Research  Open  Published:
Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces
Fixed Point Theory and Applicationsvolume 2013, Article number: 220 (2013)
Abstract
In this paper, we prove a common fixed point theorem for weakly compatible mappings under ϕcontractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716). We also indicate a minor mistake in Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716).
1 Introduction
In 1965, Zadeh [1] introduced the concept of fuzzy sets. Then many authors gave the important contribution to development of the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of a fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space, which have very important applications in quantum particle physics, particularly, in connection with both string and Einfinity theory.
Bhaskar and Lakshmikantham [4], Lakshmikantham and Ćirić [5] discussed the mixed monotone mappings and gave some coupled fixed point theorems, which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Jinxuan Fang [7] gave some common fixed point theorems for compatible and weakly compatible ϕcontractions mappings in Menger probabilistic metric spaces. XinQi Hu [8] proved a common fixed point theorem for mappings under φcontractive conditions in fuzzy metric spaces. Many authors [9–26] proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.
In this paper, we give a new coupled fixed point theorem under weaker conditions than in [8] and give an example, which shows that the result is a genuine generalization of the corresponding result in [8].
2 Preliminaries
First, we give some definitions.
Definition 2.1 (see [2])
A binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ is a continuous tnorm if ∗ satisfies the following conditions:

(1)
∗ is commutative and associative,

(2)
∗ is continuous,

(3)
$a\ast 1=a$ for all $a\in [0,1]$,

(4)
$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.
Definition 2.2 (see [27])
Let ${sup}_{0<t<1}\mathrm{\Delta}(t,t)=1$. A tnorm Δ is said to be of Htype if the family of functions ${\{{\mathrm{\Delta}}^{m}(t)\}}_{m=1}^{\mathrm{\infty}}$ is equicontinuous at $t=1$, where
The tnorm ${\mathrm{\Delta}}_{M}=min$ is an example of tnorm of Htype, but there are some other tnorms Δ of Htype [27].
Obviously, Δ is a tnorm of Htype if and only if for any $\lambda \in (0,1)$, there exists $\delta (\lambda )\in (0,1)$ such that ${\mathrm{\Delta}}^{m}(t)>1\lambda $ for all $m\in \mathbb{N}$, when $t>1\delta $.
Definition 2.3 (see [2])
A 3tuple $(X,M,\ast )$ is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous tnorm and M is a fuzzy set on ${X}^{2}\times (0,+\mathrm{\infty})$ satisfying the following conditions for each $x,y,z\in X$ and $t,s>0$,
(FM1) $M(x,y,t)>0$,
(FM2) $M(x,y,t)=1$ if and only if $x=y$,
(FM3) $M(x,y,t)=M(y,x,t)$,
(FM4) $M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s)$,
(FM5) $M(x,y,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is continuous.
We shall consider a fuzzy metric space $(X,M,\ast )$, which satisfies the following condition:
Let $(X,M,\ast )$ be a fuzzy metric space. For $t>0$, the open ball $B(x,r,t)$ with a center $x\in X$ and a radius $0<r<1$ is defined by
A subset $A\subset X$ is called open if for each $x\in A$, there exist $t>0$ and $0<r<1$ such that $B(x,r,t)\subset A$. Let τ denote the family of all open subsets of X. Then τ is called the topology on X, induced by the fuzzy metric M. This topology is Hausdorff and first countable.
Example 2.4 Let $(X,d)$ be a metric space. Define tnorm $a\ast b=ab$ or $a\ast b=min\{a,b\}$ and for all $x,y\in X$ and $t>0$, $M(x,y,t)=\frac{t}{t+d(x,y)}$. Then $(X,M,\ast )$ is a fuzzy metric space.
Definition 2.5 (see [2])
Let $(X,M,\ast )$ be a fuzzy metric space. Then

(1)
a sequence $\{{x}_{n}\}$ in X is said to be convergent to x (denoted by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$) if
$$\underset{n\to \mathrm{\infty}}{lim}M({x}_{n},x,t)=1$$for all $t>0$.

(2)
A sequence $\{{x}_{n}\}$ in X is said to be a Cauchy sequence if for any $\epsilon >0$, there exists ${n}_{0}\in \mathbb{N}$, such that
$$M({x}_{n},{x}_{m},t)>1\epsilon $$for all $t>0$ and $n,m\ge {n}_{0}$.

