To prove (I), let
\((x,y)\in\Omega\). From Lemma
2.2, we have
$$\begin{aligned}& \Vert u_{n}-x\Vert =\bigl\Vert J_{r_{n}}^{F,T}(x_{n})-J_{r_{n}}^{F,T}(x) \bigr\Vert \leq \Vert x_{n}-x\Vert , \end{aligned}$$
(3.2)
$$\begin{aligned}& \Vert v_{n}-y\Vert =\bigl\Vert J_{r_{n}}^{G,S}(y_{n})-J_{r_{n}}^{G,S}(y) \bigr\Vert \leq \Vert y_{n}-y\Vert . \end{aligned}$$
(3.3)
Since
P is a demi-contractive mapping, using the well-known identity (
2.1) and Lemma
2.4, we obtain the following estimates:
$$\begin{aligned} &\Vert x_{n+1}-x\Vert ^{2} \\ &\quad =\bigl\Vert (1-\alpha_{n}) \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)+ \alpha_{n}P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-x\bigr\Vert ^{2} \\ &\quad =\bigl\Vert (1-\alpha_{n})\bigl\{ \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\} + \alpha_{n}\bigl\{ P\bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-x\bigr\} \bigr\Vert ^{2} \\ &\quad =(1-\alpha_{n})\bigl\Vert \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\Vert ^{2}+\alpha _{n}\bigl\Vert P\bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-\alpha_{n})\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\quad \leq(1-\alpha_{n})\bigl\Vert \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\Vert ^{2}+\alpha _{n} \bigl\{ \bigl\Vert \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\Vert ^{2} \\ &\qquad {}+k_{1}\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma _{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \bigr\} \\ &\qquad {}-\alpha_{n}(1-\alpha_{n})\bigl\Vert \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\quad =\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-x\bigr\Vert ^{2} \\ &\qquad {}+k_{1}\alpha_{n}\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P\bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-\alpha_{n}) \bigl\Vert \bigl(u_{n}-\gamma _{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\quad =\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-x\bigr\Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-k_{1}-\alpha _{n})\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\quad =\Vert u_{n}-x\Vert ^{2}+\gamma_{n}^{2} \bigl\Vert A^{*}(Au_{n}-Bv_{n})\bigr\Vert ^{2}-2 \gamma_{n} \bigl\langle A^{*}(Au_{n}-Bv_{n}),u_{n}-x \bigr\rangle \\ &\qquad {}-\alpha_{n}(1-k_{1}-\alpha_{n})\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x\Vert ^{2}+\gamma_{n}^{2} \bigl\Vert A^{*}(Au_{n}-Bv_{n})\bigr\Vert ^{2}-2 \gamma_{n} \bigl\langle A^{*}(Au_{n}-Bv_{n}),u_{n}-x \bigr\rangle \\ &\qquad {}-\alpha_{n}(1-k_{1}-\alpha_{n})\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2}. \end{aligned}$$
