Let \(\mathcal{H}\) be a real Hilbert space whose norm and inner product are denoted by \(\\cdot\\) and \(\langle\cdot,\cdot\rangle\), respectively. Let K be a nonempty closed set in \(\mathcal{H}\) and let \(CB(\mathcal{H})\) be the family of all nonempty, closed, and bounded subsets of \(\mathcal {H}\). Suppose further that \(S,T:K\rightarrow CB(\mathcal{H})\) are two multivalued operators and let \(g:K\rightarrow K\) and \(\eta:K\times K\rightarrow\mathcal{H}\) be two operators. For given bifunctions \(F,G:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\), we consider the problem of finding \(u\in K\), \(\nu\in T(u)\), and \(\vartheta \in S(u)\) such that
$$\begin{aligned} F\bigl(\nu,g(v)\bigr)+G\bigl(\vartheta,\eta\bigl(g(v),g(u)\bigr) \bigr)\geq0,\quad \forall v\in K, \end{aligned}$$
(2.1)
which is called the generalized multivalued equilibriumlike problem
\((\operatorname{GMELP})\).
If \(\eta(x,y)=xy\), for all \(x,y\in K\), then the problem (2.1) reduces to the problem of finding \(u\in K\), \(\nu\in T(u)\), and \(\vartheta\in S(u)\) such that
$$\begin{aligned} F\bigl(\nu,g(v)\bigr)+G\bigl(\vartheta,g(v)g(u)\bigr)\geq0, \quad \forall v\in K, \end{aligned}$$
(2.2)
which is called the generalized multivalued hemiequilibrium problem
\((\operatorname{GMHEP})\).
It should be remarked that by taking different choices of the operators S, T, η, g, and the bifunctions F and G in the above problems, one can easily obtain the problems studied in [11, 33] and the references therein.
In the sequel, we denote by \(\operatorname{GMELP}(F,G,S,T,\eta,g,K)\) and \(\operatorname{GMHEP}(F,G,S,T,g,K)\) the set of solutions of the problems (2.1) and (2.2), respectively.
Lemma 2.1
[34]
Let
X
be a complete metric space, \(T:X\rightarrow CB(X)\)
be a multivalued mapping. Then for any
\(\varepsilon>0\)
and for any given
\(x,y\in X\), \(u\in T(x)\), there exists
\(v\in T(y)\)
such that
$$\begin{aligned} d(u,v)\leq(1+\varepsilon)M\bigl(T(x),T(y)\bigr), \end{aligned}$$
where
\(M(\cdot,\cdot)\)
is the Hausdorff metric on
\(CB(X)\)
defined by
$$\begin{aligned} M(A,B)=\max \Bigl\{ \sup_{x\in A}\inf_{y\in B}\xy \,\sup_{y\in B}\inf_{x\in A}\xy\ \Bigr\} ,\quad \forall A,B\in CB(X). \end{aligned}$$
Let F, G, S, T, η, and g be the same as in GMELP (2.1). For given \(u\in K\), \(\nu\in T(u)\), and \(\vartheta\in S(u)\), we consider the auxiliary generalized multivalued hemiequilibriumlike problem of finding \(w\in K\) such that
$$\begin{aligned} \rho F\bigl(\nu,g(v)\bigr)+\rho G\bigl(\vartheta,\eta \bigl(g(v),g(u)\bigr)\bigr)+\bigl\langle g(w)g(u),g(v)g(w)\bigr\rangle \geq0, \quad \forall v\in K, \end{aligned}$$
(2.3)
where \(\rho>0\) is a constant. Obviously, if \(w=u\), then \((w,\nu ,\vartheta)\) is a solution of GMELP (2.1). This observation and Nadler’s technique [34] enables us to suggest the following finite step predictorcorrector method for solving GMELP (2.1).
Algorithm 2.2
Let F, G, S, T, η, and g be the same as in GMELP (2.1). For given \(u_{0}\in K\), \(\nu_{0}\in T(u_{0})\), and \(\vartheta_{0}\in S(u_{0})\), compute the iterative sequences \(\{u_{n}\}\), \(\{ \nu_{n}\}\), and \(\{\vartheta_{n}\}\) by the iterative schemes
$$\begin{aligned}& \begin{aligned}[b] &\rho_{1} F\bigl(\nu_{1,n},g(v)\bigr)+ \rho_{1} G\bigl(\vartheta_{1,n},\eta \bigl(g(v),g(u_{1,n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(u_{n+1})g(y_{1,n}),g(v)g(u_{n+1}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \end{aligned}$$
(2.4)
$$\begin{aligned}& \begin{aligned}[b] &\rho_{i+1} F\bigl(\nu_{i+1,n},g(v)\bigr)+ \rho_{i+1} G\bigl(\vartheta_{i+1,n},\eta \bigl(g(v),g(u_{i+1,n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{i,n})g(y_{i+1,n}),g(v)g(y_{i,n}) \bigr\rangle \geq0,\quad i=1,2,\ldots ,q2, \forall v\in K, \end{aligned} \end{aligned}$$
(2.5)
$$\begin{aligned}& \begin{aligned}[b] &\rho_{q} F\bigl(\nu_{n},g(v)\bigr)+ \rho_{q} G\bigl(\vartheta_{n},\eta\bigl(g(v),g(u_{n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{q1,n})g(u_{n}),g(v)g(y_{q1,n}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \end{aligned}$$
(2.6)
$$\begin{aligned}& \begin{aligned}[b] &\nu_{i,n}\in T(y_{i,n}):\\nu_{i,n+1} \nu_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(y_{i,n+1}),T(y_{i,n}) \bigr),\\ &\quad i=1,2,\ldots,q1, \end{aligned} \end{aligned}$$
(2.7)
$$\begin{aligned}& \begin{aligned}[b] &\vartheta_{i,n}\in S(y_{i,n}):\\vartheta_{i,n+1} \vartheta_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M \bigl(S(y_{i,n+1}),S(y_{i,n})\bigr), \\ &\quad i=1,2, \ldots,q1, \end{aligned} \end{aligned}$$
(2.8)
$$\begin{aligned}& \nu_{n}\in T(u_{n}):\\nu_{n+1}\nu_{n}\ \leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(u_{n+1}),T(u_{n}) \bigr), \end{aligned}$$
(2.9)
$$\begin{aligned}& \vartheta_{n}\in S(u_{n}):\\vartheta_{n+1} \vartheta_{n}\\leq \bigl(1+(1+n)^{1}\bigr)M \bigl(S(u_{n+1}),S(u_{n})\bigr), \end{aligned}$$
(2.10)
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\) .
If \(\eta(x,y)=xy\), for all \(x,y\in K\), then Algorithm 2.2 reduces to the following predictorcorrector method.
