By using Theorem 3.1 and Lemma 2.1, we construct the following iterative algorithm for computing approximate solutions of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems (1).
Iterative Algorithm 4.1
For any given \((x_{1}^{0}, x_{2}^{0} )\in K_{1}\times K_{2}\), \(u_{1}^{0} \in T_{1} (x_{1}^{0} )\), \(u_{2}^{0} \in T_{2} (x_{1}^{0} )\), \(v_{1}^{0} \in S_{1} (x_{2}^{0} )\), \(v_{2}^{0} \in S_{1} (x_{2}^{0} )\), \(w_{1}^{0} \in B_{1} (x_{1}^{0},x_{2}^{0} )\), \(w_{2}^{0} \in B_{2} (x_{1}^{0},x_{2}^{0} )\) and \(z_{1}^{0} \in A_{1} (x_{1}^{0} )\), \(z_{2}^{0} \in A_{2} (x_{2}^{0} )\), compute the iterative sequences \(\{ (x_{1}^{n},x_{2}^{n} ) \} \subseteq K_{1} \times K_{2}\), \(\{u_{i}^{n} \}\), \(\{ v_{i}^{n} \}\), \(\{w_{i}^{n} \}\) and \(\{z_{i}^{n} \}\) by the following iterative schemes:
$$\begin{aligned}& \rho_{1} N_{1} \bigl(z_{1}^{n+1}, \eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr)+ \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr) + \rho _{1} \bigl\{ M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr) \\ & \quad{}+w_{1}^{n} \bigr\} ,\eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle + \rho_{1} \bigl\{ \phi_{1} \bigl(x_{1}^{n},y_{1} \bigr)-\phi_{1} \bigl(x_{1}^{n},x_{1}^{n+1} \bigr) \\& \quad{}+\alpha_{1} \bigl\Vert f_{1}(y_{1})-f_{1} \bigl(x_{1}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0,\quad\forall y_{1} \in K_{1}; \end{aligned}$$
(15)
$$\begin{aligned}& \rho_{2} N_{2} \bigl(z_{2}^{n+1}, \eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr) \bigr)+ \bigl\langle g_{2} \bigl(f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr)-g_{2} \bigl(f_{2} \bigl(x_{2}^{n} \bigr) \bigr) + \rho _{2} \bigl\{ M_{2} \bigl(u_{2}^{n},v_{2}^{n} \bigr) \\ & \quad{}+w_{2}^{n} \bigr\} ,\eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr) \bigr\rangle + \rho_{2} \bigl\{ \phi_{2} \bigl(x_{2}^{n},y_{2} \bigr)-\phi_{2} \bigl(x_{2}^{n},x_{2}^{n+1} \bigr) \\ & \quad{}+\alpha_{2} \bigl\Vert f_{2}(y_{2})-f_{2} \bigl(x_{2}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0,\quad\forall y_{2} \in K_{2}; \end{aligned}$$
(16)
$$\begin{aligned}& \textstyle\begin{cases} u_{i}^{n} \in T_{i} (x_{1}^{n} ); & \Vert u_{i}^{n+1}-u_{i}^{n}\Vert \leq (1+\frac{1}{n+1} )\mathcal{D} (T_{i} (x_{1}^{n+1} ),T_{i} (x_{1}^{n} ) );\\ v_{i}^{n} \in S_{i} (x_{2}^{n} ); & \Vert v_{i}^{n+1}-v_{i}^{n}\Vert \leq (1+\frac{1}{n+1} )\mathcal{D} (S_{i} (x_{2}^{n+1} ),S_{i} (x_{2}^{n} ) );\\ w_{i}^{n} \in B_{i} (x_{1}^{n}, x_{2}^{n} ); & \Vert w_{i}^{n+1}-w_{i}^{n}\Vert \leq (1+\frac{1}{n+1} )\mathcal {D} (B_{i} (x_{1}^{n+1},x_{2}^{n+1} ),B_{i} (x_{1}^{n},x_{2}^{n} ) );\\ z_{i}^{n} \in A_{i} (x_{1}^{n} ); & \Vert z_{i}^{n+1}-z_{i}^{n}\Vert \leq (1+\frac{1}{n+1} )\mathcal{D} (A_{i} (x_{i}^{n+1} ),A_{i} (x_{i}^{n} ) ), \end{cases}\displaystyle \end{aligned}$$
(17)
where \(n=0,1,2,\ldots\) , \(i=1,2\), and \(\rho_{1}, \rho_{2}, \alpha_{1}, \alpha _{2}>0\) are constants.
Now, we establish the following strong convergence result to obtain the solution of perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems (1).
