On a certain Köthe space determined by a differential operator (Q1390460)

The author investigates spectral properties of the fourth-order boundary value problem \[ {d^4 u\over dy^4}-2\beta\lambda^2{ d^2 u\over dy^2}+ \alpha \lambda^4 u = \varphi, \qquad u(\pm 1)=0=u'(\pm 1), \tag{*} \] where \(\alpha,\beta\) are (fixed) real constants satisfying \(\alpha-\beta^2>0\), \(\lambda\) is a (complex-valued) spectral parameter and \(\varphi\) is a certain ``averaging'' function specified explicitly. The investigation is motivated by the problem of solvability of the fourth-order partial differential equation \[ \alpha{\partial u^4\over \partial x^4}+2\beta {\partial ^4 u\over \partial x^2\partial y^2}+ {\partial ^4u\over\partial y^4}=0, \qquad \left.u(x,y)\right|_{y=\pm 1}=0= \left. {\partial u\over \partial y}\right|_{y=\pm 1}. \tag{**} \] The obtained results concerning properties of (*) are used to establish a Fourier-type representation of solutions to (**). In particular, it is proved that a certain system of eigenfunctions associated with (*) form the so-called absolute basis of the solution space to (**) and that the corresponding ``Fourier coefficients'' form a basis of a Köthe space (i.e. a space of infinite sequences endowed with a certain special norm).