Algorithm 2.3
For given \(u_{0}\in K\), \(\nu_{0}\in T(u_{0})\) and \(\vartheta_{0}\in S(u_{0})\), compute the iterative sequences \(\{u_{n}\}\), \(\{\nu_{n}\}\), and \(\{\vartheta _{n}\}\) by the iterative schemes
$$\begin{aligned}& \begin{aligned}[b] &\rho_{1} F\bigl(\nu_{1,n},g(v)\bigr)+ \rho_{1} G\bigl(\vartheta_{1,n},g(v)g(u_{1,n})\bigr) \\ &\quad{}+\bigl\langle g(u_{n+1})g(y_{1,n}),g(v)g(u_{n+1}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \\& \begin{aligned}[b] &\rho_{i+1} F\bigl(\nu_{i+1,n},g(v)\bigr)+ \rho_{i+1} G\bigl(\vartheta _{i+1,n},g(v)g(u_{i+1,n}) \bigr) \\ &\quad{}+\bigl\langle g(y_{i,n})g(y_{i+1,n}),g(v)g(y_{i,n}) \bigr\rangle \geq0,\quad i=1,2,\ldots ,q2, \forall v\in K, \end{aligned} \\& \begin{aligned}[b] &\rho_{q} F\bigl(\nu_{n},g(v)\bigr)+ \rho_{q} G\bigl(\vartheta_{n},g(v)g(u_{n})\bigr) \\ &\quad{}+\bigl\langle g(y_{q1,n})g(u_{n}),g(v)g(y_{q1,n}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \\& \nu_{i,n}\in T(y_{i,n}):\\nu_{i,n+1} \nu_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(y_{i,n+1}),T(y_{i,n}) \bigr), \quad i=1,2,\ldots,q1, \\& \vartheta_{i,n}\in S(y_{i,n}):\\vartheta_{i,n+1} \vartheta_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M \bigl(S(y_{i,n+1}),S(y_{i,n})\bigr), \quad i=1,2,\ldots,q1, \\& \nu_{n}\in T(u_{n}):\\nu_{n+1}\nu_{n}\ \leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(u_{n+1}),T(u_{n}) \bigr), \\& \vartheta_{n}\in S(u_{n}):\\vartheta_{n+1} \vartheta_{n}\\leq \bigl(1+(1+n)^{1}\bigr)M \bigl(S(u_{n+1}),S(u_{n})\bigr), \end{aligned}$$
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\) .
In order to study the convergence analysis of the iterative sequences generated by Algorithm 2.2, we need the following definitions.
Definition 2.1
Let \(T:K\rightarrow CB(\mathcal{H})\) be a multivalued operator and \(g:K\rightarrow K\) be an operator. The bifunction \(F:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\) is said to be

(a)
gmonotone with respect to T, if
$$\begin{aligned} F\bigl(w_{1},g(u_{2})\bigr)+F\bigl(w_{2},g(u_{1}) \bigr)\leq0,\quad \forall u_{1},u_{2}\in K, w_{1}\in T(u_{1}), w_{2}\in T(u_{2}); \end{aligned}$$

(b)
partially
αrelaxed
gmonotone with respect to T, if there exists a constant \(\alpha>0\) such that
$$\begin{aligned}& F\bigl(w_{1},g(u_{2})\bigr)+F\bigl(w_{2},g(z) \bigr)\leq\alpha\bigl\ g(z)g(u_{1})\bigr\ ^{2},\\& \quad\forall u_{1},u_{2},z\in K, w_{1}\in T(u_{1}), w_{2}\in T(u_{2}). \end{aligned}$$
It should be pointed out that if \(z=u_{1}\), then the partially αrelaxed gmonotonicity of the bifunction F with respect to T reduces to gmonotonicity with respect to T. Meanwhile, if \(g\equiv I\), then parts (a) and (b) of Definition 2.1 reduce to the definition of monotonicity and partially αrelaxed monotonicity of the bifunction F with respect to T, respectively.
Definition 2.2
Let \(S:K\rightarrow CB(\mathcal{H})\) be a multivalued operator and let \(g:K\rightarrow K\) and \(\eta:K\times K\rightarrow\mathcal{H}\) be two operators. The bifunction \(G:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\) is said to be

(a)
gηmonotone with respect to S, if
$$\begin{aligned}& G\bigl(w_{1},\eta\bigl(g(u_{2}),g(u_{1})\bigr) \bigr)+G\bigl(w_{2},\eta\bigl(g(u_{1}),g(u_{2}) \bigr)\bigr)\leq0,\\& \quad \forall u_{1},u_{2}\in K, w_{1}\in S(u_{1}), w_{2}\in S(u_{2}); \end{aligned}$$

(b)
partially
βrelaxed
gηmonotone with respect to S, if there exists a constant \(\beta>0\) such that
$$\begin{aligned}& G\bigl(w_{1},\eta\bigl(g(u_{2}),g(u_{1})\bigr) \bigr)+G\bigl(w_{2},\eta\bigl(g(z),g(u_{2})\bigr)\bigr)\leq \beta\bigl\ g(z)g(u_{1})\bigr\ ^{2},\\& \quad \forall u_{1},u_{2},z \in K, w_{1}\in S(u_{1}), w_{2}\in S(u_{2}). \end{aligned}$$
It should be remarked that if \(z=u_{1}\), then the partially βrelaxed gηmonotonicity of the bifunction G with respect to S reduces to gmonotonicity of the bifunction G with respect to S. Furthermore, for the case when \(\eta(x,y)=xy\), for all \(x,y\in K\), then parts (a) and (b) of Definition 2.2 reduce to the definition of gmonotonicity and partially βrelaxed gmonotonicity of the bifunction G with respect to S, respectively.
Definition 2.3
A multivalued operator \(T:\mathcal{H}\rightarrow CB(\mathcal{H})\) is said to be MLipschitz continuous with constant δ, if there exists a constant \(\delta>0\) such that
$$\begin{aligned} M\bigl(T(u),T(v)\bigr)\leq\delta\uv\,\quad \forall u,v\in\mathcal{H}, \end{aligned}$$
where \(M(\cdot,\cdot)\) is the Hausdorff metric on \(CB(\mathcal{H})\).
The next proposition plays a crucial role in the study of convergence analysis of the iterative sequences generated by Algorithm 2.2.
Proposition 2.4
Let
F, G, S, T, η, and
g
be the same as in GMELP (2.1) and let
\(\hat{u}\in K\), \(\hat{\nu}\in T(u)\), and
\(\hat{\vartheta}\in G(\hat{u})\)
be the solution of GMELP (2.1). Suppose further that
\(\{u_{n}\}\)
and
\(\{y_{i,n}\}\) (\(i=1,2,\ldots,q1\)) are the sequences generated by Algorithm
2.2. If
F
is partially
αrelaxed
gηmonotone with respect to
T, and
G
is partially
βrelaxed
gηmonotone with respect to
S, then
$$\begin{aligned} &\bigl\ g(\hat{u})g(u_{n+1})\bigr\ ^{2}\leq\bigl\ g( \hat{u})g(y_{1,n})\bigr\ ^{2} \bigl(12(\alpha+\beta) \rho_{1}\bigr)\bigl\ g(u_{n+1})g(y_{1,n})\bigr\ ^{2}, \end{aligned}$$
(2.11)
$$\begin{aligned} &\bigl\ g(\hat{u})g(y_{i,n})\bigr\ ^{2}\leq\bigl\ g( \hat{u})g(y_{i+1,n})\bigr\ ^{2}\bigl(12(\alpha +\beta) \rho_{i+1}\bigr) \bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2}, \end{aligned}$$
(2.12)
$$\begin{aligned} &\bigl\ g(\hat{u})g(y_{q1,n})\bigr\ ^{2}\leq\bigl\ g( \hat{u})g(u_{n})\bigr\ ^{2}\bigl(12(\alpha +\beta) \rho_{q}\bigr)\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2}, \end{aligned}$$
(2.13)
for all
\(n\geq0\), where
\(i=1,2,\ldots,q2\).
