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 Tαx(t)+f(t,x(t))=0 , t ∈ [ 0 , 1 ] t∈ [0,1] , subject to the boundary conditions x ( 0 ) = 0 $x(0)=0$ and x ( 1 ) = λ ∫ 0 1 x ( t ) d t x(1)= λ ∫₀¹x(t)\,dt , where the order α belongs to ( 1 , 2 ] $(1,2]$ , T α x ( t ) Tαx(t) denotes the conformable fractional derivative of a function x ( t ) $x(t)$ of order α, and f : [ 0 , 1 ] × [ 0 , ∞ ) ↦ [ 0 , ∞ ) 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.