Now, we give a new modified iterative algorithm to solve the split equality generalized mixed equilibrium problem. Moreover, we prove strong and weak convergence theorems for nonexpansive mappings in Hilbert spaces. Throughout this section, we always assume that:
-
B1.
\(H_{1}\), \(H_{2}\), and \(H_{3}\) are real Hilbert spaces, and \(C \subseteq H_{1}\) and \(Q \subseteq H_{2}\) are nonempty closed convex subsets;
-
B2.
\(F: C \times C \rightarrow \mathbb{R}\) and \(G: Q \times Q \rightarrow \mathbb{R}\) are bifunctions satisfying conditions (A1), (A2), (A4), (A5), and (A7);
-
B3.
\(T: C \rightarrow C\) and \(S: Q \times Q \rightarrow \mathbb{R}\) are mappings satisfying conditions (A3), (A6), and (A7);
-
B4.
\(\phi: C \rightarrow \mathbb{R}\cup \{ +\infty \}\) and \(\varphi: Q \rightarrow \mathbb{R}\cup \{ +\infty \}\) are proper lower semicontinuous and convex mappings such that \(C \cap \operatorname{dom}\phi \neq\emptyset\) and \(Q \cap \operatorname{dom} \varphi \neq\emptyset\);
-
B5.
\(P_{1}, P_{2}: H_{1} \rightarrow H_{1}\) and \(P_{3}, P_{4}: H_{2} \rightarrow H_{2}\) are nonexpansive mapping;
-
B6.
\(A: H_{1} \rightarrow H_{3}\) and \(B: H_{2} \rightarrow H_{3}\) are bounded linear mappings.
For an arbitrary initial value \(( x_{1}, y_{1} ) \in C \times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C \times Q\) generated by
$$ \left \{ \textstyle\begin{array}{l} F ( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u ) + \phi ( u ) - \phi ( u_{n} ) \\ \quad {}+ \langle T u_{n}, u - u_{n} \rangle + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ G ( \lambda_{2} v_{n} + ( 1- \lambda_{2} ) c, v ) + \varphi ( v ) - \varphi ( v_{n} ) \\ \quad {}+ \langle S v_{n}, v - v_{n} \rangle + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
(3.1)
for all \(u, b \in C\) and \(v, c \in Q\), where \(n \geq1\), \(\lambda_{1}, \lambda_{2} \in ( 0,1 ]\), and the sequences \(\{ \delta_{n} \}\), \(\{ \alpha_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions:
-
C1.
\(\{ \delta_{n} \}\) is a positive real sequence such that \(\delta_{n} \in ( \varepsilon, \frac{2}{\lambda_{A} + \lambda_{B}} - \varepsilon )\) for sufficiently small ε, where \(\lambda_{A}\) and \(\lambda_{B}\) are the spectral radii of \(A^{*} A\) and \(B^{*} B\), respectively;
-
C2.
\(\{ \alpha_{n} \}\) is a sequence in \(( 0,1 )\) such that, for some α, \(\beta \in ( 0,1 )\), \(0< \alpha \leq \alpha_{n} \leq \beta <1\);
-
C3.
\(\{ r_{n} \} \subset ( 0,\infty )\) is such that \(\liminf_{n \rightarrow\infty} r_{n} >0\) and \(\lim_{n \rightarrow\infty} \vert r_{n +1} - r_{n} \vert =0\).
Theorem 1
Let
\(H_{1}\), \(H_{2}\), \(H_{3}\), F, G, T, S, \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), ϕ, φ, A, and
B
satisfy conditions (B1)-(B6). Let
\(\{ ( x_{n}, y_{n} ) \}\)
be a sequence generated by (3.1). If
\(\mathcal{F} := \bigcap_{i=1}^{4} F ( P_{i} ) \cap \operatorname{SEGMEP} ( F,G, P_{i}, \phi, \varphi ) \neq\emptyset\), then:
-
(i)
the sequence
\(\{ ( x_{n}, y_{n} ) \}\)
converges weakly to a solution of problem (1.4);
-
(ii)
if
\(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence
\(\{ ( x_{n}, y_{n} ) \}\)
converges strongly to a solution of problem (1.4).
