Solvability of some two-point fractional boundary value problems under barrier strip conditions (Q1683013)

Summary: Topological techniques are used to establish existence results for a class of fractional differential equations \[ D^\alpha x(t) = f(t, x(t), D^{\alpha - 1} x(t)) \] with one of the following boundary value conditions: \[ x(0) = A \text{ and } D^{\alpha - 1} x(1) = B \text{ or } D^{\alpha - 1} x(0) = A \text{ and } x(1) = B, \] where \(1 < \alpha \leq 2\) is a real number, \(D^\alpha x(t)\) is the conformable fractional derivative, and \(f : [0,1] \times \mathbb R^2 \rightarrow \mathbb R\) is continuous. The main conditions on the nonlinear term \(f\) are sign conditions (i.e., the barrier strip conditions). The topological arguments are based on the topological transversality theorem.