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

Abstract

This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem: { − D 0 + ν y ( 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 , [ D 0 + α y ( t ) ] t = 1 = 0 , 1 ≤ α ≤ n − 2 , where D 0 + 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.