Positive Solutions for a System of Fractional Boundary Value Problems with r-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters

• Published in: Axioms Vol. 11; no. 4; p. 164
• Main Authors: Tudorache, Alexandru, Luca, Rodica
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.04.2022
• Subjects: Boundary conditions; Boundary value problems; Differential equations; existence; Fixed points (mathematics); Fractional calculus; Green's functions; Integral equations; Intervals; Laplace transforms; Mathematical analysis; nonexistence; Nonlinearity; nonlocal boundary conditions; Operators (mathematics); Parameters; positive parameters; positive solutions; Riemann–Liouville fractional differential equations; Schauder fixpoint theorem
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms11040164

In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator A, the solutions of the integral system are the fixed points of A. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator A has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution.