On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear second order differential equation with integral boundary conditions

Problems of the type presented in this article describe the operation of a harmonic oscillator under the influence of external forces, which is fixed in the extreme left position and has some mechanism at right one, that controls the displacement according to the feedback from devices measuring the displacements along parts of the oscillator. Above in the introduction, references are given to some papers in which integral boundary conditions for differential equations were considered. The paper is organized as follows: the boundary value problem considered in the paper is reduced to an equivalent integral equation, and the existence of a positive solution of the integral equation is established using the fixed point principle of an operator defined on a certain cone. An a priori estimate of such a solution is obtained, which subsequently participates in the proof of the uniqueness of a positive solution. Sufficient conditions for uniqueness follow from the uniqueness principle for 𝑢0 convex operators on a cone. At the end of the work, an example is given that demonstrates the results obtained.