 |
| [44] | Zhang, C.; Agarwal, R. P.; Bohner, M.; Li, T., New oscillation results for second-order neutral delay dynamic equations, Advances in Difference Equations, 2012 (2012) · Zbl 1377.34090 · doi:10.1186/1687-1847-2012-227 |
| [45] | Grace, S. R., On the oscillation of \(n\) th order dynamic equations on time-scales, Mediterranean Journal of Mathematics, 10, 1, 147-156 (2013) · Zbl 1262.34104 · doi:10.1007/s00009-012-0201-9 |
| [46] | Kubiaczyk, I.; Saker, S. H., Asymptotic properties of third order functional dynamic equations on time scales, Annales Polonici Mathematici, 100, 3, 203-222 (2011) · Zbl 1218.34086 · doi:10.4064/ap100-3-1 |
| [47] | Erbe, L.; Hovhannisyan, G.; Peterson, A., Asymptotic behavior of \(n\)-th order dynamic equations, Nonlinear Dynamics and Systems Theory, 12, 1, 63-80 (2012) · Zbl 1279.34106 |
| [48] | Bohner, M.; Guseinov, G. Sh.; Karpuz, B., Properties of the Laplace transform on time scales with arbitrary graininess, Integral Transforms and Special Functions, 22, 11, 785-800 (2011) · Zbl 1236.44002 · doi:10.1080/10652469.2010.548335 |
| [49] | Atici, F. M.; Biles, D. C., First- and second-order dynamic equations with impulse, Advances in Difference Equations, 2005, 2, 119-132 (2005) · Zbl 1100.39018 · doi:10.1155/ADE.2005.119 |
| [50] | Graef, J. R.; Ouahab, A., Nonresonance impulsive functional dynamic boundary value inclusions on time scales, Nonlinear Studies, 15, 4, 339-354 (2008) · Zbl 1167.34035 |
| [51] | Zhou, J.; Wang, Y.; Li, Y., Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects, Boundary Value Problems, 2012 (2012) · Zbl 1281.34134 · doi:10.1186/1687-2770-2012-148 |
| [52] | Akin-Bohner, E.; Raffoul, Y. N.; Tisdell, C., Exponential stability in functional dynamic equations on time scales, Communications in Mathematical Analysis, 9, 1, 93-108 (2010) · Zbl 1194.34173 |
| [53] | Atici, F. M.; Eloe, P. W., Fractional \(q\)-calculus on a time scale, Journal of Nonlinear Mathematical Physics, 14, 3, 333-344 (2007) · Zbl 1157.81315 · doi:10.2991/jnmp.2007.14.3.4 |
| [54] | Ahmadkhanlu, A.; Jahanshahi, M., On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bulletin of the Iranian Mathematical Society, 38, 1, 241-252 (2012) · Zbl 1279.34004 |
| [55] | Cichoń, M.; Kubiaczyk, I.; Sikorska-Nowak, A.; Yantir, A., Existence of solutions of the dynamic Cauchy problem in Banach spaces, Demonstratio Mathematica, 45, 3, 561-573 (2012) · Zbl 1272.34122 |
| [56] | Santos, I. L. D.; Silva, G. N.; Barbanti, L., Dynamical inclusions in time scales: existence of solutions under a lower semicontinuity condition, Trends in Applied and Computational Mathematics, 13, 2, 109-120 (2012) · doi:10.5540/tema.2012.013.02.0109 |
| [57] | Bohner, M.; Guseinov, G. Sh., Partial differentiation on time scales, Dynamic Systems and Applications, 13, 3-4, 351-379 (2004) · Zbl 1090.26004 |
| [58] | Sanyal, S., Stochastic dynamic equations [Ph.D. thesis] (2008), Rolla, Mo, USA: Applied Mathematics, Missouri University of Science and Technology, Rolla, Mo, USA |
| [59] | Atici, F. M.; Biles, D. C., The stochastic Itô integral on time scales, PanAmerican Mathematical Journal, 20, 4, 45-56 (2010) · Zbl 1229.60067 |
| [60] | Lungan, C.; Lupulescu, V., Random dynamical systems on time scales, Electronic Journal of Differential Equations, 2012, 86, 1-14 (2012) · Zbl 1261.34068 |
| [61] | Du, N. H.; Dieu, N. T., The first attempt on the stochastic calculus on time scale, Stochastic Analysis and Applications, 29, 6, 1057-1080 (2011) · Zbl 1238.60061 · doi:10.1080/07362994.2011.610169 |
