Variational relation problems: existence of solutions and fixed points of contraction mappings
 Abdul Latif^{1}Email author and
 Dinh The Luc^{2}
https://doi.org/10.1186/168718122013315
© Latif and Luc; licensee Springer. 2013
Received: 29 July 2013
Accepted: 29 October 2013
Published: 25 November 2013
Abstract
We propose a new approach to study variational relation problems. Namely, we apply Mizoguchi and Takahashi’s fixed point theorem of contraction mappings and an error bound of a system of linear inequalities to establish existence conditions for a variational relation problem in which the variational relation linearly depends on the decision variable. Then we develop an algorithm to compute a solution of a linear variational relation problem.
MSC:49J52, 47H10.
Keywords
1 Introduction
 (1)
$\overline{x}$ is a fixed point of S, that is, $\overline{x}\in S(\overline{x})$,
 (2)
$R(\overline{x},y)$ holds for every $y\in T(\overline{x})$,
where X and Y are nonempty sets, S is a setvalued mapping from X to itself, T is a setvalued mapping from X to Y, and $R(x,y)$ is a relation linking $x\in X$ and $y\in Y$. In an abstract setting, the relation R is represented by a subset of the product space $X\times Y$ so that $R(x,y)$ holds if and only if the point $(x,y)$ belongs to that set. In practice, however, R is often given by equality/inequality of realvalued functions, or by inclusion/intersection of setvalued mappings on $X\times Y$. For instance, given a realvalued function ϕ on $X\times Y$, a variational relation can be defined by any of the following equality and inequalities: $\varphi (x,y)=0$, $\varphi (x,y)\ne 0$, $\varphi (x,y)>0$ or $\varphi (x,y)\ge 0$. When two setvalued mappings ${G}_{1}$ and ${G}_{2}$ are given on $X\times Y$ with values in a nonempty set Z, a variational relation can be defined by any of the following inclusions and intersections: ${G}_{1}(x,y)\subseteq {G}_{2}(x,y)$, ${G}_{1}(x,y)\u2288{G}_{2}(x,y)$, ${G}_{1}(x,y)\cap {G}_{2}(x,y)=\mathrm{\varnothing}$ or ${G}_{1}(x,y)\cap {G}_{2}(x,y)\ne \mathrm{\varnothing}$. A mixture of the above relations is also possible.
The variational relation problem was introduced in [1] and studied in a number of recent works [2–13]. It encompasses a large class of problems of applied mathematics including optimization problems, variational inequalities, variational inclusions, equilibrium problems etc., and offers a unifying treatment of problems that come from different areas and have a similar structure. Existence conditions of solutions to variational relation problems were analyzed in great generality, the stability of solutions of a parametric variational relation was also studied with respect to the continuity of setvalued mappings, and very recently a numerical method was developed to solve variational relation problems when the data are linear [14]. As far as we know, all conditions established for existence of solutions of variational relation problems in the above cited papers utilize intersection theorems or fixed point theorems involving the KKM property of setvalued mappings in one or another form [15]. In the present paper, we wish to give existence conditions by exploiting Mizoguchi and Takahashi’s fixed point theorem for contraction mappings and propose an algorithm to compute solutions to (VRP), which seems to be new in the theory of variational relations. Actually, we shall study a particular model of (VRP) in which R linearly depends on the decision variable x. Throughout we assume

X and Y are nonempty closed sets in the finite dimensional Euclidean spaces ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$, respectively,

$S(x)=X$ for every $x\in X$,

$R(x,y)$ holds if and only if $Axg(y)\le 0$ with A a $k\times n$ matrix, g a vector function from Y to ${\mathbb{R}}^{k}$.
then the (VRP) above is equivalent to the fixed point problem: find $\overline{x}\in X$ such that $\overline{x}\in \mathrm{\Gamma}(\overline{x})$. Essentially we shall exploit this equivalent formulation of (VRP) to establish existence conditions and to develop a solving method.
