Stationary groups of linear differential equations. (English) Zbl 0573.34028

Neuman, František

Czech. Math. J. 34(109), 645-663 (1984).

Let \(G(P_ n)\) be a stationary group formed by all global transformations of the form \(<c| h''|^{(1-n)/2},h>\), where \(c\neq 0\) and \(h(I)=I\), \(h\in C^{n+1}(I)\), h’(x)\(\neq 0\) on I, that globally transform n-order (n\(\geq 2)\) linear differential equation \(P_ n\) into itself. For each group \(G(P_ n)\) the set of all h occurring in its elements forms the group \(G_ 0(P_ n)\) with respect to composition.
The author lists all possible groups \(G_ 0(P_ n)\) up to \(C^{n+1}\)- conjugacy, discusses the oscillatory properties for the equation \(P_ n\) if it is iterated from a second order linear differential equation. Moreover he presents all cases when \(G_ 0(P_ n)\) is \(C^{n+1}\) conjugate and specifies 5 possible types of subgroups of increasing elements of \(G_ 0(P_ n)\) with respect to the number of parameters.

Reviewer: D.Bobrowski

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30	Linear ordinary differential equations and systems
34C20	Transformation and reduction of ordinary differential equations and systems, normal forms

Keywords:

iterative equation; Kummer equation; oscillatory equation; conjugate groups; stationary group; global transformations; second order linear differential equation

References:

[1]	G. Blanton J. A. Baker: Iteration groups generated by \(C^n\) functions. Archivum Math. (Brno), 18 (1982), 121-127. · Zbl 0518.26002
[2]	O. Borůvka: Lineare Differentialtransformationen 2. Ordnung, VEB, Berlin 1967, English edition: Linear Differential Transformations of the Second Order, The English Univ. Press, London 1971. · Zbl 0153.11201
[3]	O. Borůvka: Sur une classe des groupes continus à un paramètre formés des functions réelles d’une variable. Ann. Polon Math. 42 (1982), 27-37.
[4]	G. H. Halphen: Mémoire sur la réduction des équations différentielles linéaires aux formes intégrables. Mémoires présentés par divers savants à l’académie des sciences de l’institut de France 28 (1884), 1-301.
[5]	O. Hölder: Die Axiome der Quantität und die Lehre vom Masse. Ber. Verh. Sachs. Ges. Wiss. Leipzig, Math. Phys. Cl. 53 (1901), 1-64. · JFM 32.0079.01
[6]	Z. Hustý: Die Iteration homogener linearer Differentialgleichungen. Publ. Fac. Sci. Univ. J. E. Purkyne (Brno), 449 (1964), 23-56.
[7]	F. Neuman: Categorial approach to global transformations of the n-th order linear differential equations. Časopis Pěst. Mat. 102 (1977), 350-355. · Zbl 0374.34028
[8]	F. Neuman: On solutions of the vector functional equation \(y(\xi (x))=f(x)\cdot A\cdot y(x)\). Aequationes Math. 16 (1977), 245-257. · Zbl 0375.34013 · doi:10.1007/BF01836037
[9]	F. Neuman: Criterion of global equivalence of linear differential equations. Proc. Roy. Soc. Edinburg, 97A (1984), 217-221. · Zbl 0552.34009 · doi:10.1017/S0308210500032005
[10]	F. Neuman: A survey of global properties of linear differential equations of the n-th order. Ordinary and Partial Differential Equations, Proceedings, Dundee 1982, Lecture Notes in Mathematics 964, 548-563. · Zbl 0501.34003
[11]	J. Posluszny L.A. Rubel: The motion of an ordinary differential equation. J. Diff. Equations 34 (1979), 291-302. · Zbl 0386.34020 · doi:10.1016/0022-0396(79)90011-1
[12]	P. Stāckel: Über Transformationen von Differentialgleichungen. J. Reine Angew. Math. 111 (1893), 290-302. · JFM 25.0167.01