Some order properties of solution sets of boundary value problems

A two-point boundary value problem for an ordinary differential equation with a right-hand side increasing (and not necessarily continuous) in the phase variable is investigated. This problem is written as an equivalent integral equation in the space of summable functions. Results on fixed points of monotone operators are applied to the integral equation. Thus, conditions for the existence of extremal solutions and monotone dependence on the parameters of the solution set are obtained for the boundary value problem under consideration. As an application, a differential equation for a neural network with a discontinuous neuron activation function is considered. Statements are obtained on the solvability of the two-point boundary value problem and on the well-posedness of this problem with respect to changes in the activation threshold, equation coefficients, external influence function, and boundary condition values.