On the eigenvalue problems for differential operators with coupled boundary conditions

The present paper deals with the eigenvalue problems for oneand two-dimensional second order differential operators with given nonlocal coupled boundary conditions. The corresponding finite-difference (discrete) problems have been investigated in the paper [1]. First of all, we will consider the eigenvalue problem for one-dimensional differential operator with given nonlocal coupled boundary conditions,


Introduction
The present paper deals with the eigenvalue problems for one-and two-dimensional second order differential operators with given nonlocal coupled boundary conditions.The corresponding finite-difference (discrete) problems have been investigated in the paper [1].
First of all, we will consider the eigenvalue problem for one-dimensional differential operator with given nonlocal coupled boundary conditions, where γ 0 , γ 1 ∈ R, γ 0 + γ 1 = 0. We will also briefly discuss the similar two-dimensional problem with the classical boundary conditions u(x, 0) = u(x, 1) = 0, 0 < x < 1, and the coupled boundary conditions Such values of λ that the problem (1)-( 3) or ( 4)-( 7) has the non-trivial solution are called eigenvalues, and the set of all eigenvalues is called the spectrum of the problem.Since conditions (2), ( 3) and ( 6), ( 7) are nonlocal, the corresponding differential operators are non-self-adjoint.Therefore, the analysis of the spectra of these problems leads to the problems on the existence of both real and complex eigenvalues.
Let us introduce a parameter γ, The main aim of this paper is to investigate the dependence of the qualitative structure of the spectra of the differential problems ( 1)-( 3) and ( 4)-( 7) on the parameters γ 0 , γ 1 (to be precise, on the generalized parameter γ), i.e., to formulate conditions for the existence of zero, positive, negative or complex eigenvalues, and (when it is possible) to provide analytical expressions of eigenvalues.The eigenvalue problems for differential operators with nonlocal conditions can be investigated numerically [2].We use technique and argument which are used, for example, in the papers [3,4] to investigate similar problems with other types of nonlocal conditions.
Case 1: λ < 0. If a number λ < 0 is an eigenvalue of the problem (1)-(3), the general solution of the equation ( 1) can be expressed as where c 1 and c 2 are arbitrary constants.By substituting this expression into nonlocal conditions ( 2), (3), we get a system of two linear algebraic equations with unknowns c 1 and c 2 : This system has a non-trivial solution if its determinant is equal to zero, i.e., Since α > 0, after simple rearrangements we get the equation and the following proposition is valid.
Proposition 1.The inequality γ > 1 is the necessary and sufficient condition for the existence of one and only one negative eigenvalue of the problem (1)-( 3): Case 2: λ = 0.The general solution of the equation ( 1) in this case is u Similarly as in Case 1, we get a system of two linear algebraic equations with unknowns c 1 and c 2 : There exists a non-trivial solution to this system, if Proposition 2. The number λ 0 = 0 is an eigenvalue of the problem (1)-( 3) if and only if γ = 1.
Case 3: λ > 0. In this case, the general solution of the equation ( 1) is Using the same technique as in Case 1 and Case 2, we get a system of two linear algebraic equations By equating the determinant of this system with zero, we obtain Hence, we can prove the following Proposition 3. The inequality |γ| ≤ 1 is the necessary and sufficient condition for the existence of infinitely many positive solutions to the equation (9), i.e., for the existence of infinitely many (countable set) positive eigenvalues for the problem (1)-( 3): Remark 1.When |γ| < 1, all positive eigenvalues are simple.However, when |γ| = 1, all positive eigenvalues (except λ 1 = π 2 , when γ = −1) are multiple (double).
Remark 2. We can observe the qualitative behaviour of real eigenvalues of the problem (1)-( 3) from Fig. 1, where graphs of functions γ(α) = cosh α and γ(α) = cos α, α > 0, are exhibited.Case 4: λ ∈ C. We represent the general solution of the equation (1) in the form We assume that α = 0 and β = 0.If α = 0, β = 0 or α = 0, β = 0, then this case coincides with Case 1 or Case 3, respectively.However, when α = β = 0, a situation is the same as in Case 2. By substituting the expression of general solution into nonlocal conditions ( 2) and (3), we get The determinant of this system is equal to zero, if Since cosh (iq) = cos(q), the condition for the existence of a non-trivial solution is By separating the real and imaginary parts in the latter relation, we obtain the equations Taking into account assumption that α = 0 and β = 0 allow us to prove the following statement: Proposition 4. When |γ| > 1, there exists the series of non-trivial solutions to the system (10), (α k , ±β), k ∈ Z \ {0}, where i.e., the problem (1)-( 3) has infinitely many (countable set) complex eigenvalues λ c,k .Distinct pairs of conjugate complex eigenvalues can be calculated by the formula then real parts of complex eigenvalues λ c,k are negative.

The two-dimensional problem
Now let us consider the real part of the spectrum of the two-dimensional differential eigenvalue problem (4)- (7).By separating variables, i.e., by representing the solution of the problem (4)- (7) in the form we get two one-dimensional eigenvalue problems: and where η + µ = λ.The problem (11) was considered in Section 2, while the problem ( 12) is classic.It is well-known, that all the eigenvalues of the problem ( 12) are real, positive, algebraically simple, and can be computed by the formula Let us denote It is easy to see, that the positivity of λ kl is conditioned by the positivity of η k .Therefore, the following statement is valid.
As a rule, any nonlocal condition, implies that, depending on nonlocal condition parameters, both real numbers (positive or non-positive) and complex numbers (with positive or non-positive real parts) can be the eigenvalues of the corresponding differential problem.Using the quite simple technique allow us to investigate the qualitative structure of the spectra of the differential problems (1)-( 3) and ( 4)- (7).