It is well-known that the Sturm-Liouville type boundary value problems appears in solving many important problems in physics. For example, some problems of theory of elasticity in a half-strip ,the problems of theory of vibrations of an elastic cylinder reduce to investigation of solvability of appropriate boundary value problems for Sturm-Liouville type differential equations. In the recent years, the Sturm-Liouville type boundary value problems with additional transmission conditions are investigated by many mathematical and physical researches . Note that, such type problems is very complicated because boundary value problems with additional transmission conditions may be not self-adjoint in the classical Hilbert space L2 and therefore the eigenvalues may be not real. The main goal of this study is to prove coerciveness of a new class many interval Sturm-Liouville problems with additional transmission conditions at the points of interaction. Moreover we shall establish some spectral properties and find asymptotic behaviour of the eigenvalues of the problem under consideration. Observe that this type of problems is studied in the setting of the direct sum of the Hilbert space. We also construct fundamental solutions and discuss some properties of spectrum. 

Keywords - Boundary value problem, transmission conditions, Coercive solvability, spectrum, Hilbert space.

[1] F. V. Atkinson., On bounds for the Titchmarsh-Weyl m-coecients and for spectral functions for second-order differential equations,Proc.Royal Soc. Edinburgh. (A) 97 (1984), 1-7.
[2] K. Aydemir and O. Sh. Mukhtarov, Green’s Function Method for Self- Adjoint Realization of BoundaryValue Problems with Interior Singularities, Abstract and Applied Analysis, vol. 2013, Article ID 503267,7 pages(2013), doi:10.1155/2013/503267.
[3] K. Aydemir, Boundary value problems with eigenvalue depending boundary and transmission conditions, Boundary value problems 2014:131(2014).
[4] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw-Hill Publishing Co. Ltd., New Delhi, 1972.
[5] C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edin. 77A(1977), P. 293-308.
[6] A. G. Kostyuchenko and M. B. Orazov, A problem on vibration of an elastic semicylinder and related quadratic bundles. Proceedings of I. G. Petrovsky, seminar, MGU, 6 (1981), 97-146, (in Russian).
[7] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac operators, Mathematics and its Applications, 59. Kluwer Academic Publishers Group, Dordrecht, 1991.
[8] O. Sh. Mukhtarov, H. Olgar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value ˇ Problems, Filomat, 29:7 (2015), 1671-1680.
[9] M. A. Naimark, Linear differential operators, Ungar, Newyork, 1967.
[10] P. F. Papkovich, Two problems of bend theory of thin elastic plates, PMM, 5(3) (1941), 359-374, (in Russian).
[11] P. F. Papkovich, On a form of solution of plane problem of elasticity theory for a rectangular strip, Dokl. AN SSSR, 27(4) (1940), (in Russian).
[12] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics. Oxford and New York, Pergamon, (1963).
[13] I. Titeux and Ya. Yakubov, Completeness of root functions for thermal conduction in a strip with piecewise continuous coeffcients. Math. Models Methods Appl. Sc., 7(7), pp. 1035-1050,1997.
[14] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978).
[15] Yu. A. Ustinov and Yu. I. Yudovich, On completeness of a system of elementary solutions of a biharmonic equation on a semi-strip, PMM, 37(4) (1973), 706-714, (in Russian).
[16] N. N. Voitovich, Katsenelbaum, B.Z. and Sivov, A.N., Generalized method of eigen-vibration in the theory of Diraction. Nauka, Moskow, 1997 (Russian).
[17] S. Y. Yakubov and Y. Y. Yakubov,Differantial Operator Equations (Ordinary and Partial Differential Equations), Boca Raton. Chapman and Hall/CRC, 2000.