(3.4)
From the definition of the spectral radius
\(\lambda_{A}\) of
\(A^{*}A\), we have
$$\begin{aligned} \gamma_{n}^{2}\bigl\Vert A^{*}(Au_{n}-Bv_{n})\bigr\Vert ^{2} =& \gamma_{n}^{2}\bigl\langle Au_{n}-Bv_{n},AA^{*}(Au_{n}-Bv_{n}) \bigr\rangle \\ \leq&\lambda_{A}\gamma_{n}^{2}\langle Au_{n}-Bv_{n},Au_{n}-Bv_{n}\rangle \\ =&\lambda_{A}\gamma_{n}^{2}\Vert Au_{n}-Bv_{n}\Vert ^{2}. \end{aligned}$$
(3.5)
Combining (
3.4) and (
3.5), we have
$$ \begin{aligned}[b] &\Vert x_{n+1}-x\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x\Vert ^{2}+\lambda_{A} \gamma_{n}^{2}\Vert Au_{n}-Bv_{n}\Vert ^{2}-2\gamma_{n} \langle Au_{n}-Bv_{n},Au_{n}-Ax \rangle \\ &\qquad {}-\alpha_{n}(1-k_{1}-\alpha_{n})\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2}. \end{aligned} $$
(3.6)
Similarly, the last equality of the iterative scheme (
3.1) leads to
$$\begin{aligned} &\Vert y_{n+1}-y\Vert ^{2} \\ &\quad \leq \Vert y_{n}-y\Vert ^{2}+\lambda_{B} \gamma_{n}^{2}\Vert Au_{n}-Bv_{n}\Vert ^{2}+2\gamma_{n} \langle Au_{n}-Bv_{n},Bv_{n}-By \rangle \\ &\qquad {}-\alpha_{n}(1-k_{2}-\alpha_{n})\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2}. \end{aligned}$$
(3.7)
Adding the inequalities (
3.6) and (
3.7), using
\(k=\max\{ k_{1},k_{2}\}\) and
\(Ax=By\), we get
$$\begin{aligned} &\Vert x_{n+1}-x\Vert ^{2}+\Vert y_{n+1}-y\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x\Vert ^{2}+\Vert y_{n}-y \Vert ^{2}+ \bigl(\lambda_{A}\gamma_{n}^{2}+ \lambda_{B}\gamma _{n}^{2} \bigr)\Vert Au_{n}-Bv_{n}\Vert ^{2}-2\gamma_{n} \Vert Au_{n}-Bv_{n}\Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-k-\alpha_{n}) \bigl\{ \bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\qquad {}+\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \bigr\} \\ &\quad =\Vert x_{n}-x\Vert ^{2}+\Vert y_{n}-y \Vert ^{2}-\gamma_{n} \bigl(2-\gamma_{n}( \lambda_{A}+\lambda _{B}) \bigr)\Vert Au_{n}-Bv_{n} \Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-k-\alpha_{n}) \bigl\{ \bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\qquad {}+\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.8)
Now, put
\(\Omega_{n}(x,y)=\Vert x_{n}-x\Vert ^{2}+\Vert y_{n}-y\Vert ^{2}\). Therefore from (
3.8), we have
$$\begin{aligned} \Omega_{n+1}(x,y)\leq{}&\Omega_{n}(x,y)- \gamma_{n} \bigl(2-\gamma_{n}(\lambda _{A}+ \lambda_{B}) \bigr)\Vert Au_{n}-Bv_{n}\Vert ^{2} \\ &{}-\alpha_{n}(1-k-\alpha_{n}) \bigl\{ \bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &{}+\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.9)
As
\(\alpha_{n}\in(k,1)\) and
