Solvability of singular fractional order elastic beam systems with nonlinearities depending on lower derivatives

In this article, existence results for positive solutions of a class of boundary-value problems for singular fractional order elastic beam systems with nonlinearities depending on lower derivatives are established. The nonlinearities in fractional differential equations may be singular at $t=0$ and $t=1$. A weighted function space is constructed and complete continuity of a nonlinear operator is proved. Our analysis relies on a well known fixed point theorem.