Differential equations with degenerate, depending on the unknown function operator at the derivative

B. V. Loginov ; Yu. B. Rousak ; L. R. Kim-Tyan

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, Tome 59 (2016), pp. 119-147.

Résumé

We develop the theory of generalized Jordan chains of multiparameter operator functions $A(\lambda)\colon E_1\to E_2$, $\lambda\in\Lambda$, $\dim\Lambda=k$, $\dim E_1=\dim E_2=n$, where $A_0=A(0)$ is a noninvertible operator. To simplify the notation, in 1–3 the geometric multiplicity $\lambda_0$ is set to 1, i.e. $\dim N(A_0)=1$, $N(A_0)=\operatorname{span}\{\varphi\}$, $\dim N^\ast(A_0^\ast)=1$, $N^\ast(A_0^\ast)=\operatorname{span}\{\psi\}$, and the operator function $A(\lambda)$ is supposed to be linear with respect to $\lambda$. For the polynomial dependence of $A(\lambda)$, in 4 we consider a linearization. However, the bifurcation existence theorems hold in the case of several Jordan chains as well. We consider applications to degenerate differential equations of the form $[A_{0}+R(\cdot,x)]x'=Bx$.

Bibliographie

[1] V. I. Arnol'd, Geometric Methods in the Theory of Ordinary Differential Equations, MTsNMO, Moscow, 2002 (in Russian)

[2] M. M. Vaynberg, V. A. Trenogin, Branching Theory of Solutions to Nonlinear Equations, Nauka, Moscow, 1969 (in Russian) | MR

[3] M. Golubitskiy, V. Giyemen, Stable Mappings and Their Singularities, Mir, Moscow, 1977 (in Russian) | MR

[4] M. A. Krasnosel'skij, Topological Methods in the Theory of Nonlinear Integral Equations, GITTL, Moscow, 1956 (in Russian) | MR

[5] M. A. Krasnosel'skij, P. P. Zabreyko, Geometric Methods of Nonlinear Analysis and Its Applications, Nauka, Moscow, 1975 (in Russian) | MR

[6] M. A. Krasnosel'skij, A. I. Perov, A. I. Povolotskij, and P. P. Zabreyko, Vector Fields in a Plane, Fizmatlit, Moscow, 1963 (in Russian) | MR

[7] B. V. Loginov, Yu. B. Rusak, “Generalized Jordan structure in the branching theory”, Direct and Inverse Problems for Partial Differential Equations and Their Applications, FAN, Tashkent, 1978, 113–148 (in Russian) | MR

[8] B. V. Loginov, Yu. B. Rusak, L. R. Kim-Tyan, “Normal forms of differential equations with degenerate matrix at the derivative when the Jordan chain of maximal length exists”, Some actual problems of contemporary mathematics and mathematical education (Saint-Petersburg, April 15–20, 2013), Proceedings of scientific conference “Hertsen readings”, 66, 93–109 (in Russian) | MR

[9] B. V. Loginov, Yu. B. Rusak, L. R. Kim-Tyan, “Differential Equations with Degenerate, Depending on the Unknown Function Operator at the Derivative”, Int. Conf. on Differential Equations and Dynamic Systems (Suzdal', 2014), Abstracts, MIAN, Moscow, 2014, 106–107 (in Russian)

[10] L. Nirenberg, Topics in Nonlinear Functional Analysis, Mir, Moscow, 1977 (in Russian) | MR

[11] M. M. Postnikov, Introduction to the Morse Theory, Nauka, Moscow, 1971 (in Russian) | MR

[12] Yu. B. Rusak, Generalized Jordan Structure in the Branching Theory, PhD Thesis, In-t mat. im. V. I. Romanovskogo AN UzSSR, Tashkent, 1979 (in Russian)

[13] N. A. Sidorov, General Issues of Regularization in Problems of the Branching Theory, Izd-vo Irkut. un-ta, Irkutsk, 1982 (in Russian) | MR

[14] V. A. Trenogin, Functional Analysis, Fizmatlit, Moscow, 2002 (in Russian)

[15] V. A. Trenogin, N. A. Sidorov, “Study of bifurcation points and continuous branches of solutions to nonlinear equations”, Differential and Integral Equations, 1, Irkutsk, 1972, 216–248 (in Russian) | MR

[16] V. A. Trenogin, A. F. Filippov (eds.), Nonlinear Analysis ans Nonlinear Differential Equations, Fizmatlit, Moscow, 2003 (in Russian)

[17] F. Khartman, Ordinary Differential Equations, Mir, Moscow, 1970 (in Russian) | MR

[18] Carr J., Application of centra manifold theory, Appl. Math. Sci., 35, 1981 | DOI | MR

[19] Loginov B. V., “Branching equation in the root subspaces”, Nonlinear Anal., 32:3 (1998), 439–448 | DOI | MR

[20] Loginov B. V., Rousak Yu. B., “Generalized Jordan structure in the problem of stability of bifurcation equations”, Nonlinear Anal., 17:3 (1991), 219–231 | DOI | MR

[21] Loginov B. V., Rousak Yu. B., Kim-Tyan L. R., “Differential equations with degenerated variable operator at the derivative”, Current Trends in Analysis and Its Applications, Proceedings of the 9th ISAAC Congress (Kraków 2013), Birkhäuser, Basel, 2015, 101–108 | DOI | Zbl

[22] Ma T., Wang Sh., Bifurcation theory and applications, World Scientific, Hackensack, 2005 | MR | Zbl

[23] Magnus R. J., “A generalization of multiplicity and the problem of bifurcation”, Proc. Lond. Math. Soc. (3)., 32 (1976), 251–278 | DOI | MR | Zbl

[24] Marszalek W., “Fold points and singularity induced bifurcation in inviscid transonic flow”, Phys. Lett. A, 376:28–29 (2012), 2032–2037 | DOI | Zbl

[25] Sidorov N., Loginov B., Synitsin H. V., Falaleev M. V., Lyapunov–Schmidt methods in nonlinear analysis and applications, Kluwer Academic Publ., Dordrecht, 2012