Proof
Since \(\hat{u}\in K\), \(\hat{\nu}\in T(u)\), and \(\hat{\vartheta}\in S(u)\) are the solution of GMELP (2.1), we have
$$\begin{aligned} F\bigl(\hat{\nu},g(v)\bigr)+G\bigl(\hat{\vartheta},\eta \bigl(g(v),g(\hat{u})\bigr)\bigr)\geq0,\quad \forall v\in K. \end{aligned}$$
(2.14)
Taking \(v=u_{n+1}\) in (2.14) and \(v=u\) in (2.4), we get
$$\begin{aligned} F\bigl(\hat{\nu},g(u_{n+1})\bigr)+G\bigl(\hat{\vartheta}, \eta\bigl(g(u_{n+1}),g(\hat {u})\bigr)\bigr)\geq0 \end{aligned}$$
(2.15)
and
$$\begin{aligned} &\rho_{1} F\bigl(\nu_{1,n},g(\hat{u})\bigr)+ \rho_{1} G\bigl(\vartheta_{1,n},\eta\bigl(g(\hat {u}),g(u_{1,n})\bigr)\bigr) \\ &\quad{}+\bigl\langle g(u_{n+1})g(y_{1,n}),g( \hat {u})g(u_{n+1})\bigr\rangle \geq0. \end{aligned}$$
(2.16)
By combining (2.15) and (2.16) and taking into account of the facts that the bifunction F is partially αrelaxed gmonotone with respect to T, and the bifunction G is partially βrelaxed strongly gηmonotone with respect to G, it follows that
$$\begin{aligned} &\bigl\langle g(u_{n+1})g(y_{1,n}),g( \hat{u})g(u_{n+1})\bigr\rangle \\ &\quad\geq\rho _{1} F\bigl(\nu_{1,n},g(\hat{u})\bigr) \rho_{1} G\bigl(\vartheta_{1,n},\eta\bigl(g(\hat {u}),g(u_{1,n})\bigr)\bigr) \\ &\quad\geq\rho_{1} \bigl(F\bigl(\nu_{1,n},g(\hat{u})\bigr)+F \bigl(\hat{\nu},g(u_{n+1})\bigr) \\ &\qquad{}+G\bigl(\vartheta_{1,n},\eta\bigl(g(\hat{u}),g(u_{1,n}) \bigr)\bigr) +G\bigl(\hat{\vartheta},\eta\bigl(g(u_{n+1}),g(\hat{u}) \bigr)\bigr) \bigr) \\ &\quad\geq\rho_{1}\alpha \bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2} \rho_{1}\beta\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2} \\ &\quad=\rho_{1}(\alpha+\beta)\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2}. \end{aligned}$$
(2.17)
On the other hand, letting \(x=g(\hat{u})g(u_{n+1})\) and \(y=g(u_{n+1})g(y_{1,n})\) and by utilizing the wellknown property of the inner product, we have
$$\begin{aligned} &2\bigl\langle g(u_{n+1})g(y_{1,n}),g( \hat{u})g(u_{n+1})\bigr\rangle \\ &\quad=\bigl\ g(\hat{u})g(y_{1,n})\bigr\ ^{2} \bigl\ g( \hat{u})g(u_{n+1})\bigr\ ^{2}\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2}. \end{aligned}$$
(2.18)
Applying (2.17) and (2.18), it follows that
$$\begin{aligned} \bigl\ g(\hat{u})g(u_{n+1})\bigr\ ^{2}\leq\bigl\ g(\hat{u})g(y_{1,n}) \bigr\ ^{2}\bigl(12(\alpha +\beta)\rho_{1}\bigr) \bigl\ g(u_{n+1})g(y_{1,n})\bigr\ ^{2}, \end{aligned}$$
which is the required result (2.11).
Taking \(v=y_{i,n}\) (\(i=1,2,\ldots,q2\)) in (2.14) and \(v=\hat{u}\) in (2.5), for each \(i=1,2,\ldots,q2\), we have
$$\begin{aligned} F\bigl(\hat{\nu},g(y_{i,n})\bigr)+G\bigl(\hat{\vartheta}, \eta\bigl(g(y_{i,n}),g(\hat {u})\bigr)\bigr)\geq0 \end{aligned}$$
(2.19)
and
$$\begin{aligned} &\rho_{i+1} F\bigl(\nu_{i+1,n},g(\hat{u})\bigr)+ \rho_{i+1}G\bigl(\vartheta_{i+1,n},\eta \bigl(g( \hat{u}),g(u_{i+1,n})\bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{i,n})g(y_{i+1,n}),g( \hat {u})g(y_{i,n})\bigr\rangle \geq0. \end{aligned}$$
(2.20)
Letting \(x=g(\hat{u})g(y_{i,n})\) and \(y=g(y_{i,n})g(y_{i+1,n})\) for each \(i=1,2,\ldots,q2\), and by using the wellknown property of the inner product, one has
$$\begin{aligned} &2\bigl\langle g(y_{i,n})g(y_{i+1,n}),g( \hat{u})g(y_{i,n})\bigr\rangle \\ &\quad=\bigl\ g(\hat{u})g(y_{i+1,n})\bigr\ ^{2} \bigl\ g( \hat{u})g(y_{i,n})\bigr\ ^{2}\bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2}. \end{aligned}$$
(2.21)
In a similar fashion to the preceding analysis, employing (2.19)(2.21) and considering the facts that the bifunction F is partially αrelaxed gηmonotone with respect to T, and the bifunction G is partially βrelaxed gηmonotone with respect to S, for each \(i=1,2,\ldots, q2\), one can deduce that
$$\begin{aligned} \bigl\ g(\hat{u})g(y_{i,n})\bigr\ ^{2}\leq\bigl\ g(\hat{u})g(y_{i+1,n}) \bigr\ ^{2}\bigl(12(\alpha +\beta)\rho_{i+1}\bigr) \bigl\ g(y_{i,n})g(y_{i+1,n})\bigr\ ^{2}, \end{aligned}$$
which is the required result (2.12).
Taking \(v=y_{q1,n}\) in (2.14) and \(v=\hat{u}\) in (2.6), we have
$$\begin{aligned} F\bigl(\hat{\nu},g(y_{q1,n})\bigr)+G\bigl(\hat{\vartheta}, \eta\bigl(g(y_{q1,n}),g(\hat {u})\bigr)\bigr)\geq0 \end{aligned}$$
(2.22)
and
$$\begin{aligned} &\rho_{q} F\bigl(\nu_{n},g(\hat{u})\bigr)+ \rho_{q} G\bigl(\vartheta_{n},\eta\bigl(g( \hat{u}),g(u_{n})\bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{q1,n})g(u_{n}),g( \hat{u})g(y_{q1,n})\bigr\rangle \geq0. \end{aligned}$$
(2.23)
By assuming \(x=g(\hat{u})g(y_{q1,n})\) and \(y=g(y_{q1,n})g(u_{n})\), and by utilizing the wellknown property of the inner product, we obtain
$$\begin{aligned} &2\bigl\langle g(y_{q1,n})g(u_{n}),g( \hat{u})g(y_{q1,n})\bigr\rangle \\ &\quad=\bigl\ g(\hat{u})g(u_{n})\bigr\ ^{2} \bigl\ g( \hat{u})g(y_{q1,n})\bigr\ ^{2}\bigl\ g(y_{q1,n})g(u_{n}) \bigr\ ^{2}. \end{aligned}$$
(2.24)
By a similar way to that of proof of (2.17), by using (2.22)(2.24) and in light of the facts that the bifunction F is partially αrelaxed gηmonotone with respect to T, and the bifunction G is partially βrelaxed gηmonotone with respect to S, we can show that
$$\begin{aligned} \bigl\ g(\hat{u})g(y_{q1,n})\bigr\ \leq\bigl\ g(\hat{u})g(u_{n}) \bigr\ ^{2}\bigl(12(\alpha+\beta )\rho_{q}\bigr) \bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2}, \end{aligned}$$
which is the required result (2.13). This completes the proof. □
In the next theorem, the strong convergence of the iterative sequences generated by Algorithm 2.2 to a solution of GMELP (2.1) is established.
