- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 590781, 5 pages

http://dx.doi.org/10.1155/2012/590781

## Spectral Properties of Non-Self-Adjoint Perturbations for a Spectral Problem with Involution

^{1}Department of Mathematics, M. Auezov South Kazakhstan State University, Tauke Han av., 5, 160012 Shymkent, Kazakhstan^{2}Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey^{3}Department of Mechanics and Mathematics, N.G. Chernyshevsky Saratov State University, Astrakhanskaya, 83, 410012 Saratov, Russia

Received 15 May 2012; Accepted 8 July 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Asylzat A. Kopzhassarova et al. 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

Full description of Riesz basis property for eigenfunctions of boundary value problems for first order differential equations with involutions is given.

#### 1. Introduction

Differential equations with involutions were considered firstly in [1]. They are a particular case of functional differential equations that appear in several applications (see, for instance, monographs [2, 3] and papers [4–6]). Different spectral problems for equations of this form were considered in [7, 8].

In particular, main questions about the following spectral problem: were solved in [7]. Namely, (1.1)-(1.2) is a Volterra operator if and only if ; furthermore, (1.1)-(1.2) is self-adjoint if and only if is a real number. For , the system of eigenfunctions for (1.1)-(1.2) is a Riesz basis in . Observe also that for , (1.1)-(1.2) has no associated functions, that is, all eigenvalues are simple. Note that problem (1.1)-(1.2) is an example of a generalized spectral problem of the form , with and . In general, they were considered in [9] when and are operators in a Banach space. Equiconvergence questions for two different perturbations of (1.1)-(1.2) were deeply studied in [10]. The main goal of the paper is to study questions about Riesz basis property of eigenfunctions for the following non-self-adjoint spectral problem:

Also note that problems similar to (1.3)-(1.4) appear when the Fourier method is applied for solving boundary value problems for partial differential equations with involution (see, for example, [11] and the bibliography therein).

#### 2. Results

Theorem 2.1. *If , , then the eigenfunctions system for (1.3)-(1.4) is a Riesz basis in . *

* Proof. *Before the proof, we need several facts about (1.3)-(1.4). First of all, it is easy to see that the general solution of (1.3)-(1.4) with is given by the following formula:

Next, we observe that for , eigenvalues are equal to
The related eigenfunctions are given by the following formula:

Observe also that for , the eigenvalues are
The corresponding eigenfunctions are
and for , the eigenvalues are
and the corresponding eigenfunctions are

We introduce the differential operator by
and by the boundary condition (1.4). Suppose that belongs to the domain of , . Then we consider

From the boundary condition (1.4), for we deduce that is the second order differential operator generated by the following relations:

Spectral problem (2.10)-(2.11) is a typical spectral problem for an ordinary second order differential operator. These problems are studied very well and they have numerous applications (see, for example, [12–14].) Recall [12, Chapter 2] that the following boundary conditions:
for an ordinary second order differential operator are regular if ; and they are strongly regular if additionally , where
Since , , , , we obtain , . It follows from , and that the boundary conditions (2.11) are strongly regular. It is known [12, 13] that the eigenfunctions of an operator with strongly regular boundary conditions constitute a Riesz basis in . By (2.2) numbers cannot be eigenvalues of , hence any eigenfunction of which corresponds to will be an eigenfunction which corresponds to as
Finally, we deduce the assertion of Theorem 2.1 in the case .

For the case the explicit representations of eigenfunctions (2.5) and (2.7) give the Riesz basis property for these systems directly.

*Remark 2.2. *If , then (1.3)-(1.4) coincide with the unperturbed problem (1.1)-(1.2) which is a Volterra operator for , that is, . If and , then the boundary conditions (2.7)–(2.10) are nonregular and hence the system of eigenfunctions is incomplete [12, 13]. Finally, for , (1.3) has only trivial solution.

Now, we consider other types of non-selfadjoint perturbations of (1.1)-(1.2).

Theorem 2.3. *If , then the eigenfunctions of the following spectral problem:
**
constitute a Riesz basis of . *

*Proof. *The proof is analogous to the proof of Theorem 2.1. It uses the following spectral problem:

Boundary conditions (2.17) are regular for , and nonregular for . Then, basis property for eigenfunctions of an ordinary differential second order operator with constant coefficients gives the result for .

For , boundary conditions (2.17) are strongly regular and the proof terminates analogously to the proof of Theorem 2.1.

*Remark 2.4. * The perturbation of (1.1)-(1.2) has the same form after the substitution . Hence, the result of [7] gives full description of basis properties for the following spectral problem:

#### References

- C. Babbage, “An essay towards the calculus of calculus of functions, Part II,”
*Philosophical Transactions of the Royal Society B*, vol. 106, pp. 179–256, 1816. - D. Przeworska-Rolewicz,
*Equations with Transformed Argument*, An Algebraic Approach, Elsevier-PWN, Amsterdam, The Netherlands, 1973. View at Zentralblatt MATH - J. Wiener,
*Generalized Solutions of Functional-Differential Equations*, World Scientific Publishing, Singapore, 1993. View at Zentralblatt MATH - W. Watkins, “Modified Wiener equations,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 27, no. 6, pp. 347–356, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Sh. Burlutskaya, V. P. Kurdyumov, A. S. Lukonina, and A. P. Khromov, “A functional-differential operator with involution,”
*Doklady Mathematics*, vol. 75, pp. 399–402, 2007. - W. T. Watkins, “Asymptotic properties of differential equations with involutions,”
*International Journal of Pure and Applied Mathematics*, vol. 44, no. 4, pp. 485–492, 2008. View at Zentralblatt MATH - M. A. Sadybekov and A. M. Sarsenbi, “Solution of fundamental spectral problems for all the boundary value problems for a first-order differential equation with a deviating argument,”
*Uzbek Mathematical Journal*, no. 3, pp. 88–94, 2007 (Russian). View at Zentralblatt MATH - A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator,”
*Differential Equations*, vol. 46, no. 4, pp. 506–511, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kato,
*Perturbation Theory for Linear Operators*, Springer, Berlin, Germany, 1966. - M. Sh. Burlutskaya and A. P. Khromov, “On an equiconvergence theorem on the whole interval for functional-differential operators,”
*Proceedings of Saratov University*, vol. 9:4, no. 1, pp. 3–12, 2009 (Russian). - M. Sh. Burlutskaya and A. P. Khromov, “Classical solution of a mixed problem with involution,”
*Doklady Mathematics*, vol. 82, pp. 865–868, 2010. - M. A. Naimark,
*Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators*, Frederick Ungar, New York, NY, USA, 1967. - N. Dunford and J. T. Schwartz,
*Linear Operators. Part III*, Spectral Operators, John Wiley & Sons, New York, NY, USA, 1988. - V. A. Il'in and L. V. Kritskov, “Properties of spectral expansions corresponding to non-self-adjoint differential operators,”
*Journal of Mathematical Sciences*, vol. 116, no. 5, pp. 3489–3550, 2003. View at Publisher · View at Google Scholar