On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution

: We give a deﬁnition of Green’s function of the general boundary value problems for non-self-adjoint second order differential equation with involution. The sufﬁcient conditions for the basis property of system of eigenfunctions are established in the terms of the boundary conditions. Uniform equiconvergence of spectral expansions related to the second-order differential equations with involution: − y (cid:48)(cid:48) ( x ) + α y (cid:48)(cid:48) ( − x ) + q ( x ) y ( x ) = λ y ( x ) , − 1 < x < 1, with the boundary conditions y (cid:48) ( − 1 ) + b 1 y ( − 1 ) = 0, y (cid:48) ( 1 ) + b 2 y ( 1 ) = 0, is obtained. As a corollary, it is proved that the eigenfunctions of the perturbed boundary value problems form the basis in L 2 ( − 1,1 ) for any complex-valued coefﬁcient q ( x ) ∈ L 1 ( − 1,1 ) .

Uniform equiconvergence of spectral expansions related to the operators L 0 and L given by (3), (1), respectively, is studied.
Differential equations with involution form a special class of linear functional-differential equations, with their theory having been developed since the middle of the last century. Among a variety of studies in this direction, one can mention the books [1][2][3]. The existence of a solution of the partial differential equation with involution has been studied in [2] by the separation of variables method. As in the case of classical equations, applying the Fourier method to partial differential equations with involution leads to the related spectral problems for differential operators with involution. The study of spectral problems for differential operators with involution started relatively recently. In [4][5][6][7], the spectral problems for the first-order differential operators with an involution have been studied. In [8] (see also references therein), Ref. [9], the spectral problems for differential operators with involution in the lower terms have been considered. The spectral problems related to the second-order differential operators with involution have been studied in [10][11][12][13][14]. The Green's function of the boundary value problems for the first order equations (and a system of equations) with involution have been derived in [3,[15][16][17]. In [12,13,18], the Green's functions of the second-order differential operators with involution have been investigated and theorems on basicity of eigenfunctions are proved. Theorems on basicity of eigenfunctions of the second order differential operators with involution [14] have been used to solving inverse problems in [19][20][21]. Solvability of problems for partial and ordinary differential equations with involution is discussed in [22][23][24][25][26].
The operator L defined by (1), (2) generalizes Sturm-Liouville operators, which have been studied fairly completely (see, for example, [27,28]). Spectral properties of the operator L with non self-adjoint boundary conditions in the form (2) have not been so well-studied yet, since this case is more complex for investigation. The first results about the basis property of eigenfunctions of boundary value problems for equation −y (−x) + q(x)y(x) = λy(x), −1 < x < 1, have been obtained in [12,18]. In [29], the basis property of eigenfunctions of operators (1) with periodic boundary conditions have been studied.
In this paper, the integral Cauchy method [27] (well-known in the spectral theory of ordinary differential operators) is modified for the case of differential operators with involution (1), (2) (and (3), (2)). The method is based on proving the equiconvergence of the known expansion with the eigenfunction expansion of the considered problem. We obtain our main results by developing the integral Cauchy method and by using the estimates for Green's functions.
The paper is organized as follows. In Section 2, we define the Green's function of the general boundary value problems. We give the formula for the Green's function of the operatorL 0 defined by (3), (4), and achieve the estimate for the Green's function. Section 3 is devoted to the estimate of the Green's function of the operator L 0 given by (3), (2). Finally, we discuss the basicity of eigenfunctions in Section 4.

Green's Function of the OperatorL 0 − λI
Let us introduce the definition of the Green's function of the general boundary value problem l q y = λy with boundary conditions where a ij are complex constants, λ is a complex spectral parameter. Let the boundary value problem not have a non-trivial solution. However, there can exist a function G q (x, t, λ), such that: (2) The function G q (x, t, λ) has the continuous derivative G q (x, t, λ) x for x = ∓t and satisfies the conditions: (3) The function G q (x, t, λ) has the derivative G q (x, t, λ) xx , satisfies l q y = λy (except at x = ∓t) and (5).
The function G q (x, t, λ) is called the Green's function of the considered boundary value problem (of the operator L − λI, defined by (1), (5), where I is the identity operator).
If the function G q (x, t, λ) is the Green's function of the operator L − λI, then the function gives the solution to the problem with boundary conditions (5), for any function f (x) ∈ C[−1, 1] (this statement, existence and uniqueness of the Green's function can be proved by standard methods (see [28], chapter 1).
In order to study the basis property of system of eigenfunctions of the operator L (defined by (1), (2)), we construct the Green's function G(x, t, λ) of the problem l 0 y = λy, , the linearly independent solutions of the homogeneous equation l 0 y = λy(x).