\(\gamma_{n}\in (\epsilon,\frac{2}{\lambda _{A}+\lambda_{B}}-\epsilon )\), we have
\(2-\gamma_{n}(\lambda_{A}+\lambda _{B})>0\) and
\((1-k-\alpha_{n})>0\). It follows from (
3.9) that
$$ \Omega_{n+1}(x,y)\leq\Omega_{n}(x,y). $$
Hence, the sequence
\(\{\Omega_{n}(x,y)\}\) is non-increasing and lower bounded by 0. Therefore, it converges to some finite limit, say
\(\sigma (x,y)\). So, condition (i) of Lemma
2.3 is satisfied with
\(\mu _{n}=(x_{n},y_{n})\),
\(\mu^{*}=(x,y)\), and
\(W=\Omega\). It follows from inequality (
3.9) and the convergence of the sequence
\(\{\Omega _{n}(x,y)\}\) that
$$\begin{aligned}& \lim_{n\to\infty} \Vert Au_{n}-Bv_{n} \Vert =0, \end{aligned}$$
(3.10)
$$\begin{aligned}& \lim_{n\to\infty}\bigl\Vert \bigl(u_{n}- \gamma_{n}A^{*}(Au_{n}-Bv_{n})\bigr)-P \bigl(u_{n}-\gamma _{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert =0, \end{aligned}$$
(3.11)
and
$$ \lim_{n\to\infty}\bigl\Vert \bigl(v_{n}+ \gamma_{n}B^{*}(Au_{n}-Bv_{n})\bigr)-Q \bigl(v_{n}+\gamma _{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert =0. $$
(3.12)
Moreover, as
\(\{\Omega_{n}(x,y)\}\) converges to a finite limit and
\(\Vert x_{n}-x\Vert ^{2}\leq\Omega_{n}(x,y)\),
\(\Vert y_{n}-y\Vert ^{2}\leq\Omega_{n}(x,y)\), we see that
\(\{x_{n}\}\) and
\(\{y_{n}\}\) are bounded and
\(\limsup_{n\to\infty} \Vert x_{n}-x\Vert \) and
\(\limsup_{n\to\infty} \Vert y_{n}-y\Vert \) exist. From (
3.2) and (
3.3), we get
\(\limsup_{n\to\infty} \Vert u_{n}-x\Vert \) and
\(\limsup_{n\to\infty} \Vert v_{n}-y\Vert \) also exist. Let
\(x^{*}\) and
\(y^{*}\) be weak limit points of the sequences
\(\{x_{n}\}\) and
\(\{y_{n}\}\), respectively. Also,
\(\{ u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n})\}\) weakly converges to
\(x^{*}\) and
\(\{ v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n})\}\) weakly converges to
\(y^{*}\). Using Lemma
2.4, we have
$$\begin{aligned} \Vert x_{n+1}-x_{n}\Vert ^{2} =&\Vert x_{n+1}-x-x_{n}+x\Vert ^{2} \\ =&\Vert x_{n+1}-x\Vert ^{2}-\Vert x_{n}-x \Vert ^{2}-2\langle x_{n+1}-x_{n},x_{n}-x \rangle \\ =&\Vert x_{n+1}-x\Vert ^{2}-\Vert x_{n}-x \Vert ^{2}-2\bigl\langle x_{n+1}-x^{*},x_{n}-x\bigr\rangle \\ &{}+2\bigl\langle x_{n}-x^{*},x_{n}-x\bigr\rangle . \end{aligned}$$
Therefore,
$$ \limsup_{n\to\infty} \Vert x_{n+1}-x_{n}\Vert =0. $$
Similarly, we obtain
$$ \limsup_{n\to\infty} \Vert y_{n+1}-y_{n}\Vert =0. $$
We conclude that
$$ \lim_{n\to\infty} \Vert x_{n+1}-x_{n} \Vert =0 $$
(3.13)
and
$$ \lim_{n\to\infty} \Vert y_{n+1}-y_{n} \Vert =0. $$
(3.14)
From Lemma
2.2, we have
\(u_{n}=J^{F,T}_{r_{n}}(x_{n})\) and
\(u_{n+1}=J^{F,T}_{r_{n+1}}(x_{n+1})\). Therefore, for all
\(u\in C\), we have
$$ \begin{aligned}[b] &F\bigl(\lambda_{1}u_{n}+(1- \lambda_{1})b,u\bigr)+\bigl\langle T(u_{n}),u-u_{n} \bigr\rangle \\ &\quad{}+\phi (u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle\geq0 \end{aligned} $$
(3.15)
and
$$ \begin{aligned}[b] &F\bigl(\lambda_{1}u_{n+1}+(1- \lambda_{1})b,u\bigr)+\bigl\langle T(u_{n+1}),u-u_{n+1} \bigr\rangle \\ &\quad {}+\phi(u)-\phi(u_{n+1})+\frac{1}{r_{n+1}}\langle u-u_{n+1},u_{n+1}-x_{n+1}\rangle\geq0. \end{aligned} $$
(3.16)
Putting
\(u=u_{n}\) in (
3.16) and
\(u=u_{n+1}\) in (
3.15), and adding together the resulting inequalities, we have
$$\begin{aligned} 0 \leq&F\bigl(\lambda_{1}u_{n+1}+(1-\lambda_{1})b,u_{n} \bigr)+F\bigl(\lambda_{1}u_{n}+(1-\lambda _{1})b,u_{n+1} \bigr)+\bigl\langle T(u_{n+1}),u_{n}-u_{n+1}\bigr\rangle \\ &{}+\bigl\langle T(u_{n}),u_{n+1}-u_{n}\bigr\rangle + \frac{1}{r_{n+1}}\langle u_{n}-u_{n+1},u_{n+1}-x_{n+1} \rangle+\frac{1}{r_{n}}\langle u_{n+1}-u_{n},u_{n}-x_{n} \rangle. \end{aligned}$$
By using (A2)-(A3), we have
$$\begin{aligned} 0 \leq&\frac{1}{r_{n+1}}\langle u_{n}-u_{n+1},u_{n+1}-x_{n+1} \rangle+\frac {1}{r_{n}}\langle u_{n+1}-u_{n},u_{n}-x_{n} \rangle \\ \leq& \biggl\langle u_{n+1}-u_{n},\frac{u_{n}-x_{n}}{r_{n}}- \frac {u_{n+1}-x_{n+1}}{r_{n+1}} \biggr\rangle \\ =& \biggl\langle u_{n+1}-u_{n},u_{n}-x_{n}- \frac {r_{n}}{r_{n+1}}(u_{n+1}-x_{n+1}) \biggr\rangle \\ =& \biggl\langle u_{n+1}-u_{n},u_{n}-u_{n+1}+u_{n+1}-x_{n}- \frac {r_{n}}{r_{n+1}}(u_{n+1}-x_{n+1}) \biggr\rangle \\ =& \langle u_{n+1}-u_{n},u_{n}-u_{n+1} \rangle+ \biggl\langle u_{n+1}-u_{n},x_{n+1}-x_{n}+ \biggl(1-\frac{r_{n}}{r_{n+1}} \biggr) (u_{n+1}-x_{n+1}) \biggr\rangle \\ =&-\Vert u_{n+1}-u_{n}\Vert ^{2}+ \biggl\langle u_{n+1}-u_{n},x_{n+1}-x_{n}+ \biggl(1-\frac {r_{n}}{r_{n+1}} \biggr) (u_{n+1}-x_{n+1}) \biggr\rangle , \end{aligned}$$
which implies that
$$ \Vert u_{n+1}-u_{n}\Vert ^{2}\leq \Vert u_{n+1}-u_{n}\Vert \biggl\{ \Vert x_{n+1}-x_{n} \Vert +\biggl\vert 1-\frac {r_{n}}{r_{n+1}}\biggr\vert \Vert u_{n+1}-x_{n+1}\Vert \biggr\} . $$
Thus,
$$ \Vert u_{n+1}-u_{n}\Vert \leq \Vert x_{n+1}-x_{n}\Vert +\biggl\vert 1-\frac{r_{n}}{r_{n+1}}\biggr\vert \Vert u_{n+1}-x_{n+1}\Vert . $$
(3.17)
Using (
3.13) and condition (ii) of the hypothesis, (
3.17) implies that
$$ \lim_{n\to\infty} \Vert u_{n+1}-u_{n} \Vert =0. $$
(3.18)
Similarly, using the same arguments as above, we have
$$ \lim_{n\to\infty} \Vert v_{n+1}-v_{n} \Vert =0. $$
(3.19)
From (
3.6) and (
3.7), we have