Theorem 2.5
Let
\(\mathcal{H}\)
be a finite dimensional real Hilbert space and let
\(g:K\rightarrow K\)
be a continuous and invertible operator. Suppose that the bifunction
\(F:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\)
is continuous in the first argument, the bifunction
\(G:\mathcal{H}\times \mathcal{H}\rightarrow\Bbb{R}\)
is continuous in both arguments and the operator
\(\eta:K\times K\rightarrow\mathcal{H}\)
is continuous in the second argument. Assume that the multivalued operators
\(S,T:K\rightarrow CB(\mathcal{H})\)
are
MLipschitz continuous with constants
σ
and
δ, respectively. Moreover, let all the conditions of Proposition
2.4
hold and
\(\operatorname{GMELP}(F,G,S,T,\eta,g,K)\neq\emptyset\). If
\(\rho_{i}\in(0,\frac {1}{2(\alpha+\beta)})\), for each
\(i=1,2,\ldots,q\), then the iterative sequences
\(\{u_{n}\}\), \(\{\nu_{n}\}\), and
\(\{\vartheta_{n}\}\)
generated by Algorithm
2.2
converge strongly to
\(\hat{u}\in K\), \(\hat {\nu}\in T(\hat{u})\), and
\(\hat{\vartheta}\in S(\hat{u})\), respectively, and
\((\hat{u},\hat{\nu},\hat{\vartheta})\)
is a solution of GMELP (2.1).
Proof
Let \(u\in K\), \(\nu\in T(u)\), and \(\vartheta\in S(u)\) be the solution of GMELP (2.1). In view of the fact that all the conditions of Proposition 2.4 hold, then Proposition 2.4 implies that for all \(n\geq0\)
$$\begin{aligned} &\bigl\ g(u)g(u_{n+1})\bigr\ ^{2}\leq \bigl\ g(u)g(y_{1,n})\bigr\ ^{2} \bigl(12(\alpha+\beta) \rho_{1}\bigr)\bigl\ g(u_{n+1})g(y_{1,n})\bigr\ ^{2}, \end{aligned}$$
(2.25)
$$\begin{aligned} &\bigl\ g(u)g(y_{i,n})\bigr\ ^{2}\leq \bigl\ g(u)g(y_{i+1,n})\bigr\ ^{2}\bigl(12(\alpha+\beta) \rho_{i+1}\bigr) \bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2}, \end{aligned}$$
(2.26)
$$\begin{aligned} &\bigl\ g(u)g(y_{q1,n})\bigr\ \leq\bigl\ g(u)g(u_{n}) \bigr\ ^{2}\bigl(12(\alpha+\beta)\rho_{q}\bigr)\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2}, \end{aligned}$$
(2.27)
where \(i=1,2,\ldots,q2\). From the inequalities (2.25)(2.27), it follows that the sequence \(\{\g(u_{n})g(u)\\}\) is nonincreasing and hence the sequence \(\{g(u_{n})\}\) is bounded. Since the operator g is invertible, we deduce that the sequence \(\{u_{n}\}\) is also bounded. Furthermore, by (2.25)(2.27), we have
$$\begin{aligned} & \Biggl(12(\alpha+\beta)\sum_{i=1}^{q} \rho_{i} \Biggr) \Biggl(\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2}+\sum_{i=1}^{q2}\bigl\ g(y_{i,n})g(y_{i+1,n})\bigr\ ^{2} \\ &\quad{}+\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2} \Biggr) \leq \bigl\ g(u)g(u_{n})\bigr\ ^{2}\bigl\ g(u)g(u_{n+1}) \bigr\ ^{2}, \end{aligned}$$
which implies that
$$\begin{aligned} &\sum_{n=0}^{\infty} \Biggl(12( \alpha+\beta)\sum_{i=1}^{q}\rho _{i} \Biggr) \Biggl(\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2}+\sum_{i=1}^{q2} \bigl\ g(y_{i,n})g(y_{i+1,n})\bigr\ ^{2} \\ &\quad{}+\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2} \Biggr) \leq \bigl\ g(u)g(u_{0})\bigr\ ^{2}. \end{aligned}$$
(2.28)
The inequality (2.28) guarantees that
$$\begin{aligned} \bigl\ g(u_{n+1})g(y_{1,n})\bigr\ \rightarrow0,\qquad \bigl\ g(y_{i,n})g(y_{i+1,n})\bigr\ \rightarrow0,\qquad \bigl\ g(y_{q1,n})g(u_{n})\bigr\ \rightarrow0, \end{aligned}$$
for each \(i=1,2,\ldots,q2\), as \(n\rightarrow\infty\). Let û be a cluster point of the sequence \(\{u_{n}\}\). Since \(\{u_{n}\}\) is bounded, there exists a subsequence \(\{u_{n_{j}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{j}}\rightarrow\hat{u}\), as \(j\rightarrow\infty\). Taking into consideration the fact that the multivalued operators S and T are MLipschitz continuous with constants σ and δ, respectively, in virtue of the inequalities (2.9) and (2.10), we have
$$\begin{aligned} \\nu_{n_{j}+1}\nu_{n_{j}}\&\leq \bigl(1+(1+n_{j})^{1}\bigr)M\bigl(T(u_{n_{j}+1}),T(u_{n_{j}}) \bigr) \\ &\leq\bigl(1+(1+n_{j})^{1}\bigr)\delta\u_{n_{j}+1}u_{n_{j}} \ \end{aligned}$$
(2.29)
and
$$\begin{aligned} \\vartheta_{n_{j}+1}\vartheta_{n_{j}}\&\leq \bigl(1+(1+n_{j})^{1}\bigr)M\bigl(S(u_{n_{j}+1}),S(u_{n_{j}}) \bigr) \\ &\leq\bigl(1+(1+n_{j})^{1}\bigr)\sigma\u_{n_{j}+1}u_{n_{j}} \. \end{aligned}$$
(2.30)
The inequalities (2.29) and (2.30) imply that \(\\nu _{n_{j}+1}\nu_{n_{j}}\\rightarrow0\) and \(\\vartheta_{n_{j}+1}\vartheta _{n_{j}}\\rightarrow0\), as \(j\rightarrow\infty\), that is, \(\{\nu_{n_{j}}\} \) and \(\{\vartheta_{n_{j}}\}\) are Cauchy sequences in \(\mathcal{H}\). Thus, \(\nu_{n_{j}}\rightarrow\hat{\nu}\) and \(\vartheta_{n_{j}}\rightarrow \hat{\vartheta}\) for some \(\hat{\nu},\hat{\vartheta}\in\mathcal{H}\), as \(j\rightarrow\infty\). By using (2.6), we have
$$\begin{aligned} &\rho_{q} F\bigl(\nu_{n_{j}},g(v)\bigr)+ \rho_{q} G\bigl(\vartheta_{n_{j}},\eta \bigl(g(v),g(u_{n_{j}}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{q1,n_{j}})g(u_{n_{j}}),g(v)g(y_{q1,n_{j}}) \bigr\rangle \geq0. \end{aligned}$$
(2.31)
In view of the fact that F is continuous in the first argument, G is continuous in both arguments, η is continuous in the second argument, g is continuous and \(\g(y_{q1,n})g(u_{n})\\rightarrow0\), as \(n\rightarrow\infty\), letting \(i\rightarrow\infty\) and by using (2.31), we deduce that
$$\begin{aligned} F\bigl(\hat{\nu},g(v)\bigr)+G\bigl(\hat{\vartheta},\eta\bigl(g(v),g(\hat{u}) \bigr)\bigr)\geq0,\quad \forall v\in K. \end{aligned}$$
In the meantime, from the MLipschitz continuity of T with constant δ, it follows that
$$\begin{aligned} d\bigl(\hat{\nu},T(\hat{u})\bigr)&=\inf \bigl\{ \\hat{\nu}z\:z \in T(\hat{u}) \bigr\} \\ &\leq\\hat{\nu}\nu_{n_{j}}\+d\bigl(\nu_{n_{j}},T(\hat{u})\bigr) \\ &\leq\\hat{\nu}\nu_{n_{j}}\+M\bigl(T(u_{n_{j}}),T(\hat{u}) \bigr) \\ &\leq\\hat{\nu}\nu_{n_{j}}\+\delta\u_{n_{j}}\hat{u}\. \end{aligned}$$
