 Research
 Open Access
Tripled fixed point theorem in fuzzy metric spaces and applications
 A Roldán^{1},
 J MartínezMoreno^{2}Email author and
 C Roldán^{3}
https://doi.org/10.1186/16871812201329
© Roldán et al.; licensee Springer 2013
 Received: 1 July 2012
 Accepted: 11 January 2013
 Published: 11 February 2013
Abstract
In this paper we prove an existence and uniqueness theorem for contractive type mappings in fuzzy metric spaces. In order to do that, we consider a slight modification of the concept of a tripled fixed point introduced by Berinde et al. (Nonlinear Anal. TMA 74:48894897, 2011) for nonlinear mappings. Additionally, we obtain some fixed point theorems for metric spaces. These results generalize, extend and unify several classical and very recent related results in literature. For instance, we obtain an extension of Theorem 4.1 in (Zhu and Xiao in Nonlinear Anal. TMA 74:54755479, 2011) and a version in nonpartially ordered sets of Theorem 2.2 in (Bhaskar and Lakshmikantham in Nonlinear Anal. TMA 65:13791393, 2006). As application, we solve a kind of Lipschitzian systems in three variables and an integral system. Finally, examples to support our results are also given.
Keywords
 Fixed Point Theorem
 Lipschitzian Mapping
 Cauchy Sequence
 Common Fixed Point
 Coincidence Point
Introduction
In a recent paper, Bhaskar and Lakshmikantham [1] introduced the concepts of coupled fixed point and mixed monotone property for contractive operators of the form $F:X\times X\to X$, where X is a partially ordered metric space, and then established some interesting coupled fixed point theorems. They also illustrated these important results by proving the existence and uniqueness of the solution for a periodic boundary value problem. Later, Lakshmikantham and Ćirić [2] proved coupled coincidence and coupled common fixed point results for nonlinear mappings satisfying certain contractive conditions in partially ordered complete metric spaces. After that many results appeared on coupled fixed point theory (see, e.g., [2–8]).
Fixed point theorems have been studied in many contexts, one of which is the fuzzy setting. The concept of fuzzy sets was initially introduced by Zadeh [9] in 1965. To use this concept in topology and analysis, many authors have extensively developed the theory of fuzzy sets and its applications. One of the most interesting research topics in fuzzy topology is to find an appropriate definition of fuzzy metric space for its possible applications in several areas. It is well known that a fuzzy metric space is an important generalization of the metric space. Many authors have considered this problem and have introduced it in different ways. For instance, George and Veeramani [10] modified the concept of a fuzzy metric space introduced by Kramosil and Michalek [11] and defined the Hausdorff topology of a fuzzy metric space. There exists considerable literature about fixed point properties for mappings defined on fuzzy metric spaces, which have been studied by many authors (see [10, 12–16]). Zhu and Xiao [7] and Hu [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang [3] proved some common fixed point theorems under ϕcontractions for compatible and weakly compatible mappings on Menger probabilistic metric spaces. Moreover, Elagan and Segi Rahmat [17] studied the existence of a fixed point in locally convex topology generated by fuzzy nnormed spaces.
Very recently, the concept of tripled fixed point has been introduced by Berinde and Borcut [18]. In their manuscript, some new tripled point theorems are obtained using the mixed gmonotone mapping. Their results generalize and extend the Bhaskar and Lakshmikantham’s research for nonlinear mappings. Moreover, these results could be used to study the existence of solutions of a periodic boundary value problem involving ${y}^{\mathrm{\prime}\mathrm{\prime}}=f(t,y,{y}^{\mathrm{\prime}})$. A multidimensional notion of a coincidence point between mappings and some existence and uniqueness fixed points theorems for nonlinear mappings defined on partially ordered metric spaces are studied in [19].
