Common fuzzy fixed point theorems in ordered metric spaces
Mathematical and Computer Modelling 53 (2011) 1737–1741
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Common fuzzy fixed point theorems in ordered metric spaces L. Ćirić a,∗ , M. Abbas b , B. Damjanović c , R. Saadati d a
Faculty of Mechanical Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia
b
Department of Mathematics, Lahore University of Management Sciences, Lahore - 54792, Pakistan
c
Department of Mathematics, Faculty of Agriculture, Nemanjina 6, 11 000 Belgrade, Serbia
d
Department of Mathematics, Science & Research Branch, Islamic Azad University, Post Code 14778, Ashrafi Esfahani Ave, Tehran, Islamic Republic of Iran
article
abstract
info
Article history: Received 28 July 2010 Accepted 23 December 2010
We prove the existence of fuzzy common fixed point of two mappings satisfying a generalized contractive condition in complete ordered spaces. Our results provide extension as well as substantial improvements of several well-known results in the existing literature and initiate the study of fuzzy fixed point theorems in ordered spaces. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Fuzzy mapping Fuzzy set Fuzzy fixed point
1. Introduction and preliminaries Let X be a space of points with generic element of X denoted by x and I = [0, 1]. A fuzzy subset of X is characterized by a member ship function which associates with each element in X a real number in the interval I . Let (X , d) be a metric linear space and A be a fuzzy set in X characterized by a membership function A. The α - level set of A, denoted by Aα , is defined by Aα = {x : A(x) ≥ α} if α ∈ (0, 1] A0 = {x : A(x) > 0} where B denotes the closure of the non fuzzy set B. A fuzzy set A in a metric linear space is said to be an approximate quantity if and only if Aα is compact and convex in X for each α ∈ [0, 1] and supx∈X A(x) = 1. We denote by W (X ), the family of all approximate quantities in X . Let A, B ∈ W (X ), then A is said to be more accurate than B, denoted by A ⊂ B, if and only if A(x) ≤ B(x) for each x in X , where B denotes the membership function of B. For x ∈ X , we write {x} the characteristic function of the ordinary subset {x} of X . We denote W 0 (X ) = {{x} : x ∈ X }. For α ∈ (0, 1], the fuzzy point (x)α of X is the fuzzy set of X given by xα (x) = α and α ̸= x. Let I X be the collection of all fuzzy subsets in X and W (X ) be a sub collection of all approximate quantities. For A, B ∈ W (X ), α ∈ [0, 1], define pα (A, B) =
inf
x∈Aα , y∈Bα
d(x, y),
p(A, B) = sup Pα (A, B), α
∗
Corresponding author. Fax: +381 11 3370364. E-mail addresses: [email protected] (L. Ćirić), [email protected] (M. Abbas), [email protected] (B. Damjanović), [email protected] (R. Saadati).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.12.050
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L. Ćirić et al. / Mathematical and Computer Modelling 53 (2011) 1737–1741
Dα (A, B) = H (Aα , Bα ), D(A, B) = sup Dα (A, B), α
