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					<li><a href="#abstract">Abstract</a></li>
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					<li><a href="#intro">Introduction</a></li>
					<li><a href="#preliminaries">Preliminaries </a></li>
					<li><a href="#result">Results</a></li>
					<li><a href="#proof">Proof</a></li>	
					<li><a href="#uniqueness">For uniqueness</a></li>				
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                    <p class="art-type" id="articleinfo">Research Article</p>
		    <p class="art-title">Common fixed point theorem in intuitionistic Fuzzy metric Spaces using compatible mappings of type (A)</p>
		    <p class="art-author"><?php $authors="Saurabh Manro<sup>*</sup>"; echo (stristr($authors,$coauthor))?str_replace($coauthor,"<a href='".$extpath."authors/".$courl."' target='_blank'>".$coauthor."</a>",$authors):$authors; ?></p>
<p class="art-affl">School of Mathematics and Computer Applications, Thapar University, Patiala, India</p>
		    <p class="art-aff"><b>*Corresponding author: <?php $corresponding_author="Saurabh Manro"; echo ($coauthor!="" && $coauthor==$corresponding_author)?"<a href='".$extpath."authors/".$courl."' target='_blank'>".$coauthor."</a>":$corresponding_author;?></b>,
School of Mathematics and Computer Applications, Thapar University Patiala, India,
E-mail: <a href="mailto:sauravmanro@hotmail.com">sauravmanro@hotmail.com</a></p>
<p class="art-aff"><b>Received:</b>    December 5, 2018
<b>Accepted:</b>     December 11, 2018
<b>Published:</b>   December 17, 2018</p>
<p class="art-aff"><b>Citation:</b> 
 Manro  S.  Common  fixed  point  theorem in intuitionistic Fuzzy metric Spaces using  compatible  mappings  of  type  (A).  <i>Madridge  J  Bioinform  Syst  Biol</i>.  2018;  1(1):  5-9. doi:  <a href="https://doi.org/10.18689/mjbsb-1000102">10.18689/mjbsb-1000102</a></p>
<p class="art-aff"><b>Copyright:</b> &copy;  2018 The Author(s). This work is  licensed  under  a  Creative  Commons  Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</p>
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<p class="art-subhead" id="abstract">Abstract</p>
<p class="art-para">In this paper, we prove common fixed point theorem in intuitionistic fuzzy metric space using compatible mappings of type (A).</p>

<p class="art-para"><b>Keywords:</b> Intuitionistic Fuzzy metric space; Compatible mappings of type (A); Common fixed point.</p>
<p class="art-para"><b>AMS (2010) Subject Classification:</b> 47H10, 54H25</p>

<p class="art-subhead" id="intro">Introduction</p>
<p class="art-para">Atanassove <a href="#2" id="ref2">[2]</a> introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. In 2004, Park <a href="#5" id="ref5">[5]</a> defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t-conorms. Recently, in 2006, Alaca et al.<a href="#1" id="ref1">[1]</a> using the idea of Intuitionistic fuzzy sets, defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norm and continuous t- conorms as a generalization of fuzzy metric space due to Kramosil and Michalek <a href="#3" id="ref3">[3]</a>. In this paper, we prove common fixed point theorem in intuitionistic fuzzy metric space using compatible mappings of type (A).</p>

<p class="art-subhead" id="preliminaries">Preliminaries</p>
<p class="art-para">The concepts of triangular norms (t-norms) and triangular conorms (t-conorms) are known as the axiomatic skelton that we use are characterization fuzzy intersections and union respectively. These concepts were originally introduced by Menger <a href="#4" id="ref4">[4]</a> in study of statistical metric spaces.</p>

<p class="art-para"><b>Definition</b> <a href="#6" id="ref6">[6]</a> A binary operation &#42;: [0, 1]x[0, 1] &rarr; [0, 1] is continuous t-norm if &#42; satisfies the following conditions:<br/>
(i) &#42; is commutative and associative;<br/>
(ii) &#42; is continuous;<br/>
(iii) a &#42; 1 = a for all a &isin;[0,1];<br/>
(iv) a &#42; b &le; c &#42; d whenever a &le; c and b &le; d for all a, b, c, d &isin;[0, 1] .</p>

<p class="art-para"><b>Definition <a href="#6" id="ref6">[6]</a></b> A binary operation &loz;: [0, 1] &times; [0, 1] &rarr; [0, 1] is continuous t-conorm if &loz; satisfies the following conditions:<br/>
(i) &loz; is commutative and associative;<br/>
(ii) &loz; is continuous;<br/>
(iii) a &loz; 0 = a for all a &isin;[0, 1];<br/>
(iv) a &loz; b &le; c &loz; d whenever a &le; c and b &le; d for all a, b, c,
d &isin;[0,1] .
</p>

