Geodesic
Existence of solutions of a nonlinear eigenvalue problem and their properties
Sbornik. Mathematics, Tome 215 (2024) no. 1, pp. 52-73 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An eigenvalue problem is considered for a nonlinear nonautonomous ordinary differential equation of the second order on a closed interval with conditions of the first type and an additional (local) condition. The nonlinearity in the equation is due to a monotonically increasing nonnegative function with powerlike growth at infinity. The existence of infinite numbers of negative and positive eigenvalues is shown. Asymptotic formulae for the eigenvalues and the maxima of eigenfunctions are found, and comparison theorems are established. Bibliography: 20 titles.
Keywords: nonlinear eigenvalue problem, nonlinear problem of Sturm–Liouville type, eigenvalue asymptotics, comparison theorem, nonlinearizable solutions, integral characteristic equation. 
@article{SM_2024_215_1_a2,
     author = {D. V. Valovik and S. V. Tikhov},
     title = {Existence of solutions of a~nonlinear eigenvalue problem and their properties},
     journal = {Sbornik. Mathematics},
     pages = {52--73},
     year = {2024},
     volume = {215},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2024_215_1_a2/}
}
TY  - JOUR
AU  - D. V. Valovik
AU  - S. V. Tikhov
TI  - Existence of solutions of a nonlinear eigenvalue problem and their properties
JO  - Sbornik. Mathematics
PY  - 2024
SP  - 52
EP  - 73
VL  - 215
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_2024_215_1_a2/
LA  - en
ID  - SM_2024_215_1_a2
ER  - 
%0 Journal Article
%A D. V. Valovik
%A S. V. Tikhov
%T Existence of solutions of a nonlinear eigenvalue problem and their properties
%J Sbornik. Mathematics
%D 2024
%P 52-73
%V 215
%N 1
%U http://geodesic.mathdoc.fr/item/SM_2024_215_1_a2/
%G en
%F SM_2024_215_1_a2
D. V. Valovik; S. V. Tikhov. Existence of solutions of a nonlinear eigenvalue problem and their properties. Sbornik. Mathematics, Tome 215 (2024) no. 1, pp. 52-73. http://geodesic.mathdoc.fr/item/SM_2024_215_1_a2/

[1] D. V. Valovik, “On a nonlinear eigenvalue problem related to the theory of propagation of electromagnetic waves”, Differ. Equ., 54:2 (2018), 165–177 | DOI | MR | Zbl

[2] V. Kurseeva, M. Moskaleva and D. Valovik, “Asymptotical analysis of a nonlinear Sturm–Liouville problem: linearisable and non-linearisable solutions”, Asymptot. Anal., 119:1–2 (2020), 39–59 | DOI | MR | Zbl

[3] S. V. Tikhov and D. V. Valovik, “Nonlinearizable solutions in an eigenvalue problem for Maxwell's equations with nonhomogeneous nonlinear permittivity in a layer”, Stud. Appl. Math., 149:3 (2022), 565–587 | DOI | MR

[4] L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, Course of Theoretical Physics, 8, Pergamon Press, Oxford, 1984, xiii+460 pp. | MR

[5] N. N. Akhmediev and A. Ankiewicz, Solitons, Fizmatlit, Moscow, 2003, 304 pp. (Russian)

[6] T. Cazenave, Semilinear Schrödinger equations, Courant Lect. Notes Math., 10, New York Univ., Courant Inst. Math. Sci., New York; Amer. Math. Soc., Providence, RI, 2003, xiv+323 pp. | DOI | MR | Zbl

[7] G. Fibich, The nonlinear Schrödinger equation. Singular solutions and optical collapse, Appl. Math. Sci., 192, Springer, Cham, 2015, xxxii+862 pp. | DOI | MR | Zbl

[8] P. E. Zhidkov, “Riesz basis property of the system of eigenfunctions for a non-linear problem of Sturm–Liouville type”, Sb. Math., 191:3 (2000), 359–368 | DOI | MR | Zbl

[9] B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac operators, Math. Appl. (Soviet Ser.), 59, Kluwer Acad. Publ., Dordrecht, 1991, xii+350 pp. | DOI | MR | Zbl

[10] V. A. Marchenko, Spectral theory of Sturm–Liouville operators, Naukova Dumka, Kiev, 1972, 219 pp. (Russian) | MR | Zbl

[11] D. Sansone, Equazioni differenziali nel campo reale, v. 1, 2nd ed., N. Zanichelli, Bologna, 1948, xvii+400 pp. | MR | Zbl

[12] R. Courant and D. Hilbert, Methoden der mathematischen Physik, Grundlehren Math. Wiss., I, 2. verb. Aufl., J. Springer, Berlin, 1931, xiv+469 pp. | MR | Zbl

[13] I. G. Petrovskii, Ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1966, x+232 pp. | MR | MR | Zbl | Zbl

[14] F. G. Tricomi, Differential equations, Hafner Publishing Co., New York; Blackie Son Ltd., London, 1961, x+273 pp. | MR | Zbl

[15] Yu. G. Smirnov and D. V. Valovik, “Reply to “Comment on ‘Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity’ ” ”, Phys. Rev. A (3), 92:5 (2015), 057804, 2 pp. | DOI

[16] M. A. Naimark, Linear differential operators, v. I, II, Frederick Ungar Publishing Co., New York, 1967, 1968, xiii+144 pp., xv+352 pp. | MR | MR | Zbl

[17] L. S. Pontryagin, Ordinary differential equations, ADIWES Int. Ser. in Math., Addison-Wesley Publishing Co., Inc., Reading, MA–Palo Alto, CA–London, 1962, vi+298 pp. | MR | Zbl

[18] D. V. Valovik, “On the integral characteristic function of the Sturm–Liouville problem”, Sb. Math., 211:11 (2020), 1539–1550 | DOI | MR | Zbl

[19] D. V. Valovik and G. V. Chalyshov, “Integral characteristic function of a nonlinear Sturm–Liouville problem”, Differ. Equ., 57:12 (2021), 1555–1564 | DOI | MR | Zbl

[20] H. W. Schürmann, Y. Smirnov and Y. Shestopalov, “Propagation of TE waves in cylindrical nonlinear dielectric waveguides”, Phys. Rev. E (3), 71:1 (2005), 016614, 10 pp. | DOI