Proof
(i) Let \(( x, y )\in \mathcal{F}\). So, \(x \in F ( P_{1} ) \cap F ( P_{2} )\) and \(y \in F ( P_{3} ) \cap F ( P_{4} )\). It is easy to see from Lemma 3 that
$$ \Vert u_{n} - x \Vert = \bigl\Vert J_{r_{n}}^{F, T} ( x_{n} ) - J_{r_{n}}^{F, T} ( x ) \bigr\Vert \leq \Vert x_{n} - x \Vert $$
(3.2)
and
$$ \Vert v_{n} - y \Vert = \bigl\Vert J_{r_{n}}^{F, T} ( y_{n} ) - J_{r_{n}}^{F, T} ( y ) \bigr\Vert \leq \Vert y_{n} - y \Vert . $$
(3.3)
Since \(P_{i}\), \(i =1,2,3,4\), are nonexpansive mappings and
$$\Vert x - y \Vert ^{2} = \Vert x \Vert ^{2} + \Vert y \Vert ^{2} -2 \langle y, x \rangle $$
for all \(x, y \in H\), we get from Lemma 5 that
$$\begin{aligned} \Vert x_{n +1} - x \Vert ^{2} =& \bigl\Vert ( 1- \alpha_{n} ) P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}+ \alpha_{n} P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - x \bigr\Vert ^{2} \\ =& \bigl\Vert ( 1- \alpha_{n} ) \bigl( P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - x \bigr) \\ &{}+ \alpha_{n} \bigl( P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - x \bigr) \bigr\Vert ^{2} \\ \leq& ( 1- \alpha_{n} ) \bigl\Vert u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) - x \bigr\Vert ^{2} \\ &{}+ \alpha_{n} \bigl\Vert u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) - x \bigr\Vert ^{2} \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\ =& \bigl\Vert u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) - x \bigr\Vert ^{2} \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\ =& \Vert u_{n} - x \Vert ^{2} + \delta_{n}^{2} \bigl\Vert A^{*} ( A u_{n} - B v_{n} ) \bigr\Vert ^{2} \\ &{}-2 \delta_{n} \bigl\langle A^{*} ( A u_{n} - B v_{n} ), u_{n} - x \bigr\rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\ \leq& \Vert x_{n} - x \Vert ^{2} + \delta_{n}^{2} \bigl\Vert A^{*} ( A u_{n} - B v_{n} ) \bigr\Vert ^{2} \\ &{}-2 \delta_{n} \bigl\langle A^{*} ( A u_{n} - B v_{n} ), u_{n} - x \bigr\rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.4)
On the other hand, we have
$$\begin{aligned} \delta_{n}^{2} \bigl\Vert A^{*} ( A u_{n} - B v_{n} ) \bigr\Vert ^{2} =& \delta_{n}^{2} \bigl\langle A u_{n} - B v_{n}, A A^{*} ( A u_{n} - B v_{n} ) \bigr\rangle \\ \leq& \lambda_{A} \delta_{n}^{2} \Vert A u_{n} - B v_{n} \Vert ^{2}. \end{aligned}$$
(3.5)
So, it follows from (3.4) and (3.5) that
$$\begin{aligned} \Vert x_{n +1} - x \Vert ^{2} \leq& \Vert x_{n} - x \Vert ^{2} + \lambda_{A} \delta_{n}^{2} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}-2 \delta_{n} \langle A u_{n} - B v_{n}, A u_{n} - Ax \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.6)
In a similar way, we get
$$\begin{aligned} \Vert y_{n +1} - y \Vert ^{2} \leq& \Vert y_{n} - y \Vert ^{2} + \lambda_{B} \delta_{n}^{2} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}+2 \delta_{n} \langle A u_{n} - B v_{n}, B v_{n} - By \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.7)
By adding inequalities (3.6) and (3.7) side by side and using \(Ax = By\), we obtain
$$\begin{aligned}& \Vert x_{n +1} - x \Vert ^{2} + \Vert y_{n +1} - y \Vert ^{2} \\& \quad \leq \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2} + \delta_{n}^{2} ( \lambda_{A} + \lambda_{B} ) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}-2 \delta_{n} \Vert A u_{n} - B v_{n} \Vert ^{2} - \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} + \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} \\& \quad = \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2} - \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} + \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.8)
