\theta, \lambda}: {\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}^n \to {\mathbb R}_{+}$ be a relative error tolerance function such that $$\label{eq:fphi} 
        \varphi_{\gamma, \theta, \lambda}(u, v, w)\leq \gamma \|v-u\|^2 + \theta \|w-v\|^2 +   \lambda \|w-u\|^2, \qquad \forall~ u, v, w \in \mathbb{R}^n,$$ where $ \gamma, \theta, \lambda \geq 0$ are given forcing parameters. The [*feasible inexact projection mapping*]{} relative to $u \in C$ with relative error tolerance function $\varphi_{\gamma, \theta, \lambda}$, denoted by ${\cal P}_C(\varphi_{\gamma, \theta, \lambda},u, \cdot): {\mathbb R}^n \rightrightarrows C$ is the set-valued mapping defined as follows $$\label{eq:ProjI} 
        {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v) := \left\{w\in C:~\left\langle v-w, z-w \right\rangle \leq \varphi_{\gamma, \theta, \lambda}(u, v, w), \quad \forall~ z \in C \right\}.$$ Each point $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ is called a [*feasible inexact projection of $v$ onto $C$ relative to $u$ and with relative error tolerance function $\varphi_{\gamma, \theta, \lambda}$*]{}.

In the following, we present some remarks about the definition of the feasible inexact projection mapping onto the convex set $C$.

\[rem: welldef\] Let $C\subset {\mathbb R}^n$, $u\in C$ and $\varphi_{\gamma, \theta, \lambda}$ be as in Definition \[def:InexactProj\]. Therefore, for all $v\in {\mathbb R}^n$, it follows from that ${\cal P}_C(0, u, v)$ is the exact projection of 