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In this paper, by studying the solutions of the abstract operator equation A(x,x)+B(x,x)+e=x$A(x,x)+B(x,x)+e=x$ on ordered Banach spaces, where A, B are two mixed monotone operators and e∈P$e\in P$ with θ≤e≤h$\theta \leq e\leq h$, we prove a class of boundary value problems on elastic beam equation to have a unique solution. Furthermore, we also apply our abstract result to establish the existence and uniqueness theorem of nontrivial solutions for nonlinear fractional boundary value problems. The iterative sequences to approximate unique solutions for the above two classes of problems are also obtained.

Nonlinear sum operator equations and applications to elastic beam equation and fractional differential equation