On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Integral Boundary Conditions for a Second-Order Nonlinear Functional Differential Equation

The article considers a boundary value problem with integral boundary conditions for a second-order nonlinear functional differential equation. Using Green’s function, the boundary value problem is reduced to the equivalent nonlinear Hammerstein integral equation. Next, having identified the necessary properties of  Green’s function, it is proved that the Hammerstein operator contracts the corresponding cone. The last circumstance, by virtue of the well-known Krasnoselsky theorem, guarantees the existence of at least one positive solution to the boundary value problem. Using a priori estimates and the principle of contraction mappings, the sufficient conditions for the uniqueness of a positive solution are obtained. At the end of the article, there is a nontrivial example illustrating the results obtained here.