Let \(\xi_{n} ( x, y ) = \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2}\). Thus, we have from (3.8) that
$$\begin{aligned} \xi_{n +1} ( x, y ) \leq& \xi_{n} ( x, y ) - \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} + \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.9)
Since \(\alpha_{n} \in ( 0,1 )\) and \(\delta_{n} \in ( \varepsilon, \frac{2}{\lambda_{A} + \lambda_{B}} - \varepsilon )\), we get \(2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) >0\). So, from (3.9) we obtain
$$\xi_{n +1} ( x, y ) \leq \xi_{n} ( x, y ). $$
Therefore, the sequence \(\{ \xi_{n} ( x, y ) \}\) is nonincreasing and lower bounded by 0. Hence, \(\lim_{n \rightarrow\infty} \xi_{n} ( x, y )\) exists. Let \(\lim_{n \rightarrow\infty} \xi_{n} ( x, y ) = \sigma ( x, y )\). So condition (i) of Lemma 4 is satisfied with \(\mu_{n} = ( x_{n}, y_{n} )\), \(\mu^{*} = ( x, y )\), and \(W =\mathcal{F}\). Since the sequence \(\{ \xi_{n} ( x, y ) \}\) converges to a finite limit, we have from inequality (3.9) that
$$\begin{aligned}& \lim_{n \rightarrow\infty} \Vert A u_{n} - B v_{n} \Vert =0, \end{aligned}$$
(3.10)
$$\begin{aligned}& \lim_{n \rightarrow\infty} \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert =0, \end{aligned}$$
(3.11)
and
$$ \lim_{n \rightarrow\infty} \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert =0. $$
(3.12)
Moreover, since \(\Vert x_{n} - x \Vert ^{2} \leq \xi_{n} ( x, y )\) and \(\Vert y_{n} - y \Vert ^{2} \leq \xi_{n} ( x, y )\), the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are bounded, and \(\limsup_{n \rightarrow\infty} \Vert x_{n} - x \Vert \) and \(\limsup_{n \rightarrow\infty} \Vert y_{n} - y \Vert \) exist. Also, it follows from (3.2) and (3.3) that \(\limsup_{n \rightarrow\infty} \Vert u_{n} - x \Vert \) and \(\limsup_{n \rightarrow\infty} \Vert v_{n} - y \Vert \) exist. Let us assume that the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) converge weakly to points \(x^{*}\) and \(y^{*}\), respectively. So, by (3.10), the sequence \(\{ u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \}\) converges weakly to \(x^{*}\), and \(\{ v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \}\) converges weakly to \(y^{*}\). By Lemma 5 we get
$$\begin{aligned} \Vert x_{n +1} - x_{n} \Vert ^{2} =& \Vert x_{n +1} - x - x_{n} + x \Vert ^{2} \\ =& \Vert x_{n +1} - x \Vert ^{2} - \Vert x_{n} - x \Vert ^{2} -2 \langle x_{n +1} - x_{n}, x_{n} - x \rangle \\ =& \Vert x_{n +1} - x \Vert ^{2} - \Vert x_{n} - x \Vert ^{2} \\ &{}-2 \bigl\langle x_{n +1} - x^{*}, x_{n} - x \bigr\rangle +2 \bigl\langle x_{n} - x^{*}, x_{n} - x \bigr\rangle . \end{aligned}$$
Hence, we obtain
$$ \lim_{n \rightarrow\infty} \Vert x_{n +1} - x_{n} \Vert =0 $$
(3.13)
and, similarly,
$$ \lim_{n \rightarrow\infty} \Vert y_{n +1} - y_{n} \Vert =0. $$
(3.14)
By Lemma 3, since \(u_{n} = J_{r_{n}}^{F, T} ( x_{n} )\) and \(u_{n +1} = J_{r_{n +1}}^{F, T} ( x_{n +1} )\), we have that, for all \(u \in C\),