The paper is structured as follows. In the next section, we present two preliminary results which constitute major tools of our study: a fixed point theorem by Mizoguchi and Takahashi [21], which generalizes the Nadler fixed point principle [22], and an error bound or Hoffman’s constant for a system of linear inequalities (see [17, 19, 23–28]). Section 3 is devoted to sufficient conditions for existence of solutions of (VRP). In the last section, we propose an algorithm to compute a solution of (VRP) and illustrate it by some numerical examples.
2 Preliminaries
Contraction mappings
The famous Banach contraction principle states that if $(X,d)$ is a complete metric space and if f is a real contraction function on X, then f has a fixed point. This principle was generalized to the case of setvalued mappings by Nadler [22] and Markin [29] with the help of the Hausdorff distance. Since then a lot of investigation has been made in order to weaken the contraction hypothesis (see [21, 30–34] and many references given in these). In the present paper, we are particularly interested in a theorem by Mizoguchi and Takahashi [21], which can elegantly be applied to our model. Let us recall it in details and make a discussion on its generalization.
 (i)
there exists a function $\gamma :(0,\mathrm{\infty})\to [0,1)$ such that ${lim\hspace{0.17em}sup}_{r\to {t}^{+}}\gamma (r)<1$ for each $t\in [0,\mathrm{\infty})$;
 (ii)
for every $x,y\in X$, one has $h(F(x),F(y))\le \gamma (d(x,y))d(x,y)$.
Further developments of this result can be found in [30, 35–38]. In particular the following theorem by Ciric (Theorem 2.2 [30]) is quite general: if F has closed values and if there exists a real function $\varphi :[0,\mathrm{\infty})\to [a,1)$ for some $a\in (0,1)$ such that

${lim\hspace{0.17em}sup}_{s\to {t}^{+}}\varphi (s)<1$ for every $t\ge 0$;

for every $x\in X$, there is $y\in F(x)$ satisfying$\begin{array}{c}\sqrt{\varphi (d(x,y))}d(x,y)\le d(x,F(x)),\hfill \\ d(y,F(y))\le \varphi (d(x,y))d(x,y),\hfill \end{array}$
then F admits a fixed point.
It is unfortunate that this theorem does not fully generalize Mizoguchi and Takahashi’s theorem because in the latter theorem it is not requested that the function ϕ takes its values bigger than a strictly positive number a. Another observation we can make is the fact that Mizoguchi and Takahashi’s theorem remains valid with the same argument even when F has closed values that are not necessarily bounded.
Error bound for a linear system
 (1)
I is nonempty and the family of rows ${a}^{i}$, $i\in I$ of the matrix A is linearly independent,
 (2)
there is some $y\in P$ such that I is contained in the active index set $I(y)$ at y, that is, ${a}^{i}y={b}_{i}$, $i\in I$; ${A}_{I}$ is the submatrix of A consisting of the rows ${a}^{i}$, $i\in I$, and ${A}_{I}^{T}$ is its transpose.
whenever both P and ${P}^{\prime}$ are nonempty.
3 Existence conditions
In this section we establish some sufficient conditions for existence of solutions to the variational problem (VRP) introduced in the beginning of Section 2. Mizoguchi and Takahashi’s fixed point theorem and the error bound (2.1) are the main tools we use for this purpose. Given a real function $u(x)$ on X, we define a value function (called also a marginal function) of u by $\beta (x)={inf}_{z\in T(x)}u(z)$. Stability and sensitivity of this function is one of the indispensable parts of the theory of optimization. We refer the interested reader to the books [17, 19, 25] for greater details. The next result will be needed in the sequel.
Switching the roles of x and y in the above inequality and taking into account the fact that ϵ is arbitrarily chosen, we obtain $\beta (x)\beta (y)\le \varphi (d(x,y))\psi (d(x,y)\varphi (d(x,y)))d(x,y)$ as requested. □
In the remaining part of this section, we assume that for every $x\in X$, the values ${b}_{i}(x)={inf}_{z\in T(x)}{g}_{i}(z)$, $i=1,\dots ,k$, are finite and that the system $Az\le b(x)$, $z\in X$ is consistent. The vector whose components are ${b}_{i}(x)$, $i=1,\dots ,k$, is denoted $b(x)$.
