In this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory.

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.

Criteria are established for the existence of at least two nontrivial solutions to the discrete fourth order boundary value problem $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Delta }}^{4}u(t-2)-\alpha {\mathrm{\Delta }}^{2}u(t-1)+\beta u(t)=f(t,u(t)),t\in {[1,N]}_{\mathbb{Z}},\hfill \\ u(-1)=\mathrm{\Delta }u(-1)=0,u(N+1)={\mathrm{\Delta }}^{2}u(N)=0,\hfill \end{array}\right.\end{array}$$ where $N\ge 1$ is an integer, $\alpha ,\beta \ge 0$, and $f:{[1,N]}_{\mathbb{Z}}\times \mathbb{R}\to \mathbb{R}$ is continuous in the second argument. Applications of the results to a related eigenvalue problem are also presented. The proofs are mainly based on the variational method and the classic mountain pass lemma of Ambrosetti and Rabinowitz. Examples are included to illustrate the applicability of the results.