#### 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 [46]). 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, [1214].) 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: