On a boundary-value problem with discontinuous solutions and strong nonlinearity

In this work, sufficient conditions for the existence of a solution to a second-order boundary-value problem with discontinuous solutions and strong nonlinearity are obtained. For the analysis of solutions to the boundary-value problem, we apply the pointwise approach proposed by Yu. V. Pokornyi and which has shown its effectiveness in studying second-order problems with nonsmooth solutions. Based on estimates of the Green function of the boundary-value problem obtained earlier by other authors, we show that the operator, which inverts the nonlinear problem considered, can be represented as the composition of a completely continuous operator and a continuous operator; this operator acts from the cone of nonnegative continuous functions into a narrower set. This fact allows one to prove the existence of a solution to a nonlinear boundary-value problem by using the theory of spaces with a cone.