Generalized multivalued equilibrium-like problems: auxiliary principle technique and predictor-corrector methods
- Vahid Dadashi^{1}Email author and
- Abdul Latif^{2}
https://doi.org/10.1186/s13660-016-1000-9
© Dadashi and Latif 2016
Received: 4 January 2016
Accepted: 1 February 2016
Published: 24 February 2016
Abstract
This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibrium-like problems which includes the classes of hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictor-corrector 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 equilibrium-like problem. Results obtained in this paper include several new and known results as special cases.
Keywords
MSC
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, [1–4] and the references therein. By replacing the linear term appearing in the formulation of variational inequalities by a vector-valued term, Parida et al. [5] and Yang and Chen [6] independently introduced and studied a class of variational inequalities known as variational-like inequalities or pre-variational 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 [7–10].
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, [11–15] 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, [16–26] 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 equilibrium-like (or invex equilibrium) problems as a useful and important extension of the class of equilibrium problems. It has been shown that equilibrium-like problems include equilibrium problems, variational-like 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 equilibrium-like 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, 30–32] 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 equilibrium-like problems \((\operatorname{GMELP})\), which includes hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational-like inequalities as special cases. By using the auxiliary principle technique, we suggest and analyze some predictor-corrector 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
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]
Algorithm 2.2
If \(\eta(x,y)=x-y\), for all \(x,y\in K\), then Algorithm 2.2 reduces to the following predictor-corrector method.
Algorithm 2.3
In order to study the convergence analysis of the iterative sequences generated by Algorithm 2.2, we need the following definitions.
Definition 2.1
- (a)g-monotone 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 α-relaxed g-monotone 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 α-relaxed g-monotonicity of the bifunction F with respect to T reduces to g-monotonicity 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 α-relaxed monotonicity of the bifunction F with respect to T, respectively.
Definition 2.2
- (a)g-η-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 β-relaxed g-η-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 β-relaxed g-η-monotonicity of the bifunction G with respect to S reduces to g-monotonicity of the bifunction G with respect to S. Furthermore, for the case when \(\eta(x,y)=x-y\), for all \(x,y\in K\), then parts (a) and (b) of Definition 2.2 reduce to the definition of g-monotonicity and partially β-relaxed g-monotonicity of the bifunction G with respect to S, respectively.
Definition 2.3
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
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 M-Lipschitz 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
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 \(\{y_{i,n}\}\) (\(i=1,2,\ldots ,q-1\)) be the sequences generated by Algorithm 2.3. If F is partially α-relaxed g-monotone with respect to T, and G is partially β-relaxed g-monotone 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 \(\eta(x,y)=x-y\) 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 M-Lipschitz 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 \(\eta (x,y)=x-y\) 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).
Algorithm 2.8
If \(\eta(x,y)=x-y\), for all \(x,y\in K\), then Algorithm 2.8 reduces to the following predictor-corrector method.
Algorithm 2.9
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
It should be remarked that if the operator \(\eta:K\times K\rightarrow \mathcal{H}\) is defined as \(\eta(x,y)=x-y\), for all \(x,y\in K\), then Definition 2.4 reduces to the definition of jointly g-pseudomonotonicity 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
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 M-Lipschitz 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
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 \(\{y_{i,n}\}\) (\(i=1,2,\ldots ,q-1\)) be the sequences generated by Algorithm 2.9. If the bifunctions F and G are jointly g-pseudomonotone 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 \(\eta (x,y)=x-y\) 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 M-Lipschitz 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 \(\eta(x,y)=x-y\) for all \(x,y\in K\). □
Declarations
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.
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.
Authors’ Affiliations
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