In this paper, our main aim is to obtain an existence and uniqueness theorem for contractive type mappings in the framework of fuzzy metric spaces. In order to do that, we consider a slight modification of the concept of a tripled fixed point introduced by Berinde and Borcut for nonlinear mappings. The power of this result is twofold. Firstly, we can particularize it to complete metric spaces, obtaining a BerindeBorcut type result (in nonfuzzy setting). Moreover, our result, in a unified manner, covers also coupled fixed (see Zhu and Xiao [7]) and fixed point theorems. Finally, examples to support our results are also given.
Preliminaries
Henceforth, X will denote a nonempty set and ${X}^{3}=X\times X\times X$. Subscripts will be used to indicate the arguments of a function. For instance, $F(x,y,z)$ will be denoted by ${F}_{xyz}$ and $M(x,y,t)$ will be denoted by ${M}_{xy}(t)$. Furthermore, for brevity, $g(x)$ will be denoted by gx.
From these properties, we can easily deduce that ${d}_{xy}\ge 0$ and ${d}_{yx}={d}_{xy}$ for all $x,y\in X$. The last requirement is called the triangle inequality. If d is a metric on X, we say that $(X,d)$ is a metric space (briefly, a MS).
Let $(X,d)$ be a MS. A mapping $f:X\to X$ is said to be Lipschitzian if there exists $k\ge 0$ such that $d({f}_{x},{f}_{y})\le k{d}_{xy}$ for all $x,y\in X$. The smallest k (denoted by ${k}_{f}$) for which this inequality holds is said to be the Lipschitz constant for f. A Lipschitzian mapping $f:X\to X$ is a contraction if ${k}_{f}<1$.
Theorem 1 (Banach’s contraction principle)
Every contraction from a complete metric space into itself has a unique fixed point.
If $X=\mathbb{R}$ provided with the Euclidean metric, examples of Lipschitzian mappings ${f}_{i}:\mathbb{R}\to \mathbb{R}$ are ${f}_{1}(x)=K$, ${f}_{2}(x)=\alpha x$, ${f}_{3}(x)=sinx$, ${f}_{4}(x)=cosx$, ${f}_{5}(x)=arctanx$ and ${f}_{6}(x)=1/(1+{x}^{2})$.
Definition 2 A triangular norm (also called a tnorm) is a map $\ast :{[0,1]}^{2}\to [0,1]$ that is associative, commutative, nondecreasing in both arguments and has 1 as identity. For each $a\in [0,1]$, the sequence ${\{{\ast}^{n}a\}}_{n=1}^{\mathrm{\infty}}$ is defined inductively by ${\ast}^{1}a=a$ and ${\ast}^{n}a=({\ast}^{n1}a)\ast a$. A tnorm ∗ is said to be of Htype (see [20]) if the sequence ${\{{\ast}^{n}a\}}_{n=1}^{\mathrm{\infty}}$ is equicontinuous at $a=1$, i.e., for all $\epsilon \in (0,1)$, there exists $\eta \in (0,1)$ such that if $a\in (1\eta ,1]$, then ${\ast}^{m}a>1\epsilon $ for all $m\in \mathbb{N}$.
The most important and wellknown continuous tnorm of Htype is $\ast =min$, that verifies $min(a,b)\ge ab$ for all $a,b\in [0,1]$. The following result presents a wide range of tnorms of Htype.
Lemma 3 Let $\delta \in (0,1]$ be a real number and let ∗ be a tnorm. Define ${\ast}_{\delta}$ as $x{\ast}_{\delta}y=x\ast y$, if $max(x,y)\le 1\delta $, and $x{\ast}_{\delta}y=min(x,y)$, if $max(x,y)>1\delta $. Then ${\ast}_{\delta}$ is a tnorm of Htype.
Definition 4 [11]
 (i)
${M}_{xy}(0)=0$;
 (ii)
${M}_{xy}(t)=1$ if and only if $x=y$;
 (iii)
${M}_{xy}(t)={M}_{yx}(t)$;
 (iv)
${M}_{xy}(\cdot ):[0,\mathrm{\infty})\to [0,1]$ is left continuous;
 (v)
${M}_{xy}(t)\ast {M}_{yz}(s)\le {M}_{xz}(t+s)$.