$$\begin{aligned} &\Vert x_{n+1}-x\Vert ^{2} \\ &\quad \leq \Vert u_{n}-x\Vert ^{2}+\lambda_{A} \gamma_{n}^{2}\Vert Au_{n}-Bv_{n}\Vert ^{2}-2\gamma_{n} \langle Au_{n}-Bv_{n},Au_{n}-Ax \rangle \\ &\qquad {}-\alpha_{n}(1-k_{1}-\alpha_{n})\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \end{aligned}$$
(3.20)
and
$$\begin{aligned} &\Vert y_{n+1}-y\Vert ^{2} \\ &\quad \leq \Vert v_{n}-y\Vert ^{2}+\lambda_{B} \gamma_{n}^{2}\Vert Au_{n}-Bv_{n}\Vert ^{2}+2\gamma_{n} \langle Au_{n}-Bv_{n},Bv_{n}-By \rangle \\ &\qquad {}-\alpha_{n}(1-k_{2}-\alpha_{n})\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2}. \end{aligned}$$
(3.21)
Adding the inequalities (
3.20) and (
3.21), using
\(k=\max \{k_{1},k_{2}\}\) and
\(Ax=By\), we obtain
$$\begin{aligned} &\Vert x_{n+1}-x\Vert ^{2}+\Vert y_{n+1}-y\Vert ^{2} \\ &\quad \leq \Vert u_{n}-x\Vert ^{2}+\Vert v_{n}-y \Vert ^{2}-\gamma_{n} \bigl(2-\gamma_{n}( \lambda_{A}+\lambda _{B}) \bigr)\Vert Au_{n}-Bv_{n} \Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-k-\alpha_{n}) \bigl\{ \bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\qquad {}+\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \bigr\} , \end{aligned}$$
(3.22)
where
$$\begin{aligned} \Vert u_{n}-x\Vert ^{2}=\bigl\Vert J_{r_{n}}^{F,T}(x_{n})-J_{r_{n}}^{F,T}(x) \bigr\Vert ^{2} \leq &\langle x_{n}-x,u_{n}-x \rangle \\ =&\frac{1}{2} \bigl\{ \Vert x_{n}-x\Vert ^{2}+ \Vert u_{n}-x\Vert ^{2}-\Vert x_{n}-u_{n} \Vert ^{2} \bigr\} \end{aligned}$$
(3.23)
and
$$\begin{aligned} \Vert v_{n}-y\Vert ^{2}=\bigl\Vert J_{r_{n}}^{G,S}(y_{n})-J_{r_{n}}^{G,S}(y) \bigr\Vert ^{2} \leq &\langle y_{n}-y,v_{n}-y \rangle \\ =&\frac{1}{2} \bigl\{ \Vert y_{n}-y\Vert ^{2}+ \Vert v_{n}-y\Vert ^{2}-\Vert y_{n}-v_{n} \Vert ^{2} \bigr\} . \end{aligned}$$
(3.24)
From (
3.22)-(
3.24), we conclude that
$$\begin{aligned} &\Vert x_{n}-u_{n}\Vert ^{2}+ \Vert y_{n}-v_{n}\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x\Vert ^{2}-\Vert x_{n+1}-x \Vert ^{2}+\Vert y_{n}-y\Vert ^{2}-\Vert y_{n+1}-y\Vert ^{2} \\ &\qquad {}-\gamma_{n} \bigl(2-\gamma_{n}(\lambda_{A}+ \lambda_{B}) \bigr)\Vert Au_{n}-Bv_{n}\Vert ^{2} \\ &\qquad {}-\alpha_{n}(1-k-\alpha_{n}) \bigl\{ \bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \\ &\qquad {}+\bigl\Vert \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)-Q \bigl(v_{n}+\gamma_{n}B^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.25)
By using (
3.10)-(
3.14), we have
$$\begin{aligned}& \lim_{n\to\infty} \Vert x_{n}-u_{n} \Vert =0, \end{aligned}$$
(3.26)
$$\begin{aligned}& \lim_{n\to\infty} \Vert y_{n}-v_{n} \Vert =0. \end{aligned}$$
(3.27)
Hence,
\(u_{n}\rightharpoonup x^{*}\) and
\(v_{n}\rightharpoonup y^{*}\), respectively.
