On the existence and uniqueness of a positive solution to a boundary value problem for one nonlinear functional differential equation of third order

The boundary value problem is consideredx'''(t)+f (t,(Tx )(t) )=0, 0<t<1,x(0)=x(1)=0,$$x''(0)=x''(1),$where T — linear positive continuous operator. Using the Green's function and Krasnoselsky's fixed point theorem, we formulate and prove the existence of positive solutions to the above boundary value problem for a third-order nonlinear functional differential equation. Next, in the sublinear case, using the fixed point principle, we establish the uniqueness of a positive solution to the problem under study. In addition, an example is given to illustrate the results obtained.