 (vi)
${lim}_{t\to \mathrm{\infty}}{M}_{xy}(t)=1$ for all $x,y\in X$.
Lemma 5 ${M}_{xy}(\cdot )$ is a nondecreasing function on $[0,\mathrm{\infty})$.
Definition 6 Let $(X,M)$ be a FMS under some tnorm. A sequence $\{{x}_{n}\}\subset X$ is Cauchy if, for any $\u03f5>0$ and $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that ${M}_{{x}_{n}{x}_{m}}(t)>1\u03f5$ for all $n,m\ge {n}_{0}$. A sequence $\{{x}_{n}\}\subset X$ is convergent to $x\in X$, denoted by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ if, for any $\u03f5>0$ and $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that ${M}_{{x}_{n}x}(t)>1\u03f5$ for all $n\ge {n}_{0}$. A FMS in which every Cauchy sequence is convergent is called complete.
Given any tnorm ∗, it is easy to prove that $\ast \le min$. Therefore, if $(X,M)$ is a FMS under min, then $(X,M)$ is a FMS under any (continuous or not) tnorm. This is the case in the following examples (in which, obviously, we only define ${M}_{xy}(t)$ for $t>0$ and $x\ne y$).
It is well known that $(X,{M}^{d})$ is a FMS under the product ∗ = ⋅, called the standard FMS on $(X,d)$, since it is the standard way of viewing the metric space $(X,d)$ as a FMS. However, it is also true (though lesserknown) that $(X,{M}^{d})$, $(X,{M}^{\mathrm{e}})$ and $(X,{M}^{c})$ are FMSs under min.
Furthermore, $(X,d)$ is a complete metric space if and only if $(X,{M}^{d})$ (or $(X,{M}^{c})$ or $(X,{M}^{\mathrm{e}})$) is a complete FMS. For instance, this is the case of any nonempty and closed subset (or subinterval) of ℝ provided with its Euclidean metric.
Definition 8 A function $g:X\to X$ on a FMS is said to be continuous at a point ${x}_{0}\in X$ if, for any sequence $\{{x}_{n}\}$ in X converging to ${x}_{0}$, the sequence $\{g{x}_{n}\}$ converges to $g{x}_{0}$. If g is continuous at each $x\in X$, then g is said to be continuous on X. As usual, if ${x}_{0}\in X$, we will denote ${g}^{1}({x}_{0})=\{x\in X:gx={x}_{0}\}$.
Remark 9 If $x\in [0,1]$ and $a,b\in (0,\mathrm{\infty})$, then $a\le b$ implies that ${x}^{a}\ge {x}^{b}$. We will use this fact in the following way: $0<a\le b\le 1$ implies that ${M}_{xy}{(t)}^{a}\ge {M}_{xy}{(t)}^{b}\ge {M}_{xy}(t)$.
The main result
Definition 10 Let $F:{X}^{3}\to X$ and $g:X\to X$ be two mappings.

We say that F and g are commuting if $g{F}_{xyz}={F}_{gxgygz}$ for all $x,y,z\in X$.

A point $(x,y,z)\in {X}^{3}$ is called a tripled coincidence point of the mappings F and g if ${F}_{xyz}=gx$, ${F}_{yzx}=gy$ and ${F}_{zxy}=gz$.
Then there exists a unique $x\in X$ such that $x=gx={F}_{xxx}$. In particular, F and g have, at least, one tripled coincidence point. Furthermore, $(x,x,x)$ is the unique tripled coincidence point of F and g if we assume that ${g}^{1}({x}_{0})=\{{x}_{0}\}$ only in the case that $F\equiv {x}_{0}$ is constant on ${X}^{3}$.
In this result, in order to avoid the indetermination 0^{0}, we assume that ${M}_{gxgu}{(t)}^{0}=1$ for all $t>0$ and all $x,y\in X$.