$$\begin{aligned}& F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u \bigr) + \langle T u_{n}, u - u_{n} \rangle \\& \quad {}+ \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0 \end{aligned}$$
(3.15)
and
$$\begin{aligned} \begin{aligned}[b] &F \bigl( \lambda_{1} u_{n +1} + ( 1- \lambda_{1} ) b, u \bigr) + \langle T u_{n +1}, u - u_{n +1} \rangle \\ &\quad {}+ \phi ( u ) - \phi ( u_{n +1} ) + \frac{1}{r_{n +1}} \langle u - u_{n +1}, u_{n +1} - x_{n +1} \rangle \geq0. \end{aligned} \end{aligned}$$
(3.16)
Taking \(u = u_{n}\) in (3.16) and \(u = u_{n +1}\) in (3.15) and adding the resulting inequalities side by side, we obtain
$$\begin{aligned} 0 \leq& F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u_{n +1} \bigr) + F \bigl( \lambda_{1} u_{n +1} + ( 1- \lambda_{1} ) b, u_{n} \bigr) \\ &{}+ \langle T u_{n}, u_{n +1} - u_{n} \rangle + \langle T u_{n +1}, u_{n} - u_{n +1} \rangle \\ &{}+ \frac{1}{r_{n}} \langle u_{n +1} - u_{n}, u_{n} - x_{n} \rangle + \frac{1}{r_{n +1}} \langle u_{n} - u_{n +1}, u_{n +1} - x_{n +1} \rangle. \end{aligned}$$
Using conditions (A2)-(A3), we have
$$\begin{aligned} 0 \leq& \frac{1}{r_{n +1}} \langle u_{n} - u_{n +1}, u_{n +1} - x_{n +1} \rangle + \frac{1}{r_{n}} \langle u_{n +1} - u_{n}, u_{n} - x_{n} \rangle \\ \leq& \biggl\langle u_{n +1} - u_{n}, \frac{u_{n} - x_{n}}{r_{n}} - \frac{u_{n +1} - x_{n +1}}{r_{n +1}} \biggr\rangle \\ =& \biggl\langle u_{n +1} - u_{n}, u_{n} - u_{n +1} + u_{n +1} - x_{n} - \frac{r_{n}}{r_{n +1}} ( u_{n +1} - x_{n +1} ) \biggr\rangle \\ =& \langle u_{n +1} - u_{n}, u_{n} - u_{n +1} \rangle \\ &{}+ \biggl\langle u_{n +1} - u_{n}, x_{n +1} - x_{n} + \biggl( 1- \frac{r_{n}}{r_{n +1}} \biggr) ( u_{n +1} - x_{n +1} ) \biggr\rangle \\ =&- \Vert u_{n +1} - u_{n} \Vert ^{2} \\ &{}+ \biggl\langle u_{n +1} - u_{n}, x_{n +1} - x_{n} + \biggl( 1- \frac{r_{n}}{r_{n +1}} \biggr) ( u_{n +1} - x_{n +1} ) \biggr\rangle , \end{aligned}$$
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, we get
$$ \Vert u_{n +1} - u_{n} \Vert \leq \Vert x_{n +1} - x_{n} \Vert + \biggl\vert 1- \frac{r_{n}}{r_{n +1}} \biggr\vert \Vert u_{n +1} - x_{n +1} \Vert . $$
(3.17)
Using (3.13) and (C3), from (3.17) we get
$$ \lim_{n \rightarrow\infty} \Vert u_{n +1} - u_{n} \Vert =0. $$
(3.18)
In a similar way, we get
$$ \lim_{n \rightarrow\infty} \Vert v_{n +1} - v_{n} \Vert =0. $$
(3.19)
On the other hand, from (3.6) and (3.7) we get
$$\begin{aligned} \begin{aligned}[b] \Vert x_{n +1} - x \Vert ^{2} \leq{}& \Vert u_{n} - x \Vert ^{2} + \delta_{n}^{2} \lambda_{A} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}-2 \delta_{n} \langle A u_{n} - B v_{n}, A u_{n} - Ax \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \end{aligned} \end{aligned}$$
(3.20)
and
$$\begin{aligned} \Vert y_{n +1} - y \Vert ^{2} \leq& \Vert v_{n} - y \Vert ^{2} + \delta_{n}^{2} \lambda_{B} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}+2 \delta_{n} \langle A u_{n} - B v_{n}, B v_{n} - By \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{2} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.21)
Using \(Ax = By\) and adding inequalities (3.20) and (3.21) side by side, we have
$$\begin{aligned}& \Vert x_{n +1} - x \Vert ^{2} + \Vert y_{n +1} - y \Vert ^{2} \\& \quad \leq \Vert u_{n} - x \Vert ^{2} + \Vert v_{n} - y \Vert ^{2} \\& \qquad {}- \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} , \end{aligned}$$
