We study the basis properties of systems of eigenfunctions and associated functions for one kind of generalized spectral problems for a second-order ordinary differential operator.

1. Introduction

Let us consider the partial differential equation with involution 𝑀𝑑(𝑑,π‘₯)=𝛼𝑀π‘₯π‘₯(𝑑,π‘₯)+𝑀π‘₯π‘₯(𝑑,βˆ’π‘₯),βˆ’1<π‘₯<1,𝑑>0.(1.1) If the initial conditions 𝑀(0,π‘₯)=𝑓(π‘₯)(1.2) and the boundary conditions 𝛼𝑗𝑀π‘₯(𝑑,βˆ’1)+𝛽𝑗𝑀π‘₯(𝑑,1)+𝛼𝑗1𝑀(𝑑,βˆ’1)+𝛽𝑗1𝑀(𝑑,1)=0,𝑗=1,2(1.3) are given, then the solving of this equation by Fourier’s method leads to the problem of expansion of function 𝑓(π‘₯) into series of eigenfunctions of spectral problem βˆ’π‘’ξ…žξ…ž(βˆ’π‘₯)+π›Όπ‘’ξ…žξ…žπ›Ό(π‘₯)=πœ†π‘’(π‘₯),π‘—π‘’ξ…ž(βˆ’1)+π›½π‘—π‘’ξ…ž(1)+𝛼𝑗1𝑒(βˆ’1)+𝛽𝑗𝑒(1)=0,𝑗=1,2.(1.4) If the function 𝑓(π‘₯)∈𝐿2(βˆ’1,1), then the question about basis property of eigenfunctions of spectral problem for second-order ordinary differential operator with involution raises.

Work of many researchers is devoted to the study of differential equations [1–5]. Various aspects of functionally differential equations with involution are studied in [6, 7]. The spectral problems for the double differentiation operator with involution are studied in [8–11] and the issues Riesz basis property of eigenfunctions in terms of coefficients of boundary conditions were considered.

This kind of spectral problems arises in the theory of solvability of differential equations in partial derivatives with an involution [7, page 265].

Results presented below are a continuation of studies of one of the authors in [9–11].

2. General Boundary Value Problem

In this paper, we study the spectral problem of the form πΏπ‘’β‰‘βˆ’π‘’ξ…žξ…ž(βˆ’π‘₯)+π›Όπ‘’ξ…žξ…ž(π‘₯)+π›½π‘’ξ…ž(π‘₯)+π›Ύπ‘’ξ…žπ›Ό(βˆ’π‘₯)+πœ‚π‘’(βˆ’π‘₯)=πœ†π‘’(π‘₯),(2.1)1π‘’ξ…ž(βˆ’1)+𝛽1π‘’ξ…ž(1)+𝛼11𝑒(βˆ’1)+𝛽11𝛼𝑒(1)=0,2π‘’ξ…ž(βˆ’1)+𝛽2π‘’ξ…ž(1)+𝛼21𝑒(βˆ’1)+𝛽21𝑒(1)=0,(2.2) where 𝛼,𝛽,𝛾,πœ‚,𝛼𝑖,𝛽𝑖,𝛼𝑖𝑗,𝛽𝑖𝑗 are some complex numbers.

By direct calculation, one can verify that the square of the operator is in the form 𝐿2𝑒=1+𝛼2𝑒𝐼𝑉(π‘₯)βˆ’2𝛼𝑒𝐼𝑉(βˆ’π‘₯)+2π›Όπ›Ύπ‘’ξ…žξ…žξ…ž(βˆ’π‘₯)+2π›Όπ›½π‘’ξ…žξ…žξ…ž(π‘₯)+2π›Όπœ‚π‘’ξ…žξ…ž+ξ€·(βˆ’π‘₯)βˆ’2πœ‚+𝛽2βˆ’π›Ύ2ξ€Έπ‘’ξ…žξ…ž(π‘₯)+πœ‚2𝑒(π‘₯).(2.3)

Since it is assumed that 𝐿𝑒 belongs to domain of operator 𝐿 also, then function 𝐿𝑒 satisfies boundary-value conditions (2.2) 𝛼1(𝐿𝑒)ξ…ž(βˆ’1)+𝛽1(𝐿𝑒)ξ…ž(1)+𝛼11(𝐿𝑒)(βˆ’1)+𝛽11(𝛼𝐿𝑒)(1)=0,2(𝐿𝑒)ξ…ž(βˆ’1)+𝛽2(𝐿𝑒)ξ…ž(1)+𝛼21(𝐿𝑒)(βˆ’1)+𝛽21(𝐿𝑒)(1)=0.(2.4) That is, the operator 𝐿2 is generated by previous differential expression and boundary-value conditions (2.2) and (2.4).

