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

In this article, we consider a two-point boundary value problem for a nonlinear functional differential equation of fractional order with weak nonlinearity on the interval $[0,1]$ with zero Dirichlet conditions on the boundary. The boundary value problem is reduced to an equivalent integral equation in the space of continuous functions. Using special topological tools (using the geometric properties of cones in the space of continuous functions, statements about fixed points of monotone and concave operators), the existence of a unique positive solution to the problem under consideration is proved. An example is given that illustrates the fulfillment of sufficient conditions that ensure the unique solvability of the problem. The results obtained are a continuation of the author’s research (see [Results of science and technology. Ser. Modern mat. and her appl. Subject. review, 2021, vol. 194, pp. 3–7]) devoted to the existence and uniqueness of positive solutions of boundary value problems for non-linear functional differential equations.