(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
$$\begin{aligned} \Vert v_{n} - y \Vert ^{2} =& \bigl\Vert J_{r_{n}}^{G, S} ( y_{n} ) - J_{r_{n}}^{F, T} ( y ) \bigr\Vert ^{2} \leq \langle y_{n} - y, v_{n} - y \rangle \\ =& \frac{1}{2} \bigl\{ \Vert y_{n} - y \Vert ^{2} + \Vert v_{n} - y \Vert ^{2} - \Vert y_{n} - v_{n} \Vert ^{2} \bigr\} . \end{aligned}$$
(3.24)
From (3.22)-(3.24) we conclude that
$$\begin{aligned}& \Vert x_{n} - u_{n} \Vert ^{2} + \Vert y_{n} - v_{n} \Vert ^{2} \\& \quad \leq \Vert x_{n} - x \Vert ^{2} - \Vert x_{n +1} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2} - \Vert y_{n +1} - y \Vert ^{2} \\& \qquad {}- \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.25)
Using (3.10)-(3.14), we have
$$ \lim_{n \rightarrow\infty} \Vert x_{n} - u_{n} \Vert =0 $$
(3.26)
and
$$ \lim_{n \rightarrow\infty} \Vert y_{n} - v_{n} \Vert =0. $$
(3.27)
Hence, \(u_{n} \rightharpoonup x^{*}\) and \(v_{n} \rightharpoonup y^{*}\).
Since \(P_{i}\), \(i =1,2,3,4\), are nonexpansive mappings, we obtain
$$\begin{aligned} \Vert u_{n} - P_{1} u_{n} \Vert =& \Vert u_{n} - x_{n +1} + x_{n +1} - P_{1} u_{n} \Vert \\ \leq& \Vert u_{n} - x_{n +1} \Vert + \Vert x_{n +1} - P_{1} u_{n} \Vert \\ =& \Vert u_{n} - u_{n +1} + u_{n +1} - x_{n +1} \Vert \\ &{}+ \bigl\Vert ( 1- \alpha_{n} ) P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}+ \alpha_{n} P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{1} u_{n} \bigr\Vert \\ \leq& \Vert u_{n} - u_{n +1} \Vert + \Vert u_{n +1} - x_{n +1} \Vert \\ &{}+ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{1} u_{n} \bigr\Vert \\ &{}+ \alpha_{n} \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert \\ \leq& \Vert u_{n} - u_{n +1} \Vert + \Vert u_{n +1} - x_{n +1} \Vert \\ &{}+ \vert \delta_{n} \vert \bigl\Vert A^{*} \bigr\Vert \Vert A u_{n} - B v_{n} \Vert + \alpha_{n} \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert . \end{aligned}$$
Using (3.10), (3.11), (3.18), and (3.26), we have
$$ \lim_{n \rightarrow\infty} \Vert u_{n} - P_{1} u_{n} \Vert =0. $$
(3.28)
Similarly, using the same steps as before for \(P_{2}\), \(P_{3}\), and \(P_{4}\), we get
$$ \lim_{n \rightarrow\infty} \Vert u_{n} - P_{2} u_{n} \Vert =0,\qquad \lim_{n \rightarrow\infty} \Vert v_{n} - P_{3} v_{n} \Vert =0,\quad \mbox{and} \quad \lim_{n \rightarrow\infty} \Vert v_{n} - P_{4} v_{n} \Vert =0. $$
(3.29)
Since
$$\begin{aligned} \Vert x_{n} - P_{1} x_{n} \Vert =& \Vert x_{n} - u_{n} + u_{n} - P_{1} u_{n} + P_{1} u_{n} - P_{1} x_{n} \Vert \\ \leq& \Vert x_{n} - u_{n} \Vert + \Vert u_{n} - P_{1} u_{n} \Vert + \Vert u_{n} - x_{n} \Vert \\ =&2 \Vert x_{n} - u_{n} \Vert + \Vert u_{n} - P_{1} u_{n} \Vert , \end{aligned}$$
we have from (3.26) and (3.28) that
$$ \lim_{n \rightarrow\infty} \Vert x_{n} - P_{1} x_{n} \Vert =0. $$
(3.30)
Similarly, we have
$$ \lim_{n \rightarrow\infty} \Vert x_{n} - P_{2} x_{n} \Vert =0,\qquad \lim_{n \rightarrow\infty} \Vert y_{n} - P_{3} y_{n} \Vert =0,\quad \mbox{and} \quad \lim_{n \rightarrow\infty} \Vert y_{n} - P_{4} y_{n} \Vert =0. $$
(3.31)
Since the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) converge weakly to \(x^{*}\) and \(y^{*}\), respectively, and \(( I - P_{i} )\), \(i =1,2,3,4\), are demiclosed at zero, it follows from (3.30) and (3.31) that \(x^{*} \in F ( P_{1} ) \cap F ( P_{2} )\) and \(y^{*} \in F ( P_{3} ) \cap F ( P_{4} )\). On the other hand, it is well known that every Hilbert space satisfies Opial’s condition. So, we have 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
