Abstract
Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions
Highlights
We consider the four-point coupled boundary value problem for nonlinear fractional differential equation involving the Riemann-Liouville’s derivative λf (t, u, v) u(i)(0) = v(i)(0) = 0, 0 ≤ i ≤ n − 2, u(1) = av(ξ), v(1) = bu(η), ξ, η ∈ (0, 1) (1.1)where λ is a parameter, a, b, ξ, η satisfy ξ, η ∈ (0, 1), 0 < abξη < 1, α ∈
Fractional differential equation’s modeling capabilities in engineering, science, economics, and other fields, over the last few decades has resulted in the rapid development of the theory of fractional differential equations, see
To our knowledge there are only a few papers which deal with the boundary value problem for nonlinear fractional differential equations
Summary
We consider the four-point coupled boundary value problem for nonlinear fractional differential equation involving the Riemann-Liouville’s derivative λf (t, u, v) u(i)(0) = v(i)(0) = 0, 0 ≤ i ≤ n − 2, u(1) = av(ξ), v(1) = bu(η), ξ, η ∈ (0, 1). Coupled boundary conditions arise in the study of reaction-diffusion equations and Sturm-Liouvillie problems, see [21, 22] and have wide applications in various fields of sciences and engineering, for example the heat equation [23, 24, 25] and mathematical biology [26, 27]. In [26], the authors study the blow-up properties of the positive solutions to the system of heat equations with nonlinear boundary conditions uit = △ui, i = l, · · · , k, uk+l := ul,. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnosel’skii’s fixed-point theorems
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