On the existence of a positive solution to a boundary-value problem for a nonlinear second-order functional differential equation

In this paper, we consider a boundary-value problem for a second-order nonlinear functional-differential equation with a strong nonlinearity on the interval $[0,1]$ with integral boundary conditions. Using special topological tools, we obtain sufficient conditions for the existence of a unique positive solution of the problem. The existence of a positive solution is proved by applying the well-known cone dilation theorem, and the uniqueness is established by using the uniqueness principle for convex operators. An example is given, which illustrates the fulfillment of sufficient conditions for the unique solvability of the problem.