Abstract
In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation D0+αu(t)+λf(t,u(t),v(t))=0, 0<t<1, D0+αv(t)+μg(t,u(t),v(t))=0, 0<t<1, u(0)=v(0)=0, a1D0+βu(1)=b1D0+βv(ξ), a2D0+βv(1)=b2D0+βu(η), η,ξ∈(0,1), where the coefficients ai,bi,i=1,2 are real positive constants, α∈(1,2],β∈(0,1],D0+α, D0+β are the standard Riemann-Liouville derivatives. Values of the parameters λ and μ are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.
Highlights
In recent years, fractional-order calculus has been one of the most rapidly developing areas of mathematical analysis
A great interest has been shown by many authors in the subject of fractional-order boundary value problems (BVPs), and a variety of results for BVPs equipped with different kinds of boundary conditions have been obtained; for instance, see [7–18] and the references cited therein
We investigate the existence of positive solutions for our problem (1)-(2)
Summary
Fractional-order calculus has been one of the most rapidly developing areas of mathematical analysis. D0α+ V (t) + μg (t, u (t) , V (t)) = 0, 0 < t < 1, with the coupled boundary conditions u (0) = V (0) = 0, a1D0β+ u (1) = b1D0β+ V (ξ) , (2). To the best of our knowledge, fractional-order coupled system (1) has yet to be studied with the boundary conditions (2). The existence of positive solution results for the given problem is new, though they are proved by applying the well-known fixed point theorem. As an application, we give an example to illustrate our result
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