A solution for the noncooperative equilibrium problem of two person via fixed point theory
 Tran Duc Thanh^{1},
 Aatef Hobiny^{2} and
 Erdal Karapınar^{3, 4}Email author
https://doi.org/10.1186/s1366001506793
© Duc Thanh et al.; licensee Springer. 2015
Received: 3 March 2015
Accepted: 27 April 2015
Published: 12 May 2015
Abstract
In this paper, we investigate the noncooperative equilibrium problem of two person games in the setting of game theory and propose a solution via coupled fixed point results in the context of partial metric spaces. We also realize that our coupled fixed point results can be applied to get a solution of a class of nonlinear Fredholm type integral equations.
Keywords
coupled fixed point partial metric Fcontractions noncooperative equilibriumMSC
46J10 46J15 47H101 Introduction
It is very well known fact that real world problem can be modeled as a mathematical equation. The existence of a solution of such problems has been investigated in several branches of mathematics, such as differential equations, integral equations, functional equations, partial differential equations, random differential equations, etc. and one has proposed solutions for such problems via fixed point theory. But the application area of fixed point theory is not only limited to mathematics, but also occurs in other quantitative sciences, such as, computer science, economics, biology, physics, etc. Game theory, a branch of economics, has used fixed point theory techniques and approaches to solve its own problems.
Game theory can be regarded as a formal (mathematical) way to study games. Indeed, we consider the games as conflicts where some number of individuals (called players) take part and each one tries to maximize his utility in taking part in the conflict. Games can be classified in many ways, but here we focus on the following classification: Cooperative games, in which, players are allowed to cooperate and noncooperative games, in which players are not allowed to cooperate. In the sequel, we shall demonstrate how the question of the existence of equilibria is related to the question of the existence of a fixed point. Throughout the paper, we follow the notion and notation in [1]. We recall some basic concepts.
 (1)
topological spaces \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\), the so called strategies for player 1 resp. player 2,
 (2)
a topological subspace \(U\subset \mathcal{S}_{1}\times\mathcal{S}_{2}\) of an allowed strategy pair,
 (3)a biloss operatorwhere \(L_{i}(s_{1}, s_{2})\) is the loss of player i if the strategies \(s_{1}\) and \(s_{2}\) are played.$$ \begin{aligned} L: &U\to\Bbb{R}^{2} \\ & (s_{1}, s_{2})\mapsto \bigl(L_{1}(s_{1}, s_{2});L_{2}(s_{1}, s_{2}) \bigr), \end{aligned} $$(1)
The main goal of the present work is to solve the problem of the noncooperative equilibrium of two person games. For this purpose, we shall present some coupled fixed point theorems in partial metric spaces. Our aim is to explore not only the results themselves but also their applications to nonlinear integral equations.
2 Preliminaries
The notion of a partial metric was proposed by Matthews (see [2, 3]) as a generalization of the metric concept to get better results in the branches of computer sciences: semantics and computer domain. Indeed, a partial metric is a function that is obtained from the metric by replacing the condition \(d(x,x)=0\) with the condition \(d(x,x)\leq d(x, y)\) for all x, y. In the last decade, a number of authors have brought into focus fixed point problems in the context of partial metric spaces as well as topological properties of a partial metric space; see e.g. [4–14] and the related references given therein.
We first need to recall some basic concepts and necessary results. Throughout the paper, \(\mathbb{N}\) and \(\mathbb{N}_{0}\) denote the set of positive integers and the set of nonnegative integers, respectively. Similarly, \(\mathbb{R}\), \(\mathbb{R}^{+}\), and \(\mathbb{R}^{+}_{0}\) represent the set of reals, positive reals, and nonnegative reals, respectively.
Definition 2.1
 (P1)
\(x=y\) if and only if \(p(x, x)= p(y, y)=p(x, y)\).
 (P2)
\(p(x, x)\leq p(x, y)\).
 (P3)
\(p(x, y)= p(y,x)\).
 (P4)
\(p(x, z)\leq p(x, y)+p(y,z) p(y,y)\).
Definition 2.2
 (1)
A sequence \(\{x_{n}\}\) in X converges to \(x\in X\) if and only if \(p(x,x)=\lim_{n\to\infty} p(x_{n},x)\).
 (2)
A sequence \(\{x_{n}\}\) in X is called a Cauchy sequence if and only if \(\lim_{n,m\rightarrow\infty} p(x_{n},x_{m})\) exists (and is finite).
