ORIGINAL ARTICLE

Common fixed points of Chatterjea type fuzzy mappingson closed balls

Akbar Azam • Saqib Hussain • Muhammad Arshad

Received: 2 July 2011 / Accepted: 16 March 2012 / Published online: 7 April 2012

� Springer-Verlag London Limited 2012

Abstract We establish some common fixed point theo-

rems for Chatterjea fuzzy mappings on closed balls in a

complete metric space. Our investigation is based on the

fact that fuzzy fixed point results can be obtained simply

from the fixed point theory of mappings on closed balls. In

real-world problems, there are various mathematical

models in which the mappings are contractive on the sub-

sets of a space under consideration but not on the whole

space itself. It seems that this technique of finding the fuzzy

fixed points was ignored. Our results generalize several

important results of the literature.

Keywords Fuzzy fixed point � Chatterjea mapping �Contraction � Closed ball � Continuous mapping

Mathematics Subject Classification 46S40 � 47H10 �54H25

1 Introduction and preliminaries

It is a well-known fact that the results of fixed points are

very useful for determining the existence and uniqueness of

solutions to various mathematical models. Over the period

of last 40 years, the theory of fixed points has been devel-

oped regarding the results that are related to finding the

fixed points of self and nonself nonlinear mappings in a

metric space. In 1922, Banach proved a contraction prin-

ciple which states that for a complete metric space (X, d), a

mapping T : X �! X satisfying a contraction condition

d(Tx, Ty) B kd(x, y) for all x; y 2 X; where 0 \ k \ 1 has a

unique fixed point in X. Banach contraction principle plays

a fundamental role in the emergence of modern fixed point

theory and it gains more attention because it is based on

iterations so it can be easily applied by using computer. In

1972, Chatterjea [8] a premier Indian Mathematician

proved a contraction theorem for a complete metric space

proved that a mapping T : X �! X satisfying a contraction

condition d(Tx, Ty) B k[d(x, Ty) ? (y, Tx)] for all x; y 2 X

where 0\k\ 12

has a unique fixed point in X. The Chatterjea

[8] contraction mappings need not to be continuous. Since

continuity is big requirement, this makes Chatterjea con-

traction mappings to be important with respect to applica-

tion point of view. The study of fixed points of mappings

satisfying certain contractive conditions has been at the

center of vigorous research activity, and it has a wide range

of applications in different areas such as nonlinear and

adoptive control systems, parameterize estimation prob-

lems, fractal image decoding, computing magnetostatic

fields in a nonlinear medium, and convergence of recurrent

networks, (see [13, 14, 20, 23]). In his paper, Rhoades [17]

compared different contraction conditions that become the

foundation of development of present fixed point theory.

The notion of fixed points for fuzzy mappings was

introduced by Weiss [24] and Butnariu [7]. Fixed point

theorems for fuzzy set-valued mappings have been studied

by Heilpern [11] who introduced the concept of fuzzy

contraction mappings and established Banach contraction

principle for fuzzy mappings in complete metric linear

spaces, which is a fuzzy extension of Banach fixed point

theorem and Nadler‘s [15] theorem for multivalued

A. Azam � S. Hussain

Department of Mathematics, COMSATS Institute of Information

Technology, Chack Shahzad, Islamabad 44000, Pakistan

M. Arshad (&)

Department of Mathematics, International Islamic University,

H-10, Islamabad 44000, Pakistan

e-mail: marshad_zia@yahoo.com

123

Neural Comput & Applic (2012) 21 (Suppl 1):S313–S317

DOI 10.1007/s00521-012-0907-4

mappings. Subsequently, several other authors [1, 2, 6, 9,

12, 16, 18, 19, 21, 22] studied the existence of fixed points

and common fixed points of fuzzy mappings satisfying a

contractive type condition. Frigon and O’Regan [10] and

Azam et al. [5] proved some fuzzy fixed point theorems on

closed balls. In this paper, we prove some fixed point

theorems for a pair of fuzzy mappings satisfying Chatterjea

type [8] contractive condition. Let

2X ¼ A : A is a subset of Xf g;CLð2XÞ ¼ A 2 2X : A is nonempty closed

� �

C 2X� �

¼ A 2 2X : A is nonempty and compact� �

;

