Existence and non-existence of positive solutions of BVPs for fractional order elastic beam equations with a non-Caratheodory nonlinearity

In this article, the existence and non-existence results on positive solutions of two classes of boundary value problems for nonlinear singular fractional order elastic beam equations is established. Here f depends on x  x′x′ and x″,fx″,f may be singular at t=0t=0 and t=1t=1 and f may be a non-Caratheodory function. The analysis relies on the well known Schauder’s fixed point theorem. By applying iterative techniques, results on the existence of positive solutions are obtained and the iterative scheme which starts off with zero function for approximating the solution is established. The iterative scheme obtained is very useful and feasible for computational purpose. Examples and their numerical simulation are presented to illustrate the main theorems. A conclusion section is also given at the end of this paper.