Since
P is
\(k_{1}\)-demi-contractive mapping and
\((I-P)\) is demi-closed at 0, we have
$$\begin{aligned} &\Vert u_{n}-Pu_{n}\Vert \\ &\quad =\Vert u_{n}-x_{n+1}+x_{n}{n+1}-Pu_{n} \Vert \\ &\quad \leq \Vert u_{n}-x_{n+1}\Vert +\Vert x_{n}{n+1}-Pu_{n}\Vert \\ &\quad =\Vert u_{n}-u_{n+1}+u_{n+1}-x_{n+1} \Vert \\ &\qquad {}+\bigl\Vert (1-\alpha_{n}) \bigl(u_{n}-\gamma _{n}A^{*}(Au_{n}-Bv_{n})\bigr)+\alpha_{n}P \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-Pu_{n}\bigr\Vert \\ &\quad \leq \Vert u_{n}-u_{n+1}\Vert +\Vert u_{n+1}-x_{n+1}\Vert +\bigl\Vert \bigl(u_{n}- \gamma _{n}A^{*}(Au_{n}-Bv_{n})\bigr)-Pu_{n} \bigr\Vert \\ &\qquad {}+\alpha_{n}\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma _{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert \\ &\quad \leq \Vert u_{n}-u_{n+1}\Vert +\Vert u_{n+1}-x_{n+1}\Vert +\frac{1+\sqrt{k}}{1-\sqrt {k}} \bigl\{ \vert \gamma_{n}\vert \bigl\Vert A^{*}\bigr\Vert \Vert Au_{n}-Bv_{n}\Vert \bigr\} \\ &\qquad {}+\alpha_{n}\bigl\Vert \bigl(u_{n}-\gamma_{n}A^{*}(Au_{n}-Bv_{n}) \bigr)-P \bigl(u_{n}-\gamma _{n}A^{*}(Au_{n}-Bv_{n}) \bigr)\bigr\Vert . \end{aligned}$$
Using (
3.10), (
3.11), (
3.18), and (
3.26), we have
$$ \lim_{n\to\infty} \Vert u_{n}-Pu_{n} \Vert =0. $$
(3.28)
Similarly, using the same steps as above for
Q, we have
$$ \lim_{n\to\infty} \Vert v_{n}-Qv_{n} \Vert =0. $$
(3.29)
Since
$$\begin{aligned} \Vert x_{n}-Px_{n}\Vert =&\Vert x_{n}-u_{n}+u_{n}-Pu_{n}+Pu_{n}-Px_{n} \Vert \\ \leq&\Vert x_{n}-u_{n}\Vert +\Vert u_{n}-Pu_{n}\Vert +\Vert Pu_{n}-Px_{n} \Vert \\ \leq&\Vert x_{n}-u_{n}\Vert +\Vert u_{n}-Pu_{n}\Vert +\frac{1+\sqrt{k}}{1-\sqrt{k}} \Vert u_{n}-x_{n}\Vert \\ =&\frac{2}{1-\sqrt{k}} \Vert x_{n}-u_{n}\Vert +\Vert u_{n}-Pu_{n}\Vert , \end{aligned}$$
it follows from (
3.26) and (
3.28) that
$$ \lim_{n\to\infty} \Vert x_{n}-Px_{n} \Vert =0. $$
(3.30)
Similarly, we have
$$ \lim_{n\to\infty} \Vert y_{n}-Qy_{n} \Vert =0. $$
(3.31)
As
\(\{x_{n}\}\) and
\(\{y_{n}\}\) weakly converge to
\(x^{*}\) and
\(y^{*}\), respectively, and
\((I-P)\) and
\((I-Q)\) are demi-closed at 0, it follows from (
3.30) and (
3.31) that
\(x^{*}\in \operatorname {Fix}(P)\) and
\(y^{*}\in \operatorname {Fix}(Q)\). Every Hilbert space satisfies Opial’s condition, which shows that the weakly subsequential limit of
\(\{(x_{n},y_{n})\}\) is unique.