where H is the Hausdorff metric induced by the metric d. We note that Pα is a non-decreasing function of α and D is a metric on W (X ). Let α ∈ [0, 1], then the family Wα (X ) is given by {A ∈ I X : Aα is non empty convex and compact }. Let X be an arbitrary set, Y be a metric linear space. A mapping T is called fuzzy mapping if T is a mapping from X into W (Y ), that is, Tx ∈ W (Y ) for each x in X . Thus if we characterize a fuzzy set Tx in a metric linear space Y by a member ship function Tx, then Tx(y) is the grade of member ship of y in Tx. Therefore a fuzzy mapping T is a fuzzy subset on X × Y with membership function Tx(y). A fuzzy point xα in X is called a fixed fuzzy point of the fuzzy mapping T if xα ⊂ Tx [1]. If {x} ⊂ Tx, then x is a fixed point of T . Definition 1. Let X be a nonempty set. Then (X , d, ≤) is called an ordered metric space iff. (i) (X , d) is a metric space, (ii) (X , ≤) is partial ordered. Definition 2. Let (X , ≤) be a partial ordered set. x, y ∈ X are called comparable if x ≤ y or y ≤ x holds. Following lemmas are needed in the sequel. Lemma 3 (Heilpern [2]). Let (X , d) be a metric space, x, y ∈ X and A, B ∈ W (X ): (1) if pα (x, A) = 0, then xα ⊂ A (2) pα (x, A) ≤ d(x, y)+ pα (y, B) (3) if xα ⊂ A, then pα (x, B) ≤ Dα (A, B). Lemma 4 (Lee and Cho [3]). Let (X , d) be a complete metric space, T be a fuzzy mapping from X into W (X ) and x0 ∈ X . Then there exists a x1 ∈ X such that {x1 } ⊂ Tx0 . Zadeh [4] introduced the concept of a fuzzy set which motivated a lot of mathematical activity on the generalization of the notion of a fuzzy set. Boričić in [5] considered fuzzification of propositional logics. Heilpern [2] introduced the concept of a fuzzy mappings in a metric linear space and proved a fixed point theorem for fuzzy contraction mapping which is the generalization of a fixed point theorem for multi-valued mappings of Nadler [6]. Estruch and Vidal [1] proved a fixed point theorem for fuzzy contraction mappings in a complete metric spaces which in turn generalized Heilpern fixed point theorem. Further generalization of the result given in [1] was proved in [7,8]. Recently Dutta and Choudhury [9] gave a generalization of Banach contraction principle, which in turn generalize Theorem 1 of [10] and corresponding result of [11]. Very recently Altun et al. [12] proved fixed point theorems in the frame work of ordered cone metric spaces. Bose and Shani [13] extended the result of Heilpern to pair of mappings. Kamran [14] and Sahin [15] also obtained some common fixed point theorems for fuzzy mappings in metric spaces. Recently Ðorić [16], Abbas and Ðorić [17] and Azam and Beg [18] proved common fixed point theorem for mappings which satisfy Alber and Guerr-Delabriere type contractive condition. Very recently Azam [19] established common fixed point theorems for fuzzy mappings under a ϕ -contraction condition on a metric space with the d∞ -metric. The aim of this paper is to establish the existence of a common fuzzy fixed point of generalized contractive mappings without employing any commutativity condition. Our result generalize, improve and extend many known results in the comparable literature [18,20,7]. 2. Main results We begin with the following result. Theorem 5. Let X be a complete ordered space. Suppose that T1 , T2 : X −→ Wα (X ) are two fuzzy mapping on X satisfying
ϕ(Dα (T1 x, T2 y)) ≤ ϕ(d(x, y)) − φ(d(x, y))
(1)
for all comparable elements x, y ∈ X , where, ϕ : [0, ∞) → (0, ∞) is a continuous and monotone nondecreasing functions with ϕ(t ) = 0 if and only if t = 0 and φ : [0, ∞) → (0, ∞) is lower semi continuous with ϕ(t ) = 0 if and only if t = 0. Suppose that if {y} ⊂ T1 (x0 ), then y, x0 ∈ X are comparable. Further, if x, y ∈ X are comparable, then every u ∈ (T1 x)α and every v ∈ (T2 y)α are comparable. Also suppose that if a sequence {xn } → x and its consecutive terms are comparable, then xn , x ∈ X are comparable for all n. Then there exists a point x in X such that xα ⊂ T1 x and xα ⊂ T2 x.