Theorem 4.1
For each
\(i\in I\), the mappings
\(N_{i}\), \(M_{i}\), \(A_{i}\), \(T_{i}\), \(S_{i}\), \(B_{i}\), \(\eta_{i}\), \(\phi_{i}\), and
\(f_{i}\)
satisfy the hypotheses of Theorem
3.1. Further assume that:
-
(i)
\(M_{i}\)
is
\((\mu_{i},\xi_{i} )\)-mixed Lipschitz continuous;
-
(ii)
\(g_{i}\)
is
\(\sigma_{i}\)-Lipschitz continuous with respect to
\(f_{i}\)
and
\(\eta_{i}\)
is
\(\kappa_{i}\)-Lipschitz continuous with respect to
\(f_{i}\);
-
(iii)
\(T_{i}\)
is
\(\delta_{i}\)-\(\mathcal{D}\)-Lipschitz continuous and
\(S_{i}\)
is
\(\tau_{i}\)-\(\mathcal{D}\)-Lipschitz continuous;
-
(iv)
\(B_{i}\)
is
\((\zeta_{i},\nu_{i} )\)-\(\mathcal {D}\)-Lipschitz continuous and
\(A_{i}\)
is
\(\varsigma_{i}\)-\(\mathcal {D}\)-Lipschitz continuous.
For
\(\rho_{1}, \rho_{2}>0\), if the following conditions are satisfied:
$$\begin{aligned} \textstyle\begin{cases} \frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta_{1}^{2}} \{\kappa _{1}\sigma_{1}+\rho_{1}\kappa_{1} (\mu_{1}\delta_{1}+\zeta_{1} )+\rho_{1} \gamma_{1} \}+\frac{1}{(\rho_{2} \varrho_{2}-3 \varepsilon_{2})\beta _{2}^{2}} \{\rho_{2}\kappa_{2} (\mu_{2}\delta_{2}+\zeta_{2} ) \} < 1,\\ \frac{1}{(\rho_{2}\varrho_{2}-3 \varepsilon_{2})\beta_{2}^{2}} \{\kappa _{2}\sigma_{2}+\rho_{2}\kappa_{2} (\xi_{2}\tau_{2}+\nu_{2} )+\rho_{2}\gamma _{2} \}+\frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta_{1}^{2}} \{ \rho_{1}\kappa_{1} (\xi_{1}\tau_{1}+\nu_{1} ) \}< 1, \end{cases}\displaystyle \end{aligned}$$
(18)
then there exist
\((x_{1},x_{2})\in K_{1}\times K_{2}\), \(u_{i}\in T_{i}(x_{1})\), \(v_{i}\in S_{i}(x_{2})\), \(w_{i}\in B_{i}(x_{1},x_{2})\), and
\(z_{i}\in A_{i}(x_{i})\)
such that
\((x_{1},x_{2},u_{1},u_{2},v_{1},v_{2},w_{1},w_{2},z_{1},z_{2} )\)
is the solution of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems (1) and the sequences
\(\{ x_{1}^{n} \}\), \(\{x_{2}^{n} \}\), \(\{u_{i}^{n} \}\), \(\{v_{i}^{n} \}\), \(\{w_{i}^{n} \}\), and
\(\{ z_{i}^{n} \}\)
generated by Algorithm
4.1
converge strongly to
\(x_{1}\), \(x_{2}\), \(u_{i}\), \(v_{i}\), \(w_{i}\), and
\(z_{i}\), respectively.
Proof
Firstly, from (15) of Algorithm 4.1, we have, for all \(y_{1} \in K_{1}\),
$$\begin{aligned}& \rho_{1} N_{1} \bigl(z_{1}^{n}, \eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n} \bigr) \bigr) \bigr)+ \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n-1} \bigr) \bigr) + \rho _{1} \bigl\{ M_{1} \bigl(u_{1}^{n-1},v_{1}^{n-1} \bigr) \\ & \quad{}+w_{1}^{n-1} \bigr\} ,\eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n} \bigr) \bigr) \bigr\rangle +\rho_{1} \bigl\{ \phi_{1} \bigl(x_{1}^{n-1},y_{1} \bigr)-\phi_{1} \bigl(x_{1}^{n-1},x_{1}^{n} \bigr) \\ & \quad{}+\alpha_{1} \bigl\Vert f_{1}(y_{1})-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2} \bigr\} \geq0 \end{aligned}$$
(19)
and
$$\begin{aligned}& \rho_{1} N_{1} \bigl(z_{1}^{n+1}, \eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr)+ \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr) + \rho _{1} \bigl\{ M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr) \\ & \quad{}+w_{1}^{n} \bigr\} ,\eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle +\rho_{1} \bigl\{ \phi_{1} \bigl(x_{1}^{n},y_{1} \bigr)-\phi_{1} \bigl(x_{1}^{n},x_{1}^{n+1} \bigr) \\ & \quad{}+\alpha_{1} \bigl\Vert f_{1}(y_{1})-f_{1} \bigl(x_{1}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0. \end{aligned}$$
(20)
Putting \(y_{1}=x_{1}^{n+1}\) in (19) and \(y_{1}=x_{1}^{n}\) in (20), and summing up the resulting inequalities, we obtain