(3)
A fuzzy metric space $(X,M,\ast )$ is said to be complete if and only if every Cauchy sequence in X is convergent.
Remark 2.6 (see [9])
Let $(X,M,\ast )$ be a fuzzy metric space. Then

(1)
for all $x,y\in X$, $M(x,y,\cdot )$ is nondecreasing;

(2)
if ${x}_{n}\to x$, ${y}_{n}\to y$, ${t}_{n}\to t$, then
$$\underset{n\to \mathrm{\infty}}{lim}M({x}_{n},{y}_{n},{t}_{n})=M(x,y,t);$$ 
(3)
if $M(x,y,t)>1r$ for x, y in X, $t>0$, $0<r<1$, then we can find a ${t}_{0}$, $0<{t}_{0}<t$ such that $M(x,y,{t}_{0})>1r$;

(4)
for any ${r}_{1}>{r}_{2}$, we can find a ${r}_{3}$ such that ${r}_{1}\ast {r}_{3}\ge {r}_{2}$, and for any ${r}_{4}$, we can find a ${r}_{5}$ such that ${r}_{5}\ast {r}_{5}\ge {r}_{4}$ (${r}_{1},{r}_{2},{r}_{3},{r}_{4},{r}_{5}\in (0,1)$).
Define $\mathrm{\Phi}=\{\varphi :{R}^{+}\to {R}^{+}\}$, where ${R}^{+}=[0,+\mathrm{\infty})$ and each $\varphi \in \mathrm{\Phi}$ satisfies the following conditions:
(ϕ1) ϕ is nondecreasing,
(ϕ2) ϕ is upper semicontinuous from the right,
(ϕ3) ${\sum}_{n=0}^{\mathrm{\infty}}{\varphi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$, where ${\varphi}^{n+1}(t)=\varphi ({\varphi}^{n}(t))$, $n\in \mathbb{N}$.
It is easy to prove that if $\varphi \in \mathrm{\Phi}$, then $\varphi (t)<t$ for all $t>0$.
Lemma 2.7 (see [7])
Let $(X,M,\ast )$ be a fuzzy metric space, where ∗ is a continuous tnorm of Htype. If there exists $\varphi \in \mathrm{\Phi}$ such that
for all $t>0$, then $x=y$.
Definition 2.8 (see [4])
An element $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $F:X\times X\to X$ if
Definition 2.9 (see [5])
An element $(x,y)\in X\times X$ is called a coupled coincidence point of the mappings $F:X\times X\to X$ and $g:X\to X$ if
Definition 2.10 (see [5])
An element $(x,y)\in X\times X$ is called a common coupled fixed point of the mappings $F:X\times X\to X$ and $g:X\to X$ if
Definition 2.11 (see [5])
An element $x\in X$ is called a common fixed point of the mappings $F:X\times X\to X$ and $g:X\to X$ if
Definition 2.12 (see [8])
The mappings $F:X\times X\to X$ and $g:X\to X$ are said to be compatible if
and
for all $t>0$ whenever $\{{x}_{n}\}$ and $\{{y}_{n}\}$ are sequences in X, such that
for all $x,y\in X$ are satisfied.
Definition 2.13 (see [20])
The mappings $F:X\times X\to X$ and $g:X\to X$ are called weakly compatible mappings if $F(x,y)=g(x)$, $F(y,x)=g(y)$ implies that $gF(x,y)=F(gx,gy)$, $gF(y,x)=F(gy,gx)$ for all $x,y\in X$.
Remark 2.14 It is easy to prove that if F and g are compatible, then they are weakly compatible, but the converse need not be true. See the example in the next section.
3 Main results
For simplicity, denote
for all $n\in \mathbb{N}$.
XinQi Hu [8] proved the following result.
Theorem 3.1 (see [8])
Let $(X,M,\ast )$ be a complete FMspace, where ∗ is a continuous tnorm of Htype satisfying (2.2). Let $F:X\times X\to X$ and $g:X\to X$ be two mappings, and there exists $\varphi \in \mathrm{\Phi}$ such that
for all $x,y,u,v\in X$, $t>0$.
Suppose that $F(X\times X)\subseteq g(X)$, g is continuous, F and g are compatible. Then there exist $x,y\in X$ such that $x=g(x)=F(x,x)$; that is, F and g have a unique common fixed point in X.
Now we give our main result.
Theorem 3.2 Let $(X,M,\ast )$ be a FMspace, where ∗ is a continuous tnorm of Htype satisfying (2.2). Let $F:X\times X\to X$ and $g:X\to X$ be two weakly compatible mappings, and there exists $\varphi \in \mathrm{\Phi}$ satisfying (3.1).
Suppose that $F(X\times X)\subseteq g(X)$ and $F(X\times X)$ or $g(X)$ is complete. Then F and g have a unique common fixed point in X.
Proof Let ${x}_{0},{y}_{0}\in X$ be two arbitrary points in X. Since $F(X\times X)\subseteq g(X)$, we can choose ${x}_{1},{y}_{1}\in X$ such that $g({x}_{1})=F({x}_{0},{y}_{0})$ and $g({y}_{1})=F({y}_{0},{x}_{0})$. Continuing this process, we can construct two sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in X such that
The proof is divided into 4 steps.
