In this section, we demonstrate strong convergence of the sequences acquired from the proposed iterative methods for finding a common zero of two H-accretive mappings. First, we prove the following technical lemma.
Lemma 3.1
Let
\(H:\mathcal{B}\rightarrow\mathcal{B}\)
and
\(\varPhi:\mathcal {B}\rightarrow2^{\mathcal{B}}\)
be strictly accretive and
H-accretive mappings, respectively. Then, for
\(\mu,\nu\in\mathbb {R}_{+}\)
and
\(y\in\mathcal{B}\),
$$\begin{aligned} J_{\mu,\varPhi}^{H}(y)=J_{\nu,\varPhi}^{H} \biggl( \frac{\nu}{\mu}y + \biggl(1-\frac{\nu}{\mu} \biggr)HJ_{\mu,\varPhi}^{H}(y) \biggr). \end{aligned}$$
Proof
For \(y\in\mathcal{B}\), let \(\tilde{y}=J_{\mu,\varPhi}^{H}(y)\). This implies that
$$\begin{aligned} y&=H(\tilde{y})+\mu\tilde{z}\quad\mbox{for some }\tilde{z}\in \varPhi(\tilde{y}). \end{aligned}$$
Then
$$\begin{aligned} \frac{\nu}{\mu}y+ \biggl(1-\frac{\nu}{\mu} \biggr)HJ_{\mu,\varPhi}^{H}(y) &=\frac{\nu}{\mu} \bigl(H(\tilde{y})+\mu\tilde{z} \bigr)+ \biggl(1- \frac{\nu}{\mu} \biggr)H(\tilde{y}) \\ &\in (H+\nu\varPhi ) (\tilde{y}), \end{aligned}$$
which implies that
$$\begin{aligned} J_{\nu,\varPhi}^{H} \biggl(\frac{\nu}{\mu}y + \biggl(1- \frac{\nu}{\mu} \biggr)HJ_{\mu,\varPhi}^{H}(y) \biggr)=\tilde {y}=J_{\mu,\varPhi}^{H}(y). \end{aligned}$$
□
Theorem 3.2
Let
\(\mathcal{B}\)
be a uniformly convex and uniformly smooth Banach space. Let
\(H_{1},H_{2}:\mathcal{B}\rightarrow\mathcal{B}\)
be single-valued mappings such that
\(H_{1}\)
is strongly accretive and Lipschitz continuous with constant
\(\xi_{H_{1}}\)
and
\(H_{2}\)
is strongly accretive and Lipschitz continuous with constant
\(\zeta _{H_{2}}\). Let
\(\varPhi,\varPsi:\mathcal{B}\rightarrow2^{\mathcal{B}}\)
be
\(H_{1}\)-accretive and
\(H_{2}\)-accretive mappings, respectively. Assume that
\(\varOmega:=\varPhi^{-1}(0)\cap\varPsi^{-1}(0)\neq\emptyset\). Let the sequences
\(\{y_{n}\}\)
and
\(\{x_{n}\}\)
be generated by the following iterative scheme:
$$\begin{aligned} \textstyle\begin{cases} x_{1}\in\mathcal{B}, \\ y_{n}=\kappa_{n}x_{n}+(1-\kappa_{n})J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}), \\ x_{n+1}=\rho_{n}l+\sigma_{n}x_{n}+\tau_{n}J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(y_{n}),\quad n\in\mathbb{N}, \end{cases}\displaystyle \end{aligned}$$
(3.1)
where
\(l\in\mathcal{B}\)
is an arbitrary element, sequences
\(\{\mu _{n}\}\), \(\{\nu_{n}\}\)
are in
\(\mathbb{R}_{+}\), and sequences
\(\{ \kappa_{n}\}\), \(\{\rho_{n}\}\), \(\{\sigma_{n}\}\), \(\{\tau_{n}\}\)
are in
\([0,1]\)
with
\(\rho_{n}+\sigma_{n}+\tau_{n}=1\), \(n\in\mathbb{N}\). Assume that the following conditions are fulfilled:
- (\(\mathrm{C}_{1}\)):
\(\liminf_{n\rightarrow\infty}\kappa _{n}\tau_{n}>0\), \(\lim_{n\rightarrow\infty}|\kappa_{n+1}-\kappa_{n}|=0\);
- (\(\mathrm{C}_{2}\)):
\(\lim_{n\rightarrow\infty}\rho _{n}=0\), \(\sum_{n=1}^{\infty}\rho_{n}=\infty\);
- (\(\mathrm{C}_{3}\)):
\(0<\liminf_{n\rightarrow\infty} \sigma _{n}\leq\limsup_{n\rightarrow\infty} \sigma_{n}<1\);
- (\(\mathrm{C}_{4}\)):
For some
\(\epsilon\in\mathbb{R}_{+}\)
and for all
\(n\in\mathbb{N}\), \(\mu_{n}\geq\epsilon\), \(\nu_{n}\geq \epsilon\)
and
\(\lim_{n\rightarrow\infty}|\mu_{n+1}-\mu_{n}|=0\)
and
\(\lim_{n\rightarrow\infty}|\nu_{n+1}-\nu_{n}|=0\).
Then the sequence
\(\{x_{n}\}\)
converges strongly to
\(Q_{\varOmega}l\), where
\(Q_{\varOmega}:\mathcal{B}\rightarrow\varOmega\)
is a sunny nonexpansive retraction from
\(\mathcal{B}\)
onto
Ω.
Proof
The proof will be divided into six steps.
