**1. Introduction**

Fractional differential equations have been applied in many fields of engineering, physics, biology, and chemistry see [1–4]. Moreover, to ge<sup>t</sup> a couple of developments about the theory of fractional differential equations, one can allude to the monographs of Abbas et al. [5–7], Kilbas et al. [8], Miller and Ross [9], Podlubny [10], and Zhou [11,12], as well as to the papers by Agarwal, et al. [13], Benchohra, et al. [14–16], and the references therein. In the recent past, Almeida in [17] presented a new fractional differentiation operator called by *ψ*-Caputo fractional operator. For more details see [18–23], and the references given therein.

At the present day, different kinds of fixed point theorems are widely used as fundamental tools in order to prove the existence and uniqueness of solutions for various classes of nonlinear fractional differential equations for details, we refer the reader to a series of papers [24–30] and the references therein, but here we focus on those using the monotone iterative technique, coupled with the method of upper and lower solutions. This method is a very useful tool for proving the existence and approximation of solutions to many applied problems of nonlinear differential equations and integral equations (see [31–42]). However, as far as we know, there is no work ye<sup>t</sup> reported on the existence of extremal solutions for the Cauchy problem with *ψ*-Caputo fractional derivative. Motivated

by this fact, in this paper we deal with the existence and uniqueness of extremal solutions for the following initial value problem of fractional differential equations involving the *ψ*-Caputo derivative:

$$\begin{cases} \,^cD\_{a^+}^{a;\psi} \mathbf{x}(t) = f(t, \mathbf{x}(t)), \; t \in \mathbb{J} := [a, b],\\ \mathbf{x}(a) = a^\*, \end{cases} \tag{1}$$

where *cD<sup>α</sup>*;*<sup>ψ</sup> a*+ is the *ψ*-Caputo fractional derivative of order *α* ∈ (0, 1], *f* : [*a*, *b*] × R −→ R is a given continuous function and *a*<sup>∗</sup> ∈ R.

The rest of the paper is organized as follows: in Section 2, we give some necessary definitions and lemmas. The main results are given in Section 3. Finally, an example is presented to illustrate the applicability of the results developed.
