$u$ onto $C$; see [@Bertsekas1999 Proposition 2.1.3, p. 201]. Moreover, ${\cal P}_C(0, u, v) \in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ concluding that ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)\neq \varnothing$, for all $u\in C$ and $v\in {\mathbb R}^n$. Consequently, the set-valued mapping ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, \cdot) $ is well-defined.

Next lemma is a variation of [@Reiner_Orizon_Leandro2019 Lemma 6]. It will play an important role in the remainder of this paper.

\[pr:cond\] Let $v \in {\mathbb R}^n$, $u \in C$, $\gamma, \theta, \lambda \geq 0$ and $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$. Then, there holds $$\|w-x\|^2 \leq \|v-x\|^2 + \frac{2\gamma+2\lambda}{1-2\lambda}\|v-u\|^2, \qquad \forall ~x \in C,$$ for all $ \lambda, \theta \in [0,  1/2)$.

Let $x \in C$. First note that $\|w-x\|^2 = \|v-x\|^2 - \|w-v\|^2 + 2 \langle v-w, x-w \rangle$. Since $w \in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ and $0\leq \theta < 1/2$, combining the last equality with and we obtain $$\begin{aligned}
        \|w-x\|^2 &\leq \|v-x\|^2 - (1-2\theta)\|v-w\|^2 + 2\gamma \|v-u\|^2 + 2\lambda \|w-u\|^2\notag\\
                      &\leq \|v-x\|^2  + 2\gamma \|v-u\|^2 + 2\lambda \|w-u\|^2 \label{eq:fg}.
    \end{aligned}$$ On the other hand, we also have $$\begin{aligned}
        \|w-u\|^2 &= \|v-u\|^2 + \|w-v\|^2 + 2 \langle v-w,u-v \rangle \\
                            &= \|v-u\|^2 + \|w-v\|^2 