Proof Suppose that F is constant in ${X}^{3}$, i.e., there exists ${x}_{0}\in X$ such that ${F}_{xyz}={x}_{0}$ for all $x,y,z\in X$. As F and g are commuting, we deduce that $g{x}_{0}=g{F}_{xyz}={F}_{gxgygz}={x}_{0}$. Therefore, ${x}_{0}=g{x}_{0}={F}_{{x}_{0}{x}_{0}{x}_{0}}$ and $({x}_{0},{x}_{0},{x}_{0})$ is a tripled coincidence point of F and g. Now, suppose that ${g}^{1}({x}_{0})=\{{x}_{0}\}$ and $(x,y,z)\in {X}^{3}$ is another tripled coincidence point of F and g. Then $gx={F}_{xyz}={x}_{0}$, so $x\in {g}^{1}({x}_{0})=\{{x}_{0}\}$. Similarly, $x=y=z={x}_{0}$ and $({x}_{0},{x}_{0},{x}_{0})$ is the unique tripled coincidence point of F and g.
Next, suppose that F is not constant in ${X}^{3}$. In this case, $(a,b,c)\ne (0,0,0)$ and the proof is divided into five steps. Throughout this proof, n and p will denote nonnegative integers and $t\in [0,\mathrm{\infty})$.
Step 1. Definition of the sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ and $\{{z}_{n}\}$. Let ${x}_{0},{y}_{0},{z}_{0}\in X$ be three arbitrary points of X. Since $F({X}^{3})\subseteq g(X)$, we can choose ${x}_{1},{y}_{1},{z}_{1}\in X$ such that $g{x}_{1}={F}_{{x}_{0}{y}_{0}{z}_{0}}$, $g{y}_{1}={F}_{{y}_{0}{z}_{0}{x}_{0}}$ and $g{z}_{1}={F}_{{z}_{0}{x}_{0}{y}_{0}}$. Again, from $F({X}^{3})\subseteq g(X)$, we can choose ${x}_{2},{y}_{2},{z}_{2}\in X$ such that $g{x}_{2}={F}_{{x}_{1}{y}_{1}{z}_{1}}$, $g{y}_{2}={F}_{{y}_{1}{z}_{1}{x}_{1}}$ and $g{z}_{2}={F}_{{z}_{1}{x}_{1}{y}_{1}}$. Continuing this process, we can construct sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ and $\{{z}_{n}\}$ such that, for $n\ge 0$, $g{x}_{n+1}={F}_{{x}_{n}{y}_{n}{z}_{n}}$, $g{y}_{n+1}={F}_{{y}_{n}{z}_{n}{x}_{n}}$ and $g{z}_{n+1}={F}_{{z}_{n}{x}_{n}{y}_{n}}$.
The same reasoning is also valid for ${M}_{g{y}_{n}g{y}_{n+p+1}}(t)$ and ${M}_{g{z}_{n}g{z}_{n+p+1}}(t)$. Therefore, (11) is true. This permits us to show that $\{g{x}_{n}\}$ is Cauchy. Suppose that $t>0$ and $\epsilon \in (0,1)$ are given. By the hypothesis, as ∗ is a tnorm of Htype, there exists $0<\eta <1$ such that ${\ast}^{p}a>1\epsilon $ for all $a\in (1\eta ,1]$ and for all $p\ge 1$. By (9), ${lim}_{n\to \mathrm{\infty}}{\delta}_{n}(t)=1$, so there exists ${n}_{0}\in \mathbb{N}$ such that ${\delta}_{n}(tkt)>1\eta $ for all $n\ge {n}_{0}$. Hence (11), we get ${M}_{g{x}_{n}g{x}_{n+p}}(t),{M}_{g{y}_{n}g{y}_{n+p}}(t),{M}_{g{z}_{n}g{z}_{n+p}}(t)>1\epsilon $ for all $n\ge {n}_{0}$ and $p\ge 1$. Therefore, $\{g{x}_{n}\}$ is a Cauchy sequence. Similarly, $\{g{y}_{n}\}$ and $\{g{z}_{n}\}$ are also Cauchy sequences.
