On the existence and uniquenessof a positive solution to a boundary value problem for a fourth-order nonlinear ordinary differential equation

We consider a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear fourth-order ordinary di erential equation on theinterval [0,1]. Under restrictions on the right-hand side of the equation of a supralinear nature, su cient conditions for the existence and uniqueness of a positive solutionto the problem under study are obtained. With the help of the Green’s function, theboundary value problem is reduced to an equivalent integral equation, and subsequentlythe existence of a positive solution is proved using the well-known Krasnoselsky cone extension theorem. To establish the uniqueness of the positive solution, a special principleof uniqueness for convex operators was used. In conclusion, an example is given thatillustrates the ful llment of the obtained su cient conditions for the unique solvabilityof the problem posed.