(2.32)
Notice that the right side of the above inequality tends to zero as \(j\rightarrow\infty\). Since \(T(\hat{u})\in CB(\mathcal{H})\) it follows that \(\hat{\nu}\in T(\hat{u})\). Taking into account of the fact that the multivalued operator S is MLipschitz continuous with constant σ, in a similar way to that of proof of (2.32), one can deduce that
$$\begin{aligned} d\bigl(\hat{\vartheta},S(\hat{u})\bigr)\leq\\hat{\vartheta} \vartheta_{n_{j}}\ +\sigma\u_{n_{j}}\hat{u}\, \end{aligned}$$
which relying on the fact that \(S(\hat{u})\in CB(\mathcal{H})\) implies that \(\hat{\vartheta}\in S(\hat{u})\). Hence, \(\hat{u}\in K\), \(\hat{\nu }\in T(\hat{u})\), and \(\hat{\vartheta}\in S(\hat{u})\) are the solution of GMELP (2.1). Now, the inequalities (2.11)(2.13) imply that
$$\begin{aligned} \bigl\ g(u_{n+1}g(\hat{u})\bigr\ \leq\bigl\ g(u_{n})g( \hat{u})\bigr\ ,\quad \forall n\geq0. \end{aligned}$$
(2.33)
The inequality (2.33) guarantees that \(g(u_{n})\rightarrow g(\hat {u})\), as \(n\rightarrow\infty\) and hence \(u_{n}\rightarrow\hat{u}\), as \(n\rightarrow\infty\), since g is continuous and invertible. Consequently, the sequence \(\{u_{n}\}\) has exactly one cluster point û. Considering the facts that S and T are MLipschitz continuous with constants σ and δ, respectively, by using (2.9) and (2.10), we have
$$\begin{aligned} \\nu_{n+1}\nu_{n}\&\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(u_{n+1}),T(u_{n}) \bigr) \\ &\leq\bigl(1+(1+n)^{1}\bigr)\delta\u_{n+1}u_{n}\ \end{aligned}$$
(2.34)
and
$$\begin{aligned} \\vartheta_{n+1}\vartheta_{n}\&\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(S(u_{n+1}),S(u_{n}) \bigr) \\ &\leq\bigl(1+(1+n)^{1}\bigr)\sigma\u_{n+1}u_{n} \. \end{aligned}$$
(2.35)
The inequalities (2.34) and (2.35) imply that \(\{\nu_{n}\}\) and \(\{\vartheta_{n}\}\) are Cauchy sequences in \(\mathcal{H}\). Since ν̂ and ϑ̂ are cluster points of the sequences \(\{\nu _{n}\}\) and \(\{\vartheta_{n}\}\), respectively, it follows that \(\nu _{n}\rightarrow\hat{\nu}\) and \(\vartheta_{n}\rightarrow\hat{\vartheta}\), as \(n\rightarrow\infty\), that is, the sequences \(\{\nu_{n}\}\) and \(\{ \vartheta_{n}\}\) have exactly one cluster point ν̂ and ϑ̂, respectively. The proof is completed. □
The next proposition is a main tool for studying the convergence analysis of the iterative sequences generated by Algorithm 2.3.
Proposition 2.6
Let
F, G, S, T, and
g
be the same as in GMHEP (2.2) and let
\(\hat{u}\in K\), \(\hat{\nu}\in T(u)\), and
\(\hat {\vartheta}\in G(\hat{u})\)
be the solution of GMHEP (2.2). Furthermore, let
\(\{u_{n}\}\)
and
\(\{y_{i,n}\}\) (\(i=1,2,\ldots ,q1\)) be the sequences generated by Algorithm
2.3. If
F
is partially
αrelaxed
gmonotone with respect to
T, and
G
is partially
βrelaxed
gmonotone with respect to
S, then the inequalities (2.11)(2.13) hold for all
\(n\geq0\).
Proof
It follows from Proposition 2.4 by defining the operator \(\eta :K\times K\rightarrow\mathcal{H}\) as \(\eta(x,y)=xy\) for all \(x,y\in K\). □
The next assertion provides us the required conditions under which the iterative sequences generated by Algorithm 2.3 converge strongly to a solution of GMHEP (2.2).
Corollary 2.7
Suppose that
\(\mathcal{H}\)
is a finite dimensional real Hilbert space and let
\(g:K\rightarrow K\)
be a continuous and invertible operator. Assume that the bifunction
\(F:\mathcal{H}\times\mathcal{H}\rightarrow \Bbb{R}\)
is continuous in the first argument and the bifunction
\(G:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\)
is continuous in both arguments. Let the multivalued operators
\(S,T:K\rightarrow CB(\mathcal{H})\)
be
MLipschitz continuous with constants
σ
and
δ, respectively. Furthermore, let all the conditions of Proposition
2.6
hold and
\(\operatorname{GMHEP}(F,G,S,T,g,K)\neq \emptyset\). If
\(\rho_{i}\in(0,\frac{1}{2(\alpha+\beta)})\), for each
\(i=1,2,\ldots,q\), then the iterative sequences
\(\{u_{n}\}\), \(\{\nu_{n}\}\), and
\(\{\vartheta_{n}\}\)
generated by Algorithm
2.3
converge strongly to
\(\hat{u}\in K\), \(\hat{\nu}\in T(\hat{u})\)
and
\(\hat {\vartheta}\in S(\hat{u})\), respectively, and
\((\hat{u},\hat{\nu},\hat {\vartheta})\)
is a solution of GMHEP (2.2).
Proof
By defining the operator \(\eta:K\times K\rightarrow\mathcal{H}\) as \(\eta (x,y)=xy\) for all \(x,y\in K\), the desired result follows from Theorem 2.5. □
It is well known that to implement the proximal point methods, one has to calculate the approximate solution implicitly, which is itself a difficult problem. In order to overcome this drawback, we consider another auxiliary problem and with the help of it, we construct an iterative algorithm for solving the problem (2.1).
Let S, T, F, G, η, and g be the same as in GMELP (2.1). For given \(u\in K\), \(\nu\in T(u)\), and \(\vartheta\in S(u)\), we consider the auxiliary generalized multivalued equilibriumlike problem of finding \(w\in K\), \(\xi\in T(w)\), and \(\gamma\in S(w)\) such that
$$\begin{aligned} \rho F\bigl(\xi,g(v)\bigr)+\rho G\bigl(\gamma,\eta\bigl(g(v),g(u) \bigr)\bigr)+\bigl\langle g(w)g(u),g(v)g(w)\bigr\rangle \geq0,\quad \forall v\in K, \end{aligned}$$
(2.36)
where \(\rho>0\) is a constant. It should be pointed out that the two problems (2.3) and (2.36) are quite different. If \(w=u\), then clearly \((w,\xi,\gamma)\) is a solution of GMELP (2.1). By using this observation and Nadler’s technique [34], we are able to suggest the following predictorcorrector method for solving GMELP (2.1).