$$F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u \bigr) + \langle T u_{n}, u - u_{n} \rangle + \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0 $$
for all \(b, u \in C\) and \(\lambda \in ( 0,1 ]\). From conditions (A2) and (A3) we obtain
$$\begin{aligned} \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq&- F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u \bigr) - \langle T u_{n}, u - u_{n} \rangle \\ \geq& F \bigl( \lambda_{1} u + ( 1- \lambda_{1} ) b, u_{n} \bigr) + \langle Tu, u_{n} - u \rangle, \end{aligned}$$
and hence
$$\phi ( u ) - \phi ( u_{n_{k}} ) + \frac{1}{ r_{n_{k}}} \langle u - u_{n_{k}}, u_{n_{k}} - x_{n_{k}} \rangle \geq F \bigl( \lambda_{1} u + ( 1- \lambda_{1} ) b, u_{n_{k}} \bigr) + \langle Tu, u_{n_{k}} - u \rangle. $$
From (3.26) it is easy to see that \(u_{n_{k}} \rightharpoonup x^{*}\). So, we can write \(\lim_{k \rightarrow\infty} \frac{\Vert u_{n_{k}} - x_{n_{k}} \Vert }{r_{n_{k}}} =0\), and from the lower semicontinuity of ϕ we get
$$ F \bigl( \lambda_{1} u + ( 1- \lambda_{1} ) b, x^{*} \bigr) + \bigl\langle Tu, x^{*} - u \bigr\rangle + \phi \bigl( x^{*} \bigr) - \phi ( u ) \leq0 $$
(3.32)
for all \(b, u \in C\). Set \(u_{t} = tu + ( 1- t ) x^{*}\) for \(t \in ( 0,1 ]\) and \(u \in C\). Since C is a convex set, we have \(u_{t} \in C\). 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 inequality (3.33), the convexity of ϕ, and conditions (A1)-(A4), we get
$$\begin{aligned} 0 =& F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u_{t} \bigr) + ( 1- t ) \langle T u_{t}, u_{t} - u_{t} \rangle + \phi ( u_{t} ) - \phi ( u_{t} ) \\ \leq& tF \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, x^{*} \bigr) \\ &{}+ t\phi ( u ) + ( 1- t ) \phi \bigl( x^{*} \bigr) - \phi ( u_{t} ) + ( 1- t ) \bigl\langle T u_{t}, u_{t} - x^{*} \bigr\rangle \\ &{}+ ( 1- t ) \bigl\langle T u_{t}, x^{*} - u_{t} \bigr\rangle \\ =& t \bigl\{ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T u_{t}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi ( u_{t} ) \bigr\} \\ &{}+ ( 1- t ) \bigl\{ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, x^{*} \bigr) + \bigl\langle T u_{t}, x^{*} - u_{t} \bigr\rangle + \phi \bigl( x^{*} \bigr) - \phi ( u_{t} ) \bigr\} \\ \leq& t \bigl\{ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T u_{t}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi ( u_{t} ) \bigr\} , \end{aligned}$$
which implies that
$$F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T u_{t}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi ( u_{t} ) \geq0 $$
for all \(b, u \in C\). From the definition of \(u_{t}\) it is clear that \(u_{t} \rightarrow x^{*}\) as \(t \rightarrow0\). Using conditions (A5) and (A6) and the proper lower semicontinuity of ϕ, we obtain
$$F \bigl( \lambda_{1} x^{*} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T x^{*}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi \bigl( x^{*} \bigr) \geq0 $$
for all \(b, u \in C\), which shows that \(x^{*} \in \operatorname{GMEP} ( F, T, \phi )\). By using similar steps we have that \(y^{*} \in \operatorname{GMEP} ( G, S, \varphi )\).
