Bahram A. Aliev, Nargul K. Kurbanova and Yakov Yakubov,
Solvability of the abstract Regge boundary value problem and asymptotic behavior of eigenvalues of one abstract spectral problem
Received: 14 August 2014
Accepted in revised form: 14 January 2015
Mathematics Subject Classification (2010): 47E05, 47A75, 34L15, 34G10, 35J25.
Keywords: Differential-operator equations, elliptic equations, isomorphism, spectral parameter, scattering problem, Regge problem, maximal \(L_p\)-regularity, eigenvalues.
, : National Academy of Sciences of Azerbaijan, Institute of Mathematics and Mechanics, 9, B. Vahabzade str., Baku 1141, Azerbaijan Republic
 : Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel
The third author was supported by the Israel Ministry of Absorption.
Abstract: In the paper, we give an abstract formulation of the classical Regge boundary value problem (but with a constant potential) in a Hilbert space and prove an isomorphism result for the problem. This result implies, in particular, maximal \(L_p\)-regularity for the problem. We also obtain an estimate of the solution with respect to the spectral parameter. Then, for one homogeneous abstract spectral problem, we find asymptotic behavior of its eigenvalues. A possible application of the abstract results to elliptic partial differential equations is shown at the end of the paper.
A. Aibeche, A. Favini and Ch. Mezoued,
Deficient coerciveness estimate for an abstract differential equation with a parameter dependent boundary conditions,
Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), no. 3, 535-547.
 B. A. Aliev, Asymptotic behavior of eigen-values of a boundary value problem with spectral parameter in the boundary conditions for the second order elliptic differential-operator equation, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 25 (2005), no. 7, 3-8. [MR2267195]
 B. A. Aliev, Asymptotic behavior of the eigenvalues of a boundary value problem for a second-order elliptic operator-differential equation (Russian), Ukra´n. Mat. Zh. 58 (2006), no. 8, 1146-1152. (Engl. translation in: Ukrainian Math. J. 58 (2006), no. 8, 1298-1306.) [MR2345084]
 B. A. Aliev, Solvability of the boundary-value problem for the second-order elliptic differential-operator equation with spectral parameter in the equation and boundary conditions (Russian), Ukra´n. Mat. Zh. 62 (2010), no. 1, 3-14. (Engl. translation in: Ukrainian Math. J. 62 (2010), no. 1, 1-14. [MR2888575])
 B. A. Aliev and Ya. Yakubov, Elliptic differential-operator problems with a spectral parameter in both the equation and boundary-operator conditions, Adv. Differential Equations 11 (2006), no. 10, 1081-1110. [MR2279710] (Erratum: ibid 12 (2007), no. 9, 1079. [MR2351838])
 B. A. Aliev and Ya. Yakubov, Second order elliptic differential-operator equations with unbounded operator boundary conditions in UMD Banach spaces, Integral Equations Operator Theory 69 (2011), 269-300. [MR2765589]
 M. Bairamogly and N. M. Aslanova, Distribution of eigenvalues and trace formula for the Sturm-Liouville operator equation (Russian), Ukraïn Mat. Zh. 62 (2010), no 7, 867-877. (Engl. translation in: Ukrainian Math. J. 62 (2010), no. 7, 1005-1017. [MR2888655])
 V. M. Bruk, On a class of boundary value problems with a spectral parameter in the boundary condition (Russian), Mat. Sb. (N.S.) 100 (1976), no. 2, 210-216. (Engl. translation in: Math. USSR-Sb. 29 (1976), no. 2, 186-192). [MR0415381]
 M. Denche, Abstract differential equation with a spectral parameter in the boundary conditions, Results Math. 35 (1999), 216-227. [MR1694903]
 M. V. Fedorjuk, Spectral analysis and the scattering problem for the operator \(-d^2/dx^2+A(x)\). I (Russian), Differencial'nye Uravnenija 8 (1972), 984-994. [MR0301563]
 M. V. Fedorjuk, Spectral analysis and the scattering problem for the operator \(-d^2/dx^2+A(x)\). II (Russian), Differencial'nye Uravnenija 8 (1972), 1187-1194. [MR0304763]
 V. I. Gorbachuk and M. A. Rybak, Boundary value problems for the operator Sturm-Liouville equation with a spectral parameter in the equation and in the boundary condition (Russian), Direct and inverse problems of scattering theory, vol. 148, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev 1981, 3-16. [MR0673592]
 G. M. Gubreev and V. N. Pivovarchik, Spectral analysis of the Regge problem with parameters (Russian), Funktsional. Anal. i Prilozhen. 31 (1997), no. 1, 70-74. (Engl. translation in: Funct. Anal. Appl. 31 (1997), no. 1, 54-57.) [MR1459834]
 N. B. Kerimov and Kh. R. Mamedov, On a boundary value problem with a spectral parameter in the boundary conditions (Russian), Sibirsk. Mat. Zh. 40 (1999), no. 2, 325-335. (Engl. translation in: Siberian Math. J. 40 (1999), no. 2, 281-290.) [MR1698307]
 S. G. Krein, Linear differential equations in Banach space, Providence, R.I. 1971. [MR0342804]
 K. S. Mamedov, Asymptotic behavior of distribution function of eigenvalues of abstract differential operator (Russian), Mat. Zametki 31 (1982), no. 1, 41-51. (Engl. translation in: Math. Notes 31 (1982), no. 1-2, 23-29.) [MR0646911]
 K. Maurin, Methods of Hilbert spaces, Państwowe Wydawnictwo Naukowe, Warsawa 1967. [MR0223910]
 L. A. Oleinik, Inhomogeneous boundary value problems for operator-differential equations with a spectral parameter in the boundary conditions (Russian), The spectral theory of operator-differential equations, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev 1986, 25-28. [MR0893005]
 M. A. Rybak, On asymptotic distribution of eigenvalues of certain boundary value problems for the operator Sturm-Liouville equation (Russian), Ukrain. Mat. Zh. 32 (1980), no. 2, 248-252. [MR0568843]
 H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Publishing Co., Amsterdam-New York 1978. [MR0503903]
 S. Yakubov, Solution of irregular problems by the asymptotic method, Asymptot. Anal. 22 (2000), 129-148. [MR1742531]
 S. Yakubov and Ya. Yakubov, Differential-operator equations, Ordinary and partial differential equations, Chapman and Hall/CRC, Boca Raton, FL 2000. [MR1739280]