Well-Posed and Ill-Posed Boundary Value Problems for PDEView this Special Issue
Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution
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.
Let us consider the partial differential equation with involution If the initial conditions and the boundary conditions are given, then the solving of this equation by Fourier’s method leads to the problem of expansion of function into series of eigenfunctions of spectral problem If the function , 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].
2. General Boundary Value Problem
In this paper, we study the spectral problem of the form where are some complex numbers.
By direct calculation, one can verify that the square of the operator is in the form
Since it is assumed that belongs to domain of operator also, then function satisfies boundary-value conditions (2.2) That is, the operator is generated by previous differential expression and boundary-value conditions (2.2) and (2.4).
The expression is an ordinary differential expression for .
The root vectors of operators and coincide under some conditions (see, for instance, ). Therefore, we can consider the square of the operator 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 , from here it is possible to deduce correctness of Theorem 2.1.
This technique is not applicable for a = 0 since is not an ordinary differential operator. Therefore, we consider this case separately.
3. General Solution of Special Type Equation
Let the operator be given by the differential expression with an involution and boundary conditions (2.2).
We consider the spectral problem 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 , then the general solution of equation where is the spectral parameter, has the form where and are arbitrary complex numbers.
If and , 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 can be represented as a sum of even and odd functions. Substituting this representation into (3.2) and into , we conclude that the functions and
4. The Dirichlet Problem
Consider the spectral problem (3.2) with boundary conditions Note that the spectral problem (3.2) and (4.1) is self-adjoint for real . 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.
Proof. For real values of , 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 , , is considered. Also note that every orthonormal basis is automatically a Riesz basis.
The system (4.2) does not depend on , hence Theorem 4.1 is proved.
5. Periodic and Antiperiodic Problem
Now consider the spectral problem (3.2) with the periodic boundary conditions It follows immediately from Theorem 3.1 that the eigenfunctions of the spectral problem (3.2) and (5.1) are given by They are simple and correspond to the eigenfunctions Similarly, the eigenvalues and eigenfunctions of the spectral problem with antiperiodic boundary conditions are calculated.
In this case, there are two series of eigenvalues also They correspond to the eigenfunctions
Theorem 5.1. If , then the systems of eigenfunctions of the spectral problem (3.2) with periodic or antiperiodic boundary conditions form orthonormal bases of the space .
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 .
Analogously, it is possible to check that the eigenfunctions of spectral problems (3.2), , with boundary conditions of Sturm type and with nonself-adjoint boundary conditions form orthonormal bases of .
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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