 Research
 Open access
 Published:
Generalized multivalued equilibriumlike problems: auxiliary principle technique and predictorcorrector methods
Journal of Inequalities and Applications volumeÂ 2016, ArticleÂ number:Â 73 (2016)
Abstract
This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibriumlike problems which includes the classes of hemiequilibrium problems, equilibriumlike problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictorcorrector iterative algorithms for solving them are suggested and analyzed. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in generalized multivalued equilibriumlike problem. Results obtained in this paper include several new and known results as special cases.
1 Introduction
During the last decades, the theory of variational analysis including variational inequalities \((\operatorname{VI})\) have attracted a lot of attention because of its applications in optimization, nonlinear analysis, game theory, economics, and so forth; see, for example, [1] and the references therein. Because of the importance and active impact of VI in branches of sciences, engineering, and social sciences, it has been extended and generalized in many different directions. It has been used as a tool to study different aspects of optimization problems; see, for example, [1â€“4] and the references therein. By replacing the linear term appearing in the formulation of variational inequalities by a vectorvalued term, Parida et al. [5] and Yang and Chen [6] independently introduced and studied a class of variational inequalities known as variationallike inequalities or prevariational inequalities which is an important extension of the variational inequalities. Another useful and important generalization of variational inequalities is a class of variational inequalities known as hemivariational inequalities involving the nonlinear Lipschitz continuous functions. It should be pointed out that the hemivariational inequalities are connected with nonconvex and possibly nonsmooth energy functions and have important applications in structural analysis and nonconvex optimization. For more details, we refer the reader to [7â€“10].
On the other hand, the concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, finance, optimization, image reconstruction, network, ecology, sociology, chemistry, biology, engineering sciences, transportation, and other fields. The study of equilibrium problems \((\operatorname{EP})\), which was introduced by Blum and Oettli [11] in 1994, in terms of their formulation, qualitative analysis, and computation has been the focus of much research in the past several decades and has given rise to the development of a variety of mathematical methodologies. Examples of mathematical formulations that have been used for equilibrium problems are: nonlinear equations, optimization problems, complementarity problems, fixed point problems, and, most recently, variational inequality problems; see, for example, [11â€“15] and the references therein.
In recent years, EP has received much attention by many authors due to the fact that it provides a unified model including the above mentioned problems, and various important generalizations of it have been proposed and analyzed; see, for example, [16â€“26] and the references therein. An important and useful generalization of EP is the multivalued equilibrium problems involving a nonlinear bifunction. It has been shown that a wide class of unrelated odd order and nonsymmetric free, moving, obstacle and equilibrium problems can be studied via the multivalued equilibrium problems.
Inspired and motivated by the research going in this interesting and fascinating area, Noor [27, 28] introduced and investigated the class of equilibriumlike (or invex equilibrium) problems as a useful and important extension of the class of equilibrium problems. It has been shown that equilibriumlike problems include equilibrium problems, variationallike inequalities, variational inequalities and their invariant forms as special cases. In the meanwhile, related to the hemivariational inequalities, Noor [29] introduced and studied the class of hemiequilibrium problems as a generalization of the class of equilibrium problems. It is shown that the hemiequilibrium problems include equilibrium problems, hemivariational inequalities and variational inequalities as special cases. It is worth mentioning that the hemiequilibrium problems and equilibriumlike problems are two quite different extensions of the classical equilibrium problems.
One of the most important and interesting problems in the theory of variational inequalities is the development of numerical methods which provide an efficient and implementable algorithm for solving variational inequalities and its generalizations. In the last decades, many efforts have been devoted to the development of efficiency and of implementable methods for solving variational inequalities and their extensions. Though, various numerical techniques are proposed for solving variational inequalities, but the nature of equilibrium problems does not allow us to use these methods in their present forms. As an example, projection technique, one of the main methods used in existence theory of variational inequalities, cannot be used in a similar way for equilibrium problems. The auxiliary principle technique helps to avoid these constraints and addresses the demand of nature of equilibrium problems in a right way. In this technique, a supporting (auxiliary) problem linked to the original one is considered. This actually is a way to define a mapping that relates the original problem wit the auxiliary problem. This technique was used by Glowinski et al. [30] to study the existence of a solution of mixed variational inequality and later was developed by many authors for solving various classes of variational inequalities and equilibrium problems; see, for example, [18, 30â€“32] and the references therein.
