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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 576843, 6 pages
http://dx.doi.org/10.1155/2012/576843
Research Article

Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution

Department of Mathematics, M. Auezov South Kazakhstan State University, 5 Tauke Han Avenue, 160012 Shymkent, Kazakhstan

Received 8 May 2012; Revised 13 July 2012; Accepted 6 August 2012

Academic Editor: Valery Covachev

Copyright © 2012 Asylzat Kopzhassarova and Abdizhakhan Sarsenbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 𝑤𝑡(𝑡,𝑥)=𝛼𝑤𝑥𝑥(𝑡,𝑥)+𝑤𝑥𝑥(𝑡,𝑥),1<𝑥<1,𝑡>0.(1.1) If the initial conditions 𝑤(0,𝑥)=𝑓(𝑥)(1.2) and the boundary conditions 𝛼𝑗𝑤𝑥(𝑡,1)+𝛽𝑗𝑤𝑥(𝑡,1)+𝛼𝑗1𝑤(𝑡,1)+𝛽𝑗1𝑤(𝑡,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(𝑥) into series of eigenfunctions of spectral problem 𝑢(𝑥)+𝛼𝑢𝛼(𝑥)=𝜆𝑢(𝑥),𝑗𝑢(1)+𝛽𝑗𝑢(1)+𝛼𝑗1𝑢(1)+𝛽𝑗𝑢(1)=0,𝑗=1,2.(1.4) If the function 𝑓(𝑥)𝐿2(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 [15]. 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 [811] 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 [911].

2. General Boundary Value Problem

In this paper, we study the spectral problem of the form 𝐿𝑢𝑢(𝑥)+𝛼𝑢(𝑥)+𝛽𝑢(𝑥)+𝛾𝑢𝛼(𝑥)+𝜂𝑢(𝑥)=𝜆𝑢(𝑥),(2.1)1𝑢(1)+𝛽1𝑢(1)+𝛼11𝑢(1)+𝛽11𝛼𝑢(1)=0,2𝑢(1)+𝛽2𝑢(1)+𝛼21𝑢(1)+𝛽21𝑢(1)=0,(2.2) where 𝛼,𝛽,𝛾,𝜂,𝛼𝑖,𝛽𝑖,𝛼𝑖𝑗,𝛽𝑖𝑗 are some complex numbers.

By direct calculation, one can verify that the square of the operator is in the form 𝐿2𝑢=1+𝛼2𝑢𝐼𝑉(𝑥)2𝛼𝑢𝐼𝑉(𝑥)+2𝛼𝛾𝑢(𝑥)+2𝛼𝛽𝑢(𝑥)+2𝛼𝜂𝑢+(𝑥)2𝜂+𝛽2𝛾2𝑢(𝑥)+𝜂2𝑢(𝑥).(2.3)

Since it is assumed that 𝐿𝑢 belongs to domain of operator 𝐿 also, then function 𝐿𝑢 satisfies boundary-value conditions (2.2) 𝛼1(𝐿𝑢)(1)+𝛽1(𝐿𝑢)(1)+𝛼11(𝐿𝑢)(1)+𝛽11(𝛼𝐿𝑢)(1)=0,2(𝐿𝑢)(1)+𝛽2(𝐿𝑢)(1)+𝛼21(𝐿𝑢)(1)+𝛽21(𝐿𝑢)(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 𝐿2𝑢 is an ordinary differential expression for 𝛼=0.

Therefore, applying the method in [810] 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(1,1) in the following cases:(1)𝛼1𝛽2𝛼2𝛽10; (2)𝛼1𝛽2𝛼2𝛽1=0,|𝛼1|+|𝛽1|>0,𝛼21𝛽22,𝛼221𝛽221, (3)𝛼1=𝛽1=𝛼2=𝛽2=0;𝛼11𝛽21𝛼21𝛽110.

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 [1214] 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 𝐿2𝑢 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 𝐿𝑢=𝑢(𝑥)+𝛼𝑢(𝑥),(3.1) and boundary conditions (2.2).