Now, we show that
\(x^{*}\in \operatorname {GMEP}(F,T,\phi)\) and
\(y^{*}\in \operatorname {GMEP}(G,S,\varphi )\). Since
\(u_{n}=J_{r_{n}}^{F,T}(x_{n})\), we have, for all
\(b, u\in C\) and
\(\lambda\in(0,1]\),
$$ F\bigl(\lambda_{1}u_{n}+(1-\lambda_{1})b,u\bigr)+ \bigl\langle T(u_{n}),u-u_{n}\bigr\rangle +\phi (u)- \phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n} \rangle\geq0. $$
Using (A2) and (A3), we get
$$\begin{aligned} \phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n} \rangle \geq &-F\bigl(\lambda_{1}u_{n}+(1- \lambda_{1})b,u\bigr)-\bigl\langle T(u_{n}),u-u_{n} \bigr\rangle \\ \geq&F\bigl(\lambda_{1}u+(1-\lambda_{1})b,u_{n} \bigr)+\bigl\langle T(u),u_{n}-u\bigr\rangle , \end{aligned}$$
and hence
$$ \phi(u)-\phi(u_{n_{k}})+\frac{1}{r_{n_{k}}}\langle u-u_{n_{k}},u_{n_{k}}-x_{n_{k}} \rangle\geq F\bigl(\lambda_{1}u+(1-\lambda _{1})b,u_{n_{k}} \bigr)+\bigl\langle T(u),u_{n_{k}}-u\bigr\rangle . $$
From (
3.26), we have
\(u_{n_{k}}\rightharpoonup x^{*}\). It shows that
\(\lim_{k\to\infty}\frac{\Vert u_{n_{k}}-x_{n_{k}}\Vert }{r_{n_{k}}}=0\), and from the lower semicontinuity of
ϕ, we have
$$ F\bigl(\lambda_{1}u+(1-\lambda_{1})b,x^{*} \bigr)+\bigl\langle T(u),x^{*}-u\bigr\rangle +\phi \bigl(x^{*}\bigr)-\phi(u)\leq0,\quad \forall b,u\in C. $$
(3.32)
Set
\(u_{t}=tu+(1-t)x^{*}\), for all
\(t\in(0,1]\) and
\(u\in C\). Since
C is a convex set,
\(u_{t}\in C\). Hence from (
3.32), we have
$$ F\bigl(\lambda_{1}u_{t}+(1- \lambda_{1})b,x^{*}\bigr)+\bigl\langle T(u_{t}),x^{*}-u_{t} \bigr\rangle +\phi \bigl(x^{*}\bigr)-\phi(u_{t})\leq0. $$
(3.33)
Using the conditions (A1)-(A4), convexity of
ϕ, and (
3.33), we get
$$\begin{aligned} 0 =&F\bigl(\lambda_{1}u_{t}+(1-\lambda_{1})b,u_{t} \bigr)+(1-t)\bigl\langle T(u_{t}),u_{t}-u_{t} \bigr\rangle +\phi(u_{t})-\phi(u_{t}) \\ \leq&tF\bigl(\lambda_{1}u_{t}+(1-\lambda_{1})b,u \bigr)+(1-t)F\bigl(\lambda_{1}u_{t}+(1-\lambda _{1})b,x^{*}\bigr)+t\phi(u)+(1-t)\phi\bigl(x^{*}\bigr) \\ &{}-\phi(u_{t})+(1-t)\bigl\langle T(u_{t}),u_{t}-x^{*} \bigr\rangle +(1-t)\bigl\langle T(u_{t}),x^{*}-u_{t}\bigr\rangle \\ =&t \bigl\{ F\bigl(\lambda_{1}u_{t}+(1- \lambda_{1})b,u\bigr)+(1-t)\bigl\langle T(u_{t}),u-x^{*}\bigr\rangle +\phi(u)-\phi(u_{t}) \bigr\} \\ &{}\times(1-t) \bigl\{ F\bigl(\lambda_{1}u_{t}+(1- \lambda_{1})b,x^{*}\bigr)+\bigl\langle T(u_{t}),x^{*}-u_{t} \bigr\rangle +\phi\bigl(x^{*}\bigr)-\phi(u_{t}) \bigr\} \\ \leq&t \bigl\{ F\bigl(\lambda_{1}u_{t}+(1- \lambda_{1})b,u\bigr)+(1-t)\bigl\langle T(u_{t}),u-x^{*}\bigr\rangle +\phi(u)-\phi(u_{t}) \bigr\} , \end{aligned}$$