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Proof. Let x0 be in X . By Lemma 2, there exists x1 in X such that {x1 } ⊂ T1 (x0 ) which implies that p α ( x1 , T 1 x0 ) = 0
for each α in [0, 1],
which is possible if and only if x1 ∈ (T1 x0 )α . By assumption, x0 and x1 are comparable. Since (T2 x1 )α is nonempty compact subset of X , there exists x2 ∈ (T2 x1 )α such that d(x1 , x2 ) = pα (x1 , T2 x1 ) ≤ Dα (T1 x0 , T2 x1 ). Moreover, x1 and x2 are comparable. Continuing this process, we can construct a sequence {xn } in X such that x2n+1 ∈ (T1 (x2n ))α and x2n+2 ∈ (T2 (x2n+1 ))α for all n ≥ 0, x2n and x2n+1 are comparable and d(x2n+1 , x2n+2 ) ≤ Dα (T1 x2n , T2 x2n+1 ). Since ϕ is nondecreasing, ϕ(d(x2n+1 , x2n+2 )) ≤ ϕ(Dα (T1 x2n , T2 x2n+1 )). Since x2n and x2n+1 are comparable. Thus by taking x2n for x and x2n+1 for y in the inequality (1), it follows that
ϕ(d(x2n+1 , x2n+2 )) ≤ ϕ(Dα (T1 x2n , T2 x2n+1 )) ≤ ϕ(d(x2n , x2n+1 )) − φ(d(x2n , x2n+1 )). Similarly
ϕ(d(x2n+3 , x2n+2 )) ≤ ϕ(d(x2n+2 , x2n+1 )) − φ(d(x2n+2 , x2n+1 )). Therefore, for all n
ϕ(d(xn , xn+1 )) ≤ ϕ(Dα (T1 xn−1 , T2 xn )) ≤ ϕ(d(xn−1 , xn )) − φ(d(xn−1 , xn )). Hence ϕ(d(xn , xn+1 )) ≤ ϕ(d(xn−1 , xn )). Thus, we have d(xn , xn+1 ) ≤ d(xn−1 , xn ), which shows that {d(xn , xn+1 )} is non-increasing sequence of positive real numbers which is bounded below by 0. Therefore there is a real number r ≥ 0 such that lim d(xn , xn+1 ) = r .
n→∞
Suppose that r > 0, then 0 < ϕ(r ) ≤ ϕ(d(xn , xn+1 )) ≤ ϕ(Dα (T1 xn−1 , T2 xn ))
≤ ϕ(d(xn , xn−1 )) − φ(d(xn , xn−1 )). Now by continuity of ϕ and lower semicontinuity of φ we get
ϕ(r ) ≤ lim sup ϕ(d(xn , xn−1 )) − lim inf φ(d(xn−1 , xn )) n→∞
n→∞
and hence ϕ(r ) ≤ ϕ(r ) − φ(r ) < ϕ(r ), a contradiction. Therefore r = 0 and so lim d(xn , xn+1 ) = 0.
n→∞
Following the similar arguments to those given in [21], it can be shown that {xn } is a Cauchy sequence in X . It follows from the completeness of X that xn −→ x ∈ X . Since consecutive terms of {xn } are comparable and xn ≤ x. Now, we claim that pα (x, T2 x) = 0 for each α ∈ [0, 1]. From
|pα (x, T2 x) − d(x, x2n+1 )| ≤ pα (x2n+1 , T2 x) ≤ Dα (T1 x2n , T2 x) and (1) we get
ϕ(|pα (x, T2 x) − d(x, x2n+1 )|) ≤ ϕ(Dα (T1 x2n , T2 x)) ≤ ϕ(d(x2n , x)) − φ(d(x2n , x)). Hence we obtain, as ϕ is continuous and φ is lower semicontinuous,
ϕ(pα (x, T2 x)) ≤ ϕ(0) − φ(0) = 0, that is, ϕ(pα (x, T2 x)) = 0. Hence pα (x, T2 x) = 0. Therefore xα ⊂ T2 x. Similarly, xα ⊂ T1 x.
5
Define a class of functions G = {g : R+ → R+ } satisfying the following conditions: (g1 ) g is nondecreasing in the first and 5th variables. (g2 ) If u, v, ∈ R+ are such that g (u, v, v, u, u + v) ≤ 0, or g (u, v, u, v, u + v) ≤ 0, then u ≤ hv , where 0 < h < 1 is a constant. (g3 ) If u ∈ R+ is such that g (u, 0, 0, u, u) ≤ 0, or g (u, 0, u, 0, u) ≤ 0, then u = 0.
