# Article

Full entry | PDF   (2.1 MB)
Keywords:
Differential operators; linear differential equation of third order; canonical forms; adjoint equation; cyclic permutation; oscillatory solution; Kneser solution; property $\mathrm A$
Summary:
Consider the third order differential operator $L$ given by $L(\cdot )\equiv \,\frac {1}{a_3(t)}\frac {\mbox{d}}{\mbox{d} t}\frac {1}{a_2(t)}\frac {\mbox{d}}{\mbox{d} t} \frac {1}{a_1(t)}\frac {\mbox{d}}{\mbox{d} t}\,(\cdot )$ and the related linear differential equation $L(x)(t)+x(t)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients $a_i$, $i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).
References:
[1] Bartušek M.: Asymptotic properties of oscillatory solutions of differential equations of the $n$-th order. Folia Fac. Sci. Nat. Univ. Brunensis Masarykianae (1992). MR 1271586
[2] Bartušek, M., Došlá Z.: Oscillatory criteria for nonlinear third order differential equations with quasiderivatives. Diff. equation  and Dynam. Syst. 3 (1995), 251–268. MR 1386748
[3] Cecchi M.: Oscillation criteria for a class of third order linear differential equations. Boll. Un. Mat. Ital., VI, 2–C (1983), 297–306. MR 0718377 | Zbl 0523.34029
[4] Cecchi M.: Sul comportamento delle soluzioni di una classe di equazioni differenziali lineari del terzo ordine in caso di oscillazione. Boll. Un. Mat. Ital., VI, 4–C 4 (1985), 71–85.
[5] Cecchi M., Marini M.: Oscillation properties of third order nonlinear differential equation. Nonlinear Analysis, Th. M. Appl. 15 (1990), 141–153. DOI 10.1016/0362-546X(90)90118-Z
[6] Cecchi M., Došlá Z., Marini M., Villari G.: On the qualitative behavior of solutions of third order differential equations. J. Math. Anal. Appl. 197 (1996), 749–766. DOI 10.1006/jmaa.1996.0050 | MR 1373077
[7] Cecchi M., Marini M., Villari G.: On a cyclic disconjugated operator associated to linear differential equations. Annali Mat. Pura Appl. IV CLXX (1996), 297–309. MR 1441623
[8] Coppel W.A.: Disconjugacy. Springer-Verlag 1971, Lectures Notes in Math. 220. MR 0460785 | Zbl 0224.34003
[9] Dolan J. M.: On the relationship between the oscillatory behavior of a linear third-order differential equation and its adjoint. J. Diff. Equat. 7 (1970), 367–388. DOI 10.1016/0022-0396(70)90116-6 | MR 0255908 | Zbl 0191.10001
[10] Elias U.: Nonoscillation and eventual disconjugacy. Proc. Amer.Math. Soc. 66 (1977), 269–275. DOI 10.1090/S0002-9939-1977-0460791-8 | MR 0460791 | Zbl 0367.34024
[11] Erbe L.: Oscillation, nonoscillation, and asymptotic behavior for third order nonlinear differential equations. Annali Mat. Pura Appl. IV, 110 (1976), 373–391. DOI 10.1007/BF02418014 | MR 0427738 | Zbl 0345.34023
[12] Gaudenzi M.: On the Sturm-Picone theorem for $n$th–order differential equations. Siam J. Math. Anal. 21 (1990), 980–994. DOI 10.1137/0521054 | MR 1052882
[13] Greguš M.: Third Order Linear Differential Equations. D. Reidel Publ. Comp., Dordrecht, Boston, Lancaster, Tokyo, 1987. MR 0882545
[14] Hanan M.: Oscillation criteria for third-order linear differential equation. Pacific J. Math. 11 (1961), 919-944. DOI 10.2140/pjm.1961.11.919 | MR 0145160
[15] Chanturia T.A.: On oscillatory properties of systems of nonlinear ordinary differential equations (Russian). Trudy universiteta prikladnoj matematiky, Tbilisi 14 (1983), 163–203. MR 0741463
[16] Kiguradze I. T., Chanturia T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic Publishers, Dordrecht-Boston-London (1993, 432 pp). MR 1220223
[17] Kusano T., Naito M., Tanaka K.: Oscillatory and asymptotic behavior of solutions of a class of linear ordinary differential equations. Proc. Royal Soc. Edinburgh 90A (1981), 24–40. MR 0636062
[18] Lazer A. C.: The behaviour of solutions of the differential equation $y^{\prime \prime \prime }+p(x)y^{\prime }+q(x)y=0$. Pacific J. Math. 17 (1966), 435–466. MR 0193332
[19] Ohriska J.: Oscillatory and asymptotic properties of third and fourth order linear differential equations. Czech. Math. J. 39 (114) (1989), 215–224. MR 0992128 | Zbl 0688.34018
[20] Swanson C. A.: Comparison and Oscillation Theory of Linear Differential Equations. Acad. Press, New York, 1968. MR 0463570 | Zbl 0191.09904
[21] Švec M.: Behaviour of nonoscillatory solutions of some nonlinear differential equations. Acta Math. Univ. Comenianae 34 (1980), 115–130.
[22] Trench W. F.: Canonical forms and principal systems for general disconjugate equations. TAMS 189 (1974), 319–327. DOI 10.1090/S0002-9947-1974-0330632-X | MR 0330632 | Zbl 0289.34051

Partner of