Geodesic
Disconjugacy of fourth-order equations on graphs
Sbornik. Mathematics, Tome 206 (2015) no. 12, pp. 1731-1770 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper develops the theory of disconjugacy of fourth-order equations on geometric graphs which arises in modelling rod structures. The disconjugacy of an equation is defined in terms of a special fundamental system of solutions of the homogeneous equation. The disconjugacy property is shown to be related to the positivity property of the Green's functions for certain classes of boundary value problems for a fourth-order equation on a graph. A maximum principle for a fourth-order equation on a graph is formulated, and some properties of differential inequalities are proved. Bibliography: 25 titles.
Keywords: disconjugacy, differential equation on a graph, Green's function, maximum principle
Mots-clés : conjugacy. 
@article{SM_2015_206_12_a4,
     author = {R. Ch. Kulaev},
     title = {Disconjugacy of fourth-order equations on graphs},
     journal = {Sbornik. Mathematics},
     pages = {1731--1770},
     year = {2015},
     volume = {206},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2015_206_12_a4/}
}
TY  - JOUR
AU  - R. Ch. Kulaev
TI  - Disconjugacy of fourth-order equations on graphs
JO  - Sbornik. Mathematics
PY  - 2015
SP  - 1731
EP  - 1770
VL  - 206
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2015_206_12_a4/
LA  - en
ID  - SM_2015_206_12_a4
ER  - 
%0 Journal Article
%A R. Ch. Kulaev
%T Disconjugacy of fourth-order equations on graphs
%J Sbornik. Mathematics
%D 2015
%P 1731-1770
%V 206
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2015_206_12_a4/
%G en
%F SM_2015_206_12_a4
R. Ch. Kulaev. Disconjugacy of fourth-order equations on graphs. Sbornik. Mathematics, Tome 206 (2015) no. 12, pp. 1731-1770. http://geodesic.mathdoc.fr/item/SM_2015_206_12_a4/

[1] A. Yu. Levin, “Non-oscillation of solutions of the equation $x^{(n)}+p_1(t)x^{(n-1)}+\dots+p_n(t)x=0$”, Russian Math. Surveys, 24:2 (1969), 43–99 | DOI | MR | Zbl

[2] V. Ya. Derr, “Neostsillyatsiya reshenii lineinykh differentsialnykh uravnenii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 1, 46–89

[3] A. Wintner, “One the non-existence of conjugate points”, Amer. J. Math., 73:2 (1951), 368–380 | DOI | MR | Zbl

[4] Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadiev, A. V. Borovskikh, K. P. Lazarev, S. A. Shabrov, Differentsialnye uravneniya na geometricheskikh grafakh, Fizmatlit, M., 2005, 272 pp. | MR | Zbl

[5] Yu. V. Pokornyi, Zh. I. Bakhtina, M. B. Zvereva, S. A. Shabrov, Ostsillyatsionnyi metod Shturma v spektralnykh zadachakh, Fizmatlit, M., 2009, 191 pp. | Zbl

[6] R. Ch. Kulaev, “Necessary and sufficient condition for the positivity of the Green function of a boundary value problem for a fourth-order equation on a graph”, Differ. Equ., 51:3 (2015), 303–317 | DOI | DOI | Zbl

[7] M. G. Zavgorodnii, “Variatsionnye printsipy postroeniya modelei sterzhnevykh sistem”, Matem. modelir. inform. i tekhnol. sistem, 4, Voronezh. gos. tekhnol. akademiya, Voronezh, 2000, 59–62

[8] M. G. Zavgorodnii, S. P. Maiorova, “Ob odnom uravnenii matematicheskoi fiziki chetvertogo poryadka na grafe”, Issledovaniya po differentsialnym uravneniyam i matematicheskomu modelirovaniyu, Sbornik trudov, VNTs RAN, Vladikavkaz, 2008, 88–102

[9] R. Ch. Kulaev, “On the solvability of a boundary value problem for a fourth-order equation on a graph”, Differ. Equ., 50:1 (2014), 25–32 | DOI | DOI | MR | Zbl

[10] G. Polya, “On the mean-value theorem corresponding to a given linear homogeneous differential equation”, Trans. Amer. Math. Soc., 24:4 (1922), 312–324 | DOI | MR | Zbl

[11] G. Pólya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, v. 2, Grundlehren Math. Wiss., 20, Funktionentheorie. Nullstellen. Polynome. Determinanten. Zahlentheorie, 3. bericht. Aufl., Springer-Verlag, Berlin–New York, 1964, x+407 pp. | MR | MR | Zbl

[12] R. Ch. Kulaev, “Usloviya ostsillyatsionnosti funktsii Grina razryvnoi kraevoi zadachi dlya uravneniya chetvertogo poryadka”, Vladikavk. matem. zhurn., 17:1 (2015), 47–59

[13] A. V. Borovskikh, R. Mustafokulov, K. P. Lazarev, Yu. V. Pokornyi, “A class of fourth-order differential equations on a spatial net”, Dokl. Math., 52:3 (1995), 433–435 | MR | Zbl

[14] Yu. V. Pokornyi, R. Mustafokulov, “Positive invertibility of some boundary value problems for fourth-order equation”, Differential Equations, 33:10 (1997), 1364–1371 | MR | Zbl

[15] Yu. V. Pokornyǐ, R. Mustafokulov, “On the positivity of the Green function of linear boundary value problems for fourth-order equations on a graph”, Russian Math. (Iz. VUZ), 43:2 (1999), 71–78 | MR | Zbl

[16] A. V. Borovskikh, K. P. Lazarev, “Fourth-order differential equations on geometric graphs”, J. Math. Sci. (N. Y.), 119:6 (2004), 719–738 | DOI | MR | Zbl

[17] R. Ch. Kulaev, “The Green function of the boundary-value problem on a star-shaped graph”, Russian Math. (Iz. VUZ), 57:2 (2013), 48–57 | DOI | MR | Zbl

[18] R. Ch. Kulaev, “Criterion for the positiveness of the Green function of a many-point boundary value problem for a fourth-order equation”, Differ. Equ., 51:2 (2015), 163–176 | DOI | DOI | Zbl

[19] R. Ch. Kulaev, “O znake funktsii Grina kraevoi zadachi na grafe dlya uravneniya chetvertogo poryadka”, Vladikavk. matem. zhurnal, 15:4 (2013), 19–29 | Zbl

[20] F. R. Gantmacher, M. G. Kreǐn, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Math. Lehrbucher und Monogr., V, Akademie-Verlag, Berlin, 1960, x+359 pp. | MR | MR | Zbl | Zbl

[21] A. Yu. Levin, G. D. Stepanov, “One-dimensional boundary-value problems with operators not reducing the number of changes of sign. II”, Siberian Math. J., 17:4 (1976), 612–625 | DOI | MR | Zbl

[22] G. D. Stepanov, “Effective criteria for the strong sign-regularity and the oscillation property of the Green's functions of two-point boundary-value problems”, Sb. Math., 188:11 (1997), 1687–1728 | DOI | DOI | MR | Zbl

[23] R. Ch. Kulaev, “Oscillation of the Green function of a multipoint boundary value problem for a fourth-order equation”, Differ. Equ., 51:4 (2015), 449–463 | DOI | DOI | Zbl

[24] D. R. Dunninger, “Maximum principles for fourth order ordinary differential inequalities”, J. Math. Anal. Appl., 82:2 (1981), 399–405 | DOI | MR | Zbl

[25] J. Bochenek, “On a maximum principle for fourth order ordinary differential inequalities”, Univ. Iagel. Acta Math., 27 (1988), 163–168 | MR | Zbl