The basis property of the system of eigen- and associated functions of a boundary value problem with shift for the wave equation (Q1316853)

Let \(\Omega \subset \mathbb{R}^ 2\) be the finite domain bounded by the segment \(AB : 0 \leq x \leq 1\) of the axis \(y = 0\) and by the characteristics \(AC : x - y = 0\), \(BC : x + y = 1\) of the equation \[ u_{xx} - u_{yy} = f(x,y). \tag{1} \] Problem S. Find the solution of (1), satisfying the conditions \[ au_ x + bu_ y |_{AB} = 0, \quad u \bigl( \theta_ 0 (t) \bigr) = \alpha u \bigl( \theta_ 1(t) \bigr), \quad 0 \leq t \leq 1, \] where \(\theta_ 0 (t) = (t/2, t/2)\), \(\theta_ 1 (t) = [(t + 1)/2, (1 - t)/2]\). We give a criterion for the well-posedness of the problem \(S\) and for \(\alpha \neq 0\) we prove the basis property in \(L_ 2 (\Omega)\) of the system of eigen- and associated functions.