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

A boundary value problem for a nonlinear second-order functional-differential equation on the interval $[0,1]$ with integral boundary conditions is considered. Sufficient conditions for the existence of at least one positive solution of the problem under consideration are found by using special topological tools. A nontrivial example in which the sufficient conditions for the unique solvability of the problem are satisfied is given.