We consider the spectral problem for a model second-order differential operator with an involution. The operator is given by the differential expression Lu = −u″(−x) and boundary conditions of general form. We obtain a criterion for the basis property of the systems of eigenfunctions of this operator in terms of the coefficients in the boundary conditions.

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

A class of inverse problems for restoring the right-hand side of a fractional heat equation with involution is considered. The results on existence and uniqueness of solutions of these problems are presented.

Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

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.

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

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Inverse problems for a nonlocal wave equation with an involution perturbation

For the equation αu″(-x) - u″(x) = λu(x), ™1 < x < 1, where α ∈ (™1, 1), we study the problem with the nonlocal conditions u(™1) = 0, u′(™1) = u′(1). We show that if $$r = \sqrt {\left( {1 - \alpha } \right)/\left( {1 + \alpha } \right)} $$
 is irrational, then the system of eigenfunctions is complete and minimal in L
 2(™1, 1) but is not a basis. For rational r, we indicate a method for choosing associated functions for which the system of root functions of the problem is an unconditional basis in L
 2(™1, 1).

Spectral properties of a nonlocal problem for a second-order differential equation with an involution

In the present paper, we introduce the notion of regularity of boundary conditions for a simplest second-order differential equation with a deviating argument. We prove the Riesz basis property for a system of root vectors of the corresponding generalized spectral problem with regular boundary conditions (in the sense of the introduced definition). Examples of irregular boundary conditions to which the theory of Il’in basis property can be applied are given.

Unconditional bases related to a nonclassical second-order differential operator

In the present study, the problem of a parabolic equation with involution is investigated. The stability and coercive stability estimates in Holder norms for the solution of this problem are establ...

Well-Posedness of a Parabolic Equation with Involution

We consider the differential equation αu′′(−x)− u′′(x) = λu(x), −1 < x < 1, with the nonlocal boundary conditions u(−1) = 0, u′(−1) = u′(1) where α ∈ (−1, 1). We prove that if r = p (1− α)/(1 + α) is irrational then the system of its eigenfunctions is complete and minimal in Lp(−1, 1) for any p > 1, but does not constitute a basis. In the case of a rational value of r we specify the way of choosing the associated functions which provides the system of all root functions of the problem forms a basis in Lp(−1, 1).

A. M. Sarsenbi

Papers

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

Spectral properties of a nonlocal problem for a second-order differential equation with an involution

Unconditional bases related to a nonclassical second-order differential operator

Well-Posedness of a Parabolic Equation with Involution

Basicity in Lp of root functions for differential equations with involution