Step 1. We show that \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded. Assume that \(w\in\varOmega\). By utilizing (2.1), we find that
$$\begin{aligned} J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(w)=w\in \varPhi^{-1}(0)\quad \mbox{and}\quad J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(w)=w \in\varPsi^{-1}(0). \end{aligned}$$
(3.2)
Then, from (3.1), (3.2), and Proposition 2.5, we have
$$\begin{aligned} \Vert x_{n+1}-w \Vert &= \bigl\Vert \rho_{n}l+ \sigma_{n}x_{n}+\tau_{n}J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(y_{n})-w \bigr\Vert \\ &\leq\rho_{n} \Vert l-w \Vert +\sigma_{n} \Vert x_{n}-w \Vert +\tau_{n} \bigl\Vert J_{\nu _{n},\varPsi}^{H_{2}}H_{2}(y_{n})-w \bigr\Vert \\ &\leq\rho_{n} \Vert l-w \Vert +\sigma_{n} \Vert x_{n}-w \Vert +\tau_{n} \Vert y_{n}-w \Vert . \end{aligned}$$
(3.3)
Now we calculate \(\|y_{n}-w\|\).
$$\begin{aligned}& \begin{aligned}[b] \Vert y_{n}-w \Vert &= \bigl\Vert \kappa_{n}x_{n}+(1- \kappa_{n})J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n})-w \bigr\Vert \\ &\leq\kappa_{n} \Vert x_{n}-w \Vert +(1- \kappa_{n}) \bigl\Vert J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n})-w \bigr\Vert \\ &\leq\kappa_{n} \Vert x_{n}-w \Vert +(1- \kappa_{n}) \Vert x_{n}-w \Vert , \end{aligned} \\& \quad \Rightarrow\quad \Vert y_{n}-w \Vert \leq \Vert x_{n}-w \Vert . \end{aligned}$$
(3.4)
Using (3.4) and the relation \(\rho_{n}+\sigma_{n}+\tau _{n}=1\) in (3.3), we obtain by induction
$$\begin{aligned} \Vert x_{n+1}-w \Vert &\leq\rho_{n} \Vert l-w \Vert +\sigma_{n} \Vert x_{n}-w \Vert + \tau_{n} \Vert x_{n}-w \Vert \\ &=\rho_{n} \Vert l-w \Vert +(1-\rho_{n}) \Vert x_{n}-w \Vert \\ &\leq\max\bigl\{ \Vert l-w \Vert , \Vert x_{n}-w \Vert \bigr\} \\ &\vdots \\ &\leq\max\bigl\{ \Vert l-w \Vert , \Vert x_{1}-w \Vert \bigr\} . \end{aligned}$$
Hence \(\{x_{n}\}\) is bounded. Therefore \(\{y_{n}\}\), \(\{J_{\mu _{n},\varPhi}^{H_{1}}H_{1}(x_{n})\}\), and \(\{J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(y_{n})\}\) are also bounded.
Step 2. We claim that \(\lim_{n\rightarrow\infty}\| x_{n+1}-x_{n}\|=0\) and \(\lim_{n\rightarrow\infty}\|y_{n+1}-y_{n}\|=0\).
Setting \(x_{n+1}=(1-\sigma_{n})w_{n}+\sigma_{n}x_{n}\), we see that \(w_{n}=\frac{x_{n+1}-\sigma_{n}x_{n}}{1-\sigma_{n}}\). Then
$$\begin{aligned} \Vert w_{n+1}-w_{n} \Vert &= \biggl\Vert \frac{x_{n+2}-\sigma_{n+1}x_{n+1}}{1-\sigma_{n+1}} -\frac{x_{n+1}-\sigma_{n}x_{n}}{1-\sigma_{n}} \biggr\Vert \\ &= \biggl\Vert \frac{\rho_{n+1}l+\tau_{n+1}J_{\nu_{n+1},\varPsi }^{H_{2}}H_{2}(y_{n+1})}{1-\sigma_{n+1}} -\frac{\rho_{n}l+\tau_{n}J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(y_{n})}{1-\sigma_{n}} \biggr\Vert \\ &= \biggl\Vert \frac{\rho_{n+1}l+\tau_{n+1}J_{\nu_{n+1},\varPsi }^{H_{2}}H_{2}(y_{n+1})}{1-\sigma_{n+1}} -\frac{\tau_{n}J_{\nu_{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1})}{1-\sigma_{n}} \\ &\quad{}+\frac{\tau_{n}J_{\nu_{n+1},\varPsi }^{H_{2}}H_{2}(y_{n+1})}{1-\sigma_{n}} -\frac{\rho_{n}l+\tau_{n}J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(y_{n})}{1-\sigma_{n}} \biggr\Vert \\ &\leq \biggl\vert \frac{\rho_{n+1}}{1-\sigma_{n+1}}-\frac{\rho _{n}}{1-\sigma_{n}} \biggr\vert \Vert l \Vert + \biggl\vert \frac{\tau_{n+1}}{1-\sigma_{n+1}} -\frac{\tau_{n}}{1-\sigma_{n}} \biggr\vert \bigl\Vert J_{\nu_{n+1},\varPsi }^{H_{2}}H_{2}(y_{n+1}) \bigr\Vert \\ &\quad{}+ \biggl(\frac{\tau_{n}}{1-\sigma_{n}} \biggr) \bigl\Vert J_{\nu _{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1}) -J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \bigr\Vert \\ &\leq \biggl\vert \frac{\rho_{n+1}}{1-\sigma_{n+1}}-\frac{\rho _{n}}{1-\sigma_{n}} \biggr\vert L_{1} + \bigl\Vert J_{\nu_{n+1},\varPsi }^{H_{2}}H_{2}(y_{n+1}) -J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \bigr\Vert , \end{aligned}$$
(3.5)
where \(L_{1}=\sup_{n} \{\|l\| + \|J_{\nu_{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1}) \| \}\). Now, two cases arise.
Case I. When \(\nu_{n+1}\geq\nu_{n}\), by utilizing Lemmas 3.1 and 2.2, we acquire
$$\begin{aligned} & \bigl\Vert J_{\nu_{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1}) -J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \bigr\Vert \\ &\quad = \biggl\Vert J_{\nu_{n},\varPsi}^{H_{2}} \biggl( \frac{\nu_{n}}{\nu _{n+1}}H_{2}(y_{n+1}) + \biggl(1- \frac{\nu_{n}}{\nu_{n+1}} \biggr) H_{2}J_{\nu _{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1}) \biggr) -J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \biggr\Vert \\ &\quad \leq\frac{1}{\zeta_{H_{2}}} \biggl\{ \bigl\Vert H_{2}(y_{n+1})-H_{2}(y_{n}) \bigr\Vert + \biggl\vert \frac{\nu_{n+1}-\nu_{n}}{\nu_{n+1}} \biggr\vert \bigl\Vert H_{2}J_{\nu_{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1})-H_{2}(y_{n+1}) \bigr\Vert \biggr\} \\ &\quad \leq \Vert y_{n+1}-y_{n} \Vert + \vert \nu_{n+1}-\nu_{n} \vert L_{2}, \end{aligned}$$
(3.6)
where \(L_{2}=\frac{1}{\epsilon} [\sup_{n} \{\|y_{n}\| + \|J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \| \} ]\).