Lemma 1.
If λ is not an eigenvalue of the operatorL 0 − λI, then the function is the solution of non-homogeneous problem l 0 y = λy(x) + f (x), (4) for any continuous function This Lemma 1 can be proved by direct calculations. From Lemma 1, we find the following Corollary 1. The Green's function of the operatorL 0 − λI can be represented in the form The Green's function of the operatorL 0 − λI has the following properties: The functionĜ(x, t, λ) has the continuous derivativeĜ x (x, t, λ) for x = ∓t, and satisfies the conditions: (4) The functionĜ(x, t, λ) has the derivativeĜ xx (x, t, λ), satisfies l 0 y = λy except at x = ∓t) and (4).
Further, we need an estimate of the Green's function for operatorL 0 − λI.

Lemma 2.
Let ρ / ∈ O ξ (ρ kl ) and |ρ| > 1. Then, the Green's functionĜ(x, t, λ) of the operator L 0 − λI satisfies the uniformly with respect to −1 ≤ x, t ≤ 1 the following estimate where Proof. Let us examine three cases: t ≥ x, −x ≤ t ≤ x and t ≤ −x separately. In the first case, when t ≥ x, the relation (7) can be rewritten in the form For sufficiently large |ρ|, we find the following estimate Let ρ 0 > 0 and γ be arbitrary positive number (depends only on α). For sufficiently large ρ 0 > 0, we find the relations Applying inequalities (9) to (8), we find Hence, In a similar manner, we can show that for ρ 0 < 0. Thus, for t ≥ x > 0 the Green's function satisfies the estimate (7). The completion of the proof is a result of simple computations (see [29]). Lemma 2 is proved.

Proof.
To prove Theorem 1, we consider the difference where C mj are the circles with equations ρ = ξ 2 , ρ = ρ mj + ξ 2 . By virtue of (11) there exists a constant M 1 , such that The proof of analogous to that given for inequality (26) in [29]. The proof of the theorem is complete.
From Theorem 1 derives the following result.

Corollary 2.
Suppose all assumptions of Theorem 1 hold true. Then, the system of eigenfunctions of the operator L 0 forms the basis in L 2 (−1, 1).
Here the following result holds.
Corollary 3. Suppose all assumptions of Theorem 1 hold true and b 1 , b 2 in (2) are real numbers. Then, the system of eigenfunctions of the operator L 0 is orthonormal basis in L 2 (−1, 1).
To prove this assertion it suffices to show that the operator L 0 is self-adjoint in L 2 (−1, 1). Now we consider the operator L. Let denote by G q (x, t, λ) the Green's function of the operator L − λI, where I is the identity operator. Denote by The proof is analogous to that given for Theorem 1 in [29]. From Theorem 1 follows the following:  (2), where c is a constant, 1+α 1−α = p 1 , 1−α 1+α = p 2 for any integers p 1, p 2 . It is easy to see that the system of eigenfunctions of spectral problem is simultaneously the system of eigenfunctions for the operator L 0 defined by (3), (2). Using Corollary 2 (Corollary 4), we conclude that the system of eigenfunctions of spectral problem forms the basis in L 2 (−1, 1).

Conclusions
Summarizing the investigation carried out, we note that the Green's function of the second order differential operators (3), (4) with involution has been constructed. The estimates of the Green's functions of operators (3), (4), and (3), (2) have been established. The equiconvergence theorems (Theorem 1, Theorem 2) for operators (3), (2), (1), and (2) have been proven. As a corollary, results on the basicity of eigenfunctions to the problems under consideration have been proven. These theorems might be useful in the theory of solvability of mixed problems for partial differential equations with involution. For example: Problem 1. Find a sufficiently smooth function u(x, t), satisfying the conditions: u t (x, t) = u xx (x, t) − αu xx (−x, t) − q(x)u(x, t) ; −1 < x < 1, t > 0; u(0, x) = ϕ(x), u x (−1, t) + b 1 u(−1, t) = 0, u x (1, t) + b 2 u(1, t) = 0. Problem 2. Find a sufficiently smooth function u(x, t), satisfying the conditions: The described problems are the subject of further work and we are going to consider them in our next articles. In the future, we also plan to investigate the inverse spectral problems.