CB 2X� �

¼ A 2 2X : A is nonempty closed and bounded� �

:

For A;B 2 CB 2Xð Þ;dðx;AÞ ¼ inf

y2Adðx; yÞ;

dðA;BÞ ¼ infx2A;y2B

dðx; yÞ:

Then, the Hausdorff metric dH on CB 2Xð Þ induced by

d is defined as:

dH A;Bð Þ ¼ max supa2A

d a;Bð Þ; supb2B

d A; bð Þ� �

A fuzzy set in X is a function with domain X and values

in [0, 1], IX is the collection of all fuzzy sets in X. If A is a

fuzzy set and x 2 X; then the function values A(x) is called

the grade of membership of x in A. The a-level set of a

fuzzy set A is denoted by A½ �a and is defined as:

A½ �a ¼ x : AðxÞ� a ifa 2 ð0; 1�f g;A½ �0 ¼ fx : AðxÞ[ 0g:

For x 2 X; we denote the fuzzy set v{x} by xf g unless and

until it is stated, where vA is the characteristic function of

the crisp set A.

Define some subcollections of IX as follows:

FðXÞ ¼ A 2 IX : A½ �1 is non empty and closed� �

;

For A;B 2 IX ;A � B means A(x) B B(x) for each x 2 X.

For A;B 2 F Xð Þ then define

D1ðA;BÞ ¼ dHð A½ �1; B½ �1Þ:

A point x� 2 X is called a fixed point of a fuzzy mapping

T : X ! IX if x�f g � Tx� (see, [11]).

Lemma 1 [15] Let A and B be nonempty closed and

bounded subsets of a metric space X; dð Þ. If a 2 A; then

d a;Bð Þ � dH A;Bð Þ.

Lemma 2 [15] Let A and B be nonempty closed and

bounded subsets of a metric space X; dð Þ and 0\n 2 R.

Then for a 2 A; there exists b 2 B such that d a; bð Þ �dH A;Bð Þ þ n.

Lemma 3 [4] The completeness of (X, d) implies that

(CB(2X),dH) is complete.

Theorem 4 [8] Let (X, d) be a complete metric space and

a mapping T : X �! X. Suppose there exists a constant

a 2 0; 12

� �such that

dðTx; TyÞ � a½dðx; TyÞ þ dðy; TxÞ�;

holds. Then, T has a unique fixed point in X.

2 Fixed point of Chatterjea type fuzzy mappings

on closed balls

The mappings satisfying above contractive condition in

Theorem 4 are called Chatterjea mappings. It is mentioned

that Chatterjea contractive condition does not implies that

the mapping T is continuous, which differentiates it from

Banach contractive condition. For c 2 X and 0\r 2 R; let

Sr cð Þ ¼ x 2 X j d c; xð Þ\rf g

be the ball of radius r centered at c. Also, the closure of

Sr cð Þ is denoted by Sr cð Þ. We present the existence of

common fixed point for fuzzy mappings satisfying

Chatterjea type [8] contractive condition on a closed ball.

The theorem is as follows:

Theorem 5 Let (X, d) be a complete metric space, x0 2 X

and mappings F; T : Sr x0ð Þ ! FðXÞ. Suppose there exists

a constant k 2 0; 12

� �with

D1ðFx; TyÞ � k½dðx; Ty½ �1Þ þ dðy; Fx½ �1Þ�;for all x; y 2 Sr x0ð Þ

ð1Þ

and

dðx0; Fx0½ �1Þ\1� 2k

1� k

� r; ð2Þ

holds. Then, F and T have a common fuzzy fixed point in

Sr x0ð Þ. That is there exists x� 2 Sr x0ð Þ with fx�g �Fx� \ Tx�.

