On factorizations of selfadjoint ordinary differential operators
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- by Antonio Granata PDF
- Proc. Amer. Math. Soc. 86 (1982), 260-266 Request permission
Abstract:
Consider an ordinary linear differential operator $L$, of order $n \geqslant 1$, represented by $Lu \equiv {a_n}(t){u^{(n)}} + \cdots + {a_0}(t)u\;\forall u \in {C^n}(a,b)$, with real-valued coefficients ${a_k} \in {C^k}(a,b)$, $0 \leqslant k \leqslant n$, ${a_n} \ne 0$ on $(a,b)$. According to a classical result, if $L$ is formally selfadjoint on $(a,b)$ then it has a factorization of the type $Lu \equiv {p_n}[{p_{n - 1}}( \cdots ({p_1}({p_0}u)â)â \cdots )â]â\forall u \in {C^n}(a,b)$, where the ${p_k}$âs are sufficiently-smooth and everywhere nonzero functions on $(a,b)$ such that ${p_k} = {p_{n - k}}$ $(k = 0, \ldots ,n)$. In this note we shall examine this result critically and show by means of counterexamples that the different classical proofs are either merely local or purely heuristic. A proof, which is both rigorous and global, is inferred from recent results on canonical factorizations of disconjugate operators. In addition, information is obtained on the behavior of the ${p_k}$âs at the endpoints of $(a,b)$ which may prove useful in applications.References
- W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785 G. Darboux, Leçons sur la thĂ©orie gĂ©nĂ©rale des surfaces, 2me partie, 2me edition, Gauthier-Villars, Paris, 1915. G. Frobenius, Ueber den Begriff der IrreductibilitĂ€t in der Theorie der linearen Differentialgleichungen, J. fĂŒr Math. 76 (1873), 236-270. â, Ueber die Determinante mehrerer Funktionen einer Variabeln, J. fĂŒr Math. 77 (1874), 245-257. â, Ueber die regulĂ€ren Integrale der linearen Differentialgleichungen, J. fĂŒr Math. 80 (1875), 317-333. â, Ueber adjungirte lineare DifferentialausdrĂŒcke, J. fĂŒr Math. 85 (1878), 185-213.
- Antonio Granata, Singular Cauchy problems and asymptotic behaviour for a class of $n$-th order differential equations, Funkcial. Ekvac. 20 (1977), no. 3, 193â212. MR 486834
- Antonio Granata, Corrigendum and addendum: âSingular Cauchy problems and asymptotic behaviour for a class of $n$th-order differential equationsâ [Funkcial. Ekvac. 20 (1977), no. 3, 193â212; MR 58 #6531], Funkcial. Ekvac. 22 (1979), no. 3, 351â354. MR 577852
- Antonio Granata, Canonical factorizations of disconjugate differential operators, SIAM J. Math. Anal. 11 (1980), no. 1, 160â172. MR 556506, DOI 10.1137/0511014
- E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
- A. Ju. Levin, The non-oscillation of solutions of the equation $x^{(n)}+p_{1}(t)x^{(n-1)}+\cdots +p_{n} (t)x=0$, Uspehi Mat. Nauk 24 (1969), no. 2 (146), 43â96 (Russian). MR 0254328
- G. PĂłlya, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc. 24 (1922), no. 4, 312â324. MR 1501228, DOI 10.1090/S0002-9947-1922-1501228-5 L. Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen, Band I, Teubner, Leipzig, 1895.
- William F. Trench, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 189 (1973), 319â327. MR 330632, DOI 10.1090/S0002-9947-1974-0330632-X
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 260-266
- MSC: Primary 47E05; Secondary 34B25
- DOI: https://doi.org/10.1090/S0002-9939-1982-0667285-7
- MathSciNet review: 667285