Algorithm 2.8
Let F, G, S, T, η, and g be the same as in GMELP (2.1). For given \(u_{0}\in K\), \(\xi_{0}\in T(u_{0})\), and \(\gamma_{0}\in S(u_{0})\), compute the iterative sequences \(\{u_{n}\}\), \(\{\xi _{n}\}\), and \(\{\gamma_{n}\}\) by the iterative schemes
$$\begin{aligned}& \begin{aligned}[b] &\rho_{1} F\bigl(\xi_{1,n},g(v)\bigr)+ \rho_{1} G\bigl(\gamma_{1,n},\eta \bigl(g(v),g(y_{1,n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(u_{n+1})g(y_{1,n}),g(v)g(u_{n+1}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \end{aligned}$$
(2.37)
$$\begin{aligned}& \begin{aligned}[b] &\rho_{i+1} F\bigl(\xi_{i+1,n},g(v)\bigr)+ \rho_{i+1} G\bigl(\gamma_{i+1,n},\eta \bigl(g(v),g(y_{i+1,n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{i,n})g(y_{i+1,n}),g(v)g(y_{i,n}) \bigr\rangle \geq0,\quad i=1,2,\ldots ,q2, \forall v\in K, \end{aligned} \end{aligned}$$
(2.38)
$$\begin{aligned}& \begin{aligned}[b] &\rho_{q} F\bigl(\xi_{n},g(v)\bigr)+ \rho_{q} G\bigl(\gamma_{n},\eta\bigl(g(v),g(u_{n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{q1,n})g(u_{n}),g(v)g(y_{q1,n}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \end{aligned}$$
(2.39)
$$\begin{aligned}& \begin{aligned}[b] &\xi_{i,n}\in T(y_{i,n}):\\xi_{i,n+1} \xi_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(y_{i,n+1}),T(y_{i,n}) \bigr), \\ &\quad i=1,2,\ldots,q1, \end{aligned} \end{aligned}$$
(2.40)
$$\begin{aligned}& \begin{aligned}[b] &\gamma_{i,n}\in S(y_{i,n}):\\gamma_{i,n+1} \gamma_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M \bigl(S(y_{i,n+1}),S(y_{i,n})\bigr),\\ &\quad i=1,2, \ldots,q1, \end{aligned} \end{aligned}$$
(2.41)
$$\begin{aligned}& \xi_{n}\in T(u_{n}):\\xi_{n+1}\xi_{n}\ \leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(u_{n+1}),T(u_{n}) \bigr), \end{aligned}$$
(2.42)
$$\begin{aligned}& \gamma_{n}\in S(u_{n}):\\gamma_{n+1} \gamma_{n}\\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(S(u_{n+1}),S(u_{n}) \bigr), \end{aligned}$$
(2.43)
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\) .
If \(\eta(x,y)=xy\), for all \(x,y\in K\), then Algorithm 2.8 reduces to the following predictorcorrector method.
Algorithm 2.9
For given \(u_{0}\in K\), \(\xi_{0}\in T(u_{0})\), and \(\gamma_{0}\in S(u_{0})\), compute the iterative sequences \(\{u_{n}\}\), \(\{\xi_{n}\}\), and \(\{\gamma _{n}\}\) by the iterative schemes
$$\begin{aligned}& \begin{aligned}[b] &\rho_{1} F\bigl(\xi_{1,n},g(v)\bigr)+ \rho_{1} G\bigl(\gamma _{1,n},g(v)g(y_{1,n})\bigr) \\ &\quad{}+\bigl\langle g(u_{n+1})g(y_{1,n}),g(v)g(u_{n+1}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \\& \begin{aligned}[b] &\rho_{i+1} F\bigl(\xi_{i+1,n},g(v)\bigr)+ \rho_{i+1} G\bigl(\gamma _{i+1,n},g(v)g(y_{i+1,n})\bigr) \\ &\quad{}+\bigl\langle g(y_{i,n})g(y_{i+1,n}),g(v)g(y_{i,n}) \bigr\rangle \geq0, \quad i=1,2,\ldots ,q2, \forall v\in K, \end{aligned} \\& \begin{aligned}[b] &\rho_{q} F\bigl(\xi_{n},g(v)\bigr)+ \rho_{q} G\bigl(\gamma_{n},g(v)g(u_{n})\bigr) \\ &\quad{}+\bigl\langle g(y_{q1,n})g(u_{n}),g(v)g(y_{q1,n}) \bigr\rangle \geq0,\quad \forall v\in K, \end{aligned} \\& \xi_{i,n}\in T(y_{i,n}):\\xi_{i,n+1} \xi_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(y_{i,n+1}),T(y_{i,n}) \bigr),\quad i=1,2,\ldots,q1, \\& \gamma_{i,n}\in S(y_{i,n}):\\gamma_{i,n+1} \gamma_{i,n}\\leq \bigl(1+(1+n)^{1}\bigr)M \bigl(S(y_{i,n+1}),S(y_{i,n})\bigr), \quad i=1,2,\ldots,q1, \\& \xi_{n}\in T(u_{n}):\\xi_{n+1}\xi_{n}\ \leq \bigl(1+(1+n)^{1}\bigr)M\bigl(T(u_{n+1}),T(u_{n}) \bigr), \\& \gamma_{n}\in S(u_{n}):\\gamma_{n+1} \gamma_{n}\\leq \bigl(1+(1+n)^{1}\bigr)M\bigl(S(u_{n+1}),S(u_{n}) \bigr), \end{aligned}$$
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\) .
To prove the strong convergence of the sequences generated by Algorithm 2.8 to a solution of GMELP (2.1), we need the following definition.
Definition 2.4
Let \(S,T:K\rightarrow CB(\mathcal{H})\), \(\eta:K\times K\rightarrow \mathcal{H}\), and \(g:K\rightarrow K\) be operators. The bifunctions \(F,G:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\) are said to be jointly
gηpseudomonotone with respect to S and T, if
$$\begin{aligned} &F\bigl(w_{1},g(u_{2})\bigr)+G\bigl(\vartheta_{1}, \eta\bigl(g(u_{2}),g(u_{1})\bigr)\bigr)\geq0, \quad \mbox{implies that} \\ &F\bigl(w_{2},g(u_{1})\bigr)+G\bigl(\vartheta_{2}, \eta\bigl(g(u_{1}),g(u_{2})\bigr)\bigr)\leq0, \\ &\quad\forall u_{1},u_{2}\in K, w_{1}\in T(u_{1}), w_{2}\in T(u_{2}), \vartheta_{1} \in S(u_{1}), \vartheta_{2}\in S(u_{2}). \end{aligned}$$
It should be remarked that if the operator \(\eta:K\times K\rightarrow \mathcal{H}\) is defined as \(\eta(x,y)=xy\), for all \(x,y\in K\), then Definition 2.4 reduces to the definition of jointly gpseudomonotonicity of the bifunctions F and G with respect to the multivalued operators S and T.
Before turning to the study of convergence analysis of the iterative sequences generated by Algorithm 2.8, we would like to present the following proposition which plays an important and key role in it.
