Existence and uniqueness of solutions of nonlinear fractional order problems via a fixed point theorem

Abstract

Abstract In this paper, we introduce an Caputo fractional high-order problem with a new boundary condition including two orders γ ∈ ( n 1 − 1 , n 1 ] $\gamma \in \left({n}_{1}-1,{n}_{1}\right]$ and η ∈ ( n 2 − 1 , n 2 ] $\eta \in \left({n}_{2}-1,{n}_{2}\right]$ for any n 1 , n 2 ∈ ℕ ${n}_{1},{n}_{2}\in \mathrm{ℕ}$ . We deals with existence and uniqueness of solutions for the problem. The approach is based on the Krasnoselskii’s fixed point theorem and contraction mapping principle. Moreover, we present several examples to show the clarification and effectiveness.