Fixed point iterations for a class of nonstandard Sturm–Liouville boundary value problems

The paper examines a particular class of nonlinear integro-differential equations consisting of a Sturm–Liouville boundary value problem on the half-line, where the coefficient of the differential term depends on the unknown function by means of a scalar integral operator. In order to handle the nonlinearity of the problem, we consider a fixed point iteration procedure, which is based on considering a sequence of classical Sturm–Liouville boundary value problems in the weak solution sense. The existence of a solution and the global convergence of the fixed-point iterations are stated without resorting to the Banach fixed point theorem. Moreover, the unique solvability of the problem is discussed and several examples with unique and non-unique solutions are given.