Izv. Vyssh. Uchebn. Zaved. Mat., 2024 Number 12,Pages 12–19(Mi ivm10041)
On the existence and uniqueness of a positive solution to a boundary value problem with integral boundary conditions for one nonlinear ordinary differential equation of the second order
Abstract:
The article considers a boundary value problem with integral boundary conditions for one nonlinear second-order 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, we prove 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 compressed mappings, sufficient conditions for the uniqueness of a positive solution were obtained. At the end of the article, there is a non-trivial example illustrating the results obtained here.
Keywords:positive solution, integral boundary value problem, Green's function, cone compression.
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