Step 1: We shall prove that $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ are Cauchy sequences.
Since ∗ is a tnorm of Htype, for any $\lambda >0$, there exists an $\mu >0$ such that
for all $k\in \mathbb{N}$.
Since $M(x,y,\cdot )$ is continuous and ${lim}_{t\to +\mathrm{\infty}}M(x,y,t)=1$ for all $x,y\in X$, there exists ${t}_{0}>0$ such that
On the other hand, since $\varphi \in \mathrm{\Phi}$, by condition (ϕ3), we have ${\sum}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}({t}_{0})<\mathrm{\infty}$. Then for any $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that
From condition (3.1), we have
Similarly, we have
From the inequalities above and by induction, it is easy to prove that
So, from (3.3) and (3.4), for $m>n\ge {n}_{0}$, we have
which implies that
for all $m,n\in \mathbb{N}$ with $m>n\ge {n}_{0}$ and $t>0$. So $\{g({x}_{n})\}$ is a Cauchy sequence.
Similarly, we can prove that $\{g({y}_{n})\}$ is also a Cauchy sequence.
Step 2: Now, we prove that g and F have a coupled coincidence point.
Without loss of generality, we can assume that $g(X)$ is complete, then there exist $x,y\in g(X)$, and exist $a,b\in X$ such that
From (3.1), we get
Since M is continuous, taking limit as $n\to \mathrm{\infty}$, we have
which implies that $F(a,b)=g(a)=x$.
Similarly, we can show that $F(b,a)=g(b)=y$.
Since F and g are weakly compatible, we get that $gF(a,b)=F(g(a),g(b))$ and $gF(b,a)=F(g(b),g(a))$, which implies that $g(x)=F(x,y)$ and $g(y)=F(y,x)$.
Step 3: We prove that $g(x)=y$ and $g(y)=x$.
Since ∗ is a tnorm of Htype, for any $\lambda >0$, there exists an $\mu >0$ such that
for all $k\in \mathbb{N}$.
Since $M(x,y,\cdot )$ is continuous and ${lim}_{t\to +\mathrm{\infty}}M(x,y,t)=1$ for all $x,y\in X$, there exists ${t}_{0}>0$ such that $M(gx,y,{t}_{0})\ge 1\mu $ and $M(gy,x,{t}_{0})\ge 1\mu $.
On the other hand, since $\varphi \in \mathrm{\Phi}$, by condition (ϕ3), we have ${\sum}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}({t}_{0})<\mathrm{\infty}$. Thus, for any $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that $t>{\sum}_{k={n}_{0}}^{\mathrm{\infty}}{\varphi}^{k}({t}_{0})$. Since
letting $n\to \mathrm{\infty}$, we get
Similarly, we can get
From (3.7) and (3.8), we have
From this inequality, we can get
for all $n\in \mathbb{N}$. Since $t>{\sum}_{k={n}_{0}}^{\mathrm{\infty}}{\varphi}^{k}({t}_{0})$, then, we have
Therefore, for any $\lambda >0$, we have
for all $t>0$. Hence conclude that $gx=y$ and $gy=x$.
Step 4: Now, we prove that $x=y$.
Since ∗ is a tnorm of Htype, for any $\lambda >0$, there exists an $\mu >0$ such that
for all $k\in \mathbb{N}$.
Since $M(x,y,\cdot )$ is continuous and ${lim}_{t\to +\mathrm{\infty}}M(x,y,t)=1$, there exists ${t}_{0}>0$ such that $M(x,y,{t}_{0})\ge 1\mu $.
On the other hand, since $\varphi \in \mathrm{\Phi}$, by condition (ϕ3), we have ${\sum}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}({t}_{0})<\mathrm{\infty}$. Then, for any $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that $t>{\sum}_{k={n}_{0}}^{\mathrm{\infty}}{\varphi}^{k}({t}_{0})$.
From (3.1), we have
Letting $n\to \mathrm{\infty}$ yields
Thus, we have
which implies that $x=y$.
Thus, we proved that F and g have a common fixed point in X.
The uniqueness of the fixed point can be easily proved in the same way as above. This completes the proof of Theorem 3.2. □
Taking $g=I$ (the identity mapping) in Theorem 3.2, we get the following consequence.
Corollary 3.3 Let $(X,M,\ast )$ be a FMspace, where ∗ is a continuous tnorm of Htype satisfying (2.2). Let $F:X\times X\to X$, and there exists $\varphi \in \mathrm{\Phi}$ such that
for all $x,y,u,v\in X$, $t>0$. $F(X)$ is complete.
Then there exist $x\in X$ such that $x=F(x,x)$; that is, F admits a unique fixed point in X.
Remark 3.4 Comparing Theorem 3.2 with Theorem 3.1 in [8], we can see that Theorem 3.2 is a genuine generalization of Theorem 3.2.

