Positive solutions of conformable fractional differential equations with integral boundary conditions

In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T α x ( t ) + f ( t , x ( t ) ) = 0 , t ∈ [ 0 , 1 ] , subject to the boundary conditions x ( 0 ) = 0 and x ( 1 ) = λ ∫ 0 1 x ( t ) d t , where the order α belongs to ( 1 , 2 ] , T α x ( t ) denotes the conformable fractional derivative of a function x ( t ) of order α , and f : [ 0 , 1 ] × [ 0 , ∞ ) ↦ [ 0 , ∞ ) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.