Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON THE UNIFORM CONVERGENCE OF SPECTRAL EXPANSIONS FOR A SPECTRAL PROBLEM WITH A BOUNDARY CONDITION RATIONALLY DEPENDING ON THE EIGENPARAMETER

  • Goktas, Sertac (Department of Mathematics Mersin University) ;
  • Kerimov, Nazim B. (Department of Mathematics Khazar University) ;
  • Maris, Emir A. (Department of Mathematics Mersin University)
  • Received : 2016.06.08
  • Published : 2017.07.01
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Abstract

The spectral problem $$-y^{{\prime}{\prime}}+q(x)y={\lambda}y,\;0 < x < 1, \atop y(0)cos{\beta}=y^{\prime}(0)sin{\beta},\;0{\leq}{\beta}<{\pi};\;{\frac{y^{\prime}(1)}{y(1)}}=h({\lambda})$$ is considered, where ${\lambda}$ is a spectral parameter, q(x) is real-valued continuous function on [0, 1] and $$h({\lambda})=a{\lambda}+b-\sum\limits_{k=1}^{N}{\frac{b_k}{{\lambda}-c_k}},$$ with the real coefficients and $a{\geq}0$, $b_k$ > 0, $c_1$ < $c_2$ < ${\cdots}$ < $c_N$, $N{\geq}0$. The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented.

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References

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