Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems

AbstractThis work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem: {−D0+νy(t)=f(t,y(t),y(t))+g(t,y(t)),0<t<1,n−1<ν≤n,y(i)(0)=0,0≤i≤n−2,[D0+αy(t)]t=1=0,1≤α≤n−2, where D0+ν is the standard Riemann-Liouville fractional derivative of order ν, and n∈N, n>3. Our analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main result.MSC:26A33, 34B18, 34B27.

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