 (3)
\((X,p)\) is called complete if every Cauchy sequence \(\{x_{n}\}\) in X converges to \(x\in X\).
 (4)
A mapping \(f: X\to X\) is said to be continuous at \(x_{0}\in X\) if, for every \(\varepsilon>0\), there exists \(\delta>0\) such that \(f(B(x_{0}, \delta))\subset B(f(x_{0}), \varepsilon)\).
Referring to [15], we say that a sequence \(\{x_{n}\}\) in \((X,p)\) is called a 0Cauchy sequence if \(\lim_{n,m\rightarrow\infty} p(x_{n},x_{m})=0\). Also, we say that \((X, p)\) is 0complete if every 0Cauchy sequence in X converges, with respect to the partial metric p, to a point \(x\in X\) such that \(p(x,x)=0\). Notice that if \((X,p)\) is complete, then it is 0complete, but the converse does not hold. Moreover, every 0Cauchy sequence in \((X,p)\) is Cauchy in \((X, d_{p})\).
Example 2.3
 (1)
Let \(X=[0, +\infty)\) and define \(p(x, y)=\max\{x, y\}\), for all \(x, y\in X\). Then \((X, p)\) is a complete partial metric space. It is clear that p is not a (usual) metric.
 (2)
Let \(X=[0, +\infty)\cap\Bbb{Q}\), where \(\Bbb{Q}\) is the set of rational numbers. Define \(p(x, y)=\max\{x, y\}\), for all \(x, y\in X\). Then \((X, p)\) is a 0complete partial metric space which is not complete.
Proposition 2.4
 (1)
A sequence \(\{x_{n}\}\) is a Cauchy sequence in \((X, p)\) if and only if \(\{x_{n}\}\) is a Cauchy sequence in \((X, d_{p})\).
 (2)\((X, p)\) is complete if and only if \((X, d_{p})\) complete. Moreover,$$\lim_{n\to\infty} d_{p}(x_{n}, x)=0 \quad\Leftrightarrow\quad\lim_{n\to\infty }p(x, x)= \lim_{n\to\infty}p(x_{n}, x)=\lim_{n,m\to\infty}p(x_{m}, x_{n}). $$
The following lemmas have an important role to play in the proofs of the theorems.
Lemma 2.5
Assume \(x_{n}\rightarrow z\) as \(n\rightarrow\infty\) in a PMS \((X,p)\) such that \(p(z,z)=0\). Then \(\lim_{n\rightarrow\infty} p(x_{n},y)=p(z,y)\) for every \(y \in X\).
Lemma 2.6
 (1)
If \(p(x,y)=0\) then \(x=y\).
 (2)
If \(x\neq y\), then \(p(x,y)>0\).
Lemma 2.7
Let \((X,p)\) be a PMS. If \(x_{n}\to x\) and \(y_{n}\to y\) as \(n\to\infty\) for all \(x_{n},y_{n},x,y\in X\) then \(p(x_{n},y_{n})\to p(x,y)\) as \(n\to\infty\).
The existence and uniqueness of fixed points of contractive type mappings in partially ordered metric spaces have been considered recently by several authors: Ran and Reurings [16], Nieto and RodriguezLopez [17, 18]. On the other hand, the notion of a coupled fixed point was suggested by Guo and Lakshmikantham in [19]. Following this initial result, Gnana Bhaskar and Lakshmikantham [20] proposed the notion of mixed monotone property and get coupled fixed point results in the setting of partially ordered metric spaces (see also [21–23] and the related references therein.) Later, it was reported that most of the coupled fixed point results can be derived from the existence results, and vice versa; see e.g. [24–26]. On the other hand, coupled fixed point results still have worth regarding their applications. Most of the times, using coupled fixed point theory is the most economical way to solve problems (regarding time and speed of the process). This paper can be considered as an example.
Recall that a pair \((x,y)\in X\times X \) is called a coupled fixed point of the mapping \(T: X\times X\to X\) if \(T(x,y)=x\), \(T(y,x)=y\) (see e.g. [19]).
Definition 2.8
([20])
 (F_{1}):

F is strictly increasing and continuous.
 (F_{2}):

For each sequence \((a_{n})\subset\mathbb{R}^{+}_{0}\), \(\lim_{n\to\infty} a_{n}=0\) if and only if \(\lim_{n\to\infty} F(a_{n})=\infty\).