$$\begin{aligned}& \rho_{1} \bigl\{ N_{1} \bigl(z_{1}^{n}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr),f_{1} \bigl(x_{1}^{n} \bigr) \bigr) \bigr)+N_{1} \bigl(z_{1}^{n+1}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr) \bigr\} \\& \quad{}+\bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n-1} \bigr) \bigr)+\rho_{1} \bigl\{ M_{1} \bigl(u_{1}^{n-1},v_{1}^{n-1} \bigr)+w_{1}^{n-1} \bigr\} ,\eta_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr),f_{1} \bigl(x_{1}^{n} \bigr) \bigr) \bigr\rangle \\& \quad{}+ \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr)+\rho_{1} \bigl\{ M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr)+w_{1}^{n} \bigr\} ,\eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle \\& \quad{}+\rho_{1} \bigl\{ \phi_{1} \bigl(x_{1}^{n-1},x_{1}^{n+1} \bigr)-\phi_{1} \bigl(x_{1}^{n-1},x_{1}^{n} \bigr)+\phi_{1} \bigl(x_{1}^{n},x_{1}^{n} \bigr)-\phi_{1} \bigl(x_{1}^{n},x_{1}^{n+1} \bigr) \\& \quad{}+\alpha_{1}\bigl\Vert f_{1} \bigl(x_{1}^{n+1} \bigr)-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2}+\alpha_{1}\bigl\Vert f_{1} \bigl(x_{1}^{n} \bigr)-f_{1} \bigl(x_{1}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0, \end{aligned}$$
which implies that
$$\begin{aligned}& \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n-1} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr),\eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle \\ & \qquad{}+\rho_{1} \bigl\langle M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr),\eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle +2\alpha_{1}\rho_{1}\bigl\Vert f_{1} \bigl(x_{1}^{n+1} \bigr)-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2} \\ & \qquad{}+\rho_{1} \bigl\langle w_{1}^{n}-w_{1}^{n-1}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle +\rho_{1}\phi_{1} \bigl(x_{1}^{n}-x_{1}^{n-1},x_{1}^{n}-x_{1}^{n+1} \bigr) \\ & \quad\geq \rho_{1} \bigl\{ N_{1} \bigl(z_{1}^{n}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr),f_{1} \bigl(x_{1}^{n} \bigr) \bigr) \bigr)+N_{1} \bigl(z_{1}^{n+1}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr) \bigr\} \\ & \qquad{}+ \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr),\eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle . \end{aligned}$$
Since \(N_{1}\) is \(\varrho_{1}\)-\(\eta_{1}\)-\(f_{1}\)-strongly monotone with respect to \(A_{1}\), \(g_{1}\) is \(\varepsilon_{1}\)-\(\eta_{1}\)-relaxed strongly monotone with respect to \(f_{1}\), \(\phi_{1}\) is bounded by assumption and using the Cauchy-Schwartz inequality, we have
$$\begin{aligned}& \rho_{1} \varrho_{1} \bigl\Vert f_{1} \bigl(x_{1}^{n+1} \bigr)-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2}- \varepsilon_{1} \bigl\Vert f_{1} \bigl(x_{1}^{n+1} \bigr)-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2} \\ & \quad\leq \rho_{1} \bigl\{ N_{1} \bigl(z_{1}^{n}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr),f_{1} \bigl(x_{1}^{n} \bigr) \bigr) \bigr)+N_{1} \bigl(z_{1}^{n+1}, \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr) \bigr\} \\ & \qquad {}+\bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr),\eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle \\ & \quad\leq\bigl\Vert g_{1} \bigl(f_{1} \bigl(x_{1}^{n-1} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr)\bigr\Vert \bigl\Vert \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)\bigr\Vert \\ & \qquad{}+\rho_{1}\bigl\Vert M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr)-M_{1} \bigl(u_{1}^{n-1},v_{1}^{n-1} \bigr)\bigr\Vert \bigl\Vert \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)\bigr\Vert \\ & \qquad {}+\rho_{1}\bigl\Vert w_{1}^{n}-w_{1}^{n-1} \bigr\Vert \bigl\Vert \eta_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)\bigr\Vert + \rho_{1}\gamma _{1} \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert \bigl\Vert x_{1}^{n}-x_{1}^{n+1}\bigr\Vert \\ & \qquad{}+2\alpha_{1}\rho_{1}\bigl\Vert f_{1} \bigl(x_{1}^{n+1} \bigr)-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2}. \end{aligned}$$
(21)
By using \((\mu_{1},\xi_{1} )\)-mixed Lipschitz continuity of \(M_{1}\), \(\delta_{i}\)-\(\mathcal{D}\)-Lipschitz continuity of \(T_{1}\) and \(\tau _{i}\)-\(\mathcal{D}\)-Lipschitz continuity of \(S_{1}\), it follows by Algorithm 4.1 that
$$\begin{aligned}& \bigl\Vert M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr)-M_{1} \bigl(u_{1}^{n-1},v_{1}^{n-1} \bigr)\bigr\Vert \\& \quad\leq\mu_{1}\bigl\Vert u_{1}^{n}-u_{1}^{n-1} \bigr\Vert +\xi_{1}\bigl\Vert v_{1}^{n}-v_{1}^{n-1} \bigr\Vert \\& \quad\leq\mu_{1} \biggl(1+\frac{1}{n} \biggr)\mathcal{D}\bigl\Vert T_{1} \bigl(x_{1}^{n} \bigr)-T_{1} \bigl(x_{1}^{n-1} \bigr)\bigr\Vert +\xi_{1} \biggl(1+\frac {1}{n} \biggr) \mathcal{D}\bigl\Vert S_{1} \bigl(x_{2}^{n} \bigr)-S_{1} \bigl(x_{2}^{n-1} \bigr)\bigr\Vert \\& \quad\leq\mu_{1}\delta_{1} \biggl(1+\frac{1}{n} \biggr) \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert +\xi_{1}\tau_{1} \biggl(1+\frac{1}{n} \biggr) \bigl\Vert x_{2}^{n}-x_{2}^{n-1}\bigr\Vert . \end{aligned}$$
(22)
Also by Algorithm 4.1 and \((\zeta_{1},\nu_{1} )\)-\(\mathcal{D}\)-Lipschitz continuity of \(B_{1}\), we have
$$\begin{aligned} \bigl\Vert w_{1}^{n}-w_{1}^{n-1} \bigr\Vert \leq& \biggl(1+\frac{1}{n} \biggr)\mathcal {D} \bigl(B_{1} \bigl(x_{1}^{n},x_{2}^{n} \bigr),B_{1} \bigl(x_{1}^{n-1},x_{2}^{n-1} \bigr) \bigr) \\ \leq& \biggl(1+\frac{1}{n} \biggr) \bigl(\zeta_{1}\bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert + \nu_{1}\bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert \bigr) \\ =&\zeta_{1} \biggl(1+\frac{1}{n} \biggr)\bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert +\nu _{1} \biggl(1+\frac{1}{n} \biggr)\bigl\Vert x_{2}^{n}-x_{2}^{n-1}\bigr\Vert . \end{aligned}$$
(23)
Since \(g_{1}\) is \(\sigma_{1}\)-Lipschitz continuous with respect to \(f_{1}\), \(\eta_{1}\) is \(\kappa_{1}\)-Lipschitz continuous with respect to \(f_{1}\), \(f_{1}\) is \(\beta_{1}\)-expansive with the condition \(3\varepsilon_{1}<\rho _{1}\varrho_{1}\), it follows from (21), (22), and (23) that
$$\begin{aligned}& (\rho_{1} \varrho_{1}-3 \varepsilon_{1} ) \beta_{1}^{2} \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert ^{2} \\& \quad\leq (\rho_{1}\varrho_{1}-3\varepsilon_{1})\bigl\Vert f_{1} \bigl(x_{1}^{n+1} \bigr)-f_{1} \bigl(x_{1}^{n} \bigr)\bigr\Vert ^{2} \\& \quad\leq \kappa_{1} \sigma_{1} \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n}\bigr\Vert +\rho_{1}\kappa_{1} \biggl\{ \mu_{1} \delta_{1} \biggl(1+\frac{1}{n} \biggr) \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert \\& \qquad{}+ \xi_{1} \tau_{1}\biggl(1+\frac{1}{n} \biggr)\bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert \biggr\} \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert +\kappa_{1}\rho_{1} \biggl\{ \zeta_{1} \biggl(1+\frac {1}{n} \biggr) \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert \\& \qquad{}+\nu_{1} \biggl(1+\frac{1}{n} \biggr) \bigl\Vert x_{2}^{n}-x_{2}^{n-1}\bigr\Vert \biggr\} \bigl\Vert x_{1}^{n+1}-x_{1}^{n}\bigr\Vert +\rho_{1} \gamma_{1} \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert \\& \quad= \kappa_{1}\sigma_{1} \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert +\rho_{1}\kappa_{1}\mu_{1} \delta_{1} \biggl(1+\frac {1}{n} \biggr) \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n}\bigr\Vert \\& \qquad{}+\rho_{1}\kappa_{1}\xi_{1} \tau_{1} \biggl(1+\frac{1}{n} \biggr)\bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert +\kappa_{1}\rho _{1}\zeta_{1} \biggl(1+ \frac{1}{n} \biggr) \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert \bigl\Vert x_{1}^{n+1} \\& \qquad{}-x_{1}^{n} \bigr\Vert +\kappa_{1}\rho_{1} \nu_{1} \biggl(1+ \frac{1}{n} \biggr)\bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert +\rho_{1} \gamma_{1} \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n}\bigr\Vert \\& \quad= \biggl\{ \kappa_{1}\sigma_{1}+\rho_{1} \kappa_{1}\mu_{1}\delta_{1} \biggl(1+ \frac {1}{n} \biggr)+\kappa_{1}\rho_{1} \zeta_{1} \biggl(1+\frac{1}{n} \biggr)+\rho_{1} \gamma_{1} \biggr\} \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert \\& \qquad {}+\biggl\{ \rho_{1}\kappa_{1}\xi_{1} \tau_{1} \biggl(1+\frac{1}{n} \biggr)+\kappa _{1} \rho_{1} \nu_{1} \biggl(1+\frac{1}{n} \biggr) \biggr\} \bigl\Vert x_{2}^{n}-x_{2}^{n-1}\bigr\Vert \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert , \end{aligned}$$
which implies that
$$\begin{aligned}& \bigl\Vert x_{1}^{n+1}-x_{1}^{n}\bigr\Vert \\& \quad\leq \frac{1}{ (\rho_{1}\varrho_{1}-3 \varepsilon_{1} )\beta _{1}^{2}} \biggl[ \biggl\{ \kappa_{1} \sigma_{1}+ \biggl(\rho_{1}\kappa_{1} \biggl(1+ \frac {1}{n} \biggr) \biggr) (\mu_{1}\delta_{1}+ \zeta_{1} )+\rho_{1} \gamma _{1} \biggr\} \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert \\& \qquad{}+ \biggl\{ \biggl(\rho_{1}\kappa_{1} \biggl(1+ \frac{1}{n} \biggr) \biggr) (\xi_{1}\tau_{1}+ \nu_{1} ) \biggr\} \bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert \biggr]. \end{aligned}$$
Hence,
$$ \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert \leq\theta_{1}^{n}\bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert + \vartheta_{1}^{n}\bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert , $$
(24)
where
$$\theta_{1}^{n}=\frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta_{1}^{2}} \biggl\{ \kappa_{1}\sigma_{1}+ \biggl(\rho_{1} \kappa_{1} \biggl(1+\frac{1}{n} \biggr) \biggr) (\mu_{1} \delta_{1}+\zeta_{1} )+\rho_{1} \gamma_{1} \biggr\} $$
and
$$\vartheta_{1}^{n}=\frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta_{1}^{2}} \biggl\{ \biggl( \rho_{1}\kappa_{1} \biggl(1+\frac{1}{n} \biggr) \biggr) (\xi_{1}\tau _{1}+\nu_{1} ) \biggr\} . $$
Secondly, it follows from (16) of Algorithm 4.1, for all \(y_{2} \in K_{2}\), that
$$\begin{aligned}& \rho_{2} N_{2} \bigl(z_{2}^{n}, \eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n} \bigr) \bigr) \bigr)+ \bigl\langle g_{2} \bigl(f_{2} \bigl(x_{2}^{n} \bigr) \bigr)-g_{2} \bigl(f_{2} \bigl(x_{2}^{n-1} \bigr) \bigr) + \rho _{2} \bigl\{ M_{2} \bigl(u_{2}^{n-1},v_{2}^{n-1} \bigr) \\& \quad{}+w_{2}^{n-1} \bigr\} ,\eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n} \bigr) \bigr) \bigr\rangle +\rho_{2} \bigl\{ \phi_{2} \bigl(x_{1}^{n-1},y_{2} \bigr)-\phi_{2} \bigl(x_{2}^{n-1},x_{2}^{n} \bigr) \\& \quad{}+\alpha_{2} \bigl\Vert f_{2}(y_{2})-f_{2} \bigl(x_{2}^{n} \bigr)\bigr\Vert ^{2} \bigr\} \geq0 \end{aligned}$$