Proposition 2.10
Let
F, G, S, T, η, and
g
be the same as in GMELP (2.1) and let
\(\hat{u}\in K\), \(\hat{\nu}\in T(\hat {u})\), and
\(\hat{\vartheta}\in S(\hat{u})\)
be the solution of GMELP (2.1). Assume further that
\(\{u_{n}\}\)
and
\(\{y_{i,n}\}\) (\(i=1,2,\ldots,q1\)) are the sequences generated by Algorithm
2.8. If
F
and
G
are jointly
gηpseudomonotone with respect to
S
and
T, then for all
\(n\geq0\)
$$\begin{aligned} &\bigl\ g(\hat{u})g(u_{n+1})\bigr\ ^{2}\leq\bigl\ g( \hat{u})g(y_{1,n})\bigr\ ^{2}\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2}, \end{aligned}$$
(2.44)
$$\begin{aligned} &\begin{aligned}[b] &\bigl\ g(\hat{u})g(y_{i,n})\bigr\ ^{2}\leq\bigl\ g( \hat{u})g(y_{i+1,n})\bigr\ ^{2}\bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2}, \\ &\quad i=1,2,\ldots,q2, \end{aligned} \end{aligned}$$
(2.45)
$$\begin{aligned} &\bigl\ g(\hat{u})g(y_{q1,n})\bigr\ ^{2}\leq\bigl\ g( \hat{u})g(u_{n})\bigr\ ^{2}\bigl\ g(y_{q1,n})g(u_{n}) \bigr\ ^{2}. \end{aligned}$$
(2.46)
Proof
Since \(\hat{u}\in K\), \(\hat{\nu}\in T(\hat{u})\), and \(\hat{\vartheta }\in S(\hat{u})\) are the solution of GMELP (2.1), it follows that \((\hat{u},\hat{v},\hat{\vartheta})\) satisfies (2.14). Taking \(v=y_{1,n}\) in (2.14), we have
$$\begin{aligned} F\bigl(\hat{\nu},g(y_{1,n})\bigr)+G\bigl(\hat{\vartheta}, \eta\bigl(g(y_{1,n}),g(\hat {u})\bigr)\bigr)\geq0. \end{aligned}$$
(2.47)
Taking into consideration the fact that the bifunctions F and G are jointly gηpseudomonotone with respect to S and T, from (2.47), we conclude that
$$\begin{aligned} F\bigl(\xi_{1,n},g(\hat{u})\bigr)+G\bigl( \gamma_{1,n},\eta\bigl(g(\hat{u}),g(y_{1,n})\bigr)\bigr)\leq0. \end{aligned}$$
(2.48)
Taking \(v=\hat{u}\) in (2.37), we obtain
$$\begin{aligned} &\rho_{1} F\bigl(\xi_{1,n},g(\hat{u})\bigr)+ \rho_{1} G\bigl(\gamma_{1,n},\eta\bigl(g(\hat {u}),g(y_{1,n})\bigr)\bigr) \\ &\quad{}+\bigl\langle g(u_{n+1})g(y_{1,n}),g( \hat {u})g(u_{n+1})\bigr\rangle \geq0. \end{aligned}$$
(2.49)
By combining (2.48) and (2.49), we get
$$\begin{aligned} \bigl\langle g(u_{n+1})g(y_{1,n}),g( \hat{u})g(u_{n+1})\bigr\rangle &\geq\rho_{1} F\bigl( \xi_{1,n},g(\hat{u})\bigr)\rho_{1} G\bigl( \gamma_{1,n},\eta\bigl(g(\hat {u}),g(y_{1,n})\bigr)\bigr) \\ &\geq0. \end{aligned}$$
(2.50)
Relying on (2.18) and (2.50), we get
$$\begin{aligned} \bigl\ g(\hat{u})g(u_{n+1})\bigr\ ^{2}\leq\bigl\ g(\hat{u})g(y_{1,n}) \bigr\ ^{2}\bigl\ g(u_{n+1})g(y_{1,n})\bigr\ ^{2}, \end{aligned}$$
which is the required result (2.44).
Taking \(v=y_{i+1,n}\) (\(i=1,2,\ldots,q2\)) in (2.14), we have
$$\begin{aligned} F\bigl(\hat{\nu},g(y_{i+1,n})\bigr)+G\bigl(\hat{\vartheta}, \eta\bigl(g(y_{i+1,n}),g(\hat {u})\bigr)\bigr)\geq0. \end{aligned}$$
(2.51)
Considering the fact that F and G are jointly gηpseudomonotone with respect to S and T, the inequality (2.51) implies that, for each \(i=1,2,\ldots,q2\),
$$\begin{aligned} F\bigl(\xi_{i+1,n},g(\hat{u})\bigr)+G\bigl( \gamma_{i+1,n},\eta\bigl(g(\hat {u}),g(y_{i+1,n})\bigr)\bigr)\leq0. \end{aligned}$$
(2.52)
Letting \(v=\hat{u}\) in (2.38), for each \(i=1,2,\ldots,q2\), we obtain
$$\begin{aligned} &\rho_{i+1} F\bigl(\xi_{i+1,n},g(\hat{u})\bigr)+ \rho_{i+1}G\bigl(\gamma_{i+1,n},\eta \bigl(g(\hat{u}),g(y_{i+1,n}) \bigr)\bigr) \\ &\quad{}+\bigl\langle g(y_{i,n})g(y_{i+1,n}),g( \hat{u})g(y_{i,n})\bigr\rangle \geq0. \end{aligned}$$
(2.53)
It follows from (2.18), (2.52), and (2.53) that, for each \(i=1,2,\ldots,q2\),
$$\begin{aligned} \bigl\ g(\hat{u})g(y_{i,n})\bigr\ ^{2}\leq\bigl\ g(\hat{u})g(y_{i+1,n}) \bigr\ ^{2}\bigl\ g(y_{i,n})g(y_{i+1,n})\bigr\ ^{2}, \end{aligned}$$
which is the required result (2.45).
Taking \(v=y_{q,n}\) in (2.14), we have
$$\begin{aligned} F\bigl(\hat{\nu},g(y_{q,n})\bigr)+G\bigl(\hat{\vartheta}, \eta\bigl(g(y_{q,n}),g(\hat {u})\bigr)\bigr)\geq0. \end{aligned}$$
(2.54)
In light of the fact that F and G are jointly gηpseudomotone with respect to S and T, we deduce that
$$\begin{aligned} F\bigl(\xi_{n},g(\hat{u})\bigr)+G\bigl( \gamma_{n},\eta\bigl(g(\hat{u}),g(u_{n})\bigr)\bigr)\leq0. \end{aligned}$$
(2.55)
Letting \(v=\hat{u}\) in (2.39), we get
$$\begin{aligned} \rho_{q} F\bigl(\xi_{n},g(\hat{u})\bigr)+ \rho_{q} G\bigl(\gamma_{n},\eta\bigl(g(\hat{u}),g(u_{n}) \bigr)\bigr) +\bigl\langle g(y_{q1,n})g(u_{n}),g( \hat{u})g(y_{q1,n})\bigr\rangle \geq0. \end{aligned}$$
(2.56)
Applying (2.18), (2.55) and (2.56), one can deduce that
$$\begin{aligned} \bigl\ g(\hat{u})g(y_{q1,n})\bigr\ \leq\bigl\ g(\hat{u})g(u_{n}) \bigr\ ^{2}\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2}, \end{aligned}$$
which is the required result (2.46). This completes the proof. □
Now we establish the strong convergence of the iterative sequences generated by Algorithm 2.8 to a solution of GMELP (2.1).
