Existence and Uniqueness of Solutions for a Second-Order Iterative Fractional Conformable Boundary Value Problem

Abstract

AbstractMotivated by some papers treating the iterative differential equations, we disscuss the existence and uniqueness of solution of the sencond-order iterative fractional boundary value problem $$\begin{aligned} \left\{ \begin{array}{lcr} D^{\alpha }x(t)=f(t,x(t),x^{[2]}(t)),\quad t\in ]0,1[,\\ x(0)=0,\quad x(1)=1. \end{array}\right. \end{aligned}$$where, \(x^{[2]}(t)=x(x(t)),\) \(1<\alpha < 2\) and \(D^{\alpha }\) is the conformable derivative. The main tools employed to establish our results are the Schauder and Banach fixed point theorems. We give an example in order to illustrate this situation.