The expression 𝐿2𝑒 is an ordinary differential expression for 𝛼=0.

Therefore, applying the method in [8–10] we can obtain the following statement (the result).

Theorem 2.1. If 𝛼=0, then the eigenfunctions of the generalized spectral problem (2.1) and (2.2) form a Riesz basis of the space 𝐿2(βˆ’1,1) in the following cases:(1)𝛼1𝛽2βˆ’π›Ό2𝛽1β‰ 0; (2)𝛼1𝛽2βˆ’π›Ό2𝛽1=0,|𝛼1|+|𝛽1|>0,𝛼21≠𝛽22,𝛼221≠𝛽221, (3)𝛼1=𝛽1=𝛼2=𝛽2=0;𝛼11𝛽21βˆ’π›Ό21𝛽11β‰ 0.

The root vectors of operators 𝐴 and 𝐴2 coincide under some conditions (see, for instance, [10]). Therefore, we can consider the square of the operator 𝐿 which is an ordinary differential operator. It is well known [12–14] that eigenfunctions of ordinary differential operator of even order with strongly regular boundary value conditions form a Riesz basis. As in [10], from here it is possible to deduce correctness of Theorem 2.1.

This technique is not applicable for a = 0 since 𝐿2𝑒 is not an ordinary differential operator. Therefore, we consider this case separately.

3. General Solution of Special Type Equation

Let the operator 𝐿 be given by the differential expression with an involution 𝐿𝑒=βˆ’π‘’ξ…žξ…ž(βˆ’π‘₯)+π›Όπ‘’ξ…žξ…ž(π‘₯),(3.1) and boundary conditions (2.2).

We consider the spectral problem 𝐿𝑒=πœ†π‘’(π‘₯) with periodic, antiperiodic boundary conditions, with the boundary conditions of Dirichlet and Sturm type. In these cases, it is possible to compute all the eigenvalues and eigenfunctions explicitly. The basis of our statements is the following.

Theorem 3.1. If π‘Ž2β‰ 1, then the general solution of equation βˆ’π‘’ξ…žξ…ž(βˆ’π‘₯)+π›Όπ‘’ξ…žξ…ž(π‘₯)=πœ†π‘’(π‘₯),(3.2) where πœ† is the spectral parameter, has the form 𝑒(π‘₯)=𝐴cosπœ†ξ‚™1βˆ’π›Όπ‘₯+𝐡sinπœ†βˆ’1βˆ’π›Όπ‘₯,(3.3) where 𝐴 and 𝐡 are arbitrary complex numbers.

If 𝛼2=1 and πœ†β‰ 0, then (3.2) has only the trivial solution.

Proof. It is easy to see that functions (3.3) are solutions of (3.2). Let us prove the absence of other solutions.
Any function 𝑒(π‘₯) can be represented as a sum of even and odd functions. Substituting this representation into (3.2) and into βˆ’π‘’ξ…žξ…ž(π‘₯)+π›Όπ‘’ξ…žξ…ž(βˆ’π‘₯)=πœ†π‘’(βˆ’π‘₯), we conclude that the functions 𝑒1(π‘₯) and βˆ’(1βˆ’π›Ό)𝑒1ξ…žξ…ž(π‘₯)=πœ†π‘’1(π‘₯),βˆ’(βˆ’1βˆ’π›Ό)𝑒2ξ…žξ…ž(π‘₯)=πœ†π‘’2(π‘₯).(3.4)

4. The Dirichlet Problem

Consider the spectral problem (3.2) π‘Ž2β‰ 1 with boundary conditions 𝑒(βˆ’1)=0,𝑒(1)=0.(4.1) Note that the spectral problem (3.2) and (4.1) is self-adjoint for real 𝛼. We calculate the eigenvalues and eigenfunctions of the Dirichlet problem (3.2) and (4.1). Using Theorem 3.1, it is easy to see that the spectral problem (3.2) and (4.1) has two sequences of simple eigenvalues.

