O. Sh. Mukhtarov

Sturm-Liouville problems with discontinuities at two points

In this paper we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with piecewise continuous potentials together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. By modifying some techniques of [C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293-308; E. Tunc, O.Sh. Muhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput. 157 (2004) 347-355; O.Sh. Mukhtarov, E. Tunc, Eigenvalue problems for Sturm-Liouville equations with transmission conditions, Israel J. Math. 144 (2004) 367-380] and [O.Sh. Mukhtarov, M. Kadakal, F.S. Muhtarov, Eigenvalues and normalized eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions, Rep. Math. Phys. 54 (2004) 41-56], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulae for the eigenvalues and eigenfunctions.

Coerciveness of the discontinuous initial-boundary value problem for parabolic equations

In this paper, the mixed problem for parabolic equations is investigated with the discontinuous coefficient at the highest derivative and with nonstandard boundary conditions. Namely, the boundary conditions contain values of the solution not only on the boundary points, but also on the inner points of the considered domain as well. Moreover, abstract functionals are involved in the boundary conditions. We single out a class of functional spaces in which coercive solvability occurs for the investigated problem.

Eigenvalues and eigenfunctions of discontinuous Sturm--Liouville problems with eigenparameter-dependent boundary conditions

We extend some fundamental spectral properties of classic regular Sturm--Liouville problems to discontinuous boundary-value problems with eigenvalue-dependent boundary conditions. We suggest a new approach for investigation of such type discontinuous problems.

Distribution of eigenvalues for the discontinuous boundary-value problem with functional-manypoint conditions

In this study, we investigate the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also a point of discontinuity, a finite number of internal points and abstract linear functionals. So our problem is not a pure boundary-value one.We single out a class of linear functionals and find simple algebraic conditions on the coefficients which guarantee the existence of an infinite number of eigenvalues. Also, the asymptotic formulas for the eigenvalues are found.The results obtained in this paper are new, even in the case of boundary conditions either without internal points or without linear functionals.

Eigenvalue problems for Sturm Liouville equations with transmission conditions

In this study, we consider a Sturm Liouville type boundary-value problem with eigenparameter-dependent boundary conditions and with two supplementary transmission conditions at one inner point of a finite interval under consideration. We modify some techniques of classical Sturm Liouville theory and suggest a new approach for the investigation of eigenvalues and eigenfunctions of this type of boundary-value problem.

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

Abstract In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and abstract linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

Abstract The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.

New Type of Sturm-Liouville Problems in Associated Hilbert Spaces

We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.

Multi-point transmission problems for Sturm-Liouville equation with an abstract linear operator

In this paper, we consider the spectral problem for the equation −u″(x) + (A + λI)u(x) = f(x) on the two disjoint intervals (−1, 0) and (0, 1) together with multi-point boundary conditions and supplementary transmission conditions at the point of interaction x = 0, where A is an abstract linear operator. So, our problem is not a pure differential boundary-value one. Starting with the analysis of the principal part of the problem, the coercive estimates, the Fredholmness and isomorphism are established for the main problem. The obtained results are new even in the case of boundary conditions without internal points.

Sturm-Liouville problems with eigendependent boundary and transmissions conditions

It is well-known that the sturmian theory is an important aid in solving many problems in mathematical physics. Therefore this theory is one of the most actual and extensively developing fields in spectral analysis of boundary-value problems. Basically it has been investigated boundary-value problems which consist of ordinary differential equations with continuous coefficients and end-point boundary conditions.