and
$$\begin{aligned}& \rho_{2} N_{2} \bigl(z_{2}^{n+1}, \eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr) \bigr)+ \bigl\langle g_{2} \bigl(f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr)-g_{2} \bigl(f_{2} \bigl(x_{2}^{n} \bigr) \bigr) + \rho _{2} \bigl\{ M_{2} \bigl(u_{2}^{n},v_{2}^{n} \bigr) \\& \quad{}+w_{2}^{n} \bigr\} ,\eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr) \bigr\rangle +\rho_{2} \bigl\{ \phi_{2} \bigl(x_{2}^{n},y_{2} \bigr)-\phi_{2} \bigl(x_{2}^{n},x_{2}^{n+1} \bigr) \\& \quad{}+\alpha_{2} \bigl\Vert f_{2}(y_{2})-f_{2} \bigl(x_{2}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0. \end{aligned}$$
Using the same arguments as above, the imposed conditions on \(N_{2}\), \(g_{2}\), \(\eta_{2}\), \(f_{2}\), \(A_{2}\), \(T_{2}\), \(S_{2}\), and Algorithm 4.1, we obtain
$$ \bigl\Vert x_{2}^{n+1}-x_{2}^{n} \bigr\Vert \leq\theta_{2}^{n}\bigl\Vert x_{2}^{n}-x_{2}^{n-1}\bigr\Vert + \vartheta_{2}^{n} \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert , $$
(25)
where
$$\theta_{2}^{n}=\frac{1}{(\rho_{2}\varrho_{2}-3 \varepsilon_{2})\beta_{2}^{2}} \biggl\{ \kappa_{2}\sigma_{2}+ \biggl(\rho_{2} \kappa_{2} \biggl(1+\frac{1}{n} \biggr) \biggr) (\xi_{2} \tau_{2}+\nu_{2} )+\rho_{2}\gamma_{2} \biggr\} $$
and
$$\vartheta_{2}^{n}=\frac{1}{(\rho_{2} \varrho_{2}-3 \varepsilon_{2})\beta_{2}^{2}} \biggl\{ \biggl( \rho_{2}\kappa_{2} \biggl(1+\frac{1}{n} \biggr) \biggr) (\mu_{2}\delta _{2}+\zeta_{2} ) \biggr\} . $$
Adding (24) and (25), we have
$$\begin{aligned} \bigl\Vert x_{1}^{n+1}-x_{1}^{n} \bigr\Vert +\bigl\Vert x_{2}^{n+1}-x_{2}^{n} \bigr\Vert \leq& \bigl\{ \theta_{1}^{n}+ \vartheta_{2}^{n} \bigr\} \bigl\Vert x_{1}^{n}-x_{1}^{n-1} \bigr\Vert + \bigl\{ \theta_{2}^{n}+\vartheta_{1}^{n} \bigr\} \bigl\Vert x_{2}^{n}-x_{2}^{n-1} \bigr\Vert \\ \leq& \max \bigl\{ \widetilde{\theta}_{1}^{n},\widetilde{\theta}_{2}^{n} \bigr\} \bigl\{ \bigl\Vert x_{1}^{n}-x_{1}^{n-1}\bigr\Vert +\bigl\Vert x_{2}^{n}-x_{2}^{n-1}\bigr\Vert \bigr\} , \end{aligned}$$
(26)
where
$$\begin{aligned} \widetilde{\theta}_{1}^{n}= \bigl\{ \theta_{1}^{n}+ \vartheta_{2}^{n} \bigr\} =&\frac {1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta_{1}^{2}} \biggl\{ \kappa_{1}\sigma _{1}+ \biggl(\rho_{1} \kappa_{1} \biggl(1+\frac{1}{n} \biggr) \biggr) (\mu _{1}\delta_{1}+\zeta_{1} )+\rho_{1} \gamma_{1} \biggr\} \\ &{}+\frac{1}{(\rho_{2} \varrho_{2}-3 \varepsilon_{2})\beta_{2}^{2}} \biggl\{ \biggl(\rho_{2}\kappa_{2} \biggl(1+\frac{1}{n} \biggr) \biggr) (\mu_{2}\delta _{2}+\zeta_{2} ) \biggr\} \end{aligned}$$
and
$$\begin{aligned} \widetilde{\theta}_{2}^{n}= \bigl\{ \theta_{2}^{n}+ \vartheta_{1}^{n} \bigr\} =&\frac {1}{(\rho_{2}\varrho_{2}-3 \varepsilon_{2})\beta_{2}^{2}} \biggl\{ \kappa_{2}\sigma _{2}+ \biggl(\rho_{2} \kappa_{2} \biggl(1+\frac{1}{n} \biggr) \biggr) (\xi_{2} \tau _{2}+\nu_{2} )+\rho_{2}\gamma_{2} \biggr\} \\ &{}+\frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta_{1}^{2}} \biggl\{ \biggl(\rho_{1}\kappa_{1} \biggl(1+\frac{1}{n} \biggr) \biggr) (\xi_{1}\tau_{1}+ \nu _{1} ) \biggr\} . \end{aligned}$$
Letting
$$\begin{aligned}& \widetilde{\theta}_{1}=\frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon_{1})\beta _{1}^{2}} \bigl\{ \kappa_{1} \sigma_{1}+\rho_{1}\kappa_{1} (\mu_{1} \delta_{1}+\zeta _{1} )+\rho_{1} \gamma_{1} \bigr\} +\frac{1}{(\rho_{2} \varrho_{2}-3 \varepsilon_{2})\beta_{2}^{2}} \bigl\{ \rho_{2} \kappa_{2} (\mu_{2}\delta_{2}+\zeta _{2} ) \bigr\} \end{aligned}$$
