Extended existence results of solutions for FDEs of order $1<\gamma \le 2$

The focus of our investigation was on determining the existence of solutions for fractional differential equations (FDEs) of order $1<\gamma \le 2$ involving the boundary conditions ${\kappa }_0\varphi (0)+{\eta }_0\varphi (v)={\mu }_0$, and ${\kappa }_1{\varphi }^{{}^{\prime }}(0)+{\eta }_1{\varphi }^{{}^{\prime }}(v)={\mu }_1$, for ${\kappa }_i,{\eta }_i,{\mu }_i\in {\mathbb{R}}^{+}$. The existence results were based on the Schauder fixed point theorem and the nonlinear alternative of the Leray-Schauder type. Examples were provided to illustrate the results.