| [62] | Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Mathematical Journal, 7, 82, 418-449 (1957) · Zbl 0090.30002 |
| [63] | Kurzweil, J., Generalized ordinary differential equations, Czechoslovak Mathematical Journal, 8, 83, 360-388 (1958) · Zbl 0094.05804 |
| [64] | Kurzweil, J., Unicity of solutions of generalized differential equations, Czechoslovak Mathematical Journal, 8, 83, 502-509 (1958) · Zbl 0094.05901 |
| [65] | Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Mathematical Journal, 9, 84, 564-573 (1959) · Zbl 0094.05902 |
| [66] | Kurzweil, J., Problems which lead to a generalization of the concept of an ordinary nonlinear differential equation, Differential Equations and their Applications, 65-76 (1963), New York, NY, USA: House of the Czechoslovak Academy of Sciences, Prague,Czech Republic, Academic Press, New York, NY, USA · Zbl 0151.12501 |
| [67] | Artstein, Z., Topological dynamics of ordinary differential equations and kurzweil equations, Journal of Differential Equations, 23, 2, 224-243 (1977) · Zbl 0353.34044 · doi:10.1016/0022-0396(77)90128-0 |
| [68] | Schwabik, Š., Generalized Ordinary Differential Equations. Generalized Ordinary Differential Equations, Series in Real Analysis, 5 (1992), River Edge, NJ, USA: World Scientific Publishing, River Edge, NJ, USA · Zbl 0781.34003 |
| [69] | Imaz, C.; Vorel, Z., Generalized ordinary differential equations in Banach space and applications to functional equations, Boletin Sociedad Matemática Mexicana, 11, 47-59 (1966) · Zbl 0178.44203 |
| [70] | Oliva, F.; Vorel, Z., Functional equations and generalized ordinary differential equations, Boletin Sociedad Matemática Mexicana, 11, 40-46 (1966) · Zbl 0178.44204 |
| [71] | Federson, M.; Táboas, P., Topological dynamics of retarded functional differential equations, Journal of Differential Equations, 195, 2, 313-331 (2003) · Zbl 1054.34102 · doi:10.1016/S0022-0396(03)00061-5 |
| [72] | Federson, M.; Schwabik, Š., Generalized ODE approach to impulsive retarded functional differential equations, Differential and Integral Equations, 19, 11, 1201-1234 (2006) · Zbl 1212.34251 |
| [73] | Federson, M.; Mesquita, J.; Toon, E., Stability results for measure functional differential equations via Kurzweil-equations · Zbl 1369.34050 |
| [74] | Federson, M.; Schwabik, Š., A new approach to impulsive retarded differential equations: stability results, Functional Differential Equations, 16, 4, 583-607 (2009) · Zbl 1200.34097 |
| [75] | Federson, M.; Schwabik, Š., Stability for retarded functional differential equations, Ukrains’kyi Matematychnyi Zhurnal, 60, 1, 107-126 (2008) · Zbl 1164.34530 |
| [76] | Schwabik, Š., Variational stability for generalized ordinary differential equations, Časopis Pro Pěstování Matematiky, 109, 4, 389-420 (1984) · Zbl 0574.34034 |
| [77] | Federson, M.; Mesquita, J. G.; Slavík, A., Basic results for functional differential and dynamic equations involving impulses, Mathematische Nachrichten, 286, 2-3, 181-204 (2013) · Zbl 1266.34115 · doi:10.1002/mana.201200006 |
| [78] | Federson, M.; Mesquita, J. G.; Slavík, A., Measure functional differential equations and functional dynamic equations on time scales, Journal of Differential Equations, 252, 6, 3816-3847 (2012) · Zbl 1239.34076 · doi:10.1016/j.jde.2011.11.005 |
| [79] | Federson, M.; Frasson, M.; Mesquita, J.; Tacuri, P., Neutral measure functional differential equations as generalized ODEs · Zbl 1417.34195 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
