Three-point boundary value problems for the Langevin equation with the Hilfer fractional derivative (Q2246577)

The authors consider two related problems. In the first they look at the existence and uniqueness of solutions of the three-point boundary value problem for the Langevin equation with Hilfer fractional derivative \[ ^HD^{\alpha_1,\beta_1} \left ( ^HD^{\alpha_2,\beta_2} + \lambda \right )x(t) = f(t, x(t)), \quad t \in [a, b], \] \[ x(a) = 0, \,\, x(b) = \theta x(\eta), \theta \in \mathbb{R}, \eta \in (a, b), \] where \(0 < \alpha_i < 1, 0 \leq \beta_i \leq 1\), \(\lambda \in \mathbb{R}\), and \(f:[a,b] \times \mathbb{R} \to \mathbb{R}\) is continuous. In the second section of the paper, they use the Riemann-Liouville fractional integral \[ I^\alpha u(t) = \frac{1}{\Gamma(\alpha)} \int_a^t \! (t-s)^{\alpha - 1} u(s) ds \] to define the Hilfer fractional derivative of order \(\alpha\) and parameter \(\beta\) as \[ ^HD^{\alpha, \beta}u(t) = I^{\beta(n-a)}D^nI^{(1-\beta)(n-a)}u(t), \] where \(n - 1 < \alpha < n, 0 \leq \beta \leq 1,\) and \(D = (d/dt)\). They also invert the operator and obtain an equivalent fractional integral equation. Using Krasnoselskii's fixed point theorem for the sum of a compact operator and a contraction, they establish two different sets of criteria under which the boundary value problem has a solution. They also present examples of showing applications of these theorems. The authors end this section with conditions for the uniqueness of solutions as well as an example illustrating their result. In the fourth section the authors work with the fractional inclusion problem \[ ^HD^{\alpha_1,\beta_1} \left ( ^HD^{\alpha_2,\beta_2} + \lambda \right )x(t) \in F(t, x(t)), \quad t \in [a, b] \] \[ x(a) = 0, \,\, x(b) = \theta x(\eta), \theta \in \mathbb{R}, \eta \in (a, b), \] where \(0 < \alpha_i < 1, 0 \leq \beta_i \leq 1\), \(\lambda \in \mathbb{R}\), and \(F:[a,b] \times \mathbb{R} \to \mathcal{P}(\mathbb{R})\) is a multivalued map. Their first result is based on the nonlinear alternative of Leray-Schauder type for Kakutani multivalued maps. Here they assume that \(F\) is Carathéodory and obtain an existence result under suitable condition of \(F\). They next use a fixed-point theorem for multivalued maps due to Covitz and Nadler to get another existence theorem under different conditions. They end the paper with an example.