On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition (Q1033603)

Consider the boundary-value problem \[ y^{(4)}(x)-(q(x)y^{\prime }(x))^{\prime }=\lambda y(x), \] \[ 0<x<l,\;y^{\prime }(0)\cos \alpha -y^{\prime \prime }(0)\sin \alpha =0,\quad y(0)\cos \beta +Ty(0)\sin \beta =0, \] \[ y^{\prime }(l)\cos \gamma +y^{\prime \prime }(l)\sin \gamma =0,\quad(a\lambda +b)y(l)-(c\lambda +d)Ty(l)=0, \] where \(\lambda \) is a spectral parameter; \(Ty:=y^{\prime \prime \prime }-qy^{\prime }\); \(q\) is a positive absolutely continuous function on the interval \([0,l]\); \(\alpha ,\beta ,\gamma ,a,b,c,\) and \(d\) are real constants; and \(0\leq \alpha ,\beta ,\gamma \leq \pi /2\), \(bc-ad>0\). The present paper studies the basis properties of the system of eigenfunctions of the boundary-value problem in the spaces \(L_{p}(0,l)\) (\(1<p<\infty \)).