 Research
 Open Access
Best approximation and variational inequality problems involving a simulation function
 Fairouz Tchier^{1},
 Calogero Vetro^{2}Email author and
 Francesca Vetro^{3}
https://doi.org/10.1186/s1366301605129
© Tchier et al. 2016
 Received: 3 November 2015
 Accepted: 28 February 2016
 Published: 9 March 2016
Abstract
We prove the existence of a gbest proximity point for a pair of mappings, by using suitable hypotheses on a metric space. Moreover, we establish some convergence results for a variational inequality problem, by using the variational characterization of metric projections in a real Hilbert space. Our results are applicable to classical problems of optimization theory.
Keywords
 best proximity point
 metric projection
 proximal \(\mathcal{Z}\)contraction
 variational inequality
MSC
 41A65
 47J20
1 Introduction
Let A and B be two nonempty subsets of a metric space \((X,d)\) and \(T:A\rightarrow B\) be a nonselfmapping. The equation \(Tx=x\) is known as a general fixed point equation and its solution is related to the solution of many practical situations arising in pure and applied sciences. For instance, it is well known that many problems involving differential equations may be solved by searching for the existence of a fixed point of an integral operator. But for the existence of a fixed point of T, we need that \(T ( A ) \cap A\neq\emptyset\), otherwise \(d ( x,Tx ) >0\) for all \(x\in A\). In such a situation, it is natural to search a point \(x\in A\) such that x is closest to Tx in some sense. To clarify and support this assertion, we recall the following best approximation theorem due to Ky Fan [1], in a metric version.
Theorem 1.1
([1])
Let A be a nonempty compact convex subset of a normed linear space X and \(T : A \to X\) be a continuous mapping. Then there exists \(x \in A\) such that \(\x  Tx\ = d(Tx, A)\).
This result is related to the existence of an approximate solution to the equation \(Tx=x\). Theoretical and practical aspects of this theorem have been discussed by various mathematicians; we refer the reader to [2–11].
On the other hand, very recently Khojasteh et al. [12] introduced the concept of \(\mathcal{Z}\)contraction, by using a notion of simulation function. Consequently, fixed point results involving a \(\mathcal{Z}\)contraction are established in [12]. This approach has been of great importance to discuss various fixed point problems from an unifying point of view; see for instance [13–15] and the references therein. For more contributions on the development of fixed point theorems see [16–18].
We generalize and extend many results in the existing literature, by establishing some best proximity point theorems involving \(\mathcal{Z}\)proximal contractions; see [19, 20]. In particular, we prove the existence of a unique gbest proximity point, which is a point \(x \in A\) such that \(d(gx, Tx) = d(A, B)\), where \(g : A \to A\) is a selfmapping. As an application, we give sufficient conditions to ensure the existence of a unique solution for a variational inequality problem and propose a convergent iterative algorithm to approximate this solution, by using metric projections. Our results are applicable to some classical problems of optimization theory.
2 Preliminaries
Kirk et al. gave sufficient conditions to ensure that \(A_{0}\) and \(B_{0}\) are nonempty sets; see [8]. On the other hand, Sadiq Basha and Veeramani proved that \(A_{0}\) is contained in the boundary of A; see [20].
In the sequel, we are interested in establishing results involving new types of proximal contraction and hence we recall the fundamental definitions in this direction; see [21, 22].
Definition 2.1
Definition 2.2
Definition 2.3
Many authors generalized these concepts and proved their best approximation theorems; see for instance [23–25].
In 2011, Sankar Raj [10] introduced the notion of Pproperty as follows.
Definition 2.4
([10], Definition 3)
Let A and B be two nonempty subsets of a metric space \(( X,d ) \) with \(A_{0} \neq\emptyset\). Then the pair \((A, B)\) is said to have the Pproperty if and only if \(d(x_{1}, y_{1}) = d(A, B)= d(x_{2}, y_{2})\) implies \(d(x_{1}, x_{2}) = d(y_{1}, y_{2})\) where \(x_{1}, x_{2} \in A_{0}\) and \(y_{1}, y_{2} \in B_{0}\).
By using Definition 2.4, Sankar Raj in [10] gave an extended version of the contraction mapping principle in [21]. Of course, for every nonempty subset A of X, the pair \((A, A)\) has the Pproperty. We shall consider this property in a remark of the next section.
Definition 2.5
 (i)
\(g \in\mathcal{G}_{A}\) if g is continuous and \(d(x,y) \leq d(gx,gy)\) for all \(x,y \in A\);
 (ii)
\(T \in\mathcal{T}_{g}\) if \(d(Tx,Ty) \leq d(Tgx,Tgy)\) for all \(x,y \in A\).
Finally, Khojasteh et al. in [12] defined a simulation function as follows.
Definition 2.6
 (\(\zeta_{1}\)):

