ON NONLOCAL PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS AND ON A NONLOCAL PARABOLIC TRANSMISSION PROBLEM

In the present paper we study nonlocal problems for ordinary differential 
equations with a discontinuous coefficient for the high order derivative. 
We establish sufficient conditions, known as regularity conditions, which 
guarantee the coerciveness for both the space variable and the spectral 
parameter, as well as guarantee the completeness of the system of root 
functions. The results obtained are then applied to the study of a nonlocal 
parabolic transmission problem.


Introduction
Many physical problems, the problem of heat and mass diffusion in anisotropic media, diffraction problems, and others lead to the study of equations with a discon- tinuous coefficient for the high order derivative [13].In the present paper, we start with the study of boundary value problems for ordinary differential equations with a discontinuous coefficient for the high order derivative and with boundary conditions containing abstract functionals.We establish sufficient conditions of regularity type which guarantee the coerciveness for both the space variable and the spectral para- meter, and which also guarantee the completeness of root functions.Regular pro- blems for differential operators are studied in [3, 9, 10].Completeness of the system of root functions for differential operators with functional boundary conditions is analyzed in [4, 6, 11, 14].The coefficient of the high order derivative is assumed to be constant in [6,11], whereas it is assumed to be continuous in [14].The results ob- tained are used to show the existence and uniqueness of the solution, of a mixed problem for a parabolic partial differential equation with a discontinuous coefficient for the high order derivative, and multipoint boundary values and transmission condi- tions containing abstract functionals.They are also used to show the completeness of elementary solutions of this given mixed problem.Thus, the study is reduced to a Cauchy problem for a parabolic abstract differential equation, where the analysis of the operator coefficient is given in detail in this paper.
Again, by straightforward computation, we find that each c is of the form: c 0 4-Ri(p 0, Pl + R(;o-] where 0 is obtained from 0 by replacing the column with the column formed by right-hand sides of the boundary conditions of (10) such that Ri(Po, Pl)-0, i-1,2 for Pol, Pll oo, in S e. Substituting these values in the expressions of u a and u4, we find that problem (9)-( 10) has a unique solution, given by 02 + R2(P0, Pl) 01 --/1 (/90'/91) exp -t-exp flo(X b)] Applying (16), we get that inequality (15) gives estimate (7).The uniqueness of the solution of problem ( 4)-( 6) follows from estimate (7).

Coerciveness of the General Problem
First, consider the following definition.
Definition 1: Boundary value and transmission conditions (2)-( 3) are said to be regular if the following hold. 1.
Theorem 3: Suppose that conditions 1 and 2 of Theorem 2 are satisfied.Then, for any > O, there is R e > 0 such that for any complex number A where A S e and A[ > Re, the operator 1, we get inequality (7), which implies 2 that (,) is injective Since operator B, from W (0, b) x W (b, 1)into W (0, b) x q q q 2 W-(b, 1), is compact, and since according to Theorem 2 (,)'W (0, b) Wq(b, 1)---Wq-(O,b) XWq-(b, 1)xC is a Fredholm operator, then by a Fredholm alternative () is surjective.Therefore, it is an isomorphism.

Completeness of Root Functions
In the space L2(0 b)x L2(b 1), consider the operator L defined as follows.
The root functions of operator L are root functions of the following problem: LL ,X u 0 ,,,( (17 To establish the completeness of the root functions of L, we shall use a theorem given in [14] (Theorem 3.6, with n-1).This theorem is actually a variation of the well- known theorem of N. Danford ad J.T. Schwartz [3].Consider the following.Theorem 4: Suppose that the conditions given below are satisfied. 1.
There exist two Hilbert spaces, H and Ha, with the compact embedding H C H 1, and H H-H.
The linear operator A from H 1 and H is bounded.

4.
There exists a set of rays k, in the complex plane such that angles between the neighboring rays are less than -g, and there exists a number m E N such that I I R(/, A)II B(H, H1) <--c TM, with .G k and with Then the spectrum of operator A is discrete and the system of root vectors of operator A is complete in the space H a Applying the method used in proving Theorem 2.1 in [14] and Theorem 3, we can prove the following lemma.
Theorem 5: Suppose that the conditions below hold. 1.