Case II. When \(\nu_{n+1}<\nu_{n}\), by utilizing Lemmas 3.1 and 2.2, we acquire
$$\begin{aligned} & \bigl\Vert J_{\nu_{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1}) -J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \bigr\Vert \\ &\quad = \biggl\Vert J_{\nu_{n+1},\varPsi}^{H_{2}}H_{2}(y_{n+1}) -J_{\nu_{n+1},\varPsi}^{H_{2}} \biggl(\frac{\nu_{n+1}}{\nu _{n}}H_{2}(y_{n}) + \biggl(1-\frac{\nu_{n+1}}{\nu_{n}} \biggr) H_{2}J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \biggr) \biggr\Vert \\ &\quad \leq\frac{1}{\zeta_{H_{2}}} \biggl\{ \bigl\Vert H_{2}(y_{n+1})-H_{2}(y_{n}) \bigr\Vert + \biggl\vert \frac{\nu_{n}-\nu_{n+1}}{\nu_{n}} \biggr\vert \bigl\Vert H_{2}(y_{n})-H_{2}J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \bigr\Vert \biggr\} \\ &\quad \leq \Vert y_{n+1}-y_{n} \Vert + \vert \nu_{n+1}-\nu_{n} \vert L_{2}. \end{aligned}$$
(3.7)
From (3.5), (3.6), and (3.7), we obtain
$$\begin{aligned} & \Vert w_{n+1}-w_{n} \Vert \\ &\quad\leq \biggl\vert \frac{\rho_{n+1}}{1-\sigma_{n+1}}-\frac{\rho _{n}}{1-\sigma_{n}} \biggr\vert {L}_{1} + \Vert y_{n+1}-y_{n} \Vert + \vert \nu_{n+1}-\nu_{n} \vert L_{2}. \end{aligned}$$
(3.8)
By utilizing (3.1) and Lemma 3.1, we acquire
$$\begin{aligned} & \Vert y_{n+1}-y_{n} \Vert \\ &\quad = \bigl\Vert \kappa_{n+1}x_{n+1} +(1- \kappa_{n+1})J_{\mu_{n+1},\varPhi}^{H_{1}}H_{1}(x_{n+1})- \kappa_{n}x_{n} -(1-\kappa_{n})J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n}) \bigr\Vert \\ &\quad \leq\kappa_{n+1} \Vert x_{n+1}-x_{n} \Vert + \vert \kappa_{n+1}-\kappa_{n} \vert \bigl( \Vert x_{n} \Vert + \bigl\Vert J_{\mu_{n}, \varPhi}^{H_{1}}H_{1}(x_{n}) \bigr\Vert \bigr) \\ &\qquad{}+(1-\kappa_{n+1}) \bigl\Vert J_{\mu_{n+1},\varPhi}^{H_{1}}H_{1}(x_{n+1})-J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}) \bigr\Vert \\ &\quad =\kappa_{n+1} \Vert x_{n+1}-x_{n} \Vert + \vert \kappa_{n+1}-\kappa_{n} \vert \bigl( \Vert x_{n} \Vert + \bigl\Vert J_{\mu_{n}, \varPhi}^{H_{1}}H_{1}(x_{n}) \bigr\Vert \bigr)+(1-\kappa_{n+1}) \\ &\qquad{}\cdot \biggl\Vert J_{\mu_{n},\varPhi}^{H_{1}} \biggl( \frac{\mu_{n}}{\mu _{n+1}}H_{1}(x_{n+1}) + \biggl(1- \frac{\mu_{n}}{\mu_{n+1}} \biggr)H_{1}J_{\mu_{n+1},\varPhi }^{H_{1}}H_{1}(x_{n+1}) \biggr) -J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n}) \biggr\Vert \\ &\quad \leq\kappa_{n+1} \Vert x_{n+1}-x_{n} \Vert + \vert \kappa_{n+1}-\kappa_{n} \vert \bigl( \Vert x_{n} \Vert + \bigl\Vert J_{\mu_{n}, \varPhi}^{H_{1}}H_{1}(x_{n}) \bigr\Vert \bigr) \\ &\qquad{}+\frac{(1-\kappa_{n+1})}{\xi_{H_{1}}} \biggl\{ \bigl\Vert H_{1}(x_{n+1})-H_{1}(x_{n}) \bigr\Vert \\ &\qquad{}+ \biggl\vert \frac{\mu_{n+1}-\mu_{n}}{\mu_{n+1}} \biggr\vert \bigl\Vert H_{1}J_{\mu_{n+1},\varPhi}^{H_{1}}H_{1}(x_{n+1})-H_{1}(x_{n+1}) \bigr\Vert \biggr\} \\ &\quad \leq \Vert x_{n+1}-x_{n} \Vert + \vert \kappa_{n+1}-\kappa_{n} \vert L_{3} + \vert \mu_{n+1}-\mu_{n} \vert L_{4}, \end{aligned}$$
(3.9)
where \(L_{3}=\sup_{n} \{\|x_{n}\|+ \|J_{\mu_{n}, \varPhi }^{H_{1}}H_{1}(x_{n}) \| \}\) and \(L_{4}=\frac {1}{\epsilon} [\sup_{n} \{ \|J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n}) \|+\|x_{n}\| \} ]\).