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Theorem 6. Let X be a complete ordered space. Suppose that T1 , T2 : X −→ Wα (X ) are two fuzzy mapping on X satisfying g (Dα (T1 x, T2 y), d(x, y), pα (x, T1 x), pα (y, T2 y), pα (x, T2 y) + pα (y, T1 x)) ≤ 0
(2)
for all comparable elements x, y ∈ X and for some g ∈ G. Suppose that for any y in X with {y} ⊂ T1 (x0 ) implies that y, x0 ∈ X are comparable and for comparable x, y ∈ X with u ∈ (T1 x)α and v ∈ (T2 y)α imply u, v ∈ X are comparable. Further, suppose that if a sequence {xn } → x and its consecutive terms are comparable, then xn , x ∈ X are comparable for all n. Then there exists a point x in X such that xα ⊂ T1 x and xα ⊂ T2 x. Proof. Let x0 be in X . By Lemma 2, there exists x1 in X such that {x1 } ⊂ T1 (x0 ) which implies that pα (x1 , T1 x0 ) = 0
for each α in [0, 1],
which is possible if and only if x1 ∈ (T1 x0 )α . By given assumption x0 and x1 are comparable. Since (T2 x1 )α is nonempty compact subset of X , therefore there exists x2 ∈ (T2 x1 )α such that d(x1 , x2 ) = pα (x1 , T2 x1 ) ≤ Dα (T1 x0 , T2 x1 ). Also, x1 and x2 are comaparable. Since x0 and x1 are comparable, then g (Dα (T1 x0 , T2 x1 ), d(x0 , x1 ), pα (x0 , T1 x0 ), pα (x1 , T2 x1 ), pα (x0 , T2 x1 ) + pα (x1 , T1 x0 )) ≤ 0. Since pα (x1 , T1 x0 ) = 0 and pα (x0 , T2 x1 ) ≤ d(x0 , x1 )+ pα (x1 , T2 x1 ), then g (d(x1 , x2 ), d(x0 , x1 ), d(x0 , x1 ), d(x1 , x2 ), d(x1 , x2 ) + d(x0 , x1 )) ≤ 0. Hence, as g ∈ G, d(x1 , x2 ) ≤ hd(x0 , x1 ). Similarly, we obtain that x2 and x3 are comparable and d(x2 , x3 ) ≤ hd(x1 , x2 ) ≤ h2 d(x0 , x1 ). Continuing this process, we can construct a sequence {xn } in X such that x2n+1 ∈ (T1 (x2n ))α and x2n+2 ∈ (T2 (x2n+1 ))α for all n ≥ 0, x2n and x2n+1 are comparable and d(x2n+1 , x2n+2 ) ≤ hd(x2n , x2n+1 ). Thus, by induction we have d(xn , xn+1 ) ≤ hn d(x0 , x1 ). From the proceeding inequality we conclude that {xn } is a Cauchy sequence in X . It follows from the completeness of X that xn −→ x ∈ X . Since consecutive terms of {xn } are comparable, then xn , x ∈ X are comparable for all n. Also, note that x ∈ limn→∞ (T1 x2n )α and x ∈ limn→∞ (T2 x2n+1 )α . Now, we claim that pα (x, T2 x) = 0 for each α ∈ [0, 1]. From
|pα (x, T2 x) − d(x, x2n+1 )| ≤ pα (x2n+1 , T2 x) ≤ Dα (T1 x2n , T2 x) and (2) we get, as g ∈ G, g (pα (x2n+1 , T2 x), d(x2n , x), pα (x2n , T1 x2n ), pα (x, T2 x), pα (x2n , T2 x) + pα (x, T1 x2n )) ≤ 0. Hence we obtain pα (x, T2 x) = 0. Therefore xα ⊂ T2 x. Similarly, xα ⊂ T1 x.
Acknowledgements The first and third authors accomplished research results on the project IO 174025, and resources for its implementation have been provided by Ministry of Science and Technological Development of Republic Serbia. The fourth author is grateful to the Young research Club, Islamic Azad University—Ayatollah Amoli Branch, Amol, Iran. References [1] [2] [3] [4] [5] [6] [7] [8]
V.D. Estruch, A. Vidal, A note on fixed fuzzy points for fuzzy mappings, Rend Istit. Univ. Trieste 32 (2001) 39–45. S. Heilpern, Fuzzy mappings and fuzzy fixed point theorems, Journal of Mathematical Analysis Applications 83 (1981) 566–569. B.S. Lee, S.J. Cho, A fixed point theorem for contractive type fuzzy mappings, Fuzzy Sets and Systems 61 (1994) 309–312. L.A. Zadeh, Fuzzy sets, Informations and Control 8 (1965) 103–112. B. Boričić, On fuzzification of propositional logics, Fuzzy Sets and Systems 108 (1999) 91–98. S.B Nadler Jr., Multivalued contraction mappings, Pacific Journal of Mathematics 30 (1969) 475–488. D. Turkoglu, B.E. Rhoades, A fixed fuzzy point for fuzzy mapping in complete metric spaces, Mathematical Communications 10 (2005) 115–121. I. Beg, M. Abbas, Coincidence point and invariant approximation for mapping satisfying generalized weak contractive condition, Fixed Point Theory and Applications 2006 (2006), Article ID 74503, 7 pages.