\(\zeta(0,0)=0\);
 (\(\zeta_{2}\)):

\(\zeta(t,s)< st\), for all \(t,s>0\);
 (\(\zeta_{3}\)):

if \(\{t_{n}\}\), \(\{s_{n}\}\) are sequences in \(]0,+\infty[\) such that \(\lim_{n\to\infty} t_{n} = \lim_{n\to+\infty }s_{n}=\ell\in\, ]0,+\infty[\), then \(\limsup_{n\to+\infty} \zeta(t_{n},s_{n})<0\).
Consequently, they established the existence and uniqueness of fixed point for a selfmapping defined in a complete metric space.
Theorem 2.1
([12])
Successively, Argoubi et al. [13] point out the fact that condition (\(\zeta_{1}\)) is not mentioned in the proof of Theorem 2.1. Moreover, by putting \(x=y\) in (1), it follows that \(\zeta (0,0)\geq0\) and hence, if \(\zeta(0,0)<0\), the set of mappings \(f:X\to X\) satisfying condition (1) is an empty set.
Consequently, Argoubi et al. proposed a slight modification of Definition 2.6, by removing the condition (\(\zeta_{1}\)) and retaining the rest.
Remark 2.1
Every simulation function of Khojasteh et al. is also a simulation function of Argoubi et al. However, the converse is not true.
Example 2.1
([13], Example 2.4)
In order to avoid confusion, we refer to the following definition.
Definition 2.7
A simulation function is a mapping \(\zeta: [0, +\infty[\, \times[0, +\infty[\, \to\mathbb{R}\) satisfying the conditions (\(\zeta_{2}\)) and (\(\zeta_{3}\)).
3 Best proximity point theorems
In view of Definition 2.7, we consider the following notions of proximal contractions.
Definition 3.1
Remark 3.1
If \(T:A \to B\) is a \(\mathcal{Z}\)proximal contraction of the first kind and \((A,B)\) has the Pproperty, then T is a \(\mathcal {Z}\)contraction.
Example 3.1
Definition 3.2
Example 3.2
The following lemma is useful to show that a given sequence is Cauchy; see Lemma 2.1. in [26]; see also Lemma 2.1. in [17].
Lemma 3.1
 (i)
\(n_{k} > m_{k} \geq k\), \(k \in\mathbb{N}\);
 (ii)
\(d(x_{n_{k}}, x_{m_{k}}) \geq\varepsilon\), \(d(x_{n_{k}1}, x_{m_{k}}) < \varepsilon\), \(k \in\mathbb{N}\);
 (iii)
\(\lim_{k \to+ \infty}d(x_{n_{k}}, x_{m_{k}})=\varepsilon=\lim_{k \to+ \infty}d(x_{n_{k}+1}, x_{m_{k}+1})\).
On this basis, we construct our results. Precisely, we establish some theorems of gbest proximity point for \(\mathcal{Z}\)proximal contractions and deduce some corollaries.
Theorem 3.1
 (a)
T is a \(\mathcal{Z}\)proximal contraction of the first kind;
 (b)
\(g \in\mathcal{G}_{A}\);
 (c)
\(T ( A_{0} ) \subseteq B_{0}\);
 (d)
\(A_{0}\subseteq g ( A_{0} )\).
Proof
We get the following corollary, by setting g as the identity mapping on A in Theorem 3.1.
Corollary 3.1
 (a)
T is a \(\mathcal{Z}\)proximal contraction of the first kind;
 (b)
\(T ( A_{0} ) \subseteq B_{0}\).
Example 3.3
Let X, A, B, d, T, and ζ be as in Example 3.1. Notice that \(A_{0}=A=B_{0}\) is closed and \(T(A_{0}) \subseteq B_{0}\). Thus, by an application of Corollary 3.1, the mapping \(T:A \to B\) has a unique point \(x\in A\) such that \(d(x,Tx)=0 =d( A,B)\); here \(x=0\).
From Theorem 3.1, we obtain the following corollary which is a generalization of Theorem 3.1 of [6].
Corollary 3.2
 (a)
T is a proximal contraction of the first kind;
 (b)
