On the Uniform Convergence of Fourier Series Expansions for Sturm–Liouville Problems with a Spectral Parameter in the Boundary Conditions

Abstract

In this paper, we study the uniform convergence of the spectral expansions in terms of eigenfunctions of the boundary value problem

$$\begin{aligned}&\quad \quad \quad \quad \qquad -{y}''+q\left( x \right) y=\lambda y,\quad 0<x<1,\\&\left( {{a}_{0}}\lambda +{{b}_{0}} \right) y\left( 0 \right) =\left( {{c}_{0}}\lambda +{{d}_{0}} \right) {y}'( 0 ),\quad \left( {{a}_{1}}\lambda +{{b}_{1}} \right) y\left( 1 \right) =\left( {{c}_{1}}\lambda +{{d}_{1}} \right) {y}'\left( 1 \right) \end{aligned}$$

where $\lambda$ is a spectral parameter, $q\left( x \right)$ is a real-valued continuous function on the interval $\left[ 0,1 \right]$ and ${{a}_{k}},{{b}_{k}},{{c}_{k}},{{d}_{k}} \left( k=0,1 \right)$ are real constants which satisfy the conditions

$$\begin{aligned} {{\sigma }_{k}}={{\left( -1 \right) }^{k}}\left( {{a}_{k}}{{d}_{k}}-{{b}_{k}}{{c}_{k}} \right) <0\quad \left( k=0,1 \right) . \end{aligned}$$