The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

Abstract

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

Solvability of mixed problems for the wave equation with reflection of the argument

“…The second condition in () is also satisfied, since we assume that all eigenvalues of problem () and () are simple. Then, it follows from the results of previous works 24,39 that each of the systems $$$ \left\{{X}_k(x)\right\} $$$ and $$$ \left\{{Z}_k(x)\right\} $$$ forms an unconditional basis. Since the system $$$ \left\{{X}_k(x)\right\} $$$ is normalized and $$$$ 1=\left({X}_k,{Z}_k\right)\le {\left\Vert {X}_k\right\Vert}_{L_2}{\left\Vert {Z}_k\right\Vert}_{L_2}\le {C}_0, $$$$ the system $$$ \left\{{Z}_k(x)\right\} $$$ will be almost normalized $$$ 1\le {\left\Vert {Z}_k\right\Vert}_{L_2}\le {C}_0 $$$.…”

“…$$ Let $$$ {G}^{\ast}\left(x,t\right) $$$ be Green's function of the boundary value problem () for $$$ \lambda =0 $$$. (The definition of Green's function is given in Sarsenbi 39 ). Then the spectral problem () is equivalent to the integral equation $$$$ {Z}_k(x)={\overline{\lambda}}_k\underset{-1}{\overset{1}{\int }}{G}^{\ast}\left(x,t\right){Z}_k(t) dt.$$…”

“…Theorem (Sarsenbi 39 ). Let $$$ q(x)\in C\left[-1,1\right] $$$ and all eigenvalues of the spectral problem () with periodic boundary conditions () be simple.…”

“…The theorem on the basis property of the eigenfunctions of the spectral problem (4) with periodic boundary conditions was considered in Sarsenbi 39 ; here, we formulate it in the following form.…”

The inverse problem for the heat equation with reflection of the argument and with a complex coefficient

“…Note that in the last decade, there were many papers devoted to spectral problems for differential operators with involution (see, for example, [16][17][18][19] and references therein). The basis property of eigenfunctions of the first-order differential operators with involution was studied in [16,17] (see also references therein), and in [18][19][20][21][22][23][24] the cases of the second-order operators were considered. In [25,26] the problems with operators containing an involution in lower terms are considered.…”