On discrete sequential fractional boundary value problems

In this paper, we analyze several different types of discrete sequential fractional boundary value problems. Our prototype equation is − Δ μ 1 Δ μ 2 Δ μ 3 y ( t ) = f ( t + μ 1 + μ 2 + μ 3 − 1 , y ( t + μ 1 + μ 2 + μ 3 − 1 ) ) subject to the conjugate boundary conditions y ( 0 ) = 0 = y ( b + 2 ) , where f : [ 1 , b + 1 ] N 0 × R → [ 0 , + ∞ ) is a continuous function and μ 1 , μ 2 , μ 3 ∈ ( 0 , 1 ) satisfy 1 < μ 2 + μ 3 < 2 and 1 < μ 1 + μ 2 + μ 3 < 2 . We also obtain results for delta–nabla discrete fractional boundary value problems. As an application of our analysis, we give conditions under which such problems will admit at least one positive solution.