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

G. È. Abduragimov

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 142, pp. 101-110

Résumé

The following boundary value problem is considered: \begin{align*} {0+}^\alpha x(t)+f \left (t,\left(Tx \right)(t) \right)=0,\ \ 01, \ \text{where} \ \ \alpha\in (n-1,n], \ \ n\in \mathbb{N}, \ \ n>2,\\ (0)=x'(0)=\dots x^{(n-2)}(0)=0,\\ (1)=0. \end{align*} This problem reduces to an equivalent integral equation with a monotone operator in the space $C$ of functions continuous on $[0,1]$ (the space $C$ is assumed to be an ordered cone of nonnegative functions satisfying the boundary conditions of the problem under consideration). Using the well-known Krasnosel'sky theorem about fixed points of the operator of expansion (compression) of a cone, the existence of at least one positive solution of the problem under consideration is proved. An example is given that illustrates the fulfillment of sufficient conditions that ensure the solvability of the problem. The results obtained continue the author's research (see [Russian Universities Reports. Mathematics, 27:138 (2022), 129–135]) devoted to the existence and uniqueness of positive solutions to boundary value problems for nonlinear functional-differential equations.

Détail

Keywords: functional-differential equation of fractional order, boundary value problem, Green's function.
Mots-clés : positive solution