\(g \in\mathcal{G}_{A}\);
 (c)
\(T ( A_{0} ) \subseteq B_{0}\);
 (d)
\(A_{0}\subseteq g ( A_{0} ) \).
Proof
Note that a proximal contraction of the first kind is a \(\mathcal {Z}\)proximal contraction of the first kind with respect to the simulation function \(\zeta: [0, +\infty[\, \times[0, +\infty[\, \to\mathbb {R}\) defined by \(\zeta(t,s)=kst\) for all \(t,s \in[0, +\infty[\), where \(k \in[0,1)\). □
The following theorem establishes a result of existence of a gbest proximity point for a \(\mathcal{Z}\)proximal contraction of the second kind.
Theorem 3.2
 (a)
T is a \(\mathcal{Z}\)proximal contraction of the second kind;
 (b)
T is injective on \(A_{0}\);
 (c)
\(T\in\mathcal{T}_{g}\);
 (d)
\(T ( A_{0} ) \subseteq B_{0}\);
 (e)
\(A_{0}\subseteq g ( A_{0} ) \).
Proof
We get the following corollary, by setting g as the identity mapping on A in Theorem 3.2.
Corollary 3.3
 (a)
T is a \(\mathcal{Z}\)proximal contraction of the second kind;
 (b)
T is injective on \(A_{0}\);
 (c)
\(T ( A_{0} ) \subseteq B_{0}\).
Example 3.4
It is easy to show that T is a \(\mathcal{Z}\)proximal contraction of the second kind, where the function \(\zeta_{\lambda}: [0, +\infty[\, \times[0, +\infty[\, \to\mathbb{R}\) is given in Example 2.1.
4 Variational inequality problems
Let H be a real Hilbert space, with inner product \(\langle\cdot, \cdot\rangle\) and induced norm \(\ \cdot\\). Let K be a nonempty, closed, and convex subset of H. We consider a monotone variational inequality problem as follows; see [27, 28].
Problem 4.1
Find \(u \in K\) such that \(\langle Su, vu \rangle\geq0\) for all \(v \in K\), where \(S : H \to H\) is a monotone operator (i.e., \(\langle Su  Sv, vu \rangle\geq0\) for all \(u,v \in K\)).
The theoretical background of projection and related approximation methods can be found in [30], too. Here, we need the following crucial lemmas, relating the existence of a solution for a variational inequality problem and the existence of a fixed point of a certain mapping.
Lemma 4.1
Let \(z \in H\). Then \(u \in K\) satisfies the inequality \(\langle uz,yu \rangle\geq0\), for all \(y \in K\) if and only if \(u=P_{K}z\).
Lemma 4.2
Let \(S : H \to H\) be monotone. Then \(u \in K\) is a solution of \(\langle S u, vu \rangle\geq0\), for all \(v \in K\), if and only if \(u=P_{K}(u \lambda Su)\), with \(\lambda>0\).
On this basis, we give some general convergence results on the solution of Problem 4.1.
Theorem 4.1
 (a)
\(P_{K}(I_{K} \lambda S): K \rightarrow K\) is a \(\mathcal {Z}\)contraction, with \(\lambda>0\).
Proof
Define \(T : K \to K\) by \(Tx=P_{K}(x \lambda Sx)\) for all \(x \in K\) so that, by Lemma 4.2, \(u \in K\) is a solution of \(\langle S u, vu \rangle\geq0\) for all \(v \in K\) if and only if \(u=Tu\). Clearly, the operator T satisfies all the hypotheses of Theorem 2.1 by setting \(A=B=K\). We deduce that the conclusions of Theorem 4.1 hold true as an immediate consequence of Theorem 2.1. □
Inspired by Theorem 4.1, one can consider the following algorithm to solve Problem 4.1.
Variational inequality problem solving algorithm
 Step 1 :

