Geodesic
Zeros of the Green's function for the de la Vallée-Poussin problem
Sbornik. Mathematics, Tome 199 (2008) no. 6, pp. 891-921 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Green's function for the de la Vallée-Poussin problem \begin{gather*} Lx\equiv x^{(n)}+p_1(t)x^{(n-1)}+\dots+p_n(t)x=f, \\ x(a_i)=A_i^{(0)}, \ \ x'(a_i)=A_i^{(1)}, \ \ \dots, \ \ x^{(\nu_i-1)}(a_i)=A_i^{(\nu_i-1)}, \ \ i= {1,\dots,m}, \end{gather*} where $a=a_1, $m\geqslant2$, $\sum\nu_i=n$, $p_i(\,\cdot\,)$ and $f(\,\cdot\,)\in L_1[a,b]$, is investigated. It is defined in the square $a\leqslant t,s\leqslant b$, and vanishes at the lines $t=a_i$, $i={1,\dots,m}$, $s=a$, $s=b$; it is proved that the orders of its zeros have uniform bounds. Bibliography: 27 titles. 
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Yu. V. Pokornyi. Zeros of the Green's function for the de la Vallée-Poussin problem. Sbornik. Mathematics, Tome 199 (2008) no. 6, pp. 891-921. http://geodesic.mathdoc.fr/item/SM_2008_199_6_a4/

[1] Ju. V. Pokornyi, “On estimates for the Green's function for a multipoint boundary problem”, Math. Notes, 4:5 (1968), 810–814 | DOI | MR | Zbl | Zbl

[2] M. A. Krasnosel'skii, Ju. V. Pokornyi, “Non-zero solutions of equations with strong nonlinearities”, Math. Notes, 5:2 (1969), 153–157 | DOI | MR | Zbl

[3] M. A. Krasnosel'skiĭ, P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren Math. Wiss., 263, Springer-Verlag, Berlin, 1984 | MR | MR | Zbl | Zbl

[4] Ju. V. Pokornyi, “Second solutions of a nonlinear de la Vallee Poussin problem”, Differential Equations, 6 (1970), 1215–1220 | MR | Zbl | Zbl

[5] Ju. V. Pokornyi, “Second solutions of multipoint boundary-value problems with convex nonlinearities”, Differential Equations, 11 (1975), 1345–1352 | MR | Zbl | Zbl

[6] Ju. V. Pokornyi, “Inequality for second characteristic values of positive operators of certain classes”, Math. Notes, 9:1 (1971), 17–20 | MR | Zbl | Zbl

[7] P. P. Zabreiko, M. A. Krasnosel'skii, Yu. V. Pokornyi, “On a class of linear positive operators”, Funct. Anal. Appl., 5:4 (1971), 272–279 | DOI | MR | Zbl

[8] J. Pokornas, “Some estimates for the remainder term of a Hermite generalized interpolation polynomial”, Lithuanian Math. J., 16:4 (1976), 611–620 | DOI | MR | Zbl | Zbl

[9] N. V. Azbelev, “O granitsakh primenimosti teoremy Chaplygina o differentsialnykh neravenstvakh”, Matem. sb., 39(81):2 (1956), 161–178 | MR | Zbl

[10] N. V. Azbelev, Z. B. Tsalyuk, “Ob odnom obobschenii teoremy Chaplygina o differentsialnykh neravenstvakh”, Tr. 3 Vsesoyuzn. matem. s'ezda, v. 2, M., 1956

[11] E. S. Chichkin, “Teorema o differentsialnom neravenstve dlya mnogotochechnykh kraevykh zadach”, Konf. un–ta za 1960 g., sekts. matem. n., Kazan, 1961, 40–41

[12] G. N. Zhevlakov, Yu. V. Komlenko, E. L. Tonkov, “Teoremy suschestvovaniya dlya lineinykh kraevykh zadach”, Dokl. AN BSSR, 10:9 (1966), 626–628 | MR | Zbl

[13] A. Yu. Levin, O mnogotochechnoi kraevoi zadache, Dis. ... kand. fiz.-matem. nauk, Voronezh, 1961

[14] 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 | Zbl

[15] Ph. Hartman, “On disconjugacy criteria”, Proc. Amer. Math. Soc., 24:2 (1970), 374–381 | DOI | MR | Zbl

[16] I. T. Kiguradze, Nekotorye singulyarnye kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii, Izd-vo Tbil. un-ta, Tbilisi, 1975 | MR | Zbl

[17] Yu. V. Pokornyi, “O spektre interpolyatsionnoi kraevoi zadachi”, UMN, 32:6 (1977), 263–264 | MR | Zbl

[18] V. Ya. Derr, “A generalized la Vallée Poussin problem”, Differential Equations, 23:11 (1987), 1254–1263 | MR | Zbl | Zbl

[19] A. V. Borovskikh, Yu. V. Pokornyi, “Chebyshev–Haar systems in the theory of discontinuous Kellogg kernels”, Russian Math. Surveys, 49:3 (1994), 1–42 | DOI | MR | Zbl

[20] E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen, Geest Portig, Leipzig, 1959 | MR | MR | Zbl | Zbl

[21] E. L. Ince, Ordinary differential equations, Longmans, London, 1927 | MR | Zbl

[22] E. A. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York–Toronto–London, 1955 | MR | Zbl

[23] G. Sansone, Equazioni differenziali nel campo reale, vol. 1, Nicola Zanichelli, Bologna, 1948 | MR | MR | Zbl

[24] F. V. Atkinson, Discrete and continuous boundary problems, Math. Sci. Engrg., 8, Academic Press, New York–London, 1964 | MR | MR | Zbl | Zbl

[25] Yu. V. Pokornyi, A. V. Borovskikh, “On defects of axiomatics of Green's function”, Russian Acad. Sci. Dokl. Math., 65:3 (2002), 399–403 | MR | Zbl

[26] E. F. Beckenbach, R. Bellman, Inequalities, Ergeb. Math. Grenzgeb., 30, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1961 | MR | MR | Zbl | Zbl

[27] Z. Surikova, “Sur les systèmes différentielles linéaires généraux”, Dokl. AN USSR, 1 (1936), 159–164 | Zbl