Exploring solutions to specific class of fractional differential equations of order \(3<\hat{u}\leq 4\)

Abstract

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order $3<\hat{u} \leq 4$ and includes a term $\Theta (\beta,\mathscr{Z}(\beta ))$ alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results.