Letting $n\to \mathrm{\infty}$, we have ${lim}_{n\to \mathrm{\infty}}\theta (t/{k}^{n})=1$ for all $t>0$, and this means that ${M}_{xy}(kt)={M}_{yz}(kt)={M}_{zx}(kt)=1$ for all $t>0$, i.e., $x=y=z$. The unicity of x follows from (1). □
Remark 12 The unicity of the coincidence point of F and g is not always true. For instance, if $F\equiv {x}_{0}$ is constant and $g\equiv {x}_{0}$ is also constant, then every $(x,y,z)\in {X}^{3}$ is a coincidence point of F and g.
Remark 13 In the previous theorem, we have only used the continuity of ∗ at $(1,1)$, that is, if $\{{x}_{n}\},\{{y}_{n}\}\subset [0,1]$ are sequences such that $\{{x}_{n}\}\to 1$ and $\{{y}_{n}\}\to 1$, then $\{{x}_{n}\ast {y}_{n}\}\to 1$. And this is true because $\{{x}_{n}\ast {y}_{n}\}\ge \{{x}_{n}\cdot {y}_{n}\}\to 1\cdot 1=1$.
Therefore, applying Theorem 11, we deduce that F and g have a tripled coincidence point.
Consequences
In the proof of the next result, the view of $(X,d)$ as the crisp FMS $(X,{M}^{c},min)$ is used (see Example 7). This approach allows us to deduce results for metric spaces from the corresponding result in the fuzzy setting. Moreover, Theorem 15 is just a tripled coincidence point result, similar to BerindeBorcut one, see [[18], Theorem 7] and [[21], Theorem 4], in a not necessarily partially ordered set.
 (a)
${d}_{{F}_{xyz}{F}_{uvw}}\le kmax({d}_{gxgu},{d}_{gygv},{d}_{gzgw})$ for some $k\in (0,1)$.
 (b)
${d}_{{F}_{xyz}{F}_{uvw}}\le k(\alpha {d}_{gxgu}+\beta {d}_{gygv}+\gamma {d}_{gzgw})$ for some $k\in (0,1)$ and some $\alpha ,\beta ,\gamma \in [0,1/3]$.
 (c)
${d}_{{F}_{xyz}{F}_{uvw}}\le \alpha {d}_{gxgu}+\beta {d}_{gygv}+\gamma {d}_{gzgw}$ for some $\alpha ,\beta ,\gamma \in [0,1)$ such that $\alpha +\beta +\gamma <1$.
Then there exists a unique $x\in X$ such that $x=gx={F}_{xxx}$.
Proof (a) Consider ${M}^{c}$ defined as in Example 7. As $(X,d)$ is complete, then $(X,{M}^{c},min)$ is a complete FMS. Fix $x,y,z,u,v,w\in X$ and $t>0$, and we are going to prove (1) using $a=b=c=1/3$ and $\ast =min$. If ${M}_{gxgu}^{c}(t)=0$ or ${M}_{gygv}^{c}(t)=0$ or ${M}_{gzgw}^{c}(t)=0$, then (1) is obvious. Suppose that ${M}_{gxgu}^{c}(t)=1$, ${M}_{gygv}^{c}(t)=1$ and ${M}_{gzgw}^{c}(t)=1$. This means that ${d}_{gxgu}<t$, ${d}_{gygv}<t$ and ${d}_{gzgw}<t$. Therefore, $t>max({d}_{gxgu},{d}_{gygv},{d}_{gzgw})$ and $kt>kmax({d}_{gxgu},{d}_{gygv},{d}_{gzgw})\ge {d}_{{F}_{xyz}{F}_{uvw}}$. Hence, ${M}_{{F}_{xyz}{F}_{uvw}}^{c}(kt)=1$ and (1) is also true.
