Existence of a positive solution to a class of fractional differential equations

Abstract

In this paper, we consider a (continuous) fractional boundary value problem of the form − D 0 + ν y ( t ) = f ( t , y ( t ) ) , y ( i ) ( 0 ) = 0 , [ D 0 + α y ( t ) ] t = 1 = 0 , where 0 ≤ i ≤ n − 2 , 1 ≤ α ≤ n − 2 , ν > 3 satisfying n − 1 < ν ≤ n , n ∈ N , is given, and D 0 + ν is the standard Riemann–Liouville fractional derivative of order ν . We derive the Green’s function for this problem and show that it satisfies certain properties. We then use cone theoretic techniques to deduce a general existence theorem for this problem. Certain of our results improve on recent work in the literature, and we remark on the consequences of this improvement.