Motivated and inspired by the work mentioned above, the purpose of this paper is to introduce a new class of equilibrium problems named generalized multivalued equilibriumlike problems \((\operatorname{GMELP})\), which includes hemiequilibrium problems, equilibriumlike problems, equilibrium problems, hemivariational inequalities, and variationallike inequalities as special cases. By using the auxiliary principle technique, we suggest and analyze some predictorcorrector methods for solving GMELP. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in GMELP. As special cases, one can obtain several new and known methods for solving variational inequalities and equilibrium problems. The results presented in this paper generalize and improve some recent results in this field.
2 Formulations, algorithms, and convergence results
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
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
which is called the generalized multivalued hemiequilibrium problem \((\operatorname{GMHEP})\).
It should be remarked that by taking different choices of the operators S, T, Î·, 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
where \(M(\cdot,\cdot)\) is the Hausdorff metric on \(CB(X)\) defined by
Let F, G, S, T, Î·, 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
where \(\rho>0\) is a constant. Obviously, if \(w=u\), then \((w,\nu ,\vartheta)\) is a solution of GMELP (2.1). This observation and Nadlerâ€™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, Î·, 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
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\)â€‰.
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
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\)â€‰.
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 Î±relaxed 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 Î±relaxed 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 Î±relaxed 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)
gÎ·monotone 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 Î²relaxed gÎ·monotone 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 Î²relaxed gÎ·monotonicity 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 Î²relaxed 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 Î´, if there exists a constant \(\delta>0\) such that
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, Î·, 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 Î±relaxed gÎ·monotone with respect to T, and G is partially Î²relaxed gÎ·monotone with respect to S, then
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
Taking \(v=u_{n+1}\) in (2.14) and \(v=u\) in (2.4), we get
and
By combining (2.15) and (2.16) and taking into account of the facts that the bifunction F is partially Î±relaxed gmonotone with respect to T, and the bifunction G is partially Î²relaxed strongly gÎ·monotone with respect to G, it follows that
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
Applying (2.17) and (2.18), it follows that
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
and
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
In a similar fashion to the preceding analysis, employing (2.19)(2.21) and considering the facts that the bifunction F is partially Î±relaxed gÎ·monotone with respect to T, and the bifunction G is partially Î²relaxed gÎ·monotone with respect to S, for each \(i=1,2,\ldots, q2\), one can deduce that
which is the required result (2.12).
Taking \(v=y_{q1,n}\) in (2.14) and \(v=\hat{u}\) in (2.6), we have
and
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
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 Î±relaxed gÎ·monotone with respect to T, and the bifunction G is partially Î²relaxed gÎ·monotone with respect to S, we can show that
which is the required result (2.13). This completes the proof.â€ƒâ–¡
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 Ïƒ and Î´, 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 \(n\geq0\)
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
which implies that
The inequality (2.28) guarantees that
for each \(i=1,2,\ldots,q2\), as \(n\rightarrow\infty\). Let Ã» 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 Ïƒ and Î´, respectively, in virtue of the inequalities (2.9) and (2.10), we have
and
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
In view of the fact that F is continuous in the first argument, G is continuous in both arguments, Î· 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
In the meantime, from the MLipschitz continuity of T with constant Î´, it follows that
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 Ïƒ, in a similar way to that of proof of (2.32), one can deduce that
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
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 Ã». Considering the facts that S and T are MLipschitz continuous with constants Ïƒ and Î´, respectively, by using (2.9) and (2.10), we have
and
The inequalities (2.34) and (2.35) imply that \(\{\nu_{n}\}\) and \(\{\vartheta_{n}\}\) are Cauchy sequences in \(\mathcal{H}\). Since Î½Ì‚ and Ï‘Ì‚ 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 Î½Ì‚ and Ï‘Ì‚, respectively. The proof is completed.â€ƒâ–¡
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 Î±relaxed gmonotone with respect to T, and G is partially Î²relaxed 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 \(x,y\in K\).â€ƒâ–¡
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 Ïƒ and Î´, 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.â€ƒâ–¡
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, Î·, 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
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 Nadlerâ€™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, Î·, 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
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\)â€‰.