We consider the spectral problem 𝐿𝑢=𝜆𝑢(𝑥) 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 𝑎21, then the general solution of equation 𝑢(𝑥)+𝛼𝑢(𝑥)=𝜆𝑢(𝑥),(3.2) where 𝜆 is the spectral parameter, has the form 𝑢(𝑥)=𝐴cos𝜆1𝛼𝑥+𝐵sin𝜆1𝛼𝑥,(3.3) where 𝐴 and 𝐵 are arbitrary complex numbers.

If 𝛼2=1 and 𝜆0, 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 𝑢(𝑥) can be represented as a sum of even and odd functions. Substituting this representation into (3.2) and into 𝑢(𝑥)+𝛼𝑢(𝑥)=𝜆𝑢(𝑥), we conclude that the functions 𝑢1(𝑥) and (1𝛼)𝑢1(𝑥)=𝜆𝑢1(𝑥),(1𝛼)𝑢2(𝑥)=𝜆𝑢2(𝑥).(3.4)

4. The Dirichlet Problem

Consider the spectral problem (3.2) 𝑎21 with boundary conditions 𝑢(1)=0,𝑢(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 𝛼{(8𝑘2+4𝑘+1)/(4𝑘+1)𝑘𝑍}, then corresponding eigenfunctions are given by the formulas 𝑢𝑘1𝜋(𝑥)=cos2+𝑘𝜋𝑥,𝑘=0,1,2,,𝑢𝑘2(𝑥)=sin𝑘𝜋𝑥,𝑘=1,2,.(4.2) If 𝛼(8𝑘2+4𝑘+1)/(4𝑘+1) for some 𝑘0𝑍, then the eigenfunctions of the spectral problem (3.2) and (4.1) are given by 𝑢𝑘1𝜋(𝑥)=cos2+𝑘𝜋𝑥,𝑘=0,1,2,,𝑢𝑘2(𝑥)=sin𝑘𝜋𝑥,𝑘=1,2,,𝑘𝑘0,𝑢𝑘01(𝜋𝑥)=cos2+𝑘0𝜋𝑥+sin1𝛼𝜋1𝛼2+𝑘0𝜋𝑢𝑥,𝑘02(𝑥)=sin𝑘0𝜋𝑥+cos1𝛼𝑘1𝛼0𝜋𝑥.(4.3)

Theorem 4.1. If 𝑎21, 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(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=(8𝑘20+4𝑘0+1)/(4𝑘0+1), 𝑘0𝑍, 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 conditions𝑢(1)=𝑢(1),𝑢(1)=𝑢(1).(5.1) It follows immediately from Theorem 3.1 that the eigenfunctions of the spectral problem (3.2) and (5.1) are given by𝜆𝑘12=(1+𝛼)𝑘2𝜋2,𝜆𝑘22=(1𝛼)𝑘2𝜋2.(5.2) They are simple and correspond to the eigenfunctions 𝑢𝑘1(𝑥)=sin𝑘𝜋𝑥,𝑘=0,1,2,,𝑢𝑘2(𝑥)=cos𝑘𝜋𝑥,𝑘=0,1,2,.(5.3) Similarly, the eigenvalues and eigenfunctions of the spectral problem with antiperiodic boundary conditions 𝑢(1)=𝑢(1),𝑢(1)=𝑢(1)(5.4) are calculated.

In this case, there are two series of eigenvalues also 𝜆𝑘12𝜋=(1𝛼)2𝜆+𝑘𝜋,𝑘=0,1,2,,𝑘22𝜋=(1𝛼)2+𝑘𝜋,𝑘=0,1,2,.(5.5) They correspond to the eigenfunctions𝑢𝑘1𝜋=cos2+𝑘𝜋𝑥,𝑘=1,2,,𝑢𝑘2𝜋=sin2+𝑘𝜋𝑥,𝑘=0,1,2,.(5.6)

Theorem 5.1. If 𝛼21, then the systems of eigenfunctions of the spectral problem (3.2) with periodic or antiperiodic boundary conditions form orthonormal bases of the space 𝐿2(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(1,1).

Analogously, it is possible to check that the eigenfunctions of spectral problems (3.2), 𝛼21, with boundary conditions of Sturm type 𝑢(1)=0,𝑢(1)=0(5.7) and with nonself-adjoint boundary conditions 𝑢(1)=0,𝑢(1)=𝑢(1)(5.8) form orthonormal bases of 𝐿2(1,1).

Acknowledgment

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

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