SOLVABILITY OF BOUNDARY VALUE PROBLEMS WITH TRANSMISSION CONDITIONS FOR DISCONTINUOUS ELLIPTIC DIFFERENTIAL OPERATOR EQUATIONS

. We consider nonlocal boundary value problems which includes discontinuous coefficients elliptic differential operator equations of the second order and nonlocal boundary conditions together with boundary-transmission conditions. We prove coerciveness and Fredholmness for these nonlocal boundary value problems.


Introduction
We investigated nonlocal boundary value problem with discontinuous operator coefficients for the second order elliptic differential-operator equation and boundary conditions which have transmission conditions in a Hilbert space H : . Problems of such type arise in heat and mass transfer in various physical transfer problems and in diffraction problems (see, for example, A. V. Likov [10]; A. A. Shkalikov [17] and references cited therein). Elliptic functional differential equations are closely associated with differential equations with nonlocal boundary conditions, which arises plasma theory, and boundary value problems with elliptic differential equations have some important applications such as to elasticity theory, control theory and diffusion processes (see, for example, A. L. Skubachevskii [16]). There are many papers that the spektral properties of such problem are investigated ( see, [1], [2], [4], [14], [17]). Some boundary value problems with discontinuous coefficient and eigenvalue parameter in both the differential equation and boundary conditions have been studied by O. Sh. Mukhtarov, M. Kandemir and others (see, [5]- [8], [8], [11]- [13]). In this study, we investigated coerciveness and Fredholmness of nonlocal boundary value problem with discontinuous operator coefficients and transmission conditions at point zero in [-1,1] for elliptic differential-operator equations on which S. Yakubov, G. Dore and S. Yakubov have suggestion results for nonlocal boundary value problems with elliptic differential equation in [0,1] (see, [3], [19]). Besides, we have considered methods of solution of boundary value problems for elliptic differential-operator equations, which are suggested by S. G. Krein (see, [9]).

Preliminaries
In this section, we give some definitions and auxiliary results which are used through the paper. Lemma 2.1. (     [ This theorem has been proved in ( [20], Theorem 5.4.2/1).

Homogeneous problem of transmission-boundary value problem
We will consider in this section a boundary value problem where  is a complex parameter. Denote: 1) ( 1) We will assume that k A is a closed densely defined operator in a Hilbert space H .
Theorem 3.1. Let the following conditions be satisfied: Proof. In view of the condition (1) and Theorem 2.4 for arg   , there exist semigroups which are the holomorphic and strongly continuous for (3.5) Therefore, So, we have x A I r x e r    and where, in view of the Theorem 2.2.
Hence, we can write Let us now write (3.4) the function () ux in (3.2) boundary conditions. Denote: where, () R  can be written as a matrix in the form ( 1) ( 1) ( ) : ( 1) ( 1) Hence, from (3.10), we obtain the system ( 1)  ( 1) ( 1) So, system (3.11) has a unique solution for arg   and  , and the solution can be expressed in the form of where jk C are complex numbers and in view of (3.12) for arg   and   (3.14) Therefore, we have (3.15) Substituting (3.15) into (3.4) we have Hence, for arg   and for  , we get (3.21) Therefore, we can obtain the same result for all the other terms in the right-hand side of inequality (3.17), and the inequality (3.3) has been proved too.

Nonhomogeneous transmission-boundary value problem
In this section, we will consider the boundary value problem which has the nonhomogeneous equation with a parameter and transmission conditions We will show that a solution of the problem (4.1)-(4.2) which belongs to 2 (( 1,0,1