BVP for Hadamard sequential fractional hybrid differential inclusions (Q2670277)

The paper studies the following problem \[ (D_H^{\alpha }+\lambda D_H^{\alpha -1}) \biggl(\frac{x(t)}{\rho (t,x(t),I^p_Hx(t))}\biggr)\in G(t,x(t),I^p_Hx(t)),\quad t\in (1,e) \] \[ \frac{\alpha _1x(\xi )}{\rho (\xi ,x(\xi ),I^p_Hx(\xi ))}=\alpha _2I^r_H \biggl(\frac{x(e)}{\rho (e,x(e),I^p_Hx(e))}\biggr), \] \[ \frac{\beta _1x(e)}{\rho (e,x(e),I^p_Hx(e))}=\beta _2I^r_H \biggl(\frac{x(\xi )}{\rho (\xi ,x(\xi ),I^p_Hx(\xi ))}\biggr), \] where \(\alpha \in (1,2]\), \(\xi \in (1,e)\), \(p\in (0,1)\), \(\lambda ,r>0\), \(\alpha _1,\alpha _2,\beta _1,\beta _2\in \mathbb{R}\), \(D_H\) denotes Hadamard fractional derivative, \(I_H\) denotes Hadamard fractional integral, \(\rho :[1,e]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\backslash \{0\}\) is a continuous function and \(G:[1,e]\times \mathbb{R}\times \mathbb{R}\to\mathcal{P}(\mathbb{R})\) is a set-valued map. Under certain hypotheses on the set-valued map \(G\) two existence results for this problem are provided. Their proofs are based on standard fixed point theorems.