<p class="art-para">Alaca et al. <a href="#1" id="ref1">[1]</a> using the idea of Intuitionistic fuzzy sets,
defined the notion of intuitionistic fuzzy metric space with the
help of continuous t-norm and continuous t-conorms as a
generalization of fuzzy metric space due to Kramosil and
Michalek <a href="#3" id="ref3">[3]</a> as:</p>

<p class="art-para"><b>Definition <a href="#1" id="ref1">[1]</a></b> A 5-tuple (X, M, N, &#42;, &loz;) is said to be an
intuitionistic fuzzy metric space if X is an arbitrary set, &#42; is a
continuous t-norm, &loz; is a continuous t-conorm and M, N are
fuzzy sets on X<sup>2</sup>&times; [0, &infin;) satisfying the following conditions:<br/>
(i) M(x, y, t) + N(x, y, t) &le; 1 for all x, y &isin; X and t > 0;<br/>
(ii) M(x, y, 0) = 0 for all x, y &isin; X ;<br/>
(iii) M(x, y, t) = 1 for all x, y &isin; X ; and t > 0 if and only if x = y;<br/>
(iv) M(x, y, t) = M(y, x, t) for all x, y &isin; X and t > 0;<br/>
(v) M(x, y, t) &#42; M(y, z, s) &le; M(x, z, t + s) for all x, y, z &isin; X and
s, t > 0;<br/>
(vi) for all x, y &isin; X, M(x, y, .) : [0, &infin;) &rarr; [0, 1] is left continuous;<br/>
(vii) limt&rarr;&infin;M(x, y, t) = 1 for all x, y &isin; X and t > 0;<br/>
(viii) N(x, y, 0) = 1 for all x, y &isin; X ;<br/>
(ix) N(x, y, t) = 0 for all x, y &isin; X and t > 0 if and only if x = y;<br/>
(x) N(x, y, t) = N(y, x, t) for all x, y &isin; X and t > 0;<br/>
(xi) N(x, y, t) &loz; N(y, z, s) &ge; N(x, z, t + s) for all x, y, z &isin; X and
s, t > 0;<br/>
(xii) for all x, y&isin;X, N(x, y, .) : [0, &infin;)&rarr;[0, 1] is right continuous;<br/>
(xiii) limt&rarr;&infin;N(x, y, t) = 0 for all x, y &isin; X.</p>

<p class="art-para">Then (M, N) is called an intuitionistic fuzzy metric space
on X. The functions M(x, y, t) and N(x, y, t) denote the degree
of nearness and the degree of non-nearness between x and y
w.r.t. t respectively.</p>
<p class="art-para"><i>Remark 2.1:</i> Every fuzzy metric space (X, M, &#42;) is an
intuitionistic fuzzy metric space of the form (X, M, 1-M, &#42;, &loz;)
such that t-norm &#42; and t-conorm &loz; are associated as x &loz; y =
1-((1-x) &#42; (1-y)) for all x, y &isin; X.</p>
<p class="art-para">Remark 2.2: In intuitionistic fuzzy metric space (X, M, N, &#42;,
&loz;), M (x, y, &#42;) is non-decreasing and N(x, y, &loz;) is non-increasing
for all x, y &isin; X.</p>
<p class="art-para">Alaca, Turkoglu and Yildiz <a href="#1" id="ref1">[1]</a> introduced the following
notions:</p>
<p class="art-para"><b>Definition</b> Let (X, M, N, &#42;, &loz;) be an intuitionistic fuzzy metric
space. Then<br/>
(a) a sequence {x<sub>n</sub>} in X is said to be Cauchy sequence if, for
all t > 0 and p > 0, lim<sub>n&rarr; &infin;</sub>M(x<sub>n+p</sub>, x<sub>n</sub>
, t) = 1 and lim<sub>n&rarr;&infin;</sub>N(x<sub>n+p</sub>,
x<sub>n</sub>, t) = 0.<br/>
(b) a sequence {x<sub>n</sub>} in X is said to be convergent to a point
x&isin;X if, for all t > 0, lim<sub>n&rarr;&infin;</sub>M(x<sub>n</sub>
, x, t) = 1 and lim<sub>n&rarr;&infin;</sub>N(x<sub>n</sub>, x, t) = 0.<br/>
<p class="art-para"><b>Definition <a href="#1" id="ref1">[1]</a></b> an intuitionistic fuzzy metric space (X, M, N, &#42;,
&loz;) is said to be complete if and only if every Cauchy sequence
in X is convergent.</p>