(Initialization): Select an arbitrary starting point \(u_{0} \in K\).
 Step 2 :

(Iteration): Given the current approximation point \(u_{n} \in K\), \(n \in\mathbb{N} \cup\{0\}\), computesatisfying Theorem 4.1(a).$$u_{n+1}=P_{K}(u_{n}  \lambda Su_{n}), $$
In view of the proof of Theorem 4.1 (and hence Theorem 2.1), this algorithm generates sequences converging to a unique solution of Problem 4.1.
From Corollary 3.3, we obtain the following result for the solution of a variational inequality problem.
Theorem 4.2
 (a)
\(P_{K}(I_{K} \lambda S): K \rightarrow K\) is a \(\mathcal {Z}\)proximal contraction of the second kind, with \(\lambda>0\);
 (b)
\(P_{K}(I_{K} \lambda S)\) is injective on K;
 (c)
\(P_{K}(I_{K} \lambda S)(K)\) is closed.
Inspired by Theorem 4.2, one can consider the following algorithm to solve Problem 4.1.
Variational inequality problem solving algorithm
 Step 1 :

(Initialization): Select an arbitrary starting point \(u_{0} \in K\).
 Step 2 :

(Iteration): Given the current approximation point \(u_{n} \in K\), \(n \in\mathbb{N} \cup\{0\}\), computesatisfying Theorem 4.2(a)(c).$$u_{n+1}=P_{K}(u_{n}  \lambda Su_{n}), $$
Our results apply to some fundamental problems of optimization theory. In fact, as a special case of Problem 4.1, we retrieve the following constrained minimization problem.
Problem 4.2
Find \(u \in K\) such that \(\langle\nabla fu, vu \rangle\geq0\) for all \(v \in K\), where \(f : H \to\mathbb{R}\) is a continuously differentiable function which is convex on K with ∇f denoting the gradient of f.
A second special case of Problem 4.1, is the following hierarchical variational inequality problem.
Problem 4.3
Let \(\operatorname{Fix}(g):=\{x \in K : x=gx\}\), where \(g:K \to K\) is such that \(\gxgy\ \leq\xy\\) (i.e., g is nonexpansive). Find \(u \in \operatorname{Fix}(g)\) such that \(\langle Su, vu \rangle\geq0\) for all \(v \in \operatorname{Fix}(g)\), where \(S : K \to K\) is a monotone continuous operator.
In [31], the authors proved that the set of critical points of (4) coincides with the set of solutions of a monotone variational inequality problem involving the operator S. Thus, our theory is applicable to the study of (4), which is associated to various economic problems; see again [31].
5 Conclusions
Best approximation and fixed point theories are continuously expanding topics due to their applications in many fields of pure and applied mathematics. Thus, we gave new theorems of gbest proximity point by using a notion of simulation function. This approach is useful to cover existing results in the literature from an unifying point of view. A discussion of the solvability of monotone variational inequality problems and related optimization problems supports the new theory.
Declarations
Acknowledgements
This research project was supported by a grant from the ‘Research Center of the Center for Female Scientific and Medical Colleges’, Deanship of Scientific Research, King Saud University.
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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