□
Example 16 If $X=\mathbb{R}$, $d(x,y)=xy$ for all $x,y\in \mathbb{R}$ and $a,b,c,d,M\in \mathbb{R}$ are such that $M>a+b+c$, the mappings $F:{\mathbb{R}}^{3}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$, defined as ${F}_{xyz}=(ax+by+cz+d)/M$ and $gx=x$ for all $x,y,z\in \mathbb{R}$, verify the hypothesis of Theorem 15(c). It is easy to check that $({x}_{0},{x}_{0},{x}_{0})$, where ${x}_{0}=d/(Mabc)$, is the unique tripled coincidence point of F and g and verifies $F({x}_{0},{x}_{0},{x}_{0})={x}_{0}$.
Now, we prove the existence of a coupled coincided point for $F:{X}^{2}\to X$ and g that generalizes Theorem 4.1 in [7], taking $a=b=1/2$. That is, the main result of the paper also covers the main theoretical results of Zhu and Xiao [7].
for all $x,y,u,v\in X$ and all $t>0$. Then there exists a unique $x\in X$ such that $x=gx={F}_{xx}$.
Then there exists a unique $x\in X$ such that $gx={F}_{xxx}^{\mathrm{\prime}}$. If $y\in X$ verifies ${F}_{yy}=gy$, then $gy={F}_{yy}={F}_{yyy}^{\mathrm{\prime}}$, so $x=y$. □
Corollary 18 ([[1], Theorem 2.2])
 (a)
${d}_{{F}_{xy}{F}_{uv}}\le kmax({d}_{gxgu},{d}_{gygv})$ for some $k\in (0,1)$.
 (b)
${d}_{{F}_{xy}{F}_{uv}}\le k(\alpha {d}_{gxgu}+\beta {d}_{gygv})$ for some $k\in (0,1)$ and some $\alpha ,\beta \in [0,1/2]$.
 (c)
${d}_{{F}_{xy}{F}_{uv}}\le \alpha {d}_{gxgu}+\beta {d}_{gygv}$ for some $\alpha ,\beta ,\gamma \in [0,1)$ such that $\alpha +\beta <1$.
Then there exists a unique $x\in X$ such that $x=gx={F}_{xx}$.
Proof Similar to the proof of Theorem 15. □
Remark 19 In fact, the previous result is proved for X, a partially ordered set in [1].
Moreover, from a similar procedure, we can deduce the celebrated Banach contraction principle (Theorem 1).
Applications
Lipschitzian systems
Let ${f}_{1},{f}_{2},{f}_{3}:\mathbb{R}\to \mathbb{R}$ be Lipschitzian mappings and let ${\beta}_{1},{\beta}_{2},{\beta}_{3}\in \mathbb{R}$ be real numbers. Define $h:\mathbb{R}\to \mathbb{R}$ as $h(x)={\beta}_{1}{f}_{1}(x)+{\beta}_{2}{f}_{2}(x)+{\beta}_{3}{f}_{3}(x)$ for all $x\in \mathbb{R}$. Then h is another Lipschitzian mapping and ${k}_{h}\le {\beta}_{1}{k}_{{f}_{1}}+{\beta}_{2}{k}_{{f}_{2}}+{\beta}_{3}{k}_{{f}_{3}}$. Obviously, if $K={\beta}_{1}{k}_{{f}_{1}}+{\beta}_{2}{k}_{{f}_{2}}+{\beta}_{3}{k}_{{f}_{3}}<1$, then h is a contraction, so there exists a unique ${x}_{0}\in \mathbb{R}$ such that ${h}_{{x}_{0}}={x}_{0}$.
If $K<1$, then F verifies (1) with $gx=x$ for all $x\in \mathbb{R}$.
has a unique solution, which is $({x}_{0},{x}_{0},{x}_{0})$, where ${x}_{0}$ is the only real solution of ${\beta}_{1}{f}_{1}(x)+{\beta}_{2}{f}_{2}(x)+{\beta}_{3}{f}_{3}(x)=x$.
Example 21
Finding, for example, the root by the bisection method, we get, approximately, ${x}_{0}=2.5212648363927$.
An integral system
holds for all $x\in I$, $i=1,2,3$.
(this exists as a simple application of the Banach contraction principle).
Declarations
Authors’ Affiliations
References
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