which implies that
$$ F\bigl(\lambda_{1}u_{t}+(1-\lambda_{1})b,u \bigr)+(1-t)\bigl\langle T(u_{t}),u-x^{*}\bigr\rangle +\phi (u)- \phi(u_{t})\geq0, \quad \forall u,b\in C. $$
Let
\(t\to0\) and therefore
\(u_{t}\to x^{*}\). Using the conditions (A5)-(A6) and proper lower semicontinuity of
ϕ, we have
$$ F\bigl(\lambda_{1}x^{*}+(1-\lambda_{1})b,u\bigr)+\bigl\langle T\bigl(x^{*}\bigr),u-x^{*}\bigr\rangle +\phi (u)-\phi\bigl(x^{*}\bigr)\geq0,\quad \forall u,b \in C, $$
which shows that
\(x^{*}\in \operatorname {GMEP}(F,T,\phi)\). Using the equivalent assertions to the above, we obtain
\(y^{*}\in \operatorname {GMEP}(G,S,\varphi)\).
We now prove the strong convergence conjecture (II).
Since
P and
Q are demi-compact,
\(\{x_{n}\}\) and
\(\{y_{n}\}\) are bounded, and
\(\lim_{n\to\infty} \Vert x_{n}-Px_{n}\Vert =0\),
\(\lim_{n\to\infty} \Vert y_{n}-Qy_{n}\Vert =0\), there exist (without loss of generality) subsequences
\(\{ x_{n_{k}}\}\) of
\(\{x_{n}\}\) and
\(\{y_{n_{k}}\}\) of
\(\{y_{n}\}\) such that
\(\{ x_{n_{k}}\}\) and
\(\{y_{n_{k}}\}\) converge strongly to some points
\(u^{*}\) and
\(v^{*}\), respectively. Since
\(\{x_{n_{k}}\}\) and
\(\{y_{n_{k}}\}\) converge weakly to
\(x^{*}\) and
\(y^{*}\), respectively, this implies that
\(x^{*}=u^{*}\) and
\(y^{*}=v^{*}\). It follows from the demi-closedness of
P and
Q that
\(x^{*}\in \operatorname {Fix}(P)\) and
\(y^{*}\in \operatorname {Fix}(Q)\). Using similar steps to the previous ones, we get
\(x^{*}\in \operatorname {GMEP}(F,T,\phi)\) and
\(y^{*}\in \operatorname {GMEP}(G,S,\varphi)\). Thus, we have
$$ \bigl\Vert Ax^{*}-By^{*}\bigr\Vert =\lim_{k\to\infty} \Vert Ax_{n_{k}}-By_{n_{k}}\Vert =0. $$
This implies that
\(Ax^{*}=By^{*}\). Hence
\((x^{*},y^{*})\in\Omega\). On the other hand, since
\(\Omega_{n}(x,y)=\Vert x_{n}-x\Vert ^{2}+\Vert y_{n}-y\Vert ^{2}\), for any
\((x,y)\in \Omega\), we know that
\(\lim_{k\to\infty}\Omega_{n}(x^{*},y^{*})=0\). From conjecture (I), we see that
\(\lim_{n\to\infty}\Omega_{n}(x^{*},y^{*})\) exists, therefore
\(\lim_{n\to\infty}\Omega_{n}(x^{*},y^{*}) = 0\). So, the iterative scheme (
3.1) converges strongly to a solution of problem (
1.17). This completes the proof of the conjecture (II). □