Combining (3.8) and (3.9), we acquire
$$\begin{aligned} & \Vert w_{n+1}-w_{n} \Vert - \Vert x_{n+1}-x_{n} \Vert \\ &\quad \leq \biggl\vert \frac{\rho_{n+1}}{1-\sigma_{n+1}}-\frac{\rho _{n}}{1-\sigma_{n}} \biggr\vert L_{1} + \vert \nu_{n+1}-\nu_{n} \vert L_{2} + \vert \kappa_{n+1}-\kappa_{n} \vert L_{3} + \vert \mu_{n+1}-\mu_{n} \vert L_{4}. \end{aligned}$$
(3.10)
From (\(\mathrm{C}_{1}\)), (\(\mathrm{C}_{2}\)), (\(\mathrm{C}_{4}\)), and (3.10), we acquire
$$\begin{aligned} \limsup_{n\rightarrow\infty}\bigl( \Vert w_{n+1}-w_{n} \Vert - \Vert x_{n+1}-x_{n} \Vert \bigr)\leq0. \end{aligned}$$
It follows from Lemma 2.7 that \(\lim_{n\rightarrow\infty}\| w_{n}-x_{n}\|=0\), and hence
$$\begin{aligned} \lim_{n\rightarrow\infty} \Vert x_{n+1}-x_{n} \Vert =\lim_{n\rightarrow \infty}(1-\sigma_{n}) \Vert w_{n}-x_{n} \Vert =0. \end{aligned}$$
(3.11)
From (\(\mathrm{C}_{1}\)), (\(\mathrm{C}_{4}\)), (3.9), and (3.11), we acquire
$$\begin{aligned} \lim_{n\rightarrow\infty} \Vert y_{n+1}-y_{n} \Vert =0. \end{aligned}$$
Step 3. Our claim is \(\lim_{n\rightarrow\infty}\| x_{n+1}-y_{n}\|=0\). From (3.1), (3.4), and Lemma 2.10, we have
$$\begin{aligned} & \Vert y_{n}-w \Vert ^{2} \\ &\quad =\bigl\langle \kappa_{n}(x_{n}-w)+(1- \kappa_{n}) \bigl(J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n}) -w \bigr),\mathcal{J}(y_{n}-w)\bigr\rangle ,\quad w\in\varOmega \\ &\quad \leq\kappa_{n}\bigl\langle x_{n}-w, \mathcal{J}(y_{n}-w)\bigr\rangle +(1-\kappa_{n}) \bigl\Vert J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n})-w \bigr\Vert \Vert y_{n}-w \Vert \\ &\quad \leq\frac{\kappa_{n}}{2} \bigl[ \Vert x_{n}-w \Vert ^{2}-f\bigl( \bigl\Vert (x_{n}-w)-(y_{n}-w) \bigr\Vert \bigr) + \Vert y_{n}-w \Vert ^{2} \bigr] \\ &\qquad{}+(1-\kappa_{n}) \Vert x_{n}-w \Vert \Vert y_{n}-w \Vert \\ &\quad \leq\frac{\kappa_{n}}{2} \bigl[ \Vert x_{n}-w \Vert ^{2}-f\bigl( \Vert x_{n}-y_{n} \Vert \bigr) + \Vert y_{n}-w \Vert ^{2} \bigr]+(1- \kappa_{n}) \Vert x_{n}-w \Vert ^{2} \\ &\quad = \biggl(\frac{2-\kappa_{n}}{2} \biggr) \Vert x_{n}-w \Vert ^{2}+\frac{\kappa_{n}}{2} \bigl[ \Vert y_{n}-w \Vert ^{2}-f\bigl( \Vert x_{n}-y_{n} \Vert \bigr) \bigr], \end{aligned}$$
which implies that
$$\begin{aligned} \Vert y_{n}-w \Vert ^{2}&\leq \Vert x_{n}-w \Vert ^{2}- \biggl(\frac{\kappa_{n}}{2-\kappa_{n}} \biggr) f\bigl( \Vert x_{n}-y_{n} \Vert \bigr). \end{aligned}$$
(3.12)
Then, by using (3.1) and (3.12), we acquire
$$\begin{aligned} & \Vert x_{n+1}-w \Vert ^{2} \\ &\quad \leq\rho_{n} \Vert l-w \Vert ^{2}+ \sigma_{n} \Vert x_{n}-w \Vert ^{2} + \tau_{n} \bigl\Vert J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n})-w \bigr\Vert ^{2} \\ &\quad \leq\rho_{n} \Vert l-w \Vert ^{2}+ \sigma_{n} \Vert x_{n}-w \Vert ^{2} + \tau_{n} \Vert y_{n}-w \Vert ^{2} \\ &\quad \leq\rho_{n} \Vert l-w \Vert ^{2}+ \sigma_{n} \Vert x_{n}-w \Vert ^{2} + \tau_{n} \biggl[ \Vert x_{n}-w \Vert ^{2}- \biggl(\frac{\kappa_{n}}{2-\kappa_{n}} \biggr)f\bigl( \Vert x_{n}-y_{n} \Vert \bigr) \biggr] \\ &\quad \leq\rho_{n} \Vert l-w \Vert ^{2}+ \Vert x_{n}-w \Vert ^{2}-\tau_{n} \biggl( \frac {\kappa_{n}}{2-\kappa_{n}} \biggr)f\bigl( \Vert x_{n}-y_{n} \Vert \bigr), \end{aligned}$$
which implies that
$$\begin{aligned} \biggl(\frac{\kappa_{n}\tau_{n}}{2-\kappa_{n}} \biggr)f\bigl( \Vert x_{n}-y_{n} \Vert \bigr) &\leq\rho_{n} \Vert l-w \Vert ^{2}+ \Vert x_{n}-w \Vert ^{2}- \Vert x_{n+1}-w \Vert ^{2} \\ &\leq\rho_{n} \Vert l-w \Vert ^{2}+ \Vert x_{n+1}-x_{n} \Vert \bigl( \Vert x_{n}-w \Vert + \Vert x_{n+1}-w \Vert \bigr). \end{aligned}$$
(3.13)
By utilizing (3.11), conditions (\(\mathrm{C}_{1}\))–(\(\mathrm {C}_{2}\)), and the property of f in (3.13), we obtain
$$\begin{aligned} \lim_{n\rightarrow\infty} \Vert x_{n}-y_{n} \Vert =0. \end{aligned}$$
(3.14)
Since \(\|x_{n+1}-y_{n}\|\leq\|x_{n+1}-x_{n}\|+\|x_{n}-y_{n}\|\), so from (3.11) and (3.14), we acquire that \(\lim_{n\rightarrow\infty}\|x_{n+1}-y_{n}\|=0\).