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
where \(\rho_{i}>0\) (\(i=1,2,\ldots,q\)) are constants and \(n=0,1,2,\ldots\)â€‰.
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 gÎ·pseudomonotone with respect to S and T, if
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, Î·, 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 gÎ·pseudomonotone with respect to S and T, then for all \(n\geq0\)
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
Taking into consideration the fact that the bifunctions F and G are jointly gÎ·pseudomonotone with respect to S and T, from (2.47), we conclude that
Taking \(v=\hat{u}\) in (2.37), we obtain
By combining (2.48) and (2.49), we get
Relying on (2.18) and (2.50), we get
which is the required result (2.44).
Taking \(v=y_{i+1,n}\) (\(i=1,2,\ldots,q2\)) in (2.14), we have
Considering the fact that F and G are jointly gÎ·pseudomonotone with respect to S andÂ T, the inequality (2.51) implies that, for each \(i=1,2,\ldots,q2\),
Letting \(v=\hat{u}\) in (2.38), for each \(i=1,2,\ldots,q2\), we obtain
It follows from (2.18), (2.52), and (2.53) that, for each \(i=1,2,\ldots,q2\),
which is the required result (2.45).
Taking \(v=y_{q,n}\) in (2.14), we have
In light of the fact that F and G are jointly gÎ·pseudomotone with respect to S and T, we deduce that
Letting \(v=\hat{u}\) in (2.39), we get
Applying (2.18), (2.55) and (2.56), one can deduce that
which is the required result (2.46). This completes the proof.â€ƒâ–¡
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 Ïƒ and Î´, 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
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
whence we deduce that
From (2.60), it follows that
for each \(i=1,2,\ldots,q2\), as \(n\rightarrow\infty\). Let Ã» 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 Ã», Î½Ì‚, and Ï‘Ì‚, respectively. This gives us the desired result.â€ƒâ–¡
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.â€ƒâ–¡
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 Ïƒ and Î´, 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 \(x,y\in K\).â€ƒâ–¡
References
Ansari, QH, Lalitha, CS, Mehta, M: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization. CRC Press, Boca Roton (2014)
Facchinei, F, Pang, JS: FiniteDimensional Variational Inequalities and Complementarity Problems. Volume I and II. Springer, Berlin (2003)
Giannessi, F: Constrained Optimization and Image Space Analysis: Separation of Sets and Optimality Conditions. Springer, Berlin (2005)
Goh, CJ, Yang, XQ: Duality in Optimization and Variational Inequalities. Taylor & Francis, London (2002)
Parida, J, Sahoo, M, Kumar, A: A variationallike inequality problem. Bull. Aust. Math. Soc. 39, 225231 (1989)
Yang, XQ, Chen, GY: A class of nonconvex functions and prevariational inequalities. J. Math. Anal. Appl. 169, 359373 (1992)
Panagiotopoulos, PD: Hemivariational Inequalities, Applications to Mechanics and Engineering. Springer, Berlin (1993)
Panagiotopoulos, PD: Nonconvex energy functions, hemivariational inequalities and substationarity principles. Acta Mech. 42, 160183 (1983)
Demyanov, VF, Stavroulakis, GE, Ployakova, LN, Panagiotopoulos, PD: Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics. Kluwer Academic, Dordrecht (1996)
Naniewicz, Z, Panagiotopoulos, PD: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, Boston (1995)
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123145 (1994)
AlHomidan, S, Ansari, QH, Yao, JC: Some generalizations of Ekelandtype variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 69, 126139 (2008)
Ansari, QH: Topics in Nonlinear Analysis and Optimization. World Education, Delhi (2012)
Bianchi, M, Schaible, S: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 3143 (1996)
FloresBazan, F: Existence theorems for generalized noncoercive equilibrium problems: the quasiconvex case. SIAM J. Optim. 11, 675690 (2000)
Ansari, QH, Wong, NC, Yao, JC: The existence of nonlinear inequalities. Appl. Math. Lett. 12(5), 8992 (1999)
Cho, YJ, Petrot, N: On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces. J. Inequal. Appl. 2010, Article ID 437976 (2010). doi:10.1155/2010/437976
Ding, XP: Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces. J. Optim. Theory Appl. 146, 347357 (2010)
Ding, XP: Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces. Nonlinear Anal. 73, 12641276 (2010)
Huang, NJ, Lan, HY, Cho, YJ: Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with nonmonotone setvalued mappings. J. Comput. Appl. Math. 196, 608618 (2006)
Moudafi, A: Mixed equilibrium problems: sensitivity analysis and algorithmic aspect. Comput. Math. Appl. 44, 10991108 (2002)
Moudafi, A, Thera, M: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol.Â 477, pp.Â 187201. Springer, Berlin (2002)
Zeng, LC, Ansari, QH, Schaible, S, Yao, JC: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory 12(2), 293308 (2011)
Latif, A, AlMazrooei, AE, Alofi, ASM, Yao, JC: Shrinking projection method for systems of generalized equilibria with constraints of variational inclusion and fixed point problems. Fixed Point Theory Appl. 2014, 164 (2014)
Ceng, LC, Latif, A, AlMazrooei, AE: Hybrid viscosity methods for equilibrium problems, variational inequalities, and fixed point problems. Appl. Anal. (2015). doi:10.1080/00036811.2015.1051971
Ceng, LC, Latif, A, AlMazrooei, AE: Composite viscosity methods for common solutions of general mixed equilibrium problem, variational inequalities and common fixed points. J. Inequal. Appl. 2015, 217 (2015)
Noor, MA: Invex equilibrium problems. J. Math. Anal. Appl. 302, 463475 (2005)
Noor, MA: Mixed quasi invex equilibrium problems. Int. J. Math. Math. Sci. 57, 30573067 (2004)
Noor, MA: Hemiequilibrium problems. J. Appl. Math. Stoch. Anal. 2004, 235244 (2004)
Glowinski, R, Lions, JL, Tremolieres, R: Numerical Analysis of Variational Inequalities. NorthHolland, Amsterdam (1981)
Ansari, QH, Balooee, J: Predictorcorrector methods for general regularized nonconvex variational inequalities. J.Â Optim. Theory Appl. 159, 473488 (2013)
Ding, XP: Auxiliary principle and algorithm of solutions for a new system of generalized mixed equilibrium problems in Banach spaces. J. Optim. Theory Appl. 155, 796809 (2012)
Noor, MA, Rassias, TM: On general hemiequilibrium problems. J. Math. Anal. Appl. 324, 14171428 (2006)
Nadler, SB: Multivalued contraction mapping. Pac. J. Math. 30(3), 457488 (1969)
Acknowledgements
The first author is supported by the Sari Branch, Islamic Azad University, Iran. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authorsâ€™ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Dadashi, V., Latif, A. Generalized multivalued equilibriumlike problems: auxiliary principle technique and predictorcorrector methods. J Inequal Appl 2016, 73 (2016). https://doi.org/10.1186/s1366001610009
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366001610009