Theorem 2.11
Let
\(\mathcal{H}\)
be a finite dimensional real Hilbert space and let
\(g:K\rightarrow K\)
be a continuous and invertible operator. Suppose that the bifunction
\(F:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\)
is continuous in the first argument, the bifunction
\(G:\mathcal{H}\times \mathcal{H}\rightarrow\Bbb{R}\)
is continuous in both arguments and the operator
\(\eta:K\times K\rightarrow\mathcal{H}\)
is continuous in the second argument. Assume that the multivalued operators
\(S,T:K\rightarrow CB(\mathcal{H})\)
are
MLipschitz continuous with constants
σ
and
δ, respectively. Further, let all the conditions of Proposition
2.10
hold and
\(\operatorname{GMELP}(F,G,S,T,\eta,g,K)\neq\emptyset\). Then the iterative sequences
\(\{u_{n}\}\), \(\{\xi_{n}\}\), and
\(\{\gamma_{n}\}\)
generated by Algorithm
2.8
converge strongly to
\(\hat{u}\in K\), \(\hat{\nu}\in T(\hat {u})\), and
\(\hat{\vartheta}\in S(\hat{u})\), respectively, and
\((\hat {u},\hat{\nu},\hat{\vartheta})\)
is a solution of GMELP (2.1).
Proof
Let \(u\in K\), \(\nu\in T(u)\), and \(\vartheta\in S(u)\) be the solution of GMELP (2.1). Since all the conditions of Proposition 2.10 hold, according to Proposition 2.10, for all \(n\geq0\) we have
$$\begin{aligned} &\bigl\ g(u)g(u_{n+1})\bigr\ ^{2}\leq \bigl\ g(u)g(y_{1,n})\bigr\ ^{2}\bigl\ g(u_{n+1})g(y_{1,n}) \bigr\ ^{2}, \end{aligned}$$
(2.57)
$$\begin{aligned} &\begin{aligned}[b] &\bigl\ g(u)g(y_{i,n})\bigr\ ^{2}\leq \bigl\ g(u)g(y_{i+1,n})\bigr\ ^{2}\bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2},\\ &\quad i=1,2,\ldots,q2, \end{aligned} \end{aligned}$$
(2.58)
$$\begin{aligned} &\bigl\ g(u)g(y_{q1,n})\bigr\ ^{2}\leq \bigl\ g(u)g(u_{n})\bigr\ ^{2}\bigl\ g(y_{q1,n})g(u_{n}) \bigr\ ^{2}. \end{aligned}$$
(2.59)
It is easy to see that the inequalities (2.57)(2.59) imply that the sequence \(\{\g(u_{n})g(u)\\}\) is nonincreasing and hence the sequence \(\{g(u_{n})\}\) is bounded. Considering the fact that the operator g is invertible, it follows that the sequence \(\{u_{n}\}\) is also bounded. Meanwhile, relying on (2.57)(2.59), we have
$$\begin{aligned} &\bigl\ g(u_{n+1})g(y_{1,n})\bigr\ ^{2}+\sum _{i=1}^{q2}\bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2}+\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2} \\ &\quad\leq\bigl\ g(u)g(u_{n})\bigr\ ^{2}\bigl\ g(u)g(u_{n+1}) \bigr\ ^{2}, \end{aligned}$$
whence we deduce that
$$\begin{aligned} &\sum_{n=0}^{\infty} \Biggl( \bigl\ g(u_{n+1})g(y_{1,n})\bigr\ ^{2}+\sum _{i=1}^{q2} \bigl\ g(y_{i,n})g(y_{i+1,n}) \bigr\ ^{2}+\bigl\ g(y_{q1,n})g(u_{n})\bigr\ ^{2} \Biggr) \\ &\quad\leq\bigl\ g(u)g(u_{0})\bigr\ ^{2}. \end{aligned}$$
(2.60)
From (2.60), it follows that
$$\begin{aligned} \bigl\ g(u_{n+1})g(y_{1,n})\bigr\ \rightarrow0,\qquad \bigl\ g(y_{i,n})g(y_{i+1,n})\bigr\ \rightarrow0,\qquad \bigl\ g(y_{q1,n})g(u_{n})\bigr\ \rightarrow0, \end{aligned}$$
for each \(i=1,2,\ldots,q2\), as \(n\rightarrow\infty\). Let û be a cluster point of the sequence \(\{u_{n}\}\). Since \(\{u_{n}\}\) is bounded, there exists a subsequence \(\{u_{n_{j}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{j}}\rightarrow\hat{u}\), as \(j\rightarrow\infty\). In a similar way to that of proof of Theorem 2.5, one can establish that \(\{\xi _{n_{j}}\}\) and \(\{\gamma_{n_{j}}\}\) are Cauchy sequences in \(\mathcal{H}\) and \(\xi_{n_{j}}\rightarrow\hat{\nu}\), and \(\gamma_{n_{j}}\rightarrow\hat {\vartheta}\) for some \(\hat{\nu},\hat{\vartheta}\in\mathcal{H}\), as \(j\rightarrow\infty\). Furthermore, \(\hat{u}\in K\), \(\hat{\nu}\in T(\hat {u})\), and \(\hat{\vartheta}\in S(\hat{u})\) are the solution of GMELP (2.1) and the sequences \(\{u_{n}\}\), \(\{\xi_{n}\}\), and \(\{ \gamma_{n}\}\) have exactly one cluster point û, ν̂, and ϑ̂, respectively. This gives us the desired result. □
The next proposition plays a crucial role in the study of convergence analysis of the iterative sequences generated by Algorithm 2.9.
Proposition 2.12
Let
F, G, S, T, and
g
be the same as in GMHEP (2.2) and let
\(\hat{u}\in K\), \(\hat{\nu}\in T(\hat{u})\), and
\(\hat{\vartheta}\in S(\hat{u})\)
be the solution of GMHEP (2.2). Moreover, let
\(\{u_{n}\}\)
and
\(\{y_{i,n}\}\) (\(i=1,2,\ldots ,q1\)) be the sequences generated by Algorithm
2.9. If the bifunctions
F
and
G
are jointly
gpseudomonotone with respect to
S
and
T, then the inequalities (2.44)(2.46) hold.
Proof
By defining the operator \(\eta:K\times K\rightarrow\mathcal{H}\) as \(\eta (x,y)=xy\) for all \(x,y\in K\), we get the desired result from Proposition 2.10. □
We now conclude this paper with the following result in which the strong convergence of the iterative sequence generated by Algorithm 2.9 is established.
Corollary 2.13
Assume that
\(\mathcal{H}\)
is a finite dimensional real Hilbert space and let
\(g:K\rightarrow K\)
be a continuous and invertible operator. Let the bifunction
\(F:\mathcal{H}\times\mathcal{H}\rightarrow\Bbb{R}\)
be continuous in the first argument and the bifunction
\(G:\mathcal{H}\times \mathcal{H}\rightarrow\Bbb{R}\)
be continuous in both arguments. Suppose that the multivalued operators
\(S,T:K\rightarrow CB(\mathcal{H})\)
are
MLipschitz continuous with constants
σ
and
δ, respectively. Moreover, let all the conditions of Proposition
2.12
hold and
\(\operatorname{GMHEP}(F,G,S,T,g,K)\neq\emptyset\). Then the iterative sequences
\(\{u_{n}\}\), \(\{\xi_{n}\}\), and
\(\{\gamma_{n}\}\)
generated by Algorithm
2.9
converge strongly to
\(\hat {u}\in K\), \(\hat{\nu}\in T(\hat{u})\), and
\(\hat{\vartheta}\in S(\hat {u})\), respectively, and
\((\hat{u},\hat{\nu},\hat{\vartheta})\)
is a solution of GMHEP (2.2).
Proof
We obtain the desired result from Theorem 2.11 by defining \(\eta :K\times K\rightarrow\mathcal{H}\) as \(\eta(x,y)=xy\) for all \(x,y\in K\). □