(1)
We only need the completeness of $g(X)$ or $F(X\times X)$.

(2)
The continuity of g is relaxed.

(3)
The concept of compatible has been replaced by weakly compatible.
Remark 3.5 The Example 3 in [8] is wrong, since the tnorm $a\ast b=ab$ is not the tnorm of Htype.
Next, we give an example to support Theorem 3.2.
Example 3.6 Let $X=\{0,1,\frac{1}{2},\frac{1}{3},\dots ,\frac{1}{n},\dots \}$, $\ast =min$, $M(x,y,t)=\frac{t}{xy+t}$, for all $x,y\in X$, $t>0$. Then $(X,M,\ast )$ is a fuzzy metric space.
Let $\varphi (t)=\frac{t}{2}$. Let $g:X\to X$ and $F:X\times X\to X$ be defined as
Let ${x}_{n}={y}_{n}=\frac{1}{2n}$. We have $g{x}_{n}=\frac{1}{2n+1}\to 0$, $F({x}_{n},{y}_{n})=\frac{1}{{(2n+1)}^{4}}\to 0$, but
so g and F are not compatible. From $F(x,y)=g(x)$, $F(y,x)=g(y)$, we can get $(x,y)=(0,0)$, and we have $gF(0,0)=F(g0,g0)$, which implies that F and g are weakly compatible.
The following result is easy to verify
By the definition of M and ϕ and the result above, we can get that inequality (3.1)
is equivalent to the following
Now, we verify inequality (3.11). Let $A=\{\frac{1}{2n},n\in \mathbb{N}\}$, $B=XA$. By the symmetry and without loss of generality, $(x,y)$, $(u,v)$ have 6 possibilities.
Case 1: $(x,y)\in B\times B$, $(u,v)\in B\times B$. It is obvious that (3.11) holds.
Case 2: $(x,y)\in B\times B$, $(u,v)\in B\times A$. It is obvious that (3.11) holds.
Case 3: $(x,y)\in B\times B$, $(u,v)\in A\times A$. If $u\ne v$, (3.11) holds. If $u=v$, let $u=v=\frac{1}{2n}$, then
which implies that (3.11) holds.
Case 4: $(x,y)\in B\times A$, $(u,v)\in B\times A$. It is obvious that (3.11) holds.
Case 5: $(x,y)\in B\times A$, $(u,v)\in A\times A$. If $u\ne v$, (3.11) holds. If $u=v$, let $x\in B$, $y=\frac{1}{2j}$, $u=v=\frac{1}{2n}$, then
or
(3.11) holds.
Case 6: $(x,y)\in A\times A$, $(u,v)\in A\times A$.
If $x\ne y$, $u\ne v$, (3.11) holds.
If $x\ne y$, $u=v$, let $x=\frac{1}{2i}$, $y=\frac{1}{2j}$, $i\ne j$, $u=v=\frac{1}{2n}$. Then
(3.11) holds.
If $x=y$, $u=v$, let $x=y=\frac{1}{2i}$, $u=v=\frac{1}{2n}$. Then
(3.11) holds.
Then all the conditions in Theorem 3.2 are satisfied, and 0 is the unique common fixed point of g and F.
References
 1.
Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S00199958(65)90241X
 2.
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/01650114(94)901627
 3.
George A, Veeramani P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 1997, 90: 365–368. 10.1016/S01650114(96)002072
 4.
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
 5.
Lakshmikantham V, Ćirić LjB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric space. Nonlinear Anal. TMA 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
 6.
Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal. TMA 2010, 72: 1298–1304. 10.1016/j.na.2009.08.018
 7.
Fang JX: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Anal. TMA 2009, 71: 1833–1843. 10.1016/j.na.2009.01.018
 8.
Hu XQ: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 363716 10.1155/2011/363716