 (i)
$h(T(x),T(y))\le \varphi (d(x,y))d(x,y)$ for $x,y\in X$;
 (ii)
${g}_{i}(x){g}_{i}(y)\le {\psi}_{i}(d(x,y))d(x,y)$ for $x,y\in X$ and $i=1,\dots ,k$;
 (iii)
${lim\hspace{0.17em}sup}_{s\to {t}^{+}}\varphi (s)\sqrt{{\sum}_{i=1}^{k}{\psi}_{i}{(s\varphi (s))}^{2}}<\frac{1}{\alpha}$ for all $t>0$.
Then (VRP) admits a solution.
Consider the real function $\gamma (t)=\alpha \varphi (t)\sqrt{{\sum}_{i=1}^{k}{\psi}_{i}{(t\varphi (t))}^{2}}$ for every $t\ge 0$. Then the hypotheses of Mizoguchi and Takahashi’s theorem are satisfied for the setvalued mapping Γ, by which it admits a fixed point. Consequently, (VRP) has a solution. □
When the mapping T and the function g are Lipschitz, we derive the following result.
Corollary 3.1 Assume that T is κLipschitz and ${g}_{1},\dots ,{g}_{k}$ is ℓLipschitz with $\kappa \ell <\frac{1}{\alpha \sqrt{k}}$. Then (VRP) has a solution.
Proof Set $\varphi (t)=\kappa $ and ${\psi}_{i}(t)=\ell $ for $i=1,\dots ,k$ and $t\in [0,\mathrm{\infty})$ and apply Theorem 3.1. □
Let us now consider the case when the function g is affine, that is, $g(y)=Cy+c$, where C is a $k\times m$ matrix, c is a k vector, and the graph of T is a convex polyhedral set, that is, $y\in T(x)$ if and only if x and y solve a linear system $Py\le Qx+q$, where P is a $k\times m$ matrix, Q is a $k\times n$ matrix and q is a k vector. (VRP) with such linear data is called a linear variational relation problem. It was studied in [14] in which a numerical algorithm based on Delauney’s triangulations is proposed for solving it. It is known that a linear variational problem may have no solutions (see Example 4.1). We wish to apply Theorem 3.1 to derive an existence condition for this model. The rows of the matrix C are denoted ${C}^{1},\dots ,{C}^{k}$, while the components of the vector c are denoted ${c}_{1},\dots ,{c}_{k}$. The best error bound for the system $Pz\le Qx+q$ is denoted ${\alpha}^{\prime}$.
Then (VRP) has a solution.
It remains to apply Theorem 3.1 to conclude that (VRP) has a solution. □
4 An algorithm
In this section, we consider a linear (VRP) as mentioned in the preceding section, namely we wish to find $\overline{x}\in {\mathbb{R}}^{n}$ such that $A\overline{x}\le Cy+c$ for every y solution to the system $Py\le Q\overline{x}+q$. As we have already noticed, this problem may have no solutions. Here is an example.
For every $x\in [0,2]$, $y\in T(x)$ if and only if $0\le y\le 4$. With $y=4$, it is clear that $R(x,y)$ does not hold, which means that (VRP) has no solutions.
Let us now describe an algorithm to solve (VRP).
Step 0. Choose any ${x}_{0}\in X$, a small tolerance level $\u03f5>0$. Set $r=0$.
Let ${b}_{r}$ be the vector whose components are optimal values of the above programs.
Let z be an optimal solution of this program.
Step 3. Check $\parallel {x}_{r}z\parallel \le \u03f5$. If yes, stop. The optimal solution z is considered as a solution of (VRP). Otherwise, set $r=r+1$, ${x}_{r}=z$ and return to Step 1.
We have the following convergence property of the algorithm.
