Geodesic
On a~fourth-order problem with spectral and physical parameters in the boundary condition
Izvestiya. Mathematics , Tome 68 (2004) no. 4, pp. 645-658.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the following fourth-order boundary-value problem: \begin{gather*} [(py'')'-qy']'=\lambda ry, \\ y(0)=y'(0)=y''(1)=[(py'')'-qy'](1)+\lambda my(1)=0 \end{gather*} with spectral parameter $\lambda\in\mathbb C$ and physical parameter $m\in\mathbb R$. We assign to this problem a linear pencil of bounded operators $T_m=T_m(\lambda)$ depending on the physical parameter $m$ and acting from $\mathcal H_2=\{y\mid y\in W_2^2[0,1],\ y(0)=y'(0)=0\}$ to the dual space $\mathcal H_{-2}$. We study the spectral properties of $T_m$ and use the results of this study to describe properties of the eigenvalues of the problem for various values of $m$. In particular, we establish asymptotics of these eigenvalues as $m\nearrow0$. 
@article{IM2_2004_68_4_a0,
     author = {J. Ben Amara and A. A. Vladimirov},
     title = {On a~fourth-order problem with spectral and physical parameters in the boundary condition},
     journal = {Izvestiya. Mathematics },
     pages = {645--658},
     publisher = {mathdoc},
     volume = {68},
     number = {4},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2004_68_4_a0/}
}
TY  - JOUR
AU  - J. Ben Amara
AU  - A. A. Vladimirov
TI  - On a~fourth-order problem with spectral and physical parameters in the boundary condition
JO  - Izvestiya. Mathematics 
PY  - 2004
SP  - 645
EP  - 658
VL  - 68
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2004_68_4_a0/
LA  - en
ID  - IM2_2004_68_4_a0
ER  - 
%0 Journal Article
%A J. Ben Amara
%A A. A. Vladimirov
%T On a~fourth-order problem with spectral and physical parameters in the boundary condition
%J Izvestiya. Mathematics 
%D 2004
%P 645-658
%V 68
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2004_68_4_a0/
%G en
%F IM2_2004_68_4_a0
J. Ben Amara; A. A. Vladimirov. On a~fourth-order problem with spectral and physical parameters in the boundary condition. Izvestiya. Mathematics , Tome 68 (2004) no. 4, pp. 645-658. http://geodesic.mathdoc.fr/item/IM2_2004_68_4_a0/

[1] Kollatts L., Zadachi na sobstvennye znacheniya (s tekhnicheskimi prilozheniyami), Nauka, M., 1968

[2] Timoshenko S. P., Prochnost i kolebaniya elementarnykh konstruktsii, Nauka, M., 1975

[3] Roseau F., Theory of vibration in mechanical systems, Springer-Verlag, N.Y., 1983

[4] Walter J., “Regular eigenvalue problems with eigenvalue parameter in the boundary conditions”, Math. Z., 133 (1973), 301–312 | DOI | MR | Zbl

[5] Binding P. A., Brown P. J., Seddighi K., “Sturm–Liouville problems with eigenparameter dependent boundary condition”, Proc. Edinburgh Math. Soc. (2), 37 (1993), 57–72 | DOI | MR

[6] Ben Amara Zh., Shkalikov A. A., “Zadacha Shturma–Liuvillya s fizicheskim i spektralnym parametrami v granichnom uslovii”, Matem. zametki, 66:2 (1999), 163–172 | MR | Zbl

[7] Lancaster P., Shkalikov A., Qiang Ye., “Strongly definitizable linear pencils in Hilbert space”, Integr. Equat. Oper. Th., 17 (1993), 338–360 | DOI | MR | Zbl

[8] Vladimirov A. A., “Otsenki chisla sobstvennykh znachenii samosopryazhennykh operator-funktsii”, Matem. zametki, 74:6 (2003), 839–848 | MR

[9] Krylov A. N., O nekotorykh differentsialnykh uravneniyakh matematicheskoi fiziki, imeyuschikh prilozhenie v tekhnicheskikh voprosakh, Izd-vo AN SSSR, M., 1932

[10] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[11] Glazman I. M., Pryamye metody kachestvennogo spektralnogo analiza singulyarnykh differentsialnykh operatorov, Fizmatgiz, M., 1963 | MR

[12] Shkalikov A. A., “Kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii s parametrom v granichnykh usloviyakh”, Tr. seminara im. I. G. Petrovskogo, 9 (1983), 140–166

[13] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl