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

and Applied Analysis 3 and for γ −1, the eigenvalues are λk √ 1 − α2 ( kπ π 2 ) , k 0,±1, . . . , 2.6 and the corresponding eigenfunctions are uk x √ 1 α cos ( kπ π 2 ) x √ 1 − α sin ( kπ π 2 ) x, k 0,±1, . . . 2.7 We introduce the differential operator L by Lu u′ −x αu′ x , −1 < x < 1, 2.8 and by the boundary condition 1.4 . Suppose that Lu belongs to the domain of L, Lu ∈ D L . Then we consider L2u L ( u′ −x αu′ x ) −(1 − α2)u′′ x . 2.9 From the boundary condition 1.4 , for Lu we deduce that L2 is the second order differential operator generated by the following relations: L2u − ( 1 − α2 ) u′′ x , −1 < x < 1, 2.10 ( α − γ)u′ −1 (1 − αγ)u′ 1 0, u −1 − γu 1 0. 2.11 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: a1u ′ −1 b1u′ 1 c1u −1 d1u 1 0, c0u −1 d0u 1 0, 2.12 for an ordinary second order differential operator are regular if b1c0 a1d0 / 0; and they are strongly regular if additionally θ2 0 / 4θ−1θ1, where θ−1 θ1 b1c0 a1d0, θ0 2 a1c0 b1d0 . 2.13 Since a1 α − γ , b1 1 − αγ , c0 1, d0 −γ , we obtain θ−1 θ1 γ2 − 2αγ 1, θ0 2 α − 2γ αγ2 . It follows from α2 / 1, γ2 / 1 and γ / α ± √ α2 − 1 that the boundary conditions 4 Abstract and Applied Analysis 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 L2 −1, 1 . By 2.2 numbers −λk cannot be eigenvalues of L, hence any eigenfunction of L2 which corresponds to λk will be an eigenfunction Lwhich corresponds to λk as ( L2 − λkE ) uk L λkE L − λkE uk 0. 2.14 Finally, we deduce the assertion of Theorem 2.1 in the case γ2 / 1. For the case γ2 1 the explicit representations of eigenfunctions 2.5 and 2.7 give the Riesz basis property for these systems directly. Remark 2.2. If α 0, then 1.3 1.4 coincide with the unperturbed problem 1.1 1.2 which is a Volterra operator for γ2 −1, that is, γ α ± √ α2 − 1. If α/ 0 and γ α ± √ α2 − 1, then the boundary conditions 2.7 – 2.10 are nonregular and hence the system of eigenfunctions is incomplete 12, 13 . Finally, for α2 1, 1.3 has only trivial solution. Now, we consider other types of non-selfadjoint perturbations of 1.1 1.2 . Theorem 2.3. If γ2 / ± 1, then the eigenfunctions of the following spectral problem: u′ −x αu −x λu x , −1 < x < 1, u −1 γu 1 , 2.15 constitute a Riesz basis of L2 −1, 1 . Proof. The proof is analogous to the proof of Theorem 2.1. It uses the following spectral problem: −u′′ x α2u x λu x , −1 < x < 1, 2.16 γu′ −1 − u′ 1 α ( γ2 − 1 ) u 1 0, u −1 − γu 1 0. 2.17 Boundary conditions 2.17 are regular for γ2 −1, and nonregular for γ2 1. Then, basis property for eigenfunctions of an ordinary differential second order operator with constant coefficients gives the result for γ2 1. For γ2 / ± 1, boundary conditions 2.17 are strongly regular and the proof terminates analogously to the proof of Theorem 2.1. Remark 2.4. The perturbation u′ −x αu x λu x of 1.1 1.2 has the same form after the substitution λ−α μ. Hence, the result of 7 gives full description of basis properties for the following spectral problem: u′ −x αu x λu x , −1 < x < 1, u −1 λu 1 . 2.18 Abstract and Applied Analysis 5 References 1 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. 2 D. Przeworska-Rolewicz, Equations with Transformed Argument, An Algebraic Approach, ElsevierPWN, Amsterdam, The Netherlands, 1973. 3 J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scientific Publishing, Singapore, 1993. 4 W. Watkins, “Modified Wiener equations,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 6, pp. 347–356, 2001. 5 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. 6 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. 7 M.A. Sadybekov andA.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 . 8 A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator,” Differential Equations, vol. 46, no. 4, pp. 506–511, 2010. 9 T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 1966. 10 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 . 11 M. Sh. Burlutskaya and A. P. Khromov, “Classical solution of a mixed problem with involution,” Doklady Mathematics, vol. 82, pp. 865–868, 2010. 12 M. A. Naimark, Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators, Frederick Ungar, New York, NY, USA, 1967. 13 N. Dunford and J. T. Schwartz, Linear Operators. Part III, Spectral Operators, John Wiley & Sons, New York, NY, USA, 1988. 14 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.and Applied Analysis 5 References 1 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. 2 D. Przeworska-Rolewicz, Equations with Transformed Argument, An Algebraic Approach, ElsevierPWN, Amsterdam, The Netherlands, 1973. 3 J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scientific Publishing, Singapore, 1993. 4 W. Watkins, “Modified Wiener equations,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 6, pp. 347–356, 2001. 5 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. 6 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. 7 M.A. Sadybekov andA.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 . 8 A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator,” Differential Equations, vol. 46, no. 4, pp. 506–511, 2010. 9 T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 1966. 10 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 . 11 M. Sh. Burlutskaya and A. P. Khromov, “Classical solution of a mixed problem with involution,” Doklady Mathematics, vol. 82, pp. 865–868, 2010. 12 M. A. Naimark, Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators, Frederick Ungar, New York, NY, USA, 1967. 13 N. Dunford and J. T. Schwartz, Linear Operators. Part III, Spectral Operators, John Wiley & Sons, New York, NY, USA, 1988. 14 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. 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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 γ 2 −1; furthermore, 1.1 -1.2 is self-adjoint if and only if γ is a real number.For γ 2 / − 1, the system of eigenfunctions for 1.1 -1.2 is a Riesz basis in L 2 −1, 1 .Observe also that for γ 2 / − 1, 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 Au λSu, with A d/dx and Sf x f −x .In general, they were considered in 9 when A and S 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 .
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 α 2 / 1 is given by the following formula: Next, we observe that for α 2 / 1, γ 2 / 1 eigenvalues are equal to The related eigenfunctions are given by the following formula:

2.3
Observe also that for γ 1, the eigenvalues are The corresponding eigenfunctions are and for γ −1, the eigenvalues are and the corresponding eigenfunctions are x, k 0, ±1, . . .

2.7
We introduce the differential operator L by and by the boundary condition 1.4 .Suppose that Lu belongs to the domain of L, Lu ∈ D L .Then we consider 2.9 From the boundary condition 1.4 , for Lu we deduce that L 2 is the second order differential operator generated by the following relations:

2.11
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 b 1 c 0 a 1 d 0 / 0; and they are strongly regular if additionally θ 2 0 / 4θ −1 θ 1 , where 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 L 2 −1, 1 .By 2.2 numbers −λ k cannot be eigenvalues of L, hence any eigenfunction of L 2 which corresponds to λ 2 k will be an eigenfunction L which corresponds to λ k as Finally, we deduce the assertion of Theorem 2.1 in the case γ 2 / 1.For the case γ 2 1 the explicit representations of eigenfunctions 2.5 and 2.7 give the Riesz basis property for these systems directly.
Proof.The proof is analogous to the proof of Theorem 2.1.It uses the following spectral problem:

2.17
Boundary conditions 2.17 are regular for γ 2 −1, and nonregular for γ 2 1.Then, basis property for eigenfunctions of an ordinary differential second order operator with constant coefficients gives the result for γ 2 1.
For γ 2 / ± 1, boundary conditions 2.17 are strongly regular and the proof terminates analogously to the proof of Theorem 2.1.