Existence and uniqueness theorems for the Pfaff equation with continuous coefficients

A. A. Abduganiev ; A. A. Azamov ; A. Begaliev

Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 67 (2021) no. 4, pp. 609-619.

Résumé

In this paper, the Pfaff equations with continuous coefficients are considered. Analogs of Peano's existence theorem and Kamke's theorem on the uniqueness of the solution to the Cauchy problem are established, and a method for the approximate solution of the Cauchy problem for the Pfaff equation is proposed.

Bibliographie

[1] Azamov A., Begaliev A. O., “Teorema suschestvovaniya i metod priblizhennogo resheniya dlya uravneniya Pfaffa s nepreryvnymi koeffitsientami”, Tr. IMM UrO RAN, 27, no. 3, 2021, 12–24

[2] Gaishun I. V., Vpolne razreshimye mnogomernye differentsialnye uravneniya, Editorial URSS, M., 2004

[3] Gaishun L. N., “O predstavlenii reshenii vpolne integriruemykh lineinykh sistem”, Diff. uravn., 14:4 (1978), 728–730 | MR | Zbl

[4] Kartan A., Differentsialnoe ischislenie. Differentsialnye formy, Mir, M., 1971

[5] Perov A. I., “Ob odnom obobschenii teoremy Frobeniusa”, Diff. uravn., 5:10 (1969), 1881–1884 | Zbl

[6] Agarwal R. P., Lakshmikantham V., Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific, Singapore, 1993 | Zbl

[7] Araújo J. A. C., “On uniqueness criteria for systems of ordinary differential equations”, J. Math. Anal. Appl., 281 (2003), 264–275 | DOI | MR | Zbl

[8] Arutyunov A. V., “The coincidence point problem for set-valued mappings and Ulam—Hyers stability”, Dokl. Math., 89:2 (2014), 188–191 | DOI | MR | Zbl

[9] Azamov A., Begaliyev A. O., “Existence and uniqueness of the solution of a Cauchy problem for the Pfaff equation with continuous coefficients”, Uzb. Math. J., 2019, no. 2, 18–26 | DOI | Zbl

[10] Azamov A., Suvanov Sh., Tilavov A., “Studing of behavior at infinity of vector fields on poincare's sphere: revisited”, Qual. Theory Dyn. Syst., 14:1 (2015), 2–11

[11] Bedford E., Kalka M., “Foliations and complex Monge—Ampere equations”, Commun. Pure Appl. Math., 30 (1991), 543–571 | DOI

[12] Brunella M., Gustavo M. L., “Bounding the degree of solutions to Pfaff equations”, Publ. Mat., 44:2 (2000), 593–604 | DOI | MR | Zbl

[13] Cartan É., “Sur certaines expressions différentielles et le problème de Pfaff”, Ann. Sci. Éc. Norm. Supér. (3), 16 (1899), 239–332 | DOI | Zbl

[14] Cerveau D., Lins-Neto A., “Holomorphic foliations in $CP(2)$ having an invariant algebraic curve”, Ann. Inst. Fourier (Grenoble), 41:4 (1991), 883–903 | DOI | MR

[15] Coddington E. A., Levinson N., Theory of ordinary differential equations, TATA McGRAW-Hill Publishing Co. Ltd, New Dehli, 1987

[16] Coutinho S. C., “A constructive proof of the density of algebraic Pfaff equations without algebraic solutions”, Ann. Inst. Fourier (Grenoble), 57:5 (2007), 1611–1621 | DOI | MR | Zbl

[17] Dryuma V., “On geometrical properties of the spaces defined by the Pfaff equations”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 47:1 (2005), 69–84 | MR | Zbl

[18] Hakopian H. A., Tonoyan M. G., “Partial differential analogs of ordinary differential equations and systems”, New York J. Math., 10 (2004), 89–116 | MR | Zbl

[19] Han C. K., “Pfaffian systems of Frobenius type and solvability of generic overdetermined PDE systems”, Symmetries and Overdetermined Systems of Partial Differential Equations, Springer, New York, 2008, 421–429 | Zbl

[20] Hartman Ph., Ordinary differential equations, John Willey Sons, New York, 1964 | Zbl

[21] Howard R., Methods of thermodinamics, Blaisdell Publ. Comp, New York, 1965

[22] Izobov N. A., “On the existence of linear Pfaffian systems whose set of lower characteristic vectors has a positive plane mesure”, Differ. Equ., 33:12 (1997), 1626–1632 | MR | Zbl

[23] Izobov N. A., Platonov A. S., “Construction of a linear Pfaff equation with arbitrarily given characteristics and lower characteristic sets”, Differ. Equ., 34:12 (1998), 1600–1607 | MR | Zbl

[24] Jouanolou J. P., Equations de Pfaff algébriques, Springer-Verlag, Berlin—Heidelberg, 1979

[25] Lefschetz S., Differential equations: Geometric theory, Interscience Publishers, New York—London, 1963 | Zbl

[26] Luzatto S., Türeli S., War K., Integrability of continuous bundles, 2016, arXiv: 1606.00343v2 [math.CA]

[27] Luzatto S., Türeli S., War K., “A Frobenius theorem for corank-1 continuous distributions in dimensions two and three”, arXiv: 1411.5896v5 [math.DG]

[28] Mardare S., “On Pfaff systems with $L^p$ coefficients in dimension two”, C.R. Math. Acad. Sci. Paris, 340 (2005), 879–884 | DOI | MR | Zbl

[29] Mardare S., “On Pfaff systems with $L^p$ coefficients and their applications in differential geometry”, J. Math. Pures Appl., 84 (2005), 1659–1692 | DOI | MR | Zbl

[30] Mejstrik T., “Some remarks on Nagumo's theorem”, Czech. Math. J., 62 (2012), 235–242 | DOI | MR | Zbl

[31] Mendes L. G., “Bounding the degree of solutions to Pfaff equations”, Publ. Mat., 44:2 (2000), 593–604 | MR | Zbl

[32] Musen P., On the application of Pfaff's method in the theory of variations of astronomical constants, NASA, Washington, 1964

[33] Popescu P., Popescu M., “Some aspects concerning the dynamics given by Pfaff forms”, Physics AUC, 21 (2011), 195–202

[34] Rashevskiy K. S., Geometric theory of partial differential equations, Springer, New York, 2001

[35] Siu Y. T., Partial differential equations with compatibility condition, https://www.coursehero.com/file/8864495/Lecture-notes-1/

[36] Spichekovo N. V., “On the behaviour of integral surfaces of a Pfaff equation with a nonclosed singular curve”, Differ. Equ., 41:10 (2005), 1509–1513 | DOI | MR

[37] Unni K. R., Pfaffian differential expressions and equations, Master's degree thesis, Utah State Univ., Logan, 1961, 22 pp.

[38] Vasilevich N. D., Prokhorovich T. N., “A linear Pfaff system of three equations on $C{{P}^{m}}$”, Differ. Equ., 39:6 (2003), 896–898 | DOI | MR | Zbl

[39] Źoladek H., “On algebraic solutions of algebraic Pfaff equations”, Studia Math., 114:2 (1995), 117–126 | DOI | MR