 9.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–398. 10.1016/01650114(88)900644
 10.
Ćirić LjB, Miheţ D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topol. Appl. 2009, 156(17):2838–2844. 10.1016/j.topol.2009.08.029
 11.
O’Regan D, Saadati R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 2008, 195(1):86–93. 10.1016/j.amc.2007.04.070
 12.
Jain S, Jain S, Bahadur Jain L: Compatibility of type (P) in modified intuitionistic fuzzy metric space. J. Nonlinear Sci. Appl. 2010, 3(2):96–109.
 13.
Ćirić Lj, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stoch. Anal. Appl. 2009, 27(6):1246–1259. 10.1080/07362990903259967
 14.
Ćirić Lj, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294
 15.
Aliouche A, Merghadi F, Djoudi A: A related fixed point theorem in two fuzzy metric spaces. J. Nonlinear Sci. Appl. 2009, 2(1):19–24.
 16.
Ćirić Lj: Common fixed point theorems for a family of nonself mappings in convex metric spaces. Nonlinear Anal. 2009, 71(5–6):1662–1669. 10.1016/j.na.2009.01.002
 17.
Rao KPR, Aliouche A, Babu GR: Related fixed point theorems in fuzzy metric spaces. J. Nonlinear Sci. Appl. 2008, 1(3):194–202.
 18.
Ćirić L, Cakić N: On Common fixed point theorems for nonself hybrid mappings in convex metric spaces. Appl. Math. Comput. 2009, 208(1):90–97. 10.1016/j.amc.2008.11.012
 19.
Ćirić L: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos Solitons Fractals 2009, 42: 146–154. 10.1016/j.chaos.2008.11.010
 20.
Abbas M, Ali Khan M, Radenovic S: Common coupled fixed point theorems in cone metric spaces for wcompatible mappings. Appl. Math. Comput. 2010, 217: 195–202. 10.1016/j.amc.2010.05.042
 21.
Shakeri S, Ćirić Lj, Saadati R: Common fixed point theorem in partially ordered L fuzzy metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 125082 10.1155/2010/125082
 22.
Ćirić Lj, Samet B, Vetro C: Common fixed point theorems for families of occasionally weakly compatible mappings. Math. Comput. Model. 2011, 53(5–6):631–636. 10.1016/j.mcm.2010.09.015
 23.
Ćirić Lj, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 53(9–10):1737–1741.
 24.
Ćirić L, Abbas M, Damjanović B, Saadati R: Common fuzzy fixed point theorems in ordered metric spaces. Math. Comput. Model. 2010. 10.1016/j.mcm.2010.12.050
 25.
Djoric D: Nonlinear coupled coincidence and coupled fixed point theorems for not necessary commutative contractive mappings in partially ordered probabilistic metric spaces. Appl. Math. Comput. 2013, 219: 5926–5935. 10.1016/j.amc.2012.12.047
 26.
Kamran T, Cakić NP: Hybrid tangential property and coincidence point theorems. Fixed Point Theory 2008, 9: 487–496.
 27.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.
Acknowledgements
This work of Xinqi Hu was supported by the National Natural Science Foundation of China (under grant No. 71171150). The research of B. Damjanović was supported by Grant No. 174025 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 Fixed Point Theorem
 Cauchy Sequence
 Common Fixed Point
 Unique Fixed Point
 Compatible Mapping