Since \(A: H_{1} \rightarrow H_{3}\) and \(B: H_{2} \rightarrow H_{3}\) are bounded linear mappings and \(\{ u_{n} \}\) and \(\{ v_{n} \}\) converge weakly to \(x^{*}\) and \(y^{*}\), respectively, for arbitrary \(f \in H_{3}^{*}\), we have
$$f ( A u_{n} ) \rightarrow f \bigl( A x^{*} \bigr). $$
Similarly,
$$f ( B v_{n} ) \rightarrow f \bigl( B y^{*} \bigr). $$
Hence, we get
$$A u_{n} - B v_{n} \rightharpoonup A x^{*} - B y^{*}, $$
which implies that
$$\bigl\Vert A x^{*} - B y^{*} \bigr\Vert \leq \liminf_{n \rightarrow\infty} \Vert A u_{n} - B v_{n} \Vert =0, $$
so that \(A x^{*} = B y^{*}\). So, it follows that \(( x^{*}, y^{*} ) \in \operatorname{SEGMEP} ( F, G, T, S, \phi, \varphi )\). Therefore, \(( x^{*}, y^{*} ) \in \mathcal{F}\).
Finally, we conclude that, for each \(( x^{*}, y^{*} ) \in \mathcal{F}\), \(\lim_{n \rightarrow\infty} ( \Vert x_{n} - x^{*} \Vert + \Vert y_{n} - y^{*} \Vert )\) exists and each weak cluster point of the sequence \(\Vert ( x^{*}, y^{*} ) \Vert \) belongs to \(\mathcal{F}\). Let \(H = H_{1} \times H_{2}\) with norm \(\Vert ( x, y ) \Vert = \sqrt{\Vert x \Vert ^{2} + \Vert y \Vert ^{2}}\), \(W=\mathcal{F}\), \(\mu_{n} = ( x_{n}, y_{n} )\), and \(\mu = ( x^{*}, y^{*} )\). From Lemma 4 we see that there exists \(( \overline{x}, \overline{y} ) \in \mathcal{F}\) such that \(x_{n} \rightharpoonup \overline{x}\) and \(y_{n} \rightharpoonup \overline{y}\). Therefore, the sequence \(\{ ( x_{n}, y_{n} ) \}\) generated by the iterative algorithm (3.1) converges weakly to a solution of problem (1.4) in \(\mathcal{F}\). This completes the proof.
(ii) Now, we prove the strong convergence of the sequence \(\{ ( x_{n}, y_{n} ) \}\) generated by the iterative algorithm (3.1) under the demicompact condition.