Proof. Choose x1 2 X such that fx1g � Fx0 and

dðx0; x1Þ\1� 2k

1� k

� r; ð3Þ

since [Fx0]1 = / and dðx0; Fx0½ �1Þ\ 1�2k1�k

� �r. For the sake

of simplicity, chose k ¼ k1�k. This gives us d(x0, x1) \

(1 - k)r, which implies that x1 2 Sr x0ð Þ. Now choose

�[ 0; such that

kdðx0; x1Þ þ�

1� k\kð1� kÞr: ð4Þ

Using Lemma 2, we choose x2 2 X such that fx2g � Tx1 and

dðx1; x2Þ � D1ðFx0; Tx1Þ þ �. By inequality (1), we have

S314 Neural Comput & Applic (2012) 21 (Suppl 1):S313–S317

123

dðx1; x2Þ� k½dðx0; Tx1½ �1Þ þ dðx1; Fx0½ �1Þ� þ �� k½dðx0; x2Þ þ dðx1; x1Þ� þ �� kdðx0; x2Þ þ �;

which implies that

1� kð Þdðx1; x2Þ� kdðx0; x1Þ þ �

dðx1; x2Þ�k

1� kdðx0; x1Þ þ

�

1� k

¼ kdðx0; x1Þ þ�

1� k:

By using inequality (4), we get

ðx1; x2Þ\kð1� kÞr: ð5Þ

Note that x2 2 Sr x0ð Þ; since

dðx0; x2Þ� dðx0; x1Þ þ dðx1; x2Þ;

then by using inequalities (3) and (5), we have

dðx0; x2Þ� ð1� kÞr þ kð1� kÞr�ð1� kÞr 1þ k½ �� ð1� kÞr 1þ kþ k2 þ . . .

�¼ r

Continue this process and having chosen xnð Þ in X such that

fx2kþ1g �Fx2k

and fx2kþ2g � Tx2kþ1;

with

dðx2kþ1; x2kþ2Þ\k2kþ1ð1� kÞr; for k ¼ 0; 12; 3; 4; . . .

Notice that (xn) is a Cauchy sequence in Sr x0ð Þ which is

complete. Therefore, a point x� 2 Sr x0ð Þ exists with

limn!1 xn ¼ x�: It remains to show that fx�g � Tx� and

fx�g � Fx�: Now by Lemma 1, we get

dðx�; Tx�½ �1Þ� dðx�; x2nþ1Þ þ dðx2nþ1; Tx�½ �1Þ� dðx�; x2nþ1Þ þ D1ðFx2n; Tx�Þ� dðx�; x2nþ1Þ þ k½dðx2n; Tx�½ �1Þ þ dðx�; Fx2n½ �1Þ�� dðx�; x2nþ1Þ þ kdðx2n; Tx�½ �1Þ þ kdðx�; x2nþ1Þ

which implies that

dðx�; Tx�½ �1Þ � kdðx2n; Tx�½ �1Þ� dðx�; x2nþ1Þþ kdðx�; x2nþ1Þ;

that is

ð1� kÞdðx�; Tx�½ �1Þ� 0;

Which implies that fx�g � Tx�: Similarly consider that

dðx�; Fx�½ �1Þ� dðx�; x2nþ2Þ þ dðx2nþ2; Fx�½ �1Þ

to show that fx�g � Fx�; which implies that mappings

F and T have a common fixed point in Sr x0ð Þ; i.e.

fx�g � Fx� \ Tx�. h

Corollary 6 Le (X,d) be a complete metric space, x0 2 X

and mapping F : Sr x0ð Þ ! FðXÞ. Suppose there exists a

constant k 2 0; 12

� �such that

D1ðFx;FyÞ� k½dðx; Fy½ �1Þ þ dðy; Fx½ �1Þ�;for all x; y 2 Sr x0ð Þ:

ð6Þ

Moreover

dðx0; Fx0½ �1Þ\1� 2k

1� k

� r

holds. Then, F has a fuzzy fixed point in Sr x0ð Þ. That is

there exists x� 2 Sr x0ð Þ with fx�g � Fx.

Proof. By taking T = F in Theorem 5 we get x� in Sr x0ð Þsuch that fx�g � Fx�. h

Theorem 7 Let (X, d) be a complete metric space and

mappings F; T : X ! FðXÞ. Suppose there exists a

constant k 2 0; 12

� �such that

D1ðFx; TyÞ� k½dðx; Ty½ �1Þ þ dðy; Fx½ �1Þ�;for all x; y 2 X: ð7Þ

Then, F and T have a common fuzzy fixed point in X. That is

there exists x� 2 X with fx�g � Fx� \ Tx�ðFx�ðx�Þ ¼ 1 and

Tx�ðx�Þ ¼ 1Þ.

