Sturm — Liouville problem for an equation with a discontinuous nonlinearity

On the segment $ [0, 1] $, we consider the Sturm — Liouville problem with a discontinuous nonlinearity on the right-hand side multiplied by a positive parameter. For nonnegative values of the phase variable $u$ the nonlinearity is zero, and for negative values it coincides with a continuous function on $ [0,1] \times (- \infty; 0] $. The boundary conditions are $ u (0) = a $, $ u (1) = b $, where $ a, b $ are positive numbers. The initial problem is converted to an equivalent homogeneous one, which for all positive values of the parameter has a zero solution. Its spectrum consists of those parameter values for which the boundary value problem has a nonzero solution. Assuming sublinear growth of nonlinearity at infinity for each positive value of the parameter we construct an iterative process that converges monotonically to the minimal solution. It is proved that the spectrum of the problem is of the form $ [C; + \infty) $, where $ C> 0 $, if it is non-empty.