Existence of a positive solution to systems of differential equations of fractional order

Abstract

In this paper, we consider a system of (continuous) fractional boundary value problems given by { − D 0 + ν 1 y 1 ( t ) = λ 1 a 1 ( t ) f ( y 1 ( t ) , y 2 ( t ) ) , − D 0 + ν 2 y 2 ( t ) = λ 2 a 2 ( t ) g ( y 1 ( t ) , y 2 ( t ) ) , where ν 1 , ν 2 ∈ ( n − 1 , n ] for n > 3 and n ∈ N , subject either to the boundary conditions y 1 ( i ) ( 0 ) = 0 = y 2 ( i ) ( 0 ) , for 0 ≤ i ≤ n − 2 , and [ D 0 + α y 1 ( t ) ] t = 1 = 0 = [ D 0 + α y 2 ( t ) ] t = 1 , for 1 ≤ α ≤ n − 2 , or y 1 ( i ) ( 0 ) = 0 = y 2 ( i ) ( 0 ) , for 0 ≤ i ≤ n − 2 , and [ D 0 + α y 1 ( t ) ] t = 1 = ϕ 1 ( y ) , for 1 ≤ α ≤ n − 2 , and [ D 0 + α y 2 ( t ) ] t = 1 = ϕ 2 ( y ) , for 1 ≤ α ≤ n − 2 . In the latter case, the continuous functionals ϕ 1 , ϕ 2 : C ( [ 0 , 1 ] ) → R represent nonlocal boundary conditions. We provide conditions on the nonlinearities f and g , the nonlocal functionals ϕ 1 and ϕ 2 , and the eigenvalues λ 1 and λ 2 such that the system exhibits at least one positive solution. Our results here generalize some recent results on both scalar fractional boundary value problems and systems of fractional boundary value problems, and we provide two explicit numerical examples to illustrate the generalizations that our results afford.