We denote by \(\mathcal{F}\) the family of all functions F that satisfy the conditions (F_{1})(F_{2}) (see [27]). It is easy to check that \(F(x)=\ln x\) and \(G(x)=\ln x+x\) for all \(x \in \mathbb{R}^{+}_{0}\) belong to \(\mathcal{F}\).
In [28], Wardowski introduced the new concept of an Fcontraction and proved fixed point theorems in the classical setting of metric spaces. In [27], the authors introduced the concept of an Fcontraction, a generalized Fcontraction, and they proved some fixed point theorems for multivalued mappings in the partial metric spaces (see also [6, 9, 29–32]).
Definition 2.9
([27])
3 Auxiliary results: coupled fixed points in partial metric spaces
In this section we state and prove some new coupled fixed point results for Fcontractive mappings in the context of complete partial metric spaces.
Theorem 3.1
 (1)for all \(x\leq u, y\geq v\), for some \(F\in\mathcal{F}\) and \(\tau>0\).$$ \tau+F \bigl(p \bigl(T(x,y), T(u,v) \bigr) \bigr) \leq F \bigl( \max \bigl\{ p(x, u), p(y, v) \bigr\} \bigr) $$(7)
 (2)
There are \(x_{0},y_{0}\in X\) such that \(x_{0}\leq T(x_{0},y_{0})\), \(y_{0}\geq T(y_{0},x_{0})\).
Then T has a coupled fixed point, that is, there exist \(x,y\in X\) such that \(x=T(x,y)\), \(y=T(y,x)\).
Proof
In the next theorem, we omit the continuity hypothesis of T.
Theorem 3.2
 (1)for all \(x\leq u\), \(y\geq v\), for some \(F\in\mathcal{F}\) and \(\tau>0\).$$ \tau+F \bigl(p \bigl(T(x,y), T(u,v) \bigr) \bigr) \leq F \bigl(\max \bigl\{ p(x, u), p(y, v) \bigr\} \bigr) $$(23)
 (2)
There are \(x_{0},y_{0}\in X\) such that \(x_{0}\leq T(x_{0},y_{0})\), \(y_{0}\geq T(y_{0},x_{0})\).
 (i)
If a nondecreasing sequence \(\{x_{n}\}\) in X converges to x then \(x_{n} \leq x\) for all n.
 (ii)
If a nonincreasing sequence \(\{y_{n}\}\) in X converges to y then \(y_{n} \geq y\) for all n.
Then T has a coupled fixed point, that is, there exist \(x,y\in X\) such that \(x=T(x,y)\), \(y=T(y,x)\).
Proof
We easily get the following corollary.
Corollary 3.3
 (1)for all \(x\leq u\), \(y\geq v\), for some \(F\in\mathcal{F}\) and \(\tau>0\).$$ \tau+F \bigl(p \bigl(T(x,y), T(u,v) \bigr) \bigr) \leq F \biggl( \frac{p(x, u)+p(y, v)}{2} \biggr) $$(24)
 (2)
There are \(x_{0},y_{0}\in X\) such that \(x_{0}\leq T(x_{0},y_{0})\), \(y_{0}\geq T(y_{0},x_{0})\).
 (a)
T is continuous; or
 (b)X has the properties:
 (i)
If a nondecreasing sequence \(\{x_{n}\}\) in X converges to x then \(x_{n} \leq x\) for all n.
 (ii)
If a nonincreasing sequence \(\{y_{n}\}\) in X converges to y then \(y_{n} \geq y\) for all n.
 (i)
Then T has a coupled fixed point, that is, there exist \(x,y\in X\) such that \(x=T(x,y)\), \(y=T(y,x)\).
Proof
The following corollary states that T has a fixed point under a certain condition.
Corollary 3.4
In addition to the hypotheses of Corollary 3.3, if \(x_{0}\) and \(y_{0}\) are comparable, then T has a unique fixed point, that is, there exists \(x\in X\) such that \(T(x,x)=x\).
Proof
4 Main result: noncooperative equilibrium problem for two players
In this section, by using coupled fixed point theorems, we shall show that a two person game has a noncooperative equilibrium. The reader may consult the excellent sources on general concepts of two person games in [1] and [33].