Since \(P_{i}\), \(i =1,2,3,4\), are demicompact, \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are bounded sequences, and \(\lim_{n \rightarrow\infty} \Vert x_{n} - P_{1} x_{n} \Vert =0\), \(\lim_{n \rightarrow\infty} \Vert x_{n} - P_{2} x_{n} \Vert =0\), \(\lim_{n \rightarrow\infty} \Vert y_{n} - P_{3} y_{n} \Vert =0\), and \(\lim_{n \rightarrow\infty} \Vert y_{n} - P_{4} y_{n} \Vert =0\), there exist 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. The weak convergence of \(\{ x_{n_{k}} \}\) and \(\{ y_{n_{k}} \}\) to \(x^{*}\) and \(y^{*}\), respectively, implies that \(x^{*} = u^{*}\) and \(y^{*} = v^{*}\). It follows from the demiclosedness of \(P_{i}\) that \(x^{*} \in F ( P_{1} )\cap F ( P_{2} )\) and \(y^{*} \in F ( P_{3} )\cap F ( P_{4} )\). Using similar steps to the previous ones, we get that \(x^{*} \in \operatorname{GMEP} ( F, T, \phi )\) and \(y^{*} \in \operatorname{GMEP} ( G, S, \varphi )\). Thus, we have
$$\bigl\Vert A x^{*} - B y^{*} \bigr\Vert = \lim _{k \rightarrow\infty} \Vert A x_{n_{k}} - B y_{n_{k}} \Vert =0, $$
which implies that \(A x^{*} = B y^{*}\). Hence, \(( x^{*}, y^{*} ) \in \mathcal{F}\). On the other hand, since \(\xi_{n} ( x, y ) = \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2}\) for \(( x, y )\in \mathcal{F}\), we know that \(\lim_{k \rightarrow\infty} \xi_{n_{k}} ( x^{*}, y^{*} )=0\). From conjecture (i) we see that \(\lim_{n \rightarrow\infty} \xi_{n} ( x^{*}, y^{*} )\) exists; therefore, \(\lim_{n \rightarrow\infty} \xi_{n} ( x^{*}, y^{*} )=0\). So, the iterative scheme (3.1) converges strongly to a solution of problem (1.4). This completes the proof of conjecture (ii). □
Taking \(F = G = T = S = \phi = \varphi =0\) in Theorem 1, we get the following convergence theorem for the split equality problem (1.10).
Corollary 1
Let
\(H_{1}\), \(H_{2}\), \(H_{3}\), \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), A, and
B
satisfy conditions (B1), (B5), and (B6). For an arbitrary initial value
\(( x_{1}, y_{1} ) \in C\times Q\), define the sequence
\(\{ ( x_{n}, y_{n} ) \}\)
in
\(C\times Q\)
generated by
$$\left \{ \textstyle\begin{array}{l} x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - B y_{n} ) ) + \alpha_{n} P_{2} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - B y_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( y_{n} + \delta_{n} B^{*} ( A x_{n} - B y_{n} ) ) + \alpha_{n} P_{4} ( y_{n} + \delta_{n} B^{*} ( A x_{n} - B y_{n} ) ), \end{array}\displaystyle \right . $$
where
\(n \geq1\), and the sequences
\(\{ \delta_{n} \}\)
and
\(\{ \alpha_{n} \}\)
satisfy conditions (C1) and (C2), respectively. If
\(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SEP} \neq\emptyset\), then:
-
(i)
The sequence
\(\{ ( x_{n}, y_{n} ) \}\)
converges weakly to a solution of problem (1.10);
-
(ii)
If
\(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence
\(\{ ( x_{n}, y_{n} ) \}\)
converges strongly to a solution of problem (1.10).
Taking \(B = I\) and \(H_{2} = H_{3}\) in Corollary 1, we obtain the following convergence theorem for the split feasibility problem (1.11).
Corollary 2
Let
\(H_{1}\), \(H_{2}\), \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), and
A
satisfy conditions (B1), (B5), and (B6) with
\(A: H_{1} \rightarrow H_{2}\). For an arbitrary initial value
\(( x_{1}, y_{1} ) \in C\times Q\), define the sequence
\(\{ ( x_{n}, y_{n} ) \}\)
in
\(C\times Q\)
generated by
$$\left \{ \textstyle\begin{array}{l} x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - y_{n} ) ) + \alpha_{n} P_{2} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - y_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( y_{n} + \delta_{n} ( A x_{n} - y_{n} ) ) + \alpha_{n} P_{4} ( y_{n} + \delta_{n} ( A x_{n} - y_{n} ) ), \quad n \geq1, \end{array}\displaystyle \right . $$
where
\(\{ \delta_{n} \}\)
is a positive real sequence such that
\(\delta_{n} \in ( \varepsilon, \frac{1}{\lambda_{A}} - \varepsilon )\)
for sufficiently small
ε, where
\(\lambda_{A}\)
denotes the spectral radius of
\(A^{*} A\), and
\(\{ \alpha_{n} \}\)
satisfy condition (C2). If
\(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SFP} \neq\emptyset\), then:
-
(i)
The sequence
\(\{ ( x_{n}, y_{n} ) \}\)
converges weakly to a solution of problem (1.11);
-
(ii)
If
\(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence
\(\{ ( x_{n}, y_{n} ) \}\)
converges strongly to a solution of problem (1.11).