<p class="art-para"><i>Example 2.1:</i> Let X = {1/n: n &isin; N} &cup; {0} and let * be the
continuous t-norm and &loz; be the continuous t-conorm defined
by a * b = ab and a &loz; b = min{1, a+b} respectively, for all a, b
&isin; [0,1]. For each t &isin;(0, 	&infin;) and x, y &isin; X, define (M, N) by</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq001.gif" class="img-responsive center-block"/></div>

<p class="art-para">Clearly, (X, M, N, *, &loz;) is complete intuitionistic fuzzy
metric space.<br/>
<b>Definition A</b> pair of self mappings (f, g) of a intuitionistic
fuzzy metric space (X, M, N, *, &loz;) is said to be compatible if
lim<sub>n&rarr;&infin;</sub>M(fgx<sub>n</sub>
, gfx<sub>n</sub>
, t) = 1 and lim<sub>n&rarr;&infin;</sub>N(fgx<sub>n</sub>
, gfx<sub>n</sub>
, t) = 0 for all t
> 0, whenever {x<sub>n</sub>} is a sequence in X such that lim<sub>n&rarr;&infin;</sub>fx<sub>n</sub>
 =
lim<sub>n&rarr;&infin;</sub> gx<sub>n</sub>
 = u for some u in X.<br/>
 <b>Definition A</b> pair of self mappings (f, g) of a intuitionistic
fuzzy metric space (X, M, N, *,&loz;) is said to be compatible of
type (A) iflim<sub>n&rarr;&infin;</sub>M(fgx<sub>n</sub>
, ggx<sub>n</sub>
, t) = 1, lim<sub>n&rarr;&infin;</sub>N(fgx<sub>n</sub>
, ggx<sub>n</sub>
, t) = 0
and lim<sub>n&rarr;&infin;</sub>M(gfx<sub>n</sub>
, ffx<sub>n</sub>
, t) = 1, lim<sub>n&rarr;&infin;</sub>N(gfx<sub>n</sub>
, ffx<sub>n</sub>
, t) = 0.<br/>
for all t > 0, whenever {x<sub>n</sub>
} is a sequence in X such that lim<sub>n&rarr;&infin;</sub>fx<sub>n</sub>
= lim<sub>n&rarr;&infin;</sub> gx<sub>n</sub>
 = u for some u in X.</p>
<p class="art-para"><b>Alaca <a href="#1" id="ref1">[1]</a> proved the following results:</b><br/>
<b><i>Lemma</i></b> Let (X, M, N, *, &loz;) be intuitionistic fuzzy metric space
and for all x, y in X, t > 0 and if for a number k>1 such that
M(x, y, kt) &le; M(x, y, t) and N(x, y, kt) &ge; N(x, y, t) Then x = y.<br/>
<b><i>Lemma</i></b> Let (X, M, N, *, &loz;) be intuitionistic fuzzy metric space
and for all x, y in X, t > 0 and if for a number k > 1 such that</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq002.gif" class="img-responsive center-block"/></div>

<p class="art-para">Then {y<sub>n</sub>} is a Cauchy sequence in X.</p>
<p class="art-para"><b><i>Lemma</i></b> Let f and g be compatible self mappings of type (A)
of a complete intuitionistic fuzzy metric space (X, M, N,*, &loz;)
with a *b = min{a, b} and a&loz;b = max{a, b} for all a, b &isin; [0,1] and
fu = gu for some u &isin; X . Then gfu = fgu = ffu = ggu.</p>