 (1)
\(S_{1}=S\) and \(S_{2}=S\) are strategies for player 1 and, respectively, player 2;
 (2)
the set \(U= S_{1}\times S_{2}\) of allowed strategies pairs;
 (3)we have the biloss operator$$ \begin{aligned} L: &U\to\Bbb{R}^{2}\\ &(s_{1}, s_{2})\mapsto \bigl(L_{1}(s_{1}, s_{2});L_{2}(s_{1}, s_{2}) \bigr), \end{aligned} $$(26)
Theorem 4.1
 (1)for all \(x,y\in S\) and \(y\geq x\), for some \(F\in\mathcal{F}\) and \(\tau>0\).$$ \tau+F \bigl(p \bigl(C(x), C(y) \bigr) \bigr) \leq F \bigl(p(x, y) \bigr) $$(31)
 (2)
There are \(x_{0},y_{0}\in\Bbb{R}_{0}^{+}\) such that \(x_{0}\leq C(y_{0})\), \(y_{0}\geq C(x_{0})\).
Then the two person game \(\mathcal{G}\) has a noncooperative equilibrium.
Proof
Since every metric is partial metric, we immediately obtain the following corollary.
Corollary 4.2
 (1)for all \(x,y\in\Bbb{S}\) and \(x< y\), for some \(F\in\mathcal{F}\) and \(\tau>0\).$$ \tau+F \bigl(d \bigl(C(x), C(y) \bigr) \bigr) \leq F \bigl(d(x, y) \bigr) $$(33)
 (2)
There are \(x_{0},y_{0}\in\Bbb{R}_{0}^{+}\) such that \(x_{0}\leq C(y_{0})\), \(y_{0}\geq C(x_{0})\).
Then the two person game \(\mathcal{G}\) has a noncooperative equilibrium.
Now we shall give an example to show that Corollary 4.2 is effective.
Example 4.3
5 Application to nonlinear integral equations
In this section, we study the existence of unique solution of nonlinear integral equations, as an application of the fixed point theorem proved in Section 3.
As in [34] and the references given therein, we assume that the functions \(K_{1}\), \(K_{2}\), f, g fulfill the following conditions.
Assumption 5.1
 (A)
\(f, g\in C([0, K]\times\Bbb{R})\), \(h\in X\), and \(K_{1}, K_{2}\in C([0, K]\times[0, K])\) such that \(K_{1}(t, s)\geq0\) and \(K_{2}(t, s)\leq0\) for all \(t, s\geq0\);
 (B)
\(f(t, \cdot):\Bbb{R}\to\Bbb{R}\) is increasing for all \(t\in[0, K]\); \(g(t,\cdot): \Bbb{R}\to\Bbb{R} \) is decreasing for all \(t\in[0, K]\);
 (C)there exists \(\tau\in[1, \infty)\) such thatand$$0\leq f(t, x)f(t, y)\leq\tau e^{\tau} \frac{xy}{2},\quad \forall x \geq y $$$$\tau e^{\tau} \frac{xy}{2} \leq g(t, x)g(t, y)\leq0, \quad\forall x\geq y; $$
 (D)
\(\max_{t,s\in[0, K]}  K_{1}(t, s)K_{2}(t, s)\leq1\).
Definition 5.2
Theorem 5.3
Suppose that Assumption 5.1 is fulfilled. Then the existence of a coupled normal lower and normal upper solution for (34) provides the existence of a unique solution of (34) in \(C([0,K])\).
Proof
Suppose \(\{u_{n}\}\) is a monotone nondecreasing sequence in X that converges to \(u\in X\). Then, for every \(t\in[0, K]\), the sequence of real numbers \(u_{1}(t)\leq u_{2}(t)\leq\cdots\leq u_{n}(t)\leq\cdots\) converges to \(u(t)\). Moreover, since the normed map is continuous, we can deduce that \(\ u\\leq1\) provided \(\ u_{n}\\leq1\) for all n. Therefore, for every \(t\in[0, K]\), \(n\in\Bbb{N}\), \(u_{n}(t)\leq u(t)\). Hence \(u_{n}\leq u\), for all \(n\in\Bbb{N}\).
Similarly, we can verify that the limit \(v(t)\) of a monotone nonincreasing sequence \(v_{n}(t)\) in X is a lower bound for all elements in the sequence. That is, \(v\leq v_{n}\) for all n. Hence, the condition (b) in Corollary 3.3 holds.
Declarations
Acknowledgements
The authors are grateful to the reviewers for their careful reviews and useful comments. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR 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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