Proposition 4.1 Assume that the hypothesis of Corollary 3.2 holds. Then, for any $\u03f5>0$, the algorithm terminates after a finite number of iterations. If at some iteration the optimal value of the program in Step 2 is zero, then ${x}_{r}$ is a solution of (VRP). If not, with ϵ tending to 0, the sequence of ${x}_{r}$ obtained in Step 3 converges to a solution of (VRP).
Proof It is clear that ${x}_{r+1}\in \mathrm{\Gamma}({x}_{r})$. If, for some r, the optimal value of the program in Step 2 is zero, then ${x}_{r}\in \mathrm{\Gamma}({x}_{r})$ which means that ${x}_{r}$ is a fixed point of Γ, and hence a solution of (VRP). Moreover, since under the hypothesis of Corollary 3.2 the mapping Γ is a contraction, the sequence ${\{{x}_{r}\}}_{r=0}^{\mathrm{\infty}}$ converges to a fixed point of Γ, which is also a solution of (VRP). By this the algorithm terminates after a finite number of iterations when ϵ is strictly positive. □
Below we give two examples to show how to perform the algorithm. In the first example, the algorithm terminates after two iterations and produces an exact solution. In the second example, we can obtain an approximate solution with any given strictly positive tolerance level.
The three functions to minimize are respectively $({y}_{1}+{y}_{2})/41$, 0 and 2, which implies that ${b}_{0}={(1,0,2)}^{T}$. In the next step, we minimize $z$ over the set $Az\le {b}_{0}$, $z\in X$. The optimal solution is $z=1$. As ${x}_{0}\ne z$, it is not a solution of (VRP).
Iteration 2. We set ${x}_{1}=z=1$ and return to Step 1 to solve the three above mentioned functions over the set $Py\le Q{x}_{1}+q$. The same optimal solutions are obtained and ${b}_{1}={(1,0,2)}^{T}$. In Step 2, we minimize $1z$ over the set $Az\le {b}_{1}$, $z\in X$ and obtain the optimal solution $z=1$. Since ${x}_{1}=z$, the algorithm terminates and ${x}_{1}=1$ is a solution of (VRP). It is easy to see that Γ is a (1/2)contraction mapping, that is, $h(\mathrm{\Gamma}(x),\mathrm{\Gamma}({x}^{\prime}))\le x{x}^{\prime}/2$ for every $x,{x}^{\prime}\in [0,2]$.
Iteration 1. We start the algorithm with ${x}_{0}=0$. The feasible set of the programs in Step 1 is given by the system $Py\le q$ and consists of one element $\{0\}$. Hence the vector ${b}_{0}$ is equal to ${(11/16,0,2)}^{T}$. In the next step, we minimize $z$ over the set $Az\le {b}_{0}$, $z\in X$. The optimal solution is $z=11/16$. As ${x}_{0}\ne z$, it is not a solution of (VRP).
Iteration 2. We set ${x}_{1}=z=11/16$ and return to Step 1 to solve three programs whose feasible set is given by $Py\le Q{x}_{1}+q$. We obtain ${b}_{1}={(109/128,0,2)}^{T}$. In Step 2, we minimize $11/16z$ over the set $Az\le {b}_{1}$, $z\in X$ and obtain the optimal solution $z=109/128$. Since ${x}_{1}\ne z$, the point ${x}_{1}$ is not a solution of (VRP). If we choose a priori a tolerance level $\u03f5=0.2$, then we may stop the algorithm and consider ${x}_{2}=109/128$ as an approximate solution of (VRP) because ${x}_{1}{x}_{2}<\u03f5$. If not, we continue it with ${x}_{2}$ for restarting the procedure. It can be seen that the algorithm generates the sequence ${\{{x}_{r}\}}_{r=0}^{\mathrm{\infty}}$ converging to $x=2$, which is the unique solution of (VRP).
Declarations
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. 130030D1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors thank the referees for the valuable comments and appreciation.
Authors’ Affiliations
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