and
$$\begin{aligned}& \widetilde{\theta}_{2}=\frac{1}{(\rho_{2}\varrho_{2}-3 \varepsilon_{2})\beta _{2}^{2}} \bigl\{ \kappa_{2} \sigma_{2}+\rho_{2}\kappa_{2} (\xi_{2} \tau_{2}+\nu_{2} )+\rho_{2}\gamma_{2} \bigr\} +\frac{1}{(\rho_{1} \varrho_{1}-3 \varepsilon _{1})\beta_{1}^{2}} \bigl\{ \rho_{1}\kappa_{1} ( \xi_{1}\tau_{1}+\nu_{1} ) \bigr\} , \end{aligned}$$
it can easily be seen that \(\widetilde{\theta}_{1}^{n}\rightarrow\widetilde{\theta}_{1}\) and \(\widetilde{\theta}_{2}^{n} \rightarrow\widetilde{\theta}_{2}\), as \(n\rightarrow\infty\). Taking into account the condition (18), we conclude that \(\max \{\widetilde{\theta}_{1},\widetilde{\theta}_{2} \}<1\). Hence, it follows from (26) that \(\{(x_{1}^{n},x_{2}^{n})\}\) is a Cauchy sequence in \(K_{1}\times K_{2}\); now suppose that \((x_{1}^{n},x_{2}^{n} )\rightarrow (x_{1},x_{2} )\in K_{1}\times K_{2}\), as \(n \rightarrow\infty\). By Algorithm 4.1 and \(\mathcal {D}\)-Lipschitz continuity of \(T_{i}, S_{i}, B_{i}\) and \(A_{i}\), for each \(i \in I\), we have
$$\begin{aligned}& \bigl\Vert u_{i}^{n+1}-u_{i}^{n}\bigr\Vert \leq \biggl(1+\frac{1}{n+1} \biggr)\mathcal{D} \bigl(T_{i} \bigl(x_{1}^{n+1} \bigr),T_{i} \bigl(x_{1}^{n} \bigr) \bigr) \\ & \phantom{\bigl\Vert u_{i}^{n+1}-u_{i}^{n}\bigr\Vert }\leq \biggl(1+\frac{1}{n+1} \biggr)\delta_{i}\bigl\Vert x_{1}^{n+1}-x_{i}^{n}\bigr\Vert ; \\ & \bigl\Vert v_{i}^{n+1}-v_{i}^{n}\bigr\Vert \leq \biggl(1+\frac{1}{n+1} \biggr)\mathcal{D} \bigl(S_{i} \bigl(x_{2}^{n+1} \bigr),S_{i} \bigl(x_{2}^{n} \bigr) \bigr) \\ & \phantom{\bigl\Vert v_{i}^{n+1}-v_{i}^{n}\bigr\Vert }\leq \biggl(1+\frac{1}{n+1} \biggr)\tau_{i}\bigl\Vert x_{2}^{n+1}-x_{2}^{n}\bigr\Vert ; \\ & \bigl\Vert w_{i}^{n+1}-w_{i}^{n}\bigr\Vert \leq \biggl(1+\frac{1}{n+1} \biggr)\mathcal{D} \bigl(B_{i} \bigl(x_{1}^{n+1},x_{2}^{n+1} \bigr),B_{i} \bigl(x_{1}^{n},x_{2}^{n} \bigr) \bigr) \\ & \phantom{\bigl\Vert w_{i}^{n+1}-w_{i}^{n}\bigr\Vert }\leq \biggl(1+\frac{1}{n+1} \biggr) \bigl(\zeta_{i}\bigl\Vert x_{1}^{n+1}-x_{1}^{n}\bigr\Vert +\nu_{i}\bigl\Vert x_{2}^{n+1}-x_{2}^{n} \bigr\Vert \bigr); \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert z_{i}^{n+1}-z_{i}^{n}\bigr\Vert \leq& \biggl(1+\frac{1}{n+1} \biggr)\mathcal{D} \bigl(A_{i} \bigl(x_{i}^{n+1} \bigr),A_{i} \bigl(x_{i}^{n} \bigr) \bigr) \\ \leq& \biggl(1+\frac{1}{n+1} \biggr)\varsigma_{i}\bigl\Vert x_{i}^{n+1}-x_{i}^{n}\bigr\Vert . \end{aligned}$$
Therefore, for each \(i \in I\), \(\{u_{i}^{n} \}, \{v_{i}^{n} \}, \{w_{i}^{n} \}\), and \(\{z_{i}^{n} \}\) are also Cauchy sequences; now assume that \(u_{i}^{n} \rightarrow u_{i}\), \(v_{i}^{n} \rightarrow v_{i}\), \(w_{i}^{n} \rightarrow w_{i}\), and \(z_{i}^{n} \rightarrow z_{i}\), as \(n \rightarrow\infty\). As \(u_{i}^{n} \in T_{i} (x_{1}^{n} )\), we have
$$\begin{aligned} d \bigl(u_{i},T_{i} (x_{1} ) \bigr) =&\bigl\Vert u_{i}-u_{i}^{n}\bigr\Vert +d \bigl(u_{i}^{n},T_{i} \bigl(x_{1}^{n} \bigr) \bigr)+\mathcal{D} \bigl(T_{i} \bigl(x_{1}^{n} \bigr),T_{i} (x_{1} ) \bigr) \\ \leq&\bigl\Vert u_{i}-u_{i}^{n}\bigr\Vert + \delta_{i}\bigl\Vert x_{1}^{n}-x_{1} \bigr\Vert \rightarrow0\quad \mbox{as } n\rightarrow\infty. \end{aligned}$$
Therefore, we deduce that \(u_{i} \in T_{i}(x_{1})\). Similarly, we can obtain \(v_{i} \in S_{i}(x_{2})\), \(w_{i} \in B_{i}(x_{1},x_{2})\), and \(z_{i} \in A_{i}(x_{i})\), for each \(i \in I\).
