Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative
The paper studies a fractional-order differential inclusion of the form \[ D_C^{\alpha }x(t)\in F(t,x(t),D_C^{\beta }x(t))\quad \text{for a.e. } ([0,T]) \] subject to two classes of boundary conditions (1) \(a_1x(0)+b_1D_C^{\gamma }x(0)=c_1\), \(a_2x(T)+b_2D_C^{\gamma }x(T)=c_2\); (2) \(a_1x(0)+b_1x(T)=c_1\), \(a_2D_C^{\gamma }x(0)+b_2D_C^{\gamma }x(T)=c_2\). \(F:[0,T]\times {\mathbb R}\times {\mathbb R}\to {\mathcal P}({\mathbb R})\) is a set-valued map, \(D_C^{\alpha }\) denotes the Caputo fractional derivative of order \(\alpha \), \(\alpha \in (1,2]\), \(\beta \in (0,1]\), \(\gamma \in (0,1)\) and \(a_i,b_i,c_i\in {\mathbb R}\), \(i=1,2\). Three existence results are obtained for the problems considered. The first result relies on the nonlinear alternative of Leray-Schauder type, the second result essentially uses the Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and the third result is based on the Covitz-Nadler contraction principle for set-valued maps.
nonlinear alternative of Leray-Schauder-type
scientific article; zbMATH DE number 6134087
scientific article