Existence and global asymptotic behavior of positive solutions for sublinear and superlinear fractional boundary value problems

Abstract

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form $$\left( {{P_{a,b}}} \right)\left\{ {_{u\left( 0 \right) = u\left( 1 \right) = 0,{D^{\alpha - 3}}u\left( 0 \right) = a,u'\left( 1 \right) = - b}^{{D^\alpha }u\left( x \right) + f\left( {x,u\left( x \right)} \right) = 0,x \in \left( {0,1} \right)}} \right.$$ , where 3 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in (0, 1)×[0,∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem (Pa,b), which behaves like the unique solution of the homogeneous problem corresponding to (Pa,b). Some examples are given to illustrate the existence results.