By Algorithm 4.1, we have
$$\begin{aligned}& \rho_{1} N_{1} \bigl(z_{1}^{n+1}, \eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr)+ \bigl\langle g_{1} \bigl(f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr)-g_{1} \bigl(f_{1} \bigl(x_{1}^{n} \bigr) \bigr) + \rho _{1} \bigl\{ M_{1} \bigl(u_{1}^{n},v_{1}^{n} \bigr) \\& \quad{}+w_{1}^{n} \bigr\} ,\eta_{1} \bigl(f_{1}(y_{1}),f_{1} \bigl(x_{1}^{n+1} \bigr) \bigr) \bigr\rangle +\rho_{1} \bigl\{ \phi_{1} \bigl(x_{1}^{n},y_{1} \bigr)-\phi_{1} \bigl(x_{1}^{n},x_{1}^{n+1} \bigr) \\& \quad{}+\alpha_{1} \bigl\Vert f_{1}(y_{1})-f_{1} \bigl(x_{1}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0,\quad\forall y_{1} \in K_{1}; \end{aligned}$$
(27)
and
$$\begin{aligned}& \rho_{2} N_{2} \bigl(z_{2}^{n+1}, \eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr) \bigr)+ \bigl\langle g_{2} \bigl(f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr)-g_{2} \bigl(f_{2} \bigl(x_{2}^{n} \bigr) \bigr) + \rho _{2} \bigl\{ M_{2} \bigl(u_{2}^{n},v_{2}^{n} \bigr) \\& \quad{}+w_{2}^{n} \bigr\} ,\eta_{2} \bigl(f_{2}(y_{2}),f_{2} \bigl(x_{2}^{n+1} \bigr) \bigr) \bigr\rangle +\rho_{2} \bigl\{ \phi_{2} \bigl(x_{2}^{n},y_{2} \bigr)-\phi_{2} \bigl(x_{2}^{n},x_{2}^{n+1} \bigr) \\& \quad{}+\alpha_{2} \bigl\Vert f_{2}(y_{2})-f_{2} \bigl(x_{2}^{n+1} \bigr)\bigr\Vert ^{2} \bigr\} \geq0,\quad\forall y_{2} \in K_{2}. \end{aligned}$$
(28)
By using the continuity of \(N_{i}\), \(M_{i}\), \(g_{i}\), \(\phi_{i}\), \(f_{i}\), and \(\eta_{i}\), for each \(i \in I\), and since \(u_{i}^{n} \rightarrow u_{i}\), \(v_{i}^{n} \rightarrow v_{i}\), \(w_{i}^{n} \rightarrow w_{i}\), \(z_{i}^{n} \rightarrow z_{i}\), and \(x_{i}^{n} \rightarrow x_{i}\) for \(n \rightarrow\infty\), from (27) and (28), we have, for \(\rho_{i} >0\),
$$\begin{aligned}& N_{1}\bigl(z_{1},\eta_{1}\bigl(f_{1}(y_{1}),f_{1}(x_{1}) \bigr)\bigr)+\bigl\langle M_{1}(u_{1},v_{1})+w_{1}, \eta _{1}\bigl(f_{1}(y_{1}),f_{1}(x_{1}) \bigr)\bigr\rangle \\& \quad{}+\phi_{1}(x_{1},y_{1})-\phi_{1}(x_{1},x_{1})+ \alpha_{1}\bigl\| f_{1}(y_{1})-f_{1}(x_{1}) \bigr\| ^{2}\geq 0,\quad\forall y_{1}\in K_{1}, \end{aligned}$$
and
$$\begin{aligned}& N_{2}\bigl(z_{2},\eta_{2}\bigl(f_{2}(y_{2}),f_{2}(x_{2}) \bigr)\bigr)+\bigl\langle M_{2}(u_{2},v_{2})+w_{2}, \eta _{2}\bigl(f_{2}(y_{2}),f_{2}(x_{2}) \bigr)\bigr\rangle \\& \quad{}+\phi_{2}(x_{2},y_{2})-\phi_{2}(x_{2},x_{2})+ \alpha_{2}\bigl\| f_{2}(y_{2})-f_{2}(x_{2}) \bigr\| ^{2}\geq 0,\quad\forall y_{2}\in K_{2}. \end{aligned}$$
Therefore \((x_{1},x_{2},u_{1},u_{2},v_{1},v_{2},z_{1},z_{2},w_{1},w_{2} )\) is the solution of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems (1). This completes the proof. □