Proof. Fix x0 2 X: Choose r [ 0 so that

dðx0; Fx0½ �1Þ\1� 2k

1� k

� r:

Now Theorem 5 guarantees that there exists x� 2 Sr x0ð Þwith fx�g � Fx� \ Tx�. h

Corollary 8 Let (X,d) be a complete metric space and

mappings F : X ! FðXÞ. Suppose there exists a constant

k 2 0; 12

� �such that

D1ðFx;FyÞ� k½dðx; Fy½ �1Þ þ dðy; Fx½ �1Þ�;for all x; y 2 X:

ð8Þ

Then, F has a fuzzy fixed point in X. That is there exists

x� 2 X with fx�g � Fx�.

Proof. In Theorem 7, take T = F. h

Example 9 Let X ¼ ff1; 2; 3g; 1f g; 2f g; 3f gg be crisp

sets. Define d : X X ! R as follows:

dðx; yÞ ¼

0 if x ¼ y5

70if x 6¼ y and x; y 2 X � f2g

170

if x 6¼ y and x; y 2 X � f3g4

70if x 6¼ y and x; y 2 X � f1g:

8>><

>>:

Neural Comput & Applic (2012) 21:S313–S317 S315

123

Define fuzzy mappings T : X ! IX as follows:

Tð1Þ tð Þ ¼ Tð3Þ tð Þ ¼1 if t ¼ 11

20if t ¼ 2

0 if t ¼ 3

8<

:

and

Tð2Þ tð Þ ¼0 if t ¼ 11

30if t ¼ 2

1 if t ¼ 3:

8<

:

Then,

Tx½ �1¼ t : TðxÞðtÞ ¼ 1f g ¼ 1f g if x 6¼ 2

3f g if x ¼ 2:

�

Now,

D1ðT 3ð Þ; T 2ð ÞÞ ¼ dH T3½ �1; T2½ �1� �

¼ max supa2 1f g

d a; 3f gð Þ; supb2 3f g

d 1f g; bð Þ( )

¼ max d 1; 3ð Þ; d 1; 3ð Þf g ¼ 5

70;

dð3; 2Þ ¼ 4

70:

Since D1ðT 3ð Þ; T 2ð ÞÞ[ adð3; 2Þ for each a\ 1 (and X is

not linear), therefore the existing results (e.g. see [1, 2, 3, 6,

8, 9, 11, 12, 15, 19] are not applicable to find 1 2 T1½ �1. In

order to apply the Theorem 5, consider the fuzzy mapping

F : X ! IX

F 1ð Þ tð Þ ¼ F 2ð Þ tð Þ ¼ F 3ð Þ tð Þ ¼1 if t ¼ 11

40if t ¼ 2

0 if t ¼ 3

8<

:

½Fx�1 ¼ t : FðxÞðtÞ ¼ 1f g ¼ f1g for all x 2 X

dHð½Fx�1; ½Ty�1Þ ¼ dHðFx; TyÞ

¼dHðf1g; f1gÞ ¼ 0 if y 6¼ 2

dHðf1g; f3gÞ ¼ 570

if y ¼ 2:

(

D1ðFð3Þ; Tð2ÞÞ ¼ dHð½F3�1; ½T2�1Þ ¼5

70;

dð3; ½T2�1Þ þ dð2; ½F3�1Þ ¼1

70:

It follows that the inequality (1) is not satisfied by map-

pings F and T for all x; y 2 X; whereas this condition is

satisfied for every x; y 2 S 171

1ð Þ. Hence, for k ¼ 13; r ¼

171; x0 ¼ 1 all conditions of Theorem 5 are satisfied to

obtain 1 2 F1½ �1\ T1½ �1.

3 Conclusion

As for as the application of contraction mapping is con-

cerned, the situation is not yet fully exploited. It is quite

possible that a contraction mapping T is defined on the

whole space X but it is contractive on the subset Y of the

space rather than on the whole space X. Moreover,

the contraction mapping under consideration may not be

continues. If Y is closed, then it is complete, so that the

mapping T has a fixed x in Y, and xn �! x as in the case of

the whole space X, provided we impose a simple restriction

on the choice of xo, so that x0ms remains in Y. In this paper,

we have discussed this concept for fuzzy Chatterjea [8]

mappings on a complete metric space X, which generalizes/

improves several classical fixed point results.

Acknowledgments The present version owes much to the precise

and kind remarks of anonymous referees.

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