Step 4. We claim that \(\lim_{n\rightarrow\infty}\| W(x_{n})-x_{n}\|=0\), where \(W=\frac{1}{2} (J_{\vartheta,\varPhi }^{H_{1}}H_{1}+J_{\vartheta,\varPsi}^{H_{2}}H_{2} )\) and \(0<\vartheta<\epsilon\).
$$\begin{aligned} \bigl\Vert W(x_{n})-x_{n} \bigr\Vert \leq\frac{1}{2} \bigl( \bigl\Vert x_{n}-J_{\vartheta ,\varPhi}^{H_{1}}H_{1}(x_{n}) \bigr\Vert + \bigl\Vert x_{n}-J_{\vartheta,\varPsi}^{H_{2}}H_{2}(x_{n}) \bigr\Vert \bigr). \end{aligned}$$
(3.15)
First, we compute \(\|x_{n}-J_{\vartheta,\varPhi }^{H_{1}}H_{1}(x_{n}) \|\). From Lemma 3.1 and (3.1), we have
$$\begin{aligned} & \bigl\Vert x_{n}-J_{\vartheta,\varPhi}^{H_{1}}H_{1}(x_{n}) \bigr\Vert \\ &\quad \leq \Vert x_{n}-y_{n} \Vert + \bigl\Vert y_{n}-J_{\vartheta,\varPhi }^{H_{1}}H_{1}(y_{n}) \bigr\Vert + \bigl\Vert J_{\vartheta,\varPhi}^{H_{1}}H_{1}(x_{n})-J_{\vartheta,\varPhi }^{H_{1}}H_{1}(y_{n}) \bigr\Vert \\ &\quad \leq2 \Vert x_{n}-y_{n} \Vert + \kappa_{n} \bigl\Vert x_{n}-J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}) \bigr\Vert + \bigl\Vert J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n})-J_{\vartheta,\varPhi }^{H_{1}}H_{1}(y_{n}) \bigr\Vert \\ &\quad = 2 \Vert x_{n}-y_{n} \Vert + \kappa_{n} \bigl\Vert x_{n}-J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}) \bigr\Vert \\ &\qquad{} + \biggl\Vert J_{\vartheta,\varPhi}^{H_{1}} \biggl( \frac{\vartheta}{\mu _{n}}H_{1}(x_{n}) + \biggl(1- \frac{\vartheta}{\mu_{n}} \biggr)H_{1}J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}) \biggr) -J_{\vartheta,\varPhi}^{H_{1}}H_{1}(y_{n}) \biggr\Vert \\ &\quad \leq2 \Vert x_{n}-y_{n} \Vert + \kappa_{n} \bigl\Vert x_{n}-J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}) \bigr\Vert \\ &\qquad{}+\frac{1}{\xi_{H_{1}}} \biggl\{ \bigl\Vert H_{1}(x_{n})-H_{1}(y_{n}) \bigr\Vert + \biggl\vert \frac{\mu_{n}-\vartheta}{\mu_{n}} \biggr\vert \bigl\Vert H_{1}J_{\mu _{n},\varPhi}^{H_{1}}H_{1}(x_{n}) -H_{1}(x_{n}) \bigr\Vert \biggr\} \\ &\quad \leq3 \Vert x_{n}-y_{n} \Vert + \biggl( \frac{\mu_{n}-\vartheta}{\mu _{n}}+\kappa_{n} \biggr) \bigl\Vert J_{\mu_{n},\varPhi}^{H_{1}}H_{1}(x_{n})-x_{n} \bigr\Vert \\ &\quad = \biggl\{ 3+\frac{1}{(1-\kappa_{n})} \biggl(\frac{\mu_{n}-\vartheta }{\mu_{n}}+ \kappa_{n} \biggr) \biggr\} \Vert x_{n}-y_{n} \Vert . \end{aligned}$$
Since \(\|x_{n}-y_{n}\|\rightarrow0\) as \(n\rightarrow\infty\) (from (3.14)), we acquire that \(\lim_{n\rightarrow\infty} \| x_{n}-J_{\vartheta,\varPhi}^{H_{1}}H_{1}(x_{n}) \|=0\).
Next, we compute \(\|x_{n}-J_{\vartheta,\varPsi }^{H_{2}}H_{2}(x_{n}) \|\).
$$\begin{aligned} & \bigl\Vert x_{n}-J_{\vartheta,\varPsi}^{H_{2}}H_{2}(x_{n}) \bigr\Vert \\ &\quad \leq \bigl\Vert x_{n}-J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(x_{n}) \bigr\Vert + \bigl\Vert J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(x_{n})-J_{\vartheta,\varPsi }^{H_{2}}H_{2}(x_{n+1}) \bigr\Vert \\ &\qquad{}+ \bigl\Vert J_{\vartheta,\varPsi}^{H_{2}}H_{2}(x_{n+1})-J_{\vartheta ,\varPsi}^{H_{2}}H_{2}(x_{n}) \bigr\Vert \\ &\quad \leq \Vert x_{n+1}-x_{n} \Vert + \bigl\Vert x_{n}-J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(x_{n}) \bigr\Vert \\ &\qquad{}+ \biggl\Vert J_{\vartheta,\varPsi}^{H_{2}} \biggl( \frac{\vartheta}{\nu _{n}}H_{2}(x_{n}) + \biggl(1- \frac{\vartheta}{\nu_{n}} \biggr)H_{2}J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(x_{n}) \biggr) -J_{\vartheta,\varPsi}^{H_{2}}H_{2}(x_{n+1}) \biggr\Vert \\ &\quad \leq2 \Vert x_{n+1}-x_{n} \Vert + \biggl(2- \frac{\vartheta}{\nu_{n}} \biggr) \bigl\Vert x_{n}-J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(x_{n}) \bigr\Vert \\ &\quad \leq2 \Vert x_{n+1}-x_{n} \Vert \\ &\qquad{}+ \biggl(2-\frac{\vartheta}{\nu_{n}} \biggr) \bigl( \bigl\Vert x_{n}-J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) \bigr\Vert + \bigl\Vert J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(x_{n})-J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(y_{n}) \bigr\Vert \bigr) \\ &\quad \leq2 \Vert x_{n+1}-x_{n} \Vert + \biggl(2- \frac{\vartheta}{\nu_{n}} \biggr) \biggl\{ \biggl\Vert x_{n}- \biggl( \frac{x_{n+1}-\rho_{n}l-\sigma _{n}x_{n}}{\tau_{n}} \biggr) \biggr\Vert + \Vert x_{n}-y_{n} \Vert \biggr\} \\ &\quad \leq2 \Vert x_{n+1}-x_{n} \Vert + \biggl(2- \frac{\vartheta}{\nu_{n}} \biggr) \biggl\{ \frac{1}{\tau_{n}} \bigl( \Vert x_{n+1}-x_{n} \Vert +\rho_{n} \Vert x_{n}-l \Vert \bigr)+ \Vert x_{n}-y_{n} \Vert \biggr\} . \end{aligned}$$
Since \(\|x_{n+1}-x_{n}\|\rightarrow0\) (from (3.11)), \(\| x_{n}-y_{n}\|\rightarrow0\) (from (3.14)), and \(\rho _{n}\rightarrow0\) (from (\(\mathrm{C}_{2}\))) as \(n\rightarrow\infty \), we acquire that \(\lim_{n\rightarrow\infty} \| x_{n}-J_{\vartheta,\varPsi}^{H_{2}}H_{2}(x_{n}) \|=0\). Hence it follows from (3.15) that \(\lim_{n\rightarrow\infty}\| W(x_{n})-x_{n}\|=0\).