<p class="art-subhead" id="result">Results</p>
<p class="art-para"><b>Theorem:</b> Let (X, M, N,*, &loz;) be a complete intuitionistic fuzzy
metric space with a *b = min{a, b} and a&loz;b = max{a, b} for all
a, b &isin;[0, 1]. Let A, B, S, T, P and Q be mappings from X into
itself such that the following conditions are satisfied:<br/>
(3.1) P(X) 	&sube; ST (X), Q(X) &isin; AB(X),<br/>
(3.2) AB = BA, ST = TS, PB = BP, QT = TQ,<br/>
(3.3) P or AB is continuous,<br/>
(3.4) (P, AB) and (Q, ST) are pairs of compatible mappings of
type (A),<br/>
(3.5) there exist k &isin; (0,1) such that for every x, y &isin; X and t > 0<br/>
M (Px,Qy, kt) &ge; M (ABx, STy,t) * M (Px, ABx, t) * M (Qy, STy, t) *
M (Px, STy, t)<br/>
N (Px,Qy, kt) &le; N (ABx, STy, t)&loz;N (Px, ABx, t) &loz;N (Qy, STy, t)&loz;N (Px, STy, t)</p>
<p class="art-para">Then A, B, S, T, P and Q have a unique common fixed point in
X.</p>
<p class="art-subhead" id="references">Proof</p>
<p class="art-para"><b>Forexistence:</b><br/>
Let x<sub>0</sub> &isin;X<sub>2</sub>
, from (3.1), there exist x<sub>1</sub>
, x<sub>2</sub> &isin; X such that</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq003.gif" class="img-responsive center-block"/></div>

<p class="art-para">Thus, we have</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq004.gif" class="img-responsive center-block"/></div>
<p class="art-para">Therefore, we have</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq005.gif" class="img-responsive center-block"/></div>

<p class="art-para">when n&rarr;&infin;.</p>
<p class="art-para">For each &isin;> 0 and t > 0, we can choose n<sub>0</sub> &isin; N such that</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq006.gif" class="img-responsive center-block"/></div>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq007.gif" class="img-responsive center-block"/></div>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq008.gif" class="img-responsive center-block"/></div>

<p class="art-para">and hence {y<sub>n</sub>} is a Cauchy sequence in X. As X is complete, {y<sub>n</sub>}
converges to some point</p>
<p class="art-para">z &isin; X . Also, its subsequences converge to this point z &isin; X.</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq009.gif" class="img-responsive center-block"/></div>

<p class="art-para">Suppose AB is continuous.</p>
<p class="art-para">As AB is continuous, we have <img src="<?php echo $imgpath;?>images/mjbsb-102-eq010.gif"/></p>
<p class="art-para">As (P, AB) is compatible pair of type (A), we have {PABx<sub>2n</sub>} &rarr;
ABz.</p>
<p class="art-para">Take x <sub>-</sub> ABx<sub>2n</sub>
, y <sub>-</sub> x<sub>2n+1</sub> in (3.5), we get</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq011.gif" class="img-responsive center-block"/></div>

<p class="art-para"><b>By lemma</b>, ABz = z.<br/>
Next, we show that Pz = z.<br/>
Put x = z and y <sub>-</sub> x<sub>2n</sub> in (3.5), we get</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq012.gif" class="img-responsive center-block"/></div>

<p class="art-para">Therefore, ABz = z = Pz.<br/>
Now, we show that Bz =z.<br/>
Put x = Bz and y <sub>_</sub> x<sub>2n-1</sub> in (3.5), we get</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq013.gif" class="img-responsive center-block"/></div>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq014.gif" class="img-responsive center-block"/></div>

<p class="art-para">As BP = PB and AB = BA, so that<br/>
P(Bz) = (PB)z = BPz = Bz and (AB)(Bz) = (BA)(Bz) = B(AB)z = Bz.<br/>
Taking, n &rarr;&infin;, we get<br/>
M (Bz, z, kt) &ge; M (Bz, z, t) * M (Bz, Bz, t) * M (z, z, t) * M (Bz, z, t)<br/>
M (Bz, z, kt) &ge; M (Bz, z, t)<br/>
And<br/>
N (Bz, z, kt) &le; N (Bz, z, t)&loz;N (Bz, Bz, t)&loz;N (z, z, t)&loz;N (Bz, z, t)<br/>
N (Bz, z, kt) &le; N (Bz, z, t).<br/>
Therefore, by using lemma, we get Bz = z and also we have,<br/>
ABz = z. Therefore, Az = Bz = Pz = z.<br/>
As P(X) &sube; ST (X), there exist u &isin; X such that z = Pz = STu.<br/>
Putting, x = x<sub>2n</sub>, y = u in (3.5), we get</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq015.gif" class="img-responsive center-block"/></div>

<p class="art-para">By using lemma, we get Qu = z. Hence, STu = z = Qu.<br/>
Since (Q, ST) is compatible pair of type (A), therefore, by
lemma, we have QSTu = STQu. Therefore, Qz = STz.<br/>
Now, we show that Qz = z.<br/>
Take x = x<sub>2n</sub>, y = z in (3.5), we get</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq016.gif" class="img-responsive center-block"/></div>

