Page 630 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 630
AMICAL SYSTEMS AND ORDINARY DIFFERENTIAL EQUATIONS AND
APPLICATIONS
The minimality of Sturm-Liouville problems with a boundary condition
depending quadratically on the eigenparameter
Yagub Aliyev, yaliyev@ada.edu.az
ADA University, Azerbaijan
We study minimality of root functions of the following Sturm-Liouville problem
−y + q(x)y = λy, 0 < x < 1, (0.1)
y(0) cos β = y (0) sin β, 0 ≤ β < π, (0.2)
y(1) = (aλ2 + bλ + c)y (1), a = 0, (0.3)
where λ is the spectral parameter, q(x) is a real valued and continuous function on the inter-
val [0, 1], and a, b, c are real. It is known that the eigenvalues of (0.1)-(0.3) form an infinite
sequence, accumulating only at +∞, and one of the following cases are possible:
(a) All the eigenvalues are real and simple;
(b) All the eigenvalues are simple and all, except a conjugate pair of non-real, are real;
(c) All the eigenvalues are real and all, except one double, are simple;
(d) All the eigenvalues are real and all, except one triple, are simple.
By constructing the biorthogonal system explicitly, it is possible to show that in the case (a)
the system of eigenfunctions with any two eigenfunctions excluded forms a minimal system in
space L2(0, 1). It is also possible to give similar results in the cases (b), (c) and (d). In particular,
for the case (b) one can prove that the system of eigenfunctions without the eigenfunction,
corresponding to the double eigenvalue, is a minimal system. Similarly, in the case (c) one
can prove that the system of eigenfunctions with no excluded functions is a minimal system.
Finally, in the case (d) it is possible to prove that the system of eigenfunctions, without the two
eigenfunctions, corresponding to non-real eigenvalues, is minimal. These minimality results
can then be extended to basis properties in L2(0, 1) and Lp(0, 1) (1 < p < ∞). But the study of
these properties wouldn’t be complete if we will not consider similar questions for the system of
root functions which contain beside eigenfunctions some associated functions. If the associated
functions are not excluded then these minimality properties do not always hold true. There
are associated functions which when included make the system of root functions not minimal
anymore. In the current work, we will give necessary and sufficient conditions for the system of
root functions to be minimal in L2(0, 1) for all possible choices of the two excluded functions,
including the cases when the excluded functions are eigenfunction and when they are associated
functions.
On Linear Inhomogeneous Boundary-Value Problems for Differential
Systems in Sobolev Spaces
Olena Atlasiuk, hatlasiuk@gmail.com
Institute of Mathematics of the National Academy of Sciences of Ukraine, Ukraine
For the systems of ordinary differential equations of an arbitrary order on a compact interval,
we study a character of solvability of the most general linear boundary-value problems in the
Sobolev spaces Wpn, with n ∈ N and 1 ≤ p ≤ ∞. We find the indices of these Fredholm
problems and obtain a criterion of their well-posedness. Each of these boundary-value problems
relates to a certain rectangular numerical characteristic matrix with kernel and cokernel of the
628
APPLICATIONS
The minimality of Sturm-Liouville problems with a boundary condition
depending quadratically on the eigenparameter
Yagub Aliyev, yaliyev@ada.edu.az
ADA University, Azerbaijan
We study minimality of root functions of the following Sturm-Liouville problem
−y + q(x)y = λy, 0 < x < 1, (0.1)
y(0) cos β = y (0) sin β, 0 ≤ β < π, (0.2)
y(1) = (aλ2 + bλ + c)y (1), a = 0, (0.3)
where λ is the spectral parameter, q(x) is a real valued and continuous function on the inter-
val [0, 1], and a, b, c are real. It is known that the eigenvalues of (0.1)-(0.3) form an infinite
sequence, accumulating only at +∞, and one of the following cases are possible:
(a) All the eigenvalues are real and simple;
(b) All the eigenvalues are simple and all, except a conjugate pair of non-real, are real;
(c) All the eigenvalues are real and all, except one double, are simple;
(d) All the eigenvalues are real and all, except one triple, are simple.
By constructing the biorthogonal system explicitly, it is possible to show that in the case (a)
the system of eigenfunctions with any two eigenfunctions excluded forms a minimal system in
space L2(0, 1). It is also possible to give similar results in the cases (b), (c) and (d). In particular,
for the case (b) one can prove that the system of eigenfunctions without the eigenfunction,
corresponding to the double eigenvalue, is a minimal system. Similarly, in the case (c) one
can prove that the system of eigenfunctions with no excluded functions is a minimal system.
Finally, in the case (d) it is possible to prove that the system of eigenfunctions, without the two
eigenfunctions, corresponding to non-real eigenvalues, is minimal. These minimality results
can then be extended to basis properties in L2(0, 1) and Lp(0, 1) (1 < p < ∞). But the study of
these properties wouldn’t be complete if we will not consider similar questions for the system of
root functions which contain beside eigenfunctions some associated functions. If the associated
functions are not excluded then these minimality properties do not always hold true. There
are associated functions which when included make the system of root functions not minimal
anymore. In the current work, we will give necessary and sufficient conditions for the system of
root functions to be minimal in L2(0, 1) for all possible choices of the two excluded functions,
including the cases when the excluded functions are eigenfunction and when they are associated
functions.
On Linear Inhomogeneous Boundary-Value Problems for Differential
Systems in Sobolev Spaces
Olena Atlasiuk, hatlasiuk@gmail.com
Institute of Mathematics of the National Academy of Sciences of Ukraine, Ukraine
For the systems of ordinary differential equations of an arbitrary order on a compact interval,
we study a character of solvability of the most general linear boundary-value problems in the
Sobolev spaces Wpn, with n ∈ N and 1 ≤ p ≤ ∞. We find the indices of these Fredholm
problems and obtain a criterion of their well-posedness. Each of these boundary-value problems
relates to a certain rectangular numerical characteristic matrix with kernel and cokernel of the
628