Step 5. Our claim is \(\limsup_{n\rightarrow\infty} \langle(I-Q_{\varOmega})l,\mathcal{J}(x_{n}-Q_{\varOmega}l) \rangle \leq0\). Define a sequence \(\{x_{t}\}\) by \(x_{t}=tl+(1-t)W(x_{t})\), \(t\in (0,1)\). Then Theorem 2.4 ensures the strong convergence of \(\{ x_{t}\}\) to \(Q_{\varOmega}l\in\mathcal{F}(W)=\varOmega\). Now
$$\begin{aligned} & \Vert x_{t}-x_{n} \Vert ^{2} \\ &\quad = \bigl\Vert tl+(1-t)W(x_{t})-x_{n} \bigr\Vert ^{2} \\ &\quad = \bigl\Vert t(l-x_{n})+(1-t) \bigl(W(x_{t})-x_{n} \bigr) \bigr\Vert ^{2} \\ &\quad = \bigl\langle t(l-x_{n})+(1-t) \bigl(W(x_{t})-x_{n} \bigr),\mathcal {J}(x_{t}-x_{n}) \bigr\rangle \\ &\quad =t\bigl\langle l-x_{t},\mathcal{J}(x_{t}-x_{n}) \bigr\rangle +t\bigl\langle x_{t}-x_{n}, \mathcal{J}(x_{t}-x_{n})\bigr\rangle \\ &\qquad{}+(1-t) \bigl\{ \bigl\langle W(x_{t})-W(x_{n}), \mathcal{J}(x_{t}-x_{n})\bigr\rangle +\bigl\langle W(x_{n})-x_{n}, \mathcal{J}(x_{t}-x_{n}) \bigr\rangle \bigr\} \\ &\quad \leq t\bigl\langle l-x_{t},\mathcal{J}(x_{t}-x_{n}) \bigr\rangle +t \Vert x_{t}-x_{n} \Vert ^{2} \\ &\qquad{}+(1-t) \bigl\{ \bigl\Vert W(x_{t})-W(x_{n}) \bigr\Vert \Vert x_{t}-x_{n} \Vert + \bigl\Vert W(x_{n})-x_{n} \bigr\Vert \Vert x_{t}-x_{n} \Vert \bigr\} , \end{aligned}$$
which implies that
$$\begin{aligned} t&\bigl\langle l-x_{t},\mathcal{J}(x_{n}-x_{t}) \bigr\rangle \\ &\leq(1-t) \bigl\{ \bigl\Vert W(x_{t})-W(x_{n}) \bigr\Vert \Vert x_{t}-x_{n} \Vert - \Vert x_{t}-x_{n} \Vert ^{2} + \bigl\Vert W(x_{n})-x_{n} \bigr\Vert \Vert x_{t}-x_{n} \Vert \bigr\} \\ &\leq(1-t) \bigl\{ \Vert x_{t}-x_{n} \Vert ^{2}- \Vert x_{t}-x_{n} \Vert ^{2} + \bigl\Vert W(x_{n})-x_{n} \bigr\Vert \Vert x_{t}-x_{n} \Vert \bigr\} \\ &\leq \bigl\Vert W(x_{n})-x_{n} \bigr\Vert \Vert x_{t}-x_{n} \Vert . \end{aligned}$$
Let \(L_{5}=\sup\{\|x_{t}-x_{n}\|:t\in(0,1),n\in\mathbb{N}\}\). Then
$$\begin{aligned} \bigl\langle l-x_{t},\mathcal{J}(x_{n}-x_{t}) \bigr\rangle \leq\frac{L_{5}}{t} \bigl\Vert W(x_{n})-x_{n} \bigr\Vert . \end{aligned}$$
Since \(\|W(x_{n})-x_{n}\|\rightarrow0\) as \(n\rightarrow\infty\), we acquire
$$\begin{aligned} \limsup_{n\rightarrow\infty}\bigl\langle l-x_{t},\mathcal {J}(x_{n}-x_{t})\bigr\rangle \leq0. \end{aligned}$$
(3.16)
By utilizing the fact that \(x_{t}\rightarrow Q_{\varOmega}l\) as \(t\rightarrow0^{+}\) and \(\mathcal{J}\) is norm-to-weak∗ uniformly continuous on bounded subsets of \(\mathcal{B}\), we acquire
$$\begin{aligned} &\bigl\vert \bigl\langle (I-Q_{\varOmega})l,\mathcal{J}(x_{n}-Q_{\varOmega }l) \bigr\rangle -\bigl\langle l-x_{t},\mathcal{J}(x_{n}-x_{t}) \bigr\rangle \bigr\vert \\ &\quad = \bigl\vert \bigl\langle l-Q_{\varOmega}l,\mathcal{J}(x_{n}-Q_{\varOmega }l)- \mathcal{J}(x_{n}-x_{t}) \bigr\rangle + \bigl\langle x_{t}-Q_{\varOmega}l,\mathcal{J}(x_{n}-x_{t}) \bigr\rangle \bigr\vert \\ &\quad \leq \bigl\vert \bigl\langle l-Q_{\varOmega}l,\mathcal{J}(x_{n}-Q_{\varOmega }l)- \mathcal{J}(x_{n}-x_{t}) \bigr\rangle \bigr\vert +L_{5} \Vert x_{t}-Q_{\varOmega}l \Vert \\ &\quad \longrightarrow0 \quad \mbox{as } h\rightarrow0^{+}. \end{aligned}$$
For \(\varepsilon>0\), there is \(0<\delta<1\) such that
$$\begin{aligned} \bigl\langle (I-Q_{\varOmega})l,\mathcal{J}(x_{n}-Q_{\varOmega}l) \bigr\rangle < \bigl\langle l-x_{t},\mathcal{J}(x_{n}-x_{t}) \bigr\rangle +\varepsilon ,\quad \forall0< t< \delta. \end{aligned}$$
From (3.16), we acquire
$$\begin{aligned} \limsup_{n\rightarrow\infty} \bigl\langle (I-Q_{\varOmega})l,\mathcal {J}(x_{n}-Q_{\varOmega}l) \bigr\rangle \leq\limsup _{n\rightarrow\infty}\bigl\langle l-x_{t},\mathcal {J}(x_{n}-x_{t})\bigr\rangle +\varepsilon\leq \varepsilon. \end{aligned}$$
Since ε is arbitrary, we acquire that \(\limsup_{n\rightarrow\infty} \langle(I-Q_{\varOmega})l,\mathcal {J}(x_{n}-Q_{\varOmega}l) \rangle \leq0\).