<p class="art-para">N (z, Qz, kt) = N (z, Qz, t)<br/>
Therefore, by using lemma, Qz = z.<br/>
As QT = TQ, ST = TS, we have QTz = TQz = Tz and STTz = TSTz
= TQz = Tz.<br/>
Next, we claim that Tz = z.<br/>
For this, take = x<sub>2n</sub>, y = Tz in (3.5), we get</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq017.gif" class="img-responsive center-block"/></div>

<p class="art-para">therefore, by lemma, we get Tz = z. as STz = Qz = z = Tz. This
gives, Sz =z. Hence, Az = Bz = Pz = Qz = Sz = Tz = z. Hence,
z is a common fixed point of A, B, S, T, P and Q. The proof is
similar P is continuous.</p>
<p class="art-subhead" id="uniqueness">For uniqueness</p>
<p class="art-para">Let u is another fixed point of A, B, S, T, P and Q. Therefore,
take x = z and y = u in (3.5), we get</p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq018.gif" class="img-responsive center-block"/></div>

<p class="art-para">By lemma, we get z = u. Hence, z is a unique common fixed
point of A, B, S, T, P and Q. Take B = T = I (Identity map), then
theorem 3.1 becomes:</p>
<p class="art-para"><b>Corollary 3.1:</b> Let (X, M, N,*, &loz;) be a complete intuitionistic
fuzzy metric space with a *b = min{a, b} and a&loz;b = max {a, b}
for all a, b &isin; [0,1]. Let A, S, P and Q be mappings from X in to
itself such that the following conditions are satisfied:<br/>
(3.6) P(X) 	&sube; S(X), Q(X) &sube; A(X),<br/>
(3.7) P or A is continuous,<br/>
(3.8) (P, A) and (Q, S) are pairs of compatible mappings of type
(A),<br/>
(3.9) there exist k &isin;(0,1) such that for every x, y &isin; X and t > 0</p>

<div class="art-img">
<img src="<?php echo $imgpath;?>images/mjbsb-102-eq019.gif" class="img-responsive center-block"/></div>

<p class="art-para">Then A, S, P and Q have a unique common fixed point in X.</p>

<p class="art-subhead" id="references">References</p>
<ol>
<li class="ref"><div id="1"> Alaca C, Turkoglu D, Yildiz C. <a href="https://www.sciencedirect.com/science/article/pii/S0960077905007150" target="_blank">Fixed points in Intuitionistic fuzzy metric spaces.</a><i> Chaos, Solitons & Fractals</i>. 2006; 29: 1073-1078. doi: 10.1016/j.chaos.2005.08.066<a href="#ref1"><i class="fa fa-level-up"></i></a></div></li>
<li class="ref"><div id="2"> Atanassov K, <a href="https://www.sciencedirect.com/science/article/pii/S0165011486800343" target="_blank">Intuitionistic Fuzzy sets.</a><i> Fuzzy sets and system</i>. 1986; 20(1): 87-96. doi: 10.1016/S0165-0114(86)80034-3<a href="#ref2"><i class="fa fa-level-up"></i></a></div></li>
<li class="ref"><div id="3"> Kramosil I, Michalek J. <a href="https://www.researchgate.net/publication/264960523_Fuzzy_metric_and_statistical_metric_space" target="_blank">Fuzzy metric and Statistical metric spaces.</a><i> Kybernetica</i>. 1975; 11: 326-334.<a href="#ref3"><i class="fa fa-level-up"></i></a></div></li>
<li class="ref"><div id="4"> Menger K.<a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078534/" target="_blank"> Statistical metrics.</a><i> Proc. Nat. Acad. Sci.</i> 1942; 28(12): 535- 537.<a href="#ref4"><i class="fa fa-level-up"></i></a></div></li>
<li class="ref"><div id="5"> Park JH. <a href="https://www.sciencedirect.com/science/article/pii/S0960077904000955" target="_blank">Intuitionistic fuzzy metric spaces.</a> Chaos, Solitons & Fractals. 2004; 22: 1039-1046. doi: 10.1016/j.chaos.2004.02.051<a href="#ref5"><i class="fa fa-level-up"></i></a></div></li>
<li class="ref"><div id="6">Schweizer B, Sklar A. <a href="https://www.researchgate.net/publication/276060187_Probabilistic_Metric_Spaces" target="_blank">Probabilistic Metric Spaces.</a><i> North Holland Amsterdam</i>. 1983.<a href="#ref6"><i class="fa fa-level-up"></i></a></div></li>
</ol>
</div>

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