New existence results for differential inclusions involving Langevin equation with two indicess (Q2843769)

Dirichlet boundary value problem for fractional differential inclusions with two different differential orders of the form NEWLINENEWLINE\[NEWLINE^{c}D^{\beta}(\,^{c}D^{\alpha}+\lambda)x(t)\in F(t,x(t)), \, 0<t<1, ), \,0<\alpha, \beta \leq 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)=\gamma_ 1, \quad x(1)= \gamma_ 2,NEWLINE\]NEWLINE NEWLINEwhere \(^{c}D\) is the Caputo fractional derivative, \(F:[0,1]\times \mathbb{R}\to {\mathcal P}(\mathbb{R})\) is a compact multivalued map, and \({\mathcal P}(\mathbb{R})\) is a family of all subsets of \(\mathbb{R}\) is studied and well discussed. Under certain conditions, the authors established the existence of solutions of the above problem when the right hand side is convex as well as nonconvex valued. The results obtained are based on the nonlinear alternative of Leray-Schauder type and some suitable results from fixed point theory for multivalued maps. Finally, an example is given to convey the importance of the results obtained.