Step 6. Finally, our claim is \(x_{n}\rightarrow Q_{\varOmega}l\) as \(n\rightarrow\infty\). From (3.1), (3.4), Lemmas 2.6 and 2.9, it follows that
$$\begin{aligned} & \Vert x_{n+1}-Q_{\varOmega}l \Vert ^{2} \\ &\quad = \bigl\Vert \rho_{n}l+\sigma_{n}x_{n} +\tau_{n}J_{\nu_{n},\varPhi}^{H_{2}}H_{2}(y_{n})-Q_{\varOmega}l \bigr\Vert ^{2} \\ &\quad \leq \bigl\Vert \sigma_{n}(x_{n}-Q_{\varOmega}l)+ \tau_{n} \bigl(J_{\nu _{n},\varPsi}^{H_{2}}H_{2}(y_{n}) -Q_{\varOmega}l \bigr) \bigr\Vert ^{2} +2\rho_{n} \bigl\langle l-Q_{\varOmega}l,\mathcal{J}(x_{n+1}-Q_{\varOmega }l) \bigr\rangle \\ &\quad \leq(1-\sigma_{n}) \biggl\Vert \frac{\tau_{n}}{(1-\sigma_{n})} \bigl(J_{\nu_{n},\varPsi}^{H_{2}}H_{2}(y_{n}) -Q_{\varOmega}l \bigr) \biggr\Vert ^{2}+\sigma_{n} \Vert x_{n}-Q_{\varOmega}l \Vert ^{2} \\ &\qquad{}+2\rho_{n}\bigl\langle l-Q_{\varOmega}l, \mathcal{J}(x_{n+1}-Q_{\varOmega }l)\bigr\rangle \\ &\quad \leq\frac{\tau_{n}^{2}}{(1-\sigma_{n})} \Vert y_{n}-Q_{\varOmega}l \Vert ^{2}+\sigma_{n} \Vert x_{n} -Q_{\varOmega}l \Vert ^{2} +2\rho_{n}\bigl\langle l-Q_{\varOmega}l,\mathcal{J}(x_{n+1}-Q_{\varOmega }l)\bigr\rangle \\ &\quad \leq \biggl\{ \frac{\tau_{n}^{2}}{(1-\sigma_{n})}+\sigma_{n} \biggr\} \Vert x_{n}-Q_{\varOmega}l \Vert ^{2} +2 \rho_{n}\bigl\langle l-Q_{\varOmega}l,\mathcal{J}(x_{n+1}-Q_{\varOmega }l) \bigr\rangle \\ &\quad = \biggl\{ (1-\rho_{n})+\frac{\rho_{n}^{2}}{(1-\sigma_{n})}-\rho _{n} \biggr\} \Vert x_{n}-Q_{\varOmega}l \Vert ^{2} +2\rho_{n}\bigl\langle l-Q_{\varOmega }l, \mathcal{J}(x_{n+1}-Q_{\varOmega}l)\bigr\rangle \\ &\quad =(1-\rho_{n}) \Vert x_{n}-Q_{\varOmega}l \Vert ^{2} \\ &\qquad{}+\rho_{n} \biggl\{ \biggl(\frac{\rho_{n}}{1-\sigma_{n}}-1 \biggr) \Vert x_{n}-Q_{\varOmega}l \Vert ^{2} +2\bigl\langle l-Q_{\varOmega}l,\mathcal{J}(x_{n+1}-Q_{\varOmega}l) \bigr\rangle \biggr\} \\ &\quad =(1-\rho_{n}) \Vert x_{n}-Q_{\varOmega}l \Vert ^{2}+\eta_{n}. \end{aligned}$$
Evidently, \(\sum_{n=1}^{\infty}\rho_{n}=\infty\), \(\{\rho_{n}\} \subset(0,1)\) and \(\limsup_{n\rightarrow\infty}\frac{\eta _{n}}{\rho_{n}}\leq0\). Hence, by Lemma 2.8, we acquire that \(x_{n}\rightarrow Q_{\varOmega}l\) as \(n\rightarrow\infty\). Thus, the proof is completed. □
In the next theorem, we prove the strong convergence of the sequence generated by the following Mann type viscosity approximation method:
$$\begin{aligned} \textstyle\begin{cases} u_{1}\in\mathcal{B}, \\ v_{n}=\kappa_{n}u_{n}+(1-\kappa_{n})J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(u_{n}), \\ u_{n+1}=\rho_{n}g(v_{n})+\sigma_{n}u_{n}+\tau_{n}J_{\nu_{n},\varPsi }^{H_{2}}H_{2}(v_{n}), \quad n\in\mathbb{N}. \end{cases}\displaystyle \end{aligned}$$
(3.17)
Theorem 3.3
Let
\(\mathcal{B}\)
be a uniformly convex and uniformly smooth Banach space. Let
\(H_{1},H_{2}:\mathcal{B}\rightarrow\mathcal{B}\)
be single-valued mappings such that
\(H_{1}\)
is strongly accretive and Lipschitz continuous with constant
\(\xi_{H_{1}}\)
and
\(H_{2}\)
is strongly accretive and Lipschitz continuous with constant
\(\zeta _{H_{2}}\). Let
\(\varPhi, \varPsi:\mathcal{B}\rightarrow2^{\mathcal{B}}\)
be
\(H_{1}\)-accretive and
\(H_{2}\)-accretive mappings, respectively; let
\(g:\mathcal{B}\rightarrow\mathcal{B}\)
be a
ϑ-contraction mapping, where
\(0< \vartheta<1\). Assume that
\(\varOmega:=\varPhi ^{-1}(0)\cap\varPsi^{-1}(0)\neq\emptyset\). Let the sequences
\(\{v_{n}\} \)
and
\(\{u_{n}\}\)
be generated by (3.17). Assume that conditions (\(\mathrm{C}_{1}\))–(\(\mathrm {C}_{4}\)) of Theorem 3.2
are fulfilled. Then the sequence
\(\{ u_{n}\}\)
converges strongly to
\(\tilde{u}=Q_{\varOmega}g(\tilde{u})\in \varOmega\).
