Using Schauder's fixed-point theorem, we establish sufficient conditions for the existence and uniqueness of solutions to the nonlinear fractional boundary value problem: cases ₂D^ (x) + f (x, (x), I^ (x) ) = 0, x I = 0, 1, 1 0, \\? (0) = 0, (1) = (), cases (0. 1) where is a functional defined on C (I, R). By constructing an appropriate Green''s function, we derive a Lyapunov-type inequality for a special case of the problem (0. 1): cases ₂D^ (x) + (x) I^ (x) = (x, (x) ), x I = 0, 1, 1 0, \\? (0) = 0, (1) = (). cases (0. 2) We further make an analysis for equation (0. 2) by applying the inverse operator method and the Mittag-Leffler function with illustrative examples demonstrating applications obtained. Finally, we construct an analytic solution to the following generalized fractional heat equation with an initial condition in n dimensions based on an inverse operator: cases ₂Dₓ^u (t, x) = ₀_₁ (x₁), , a₍ (x₍) u (t, x) + f (t, x), (t, x) R^+ R^n, 0 < 1, \ (0, x) = (x), cases (0. 3) where ₀_₁ (x₁), , a₍ (x₍) = a₁ (x₁) ^2 x₁^{2} + + a₍ (x₍) ^2 x₍^{2}.
Li et al. (Wed,) studied this question.