On the basis property of the eigenfunctions of the Frankl problem with nonlocal evenness and oddness conditions of the second kind

Moiseev, E. I.; Ambartsumyan, V. E.

doi:10.1134/s1064562410030257

Cited by 6 publications

(6 citation statements)

References 3 publications

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“…Sufficiency. A according to account (25), (26), for arbitrary function h ∈ L 2 (0, 1) we obtain the following relations…”

Section: Lemma 3 the System Of Functionsmentioning

confidence: 99%

“…The set of operators B p,Θ , eigenfunctions of which are determined by formulas (25), (26) is denoted by Γ p (L 0 ). The corresponding set of transformation operators R(B p,Θ ) = E + S(B p,Θ ), we denote by Φ p (L 0 ).…”

Section: Lemma 3 the System Of Functionsmentioning

confidence: 99%

“…Introduction and main results. During recent years the number of publications with the use of an involution operator in the theory of P T -symmetric operators, P T -symmetric quantum theory (see [11], [12]), various sections of the theory of ordinary differential equations (see [1], [4], [15], [17], [19], [21]- [24], [27], [31], [32]), partial differential equations (see [6], [10], [14], [20], [25]) and differential equations with operator coefficients (see [2], [3], [7], [8]) increased significantly.…”

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confidence: 99%

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The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

Baranetskij¹,

Kalenyuk²,

Kopach³

et al. 2020

Mat. Stud.

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The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

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“…Sufficiency. A according to account (25), (26), for arbitrary function h ∈ L 2 (0, 1) we obtain the following relations…”

Section: Lemma 3 the System Of Functionsmentioning

confidence: 99%

Section: Lemma 3 the System Of Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

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The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

Baranetskij¹,

Kalenyuk²,

Kopach³

et al. 2020

Mat. Stud.

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“…[1,2,7,9,11,12,14]), диференціальних рівнянь з частковими похідними (див. [6,10,13,15,16]) та диференціальних рівнянь з операторними коефіцієнтами (див.…”

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Dirichlet Problem for Differential Equations of Even Order Operator Coefficients That Contain an Involution

Baranetskij

2018

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We study a problem with Dirichlet conditions for a differential equation of order 2n, whose coefficients are non-self-adjoint operators. It is established that the task operator has two subspaces generated by the involution operator, and two subsystems of the system of eigenfunctions, which are Riesz bases in each of the subspaces. Eigenvalues and eigenfunctions are defined. Sufficient conditions are obtained under which the system of eigenfunctions is the Rees base. The conditions for the existence of unity of the solution of the problem with homogeneous boundary conditions, constructed only as a series on the system of eigenfunctions, are established.

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“…During recent years, the number of publications with the use of an involution operator in various sections of the theory of ordinary differential equations (see [4,10,12,15,16,19]), of partial differential equations (see [3,7,9,14,16,17,20,21]), of linear operators, T-invariant with respect to some group of homeomorphisms (see [8]), differential equations with operator coefficients (see [5][6][7]), PT-symmetric operators (see [1,2]) increased significantly.…”

Section: Introductionmentioning

confidence: 99%

The nonlocal problem for the differential-operator equation of the even order with the involution

Baranetskij¹,

Kalenyuk²,

Kolyasa³

et al. 2018

Carpathian Math. Publ.

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In this paper, the problem with boundary non-self-adjoint conditions for differential-operator equations of the order 2n with involution is studied. Spectral properties of operator of the problem is investigated.By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence {L k } ∞ k=1 of operators of boundary value problems for ordinary differential equations of even order. It is established that each element L k of this sequence is an isospectral perturbation of the self-adjoint operator L 0,k of the boundary value problem for some linear differential equation of order 2n.We construct a commutative group of transformation operators whose elements reflect the system V(L 0,k ) of the eigenfunctions of the operator L 0,k in the system V(L k ) of the eigenfunctions of the operators L k . The eigenfunctions of the operator L of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators L 0,k .The conditions under which the system of eigenfunctions of the operator L of the studied problem is a Riesz basis is established.

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On the basis property of the eigenfunctions of the Frankl problem with nonlocal evenness and oddness conditions of the second kind

Cited by 6 publications

References 3 publications

The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

Dirichlet Problem for Differential Equations of Even Order Operator Coefficients That Contain an Involution

The nonlocal problem for the differential-operator equation of the even order with the involution

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