Proof
Assume that ũ is a unique fixed point of \(Q_{\varOmega}g\). Then \(Q_{\varOmega}g(\tilde{u})=\tilde{u}\). If we put \(g(\tilde{u})\) in place of l in (3.1), then Theorem 3.2 ensures the strong convergence of \(\{x_{n}\}\) to \(Q_{\varOmega}g(\tilde{u})=\tilde {u}\), i.e., \(\lim_{n\rightarrow\infty}\|x_{n}-\tilde{u}\|=0\).
First, we demonstrate that \(\lim_{n\rightarrow\infty}\|u_{n}-x_{n}\| =0\). We are assuming on the contrary that
$$\begin{aligned} \limsup_{n\rightarrow\infty} \Vert u_{n}-x_{n} \Vert >0. \end{aligned}$$
Then we can pick ε such that \(0<\varepsilon<\limsup_{n\rightarrow\infty}\|u_{n}-x_{n}\|\). As \(\{x_{n}\}\) is strongly convergent to ũ, so there exists \(m^{\prime}\in\mathbb {N}\) such that
$$\begin{aligned} \Vert x_{n}-\tilde{u} \Vert < \biggl(\frac{1-\vartheta}{\vartheta} \biggr)\varepsilon,\quad \forall n\geq m^{\prime}. \end{aligned}$$
Now two possibilities arise.
- (P1)
There exists \(m\in\mathbb{N}\) with \(m\geq m^{\prime}\) and \(\|u_{m}-x_{m}\|\leq\varepsilon\);
- (P2)
\(\|u_{n}-x_{n}\|>\varepsilon\), \(\forall n\geq m^{\prime}\).
In possibility (P1),
$$\begin{aligned} & \Vert u_{m+1}-x_{m+1} \Vert \\ &\quad = \bigl\Vert \rho_{m}g(v_{m})+\sigma_{m}u_{m} +\tau_{m}J_{\nu_{m},\varPsi}^{H_{2}}H_{2}(v_{m}) -\rho_{m}g(\tilde{u})-\sigma_{m}x_{m} - \tau_{m}J_{\nu_{m},\varPsi}^{H_{2}}H_{2}(y_{m}) \bigr\Vert \\ &\quad \leq\rho_{m}\vartheta \Vert v_{m}-\tilde{u} \Vert +\sigma_{m} \Vert u_{m}-x_{m} \Vert + \tau_{m} \Vert v_{m}-y_{m} \Vert \\ &\quad =\rho_{m}\vartheta \bigl\Vert \kappa_{m}u_{m}+(1- \kappa_{m})J_{\mu_{m},\varPhi }^{H_{1}}H_{1}(u_{m})- \tilde{u} \bigr\Vert +\sigma_{m} \Vert u_{m}-x_{m} \Vert \\ &\qquad{}+\tau_{m} \bigl\Vert \kappa_{m}u_{m}+(1- \kappa_{m})J_{\mu_{m},\varPhi }^{H_{1}}H_{1}(u_{m}) -\kappa_{m}x_{m}-(1-\kappa_{m})J_{\mu_{m},\varPhi }^{H_{1}}H_{1}(x_{m}) \bigr\Vert \\ &\quad \leq\rho_{m}\vartheta \Vert u_{m}-\tilde{u} \Vert +(\sigma_{m}+\tau_{m}) \Vert u_{m}-x_{m} \Vert \\ &\quad \leq\rho_{m}\vartheta \Vert x_{m}-\tilde{u} \Vert + \bigl(1-\rho _{m}(1-\vartheta) \bigr) \Vert u_{m}-x_{m} \Vert \\ &\quad \leq\rho_{m}(1-\vartheta)\varepsilon+ \bigl(1- \rho_{m}(1-\vartheta ) \bigr)\varepsilon=\varepsilon. \end{aligned}$$
By induction,
$$\begin{aligned} \Vert u_{n+1}-x_{n+1} \Vert \leq\varepsilon,\quad \forall n\geq m, \end{aligned}$$
which is a contradiction to \(\varepsilon< \limsup_{n\rightarrow \infty}\|u_{n}-x_{n}\|\).
In possibility (P2), for all \(n\geq m^{\prime}\), we acquire
$$\begin{aligned} &\Vert u_{n+1}-x_{n+1} \Vert \\ &\quad \leq\rho_{n}\vartheta \Vert v_{n}-\tilde{u} \Vert +\sigma_{n} \Vert u_{n}-x_{n} \Vert + \tau_{n} \Vert v_{n}-y_{n} \Vert \\ &\quad =\rho_{n}\vartheta \bigl\Vert \kappa_{n}u_{n}+(1- \kappa_{n})J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(u_{n})- \tilde{u} \bigr\Vert +\sigma_{n} \Vert u_{n}-x_{n} \Vert \\ &\qquad{}+\tau_{n} \bigl\Vert \kappa_{n}u_{n}+(1- \kappa_{n})J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(u_{n}) -\kappa_{n}x_{n}-(1-\kappa_{n})J_{\mu_{n},\varPhi }^{H_{1}}H_{1}(x_{n}) \bigr\Vert \\ &\quad \leq\rho_{n}\vartheta \Vert u_{n}-\tilde{u} \Vert +(\sigma_{n}+\tau_{n}) \Vert u_{n}-x_{n} \Vert \\ &\quad \leq \bigl(1-\rho_{n}(1-\vartheta) \bigr) \Vert u_{n}-x_{n} \Vert +\rho _{n}\vartheta \Vert x_{n}-\tilde{u} \Vert . \end{aligned}$$
By Lemma 2.8, we acquire that \(\lim_{n\rightarrow\infty}\| u_{n}-x_{n}\|=0\), which is a contradiction. Therefore \(\lim_{n\rightarrow\infty}\|u_{n}-x_{n}\|=0\), and hence
$$\begin{aligned} \lim_{n\rightarrow\infty} \Vert u_{n}-\tilde{u} \Vert \leq \lim_{n\rightarrow\infty} \Vert u_{n}-x_{n} \Vert +\lim_{n\rightarrow\infty} \Vert x_{n}-\tilde{u} \Vert =0. \end{aligned}$$
Thus, the proof is completed. □