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

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...

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).

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.

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...

In the present study, the problem of a parabolic equation with nonlocal condition is investigated. The positivity of the second order differential operator A x defined by the formula