If π›Όβˆ‰{(8π‘˜2+4π‘˜+1)/(4π‘˜+1)βˆΆπ‘˜βˆˆπ‘}, then corresponding eigenfunctions are given by the formulas π‘’π‘˜1ξ‚€πœ‹(π‘₯)=cos2+π‘˜πœ‹π‘₯,π‘˜=0,1,2,…,π‘’π‘˜2(π‘₯)=sinπ‘˜πœ‹π‘₯,π‘˜=1,2,….(4.2) If π›Όβˆ‰(8π‘˜2+4π‘˜+1)/(4π‘˜+1) for some π‘˜0βˆˆπ‘, then the eigenfunctions of the spectral problem (3.2) and (4.1) are given by π‘’π‘˜1ξ‚€πœ‹(π‘₯)=cos2+π‘˜πœ‹π‘₯,π‘˜=0,1,2,…,π‘’π‘˜2(π‘₯)=sinπ‘˜πœ‹π‘₯,π‘˜=1,2,…,π‘˜β‰ π‘˜0,π‘’π‘˜01(ξ‚€πœ‹π‘₯)=cos2+π‘˜0πœ‹ξ‚ξ‚™π‘₯+sin1βˆ’π›Όξ‚€πœ‹βˆ’1βˆ’π›Ό2+π‘˜0πœ‹ξ‚π‘’π‘₯,π‘˜02(π‘₯)=sinπ‘˜0ξ‚™πœ‹π‘₯+cosβˆ’1βˆ’π›Όπ‘˜1βˆ’π›Ό0πœ‹π‘₯.(4.3)

Theorem 4.1. If π‘Ž2β‰ 1, then the system of eigenfunctions of the spectral problem (3.2) and (4.1), which is given above, forms an orthonormal basis of the space 𝐿2(βˆ’1,1).

Proof. For real values of 𝛼, the spectral problem (3.2) and (4.1) is self-adjoint. Therefore, the system (4.1), as a system of eigenfunctions self-adjoint operator, is an orthonormal. Analogously, the case 𝛼=(8π‘˜20+4π‘˜0+1)/(4π‘˜0+1), π‘˜0βˆˆπ‘, is considered. Also note that every orthonormal basis is automatically a Riesz basis.
The system (4.2) does not depend on 𝛼, hence Theorem 4.1 is proved.

5. Periodic and Antiperiodic Problem

Now consider the spectral problem (3.2) with the periodic boundary conditions𝑒(βˆ’1)=𝑒(1),π‘’ξ…ž(βˆ’1)=π‘’ξ…ž(1).(5.1) It follows immediately from Theorem 3.1 that the eigenfunctions of the spectral problem (3.2) and (5.1) are given byξ€·πœ†π‘˜1ξ€Έ2=βˆ’(1+𝛼)π‘˜2πœ‹2,ξ€·πœ†π‘˜2ξ€Έ2=(1βˆ’π›Ό)π‘˜2πœ‹2.(5.2) They are simple and correspond to the eigenfunctions π‘’π‘˜1(π‘₯)=sinπ‘˜πœ‹π‘₯,π‘˜=0,1,2,…,π‘’π‘˜2(π‘₯)=cosπ‘˜πœ‹π‘₯,π‘˜=0,1,2,….(5.3) Similarly, the eigenvalues and eigenfunctions of the spectral problem with antiperiodic boundary conditions 𝑒(βˆ’1)=βˆ’π‘’(1),π‘’ξ…ž(βˆ’1)=βˆ’π‘’ξ…ž(1)(5.4) are calculated.

In this case, there are two series of eigenvalues also ξ€·πœ†π‘˜1ξ€Έ2ξ‚€πœ‹=(1βˆ’π›Ό)2ξ‚ξ€·πœ†+π‘˜πœ‹,π‘˜=0,1,2,…,π‘˜2ξ€Έ2ξ‚€πœ‹=(βˆ’1βˆ’π›Ό)2+π‘˜πœ‹,π‘˜=0,1,2,….(5.5) They correspond to the eigenfunctionsπ‘’π‘˜1ξ‚€πœ‹=cos2+π‘˜πœ‹π‘₯,π‘˜=1,2,…,π‘’π‘˜2ξ‚€πœ‹=sin2+π‘˜πœ‹π‘₯,π‘˜=0,1,2,….(5.6)

Theorem 5.1. If 𝛼2β‰ 1, then the systems of eigenfunctions of the spectral problem (3.2) with periodic or antiperiodic boundary conditions form orthonormal bases of the space 𝐿2(βˆ’1,1).

The proof is analogous to the proof of Theorem 4.1. Also note that for periodic conditions the eigenfunctions form the classical orthonormal basis of 𝐿2(βˆ’1,1).

Analogously, it is possible to check that the eigenfunctions of spectral problems (3.2), 𝛼2β‰ 1, with boundary conditions of Sturm type π‘’ξ…ž(βˆ’1)=0,π‘’ξ…ž(1)=0(5.7) and with nonself-adjoint boundary conditions 𝑒(βˆ’1)=0,π‘’ξ…ž(βˆ’1)=π‘’ξ…ž(1)(5.8) form orthonormal bases of 𝐿2(βˆ’1,1).


The work is carried out under the auspices of Ministry of Education and Science of the Republic of Kazakhstan (0264/SF, 0753/SF).