This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c $D_{0^{+}}^{q}u\left(t\right)=f(t,u\left(t\right)),t\in [0,T]$ , ${\displaystyle u^{\left(k\right)}\left(0\right)={\xi }_k,u\left(T\right)=\sum _{i=1}^m{{\beta }_i}_{RL}{I}_{0^{+}}^{p_i}u\left({\eta }_i\right),}$ where $n-1<q<n,n\ge 2,m,n\in \mathbb{N}$ , ${\xi }_k,{\beta }_i\in \mathbb{R}$ , $k=0,1,\dots ,n-2$ , $i=1,2,\dots ,m$ , and ^c $D_{0^{+}}^q$ is the Caputo fractional derivatives, $f:[0,T]\times C\left(\right[0,T],E)\to E$ , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be $\mathbb{R}$ (with the absolute value) or $C\left(\right[0,T],\mathbb{R})$ with the supremum-norm. _RL $I_{0^{+}}^{p_i}$ is the Riemann–Liouville fractional integral of order $p_i>0$ , ${\eta }_i\in (0,T)$ , and ${\displaystyle \sum _{i=1}^m{\beta }_i{\eta }_i^{p_i+n-1}\frac{\Gamma \left(n\right)}{\Gamma (n+p_i)}\ne T^{n-1}}$ . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

A class of generalized $(\psi ,\alpha ,\beta )$ —weak contraction is introduced and some fixed-point theorems in a framework of partially ordered metric spaces are proved. The main result of this paper is applied to a first-order ordinary differential equation to find its solution.

This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.