ON THE EXISTENCE AND UNIQUENESS OF A POSITIVE SOLUTION TO A BOUNDARY VALUE PROBLEM FOR A FOURTH–ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATION

Abstract: We consider a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear fourth-order ordinary differential equation on the interval [0,1]. Under restrictions on the right-hand side of the equation of a supralinear nature, sufficient conditions for the existence and uniqueness of a positive solution to the problem under study are obtained. With the help of the Green’s function, the boundary value problem is reduced to an equivalent integral equation, and subsequently the 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 principle of uniqueness for convex operators was used. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem posed.

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