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[9] P.N. Dutta, B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory and Applications (2008), Article ID 406368, 8 pages. [10] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis 47 (2001) 2683–2693. [11] Ya.I. Alber, S. Guerre-Delabriere, ‘‘Principle of weakly contractive maps in Hilbert spaces’’ in new results in operator theory and its applications, in: I. Gohberg, Y. Lyubich (Eds.), in: Operator Theory: Advances and Applications, vol. 98, Birkhäuser, Basel, Switzerland, 1997, pp. 7–22. [12] I. Altun, B. Damjanović, D. Djori ć, Fixed point and common fixed point theorems on ordered cone metric spaces, Applied Mathematics Letters 23 (2010) 310–316. [13] R.K. Bose, D. Shani, Fuzzy mappings and fixed point theorems, Fuzzy Sets and Systems 21 (1987) 53–58. [14] T. Kamran, Common fixed points theorems for fuzzy mappings, Chaos, Solitons and Fractals 38 (2008) 1378–1382. [15] I. Sahin, H. Karayilan, M. Telci, Common fixed point theorems for fuzzy mappings in quasi pseudo metric spaces, Turkish Journal of Mathematics 29 (2005) 129–140. [16] D. Ðorić, Common fixed point for generalized (ϕ; φ ′ )-weak contractions, Applied Mathematics Letters 22 (2009) 1896–1900. [17] M. Abbas, D. Ðorić, Common fixed point theorem for four mappings satisfying generalized weak contractive contraction, Filomat 24 (2) (2010) 1–10. [18] A. Azam, I. Beg, Common fixed points of fuzzy maps, Mathematical and Computer Modelling 49 (2009) 1331–1336. [19] A. Azam, M. Arshad, P. Vetro, On a pair of fuzzy ϕ -contractive mappings, Mathematical and Computer Modelling 52 (1–2) (2010) 207–214. [20] V. Berinde, General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces, Carpathian Journal of Mathematics 24 (2) (2008) 10–19. [21] M. Abbas, M. Ali Khan, Common fixed point theorem of two mappings satisfying a generalized weak contractive condition, International Journal of Mathematics and Mathematical Sciences 2009 Article ID 131068, 9 pages.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Common fuzzy fixed point theorems in ordered metric spaces L. Ćirić a,∗ , M. Abbas b , B. Damjanović c , R. Saadati d a
Faculty of Mechanical Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia
b
Department of Mathematics, Lahore University of Management Sciences, Lahore - 54792, Pakistan
c
Department of Mathematics, Faculty of Agriculture, Nemanjina 6, 11 000 Belgrade, Serbia
d
Department of Mathematics, Science & Research Branch, Islamic Azad University, Post Code 14778, Ashrafi Esfahani Ave, Tehran, Islamic Republic of Iran
article
abstract
info
Article history: Received 28 July 2010 Accepted 23 December 2010
We prove the existence of fuzzy common fixed point of two mappings satisfying a generalized contractive condition in complete ordered spaces. Our results provide extension as well as substantial improvements of several well-known results in the existing literature and initiate the study of fuzzy fixed point theorems in ordered spaces. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Fuzzy mapping Fuzzy set Fuzzy fixed point
1. Introduction and preliminaries Let X be a space of points with generic element of X denoted by x and I = [0, 1]. A fuzzy subset of X is characterized by a member ship function which associates with each element in X a real number in the interval I . Let (X , d) be a metric linear space and A be a fuzzy set in X characterized by a membership function A. The α - level set of A, denoted by Aα , is defined by Aα = {x : A(x) ≥ α} if α ∈ (0, 1] A0 = {x : A(x) > 0} where B denotes the closure of the non fuzzy set B. A fuzzy set A in a metric linear space is said to be an approximate quantity if and only if Aα is compact and convex in X for each α ∈ [0, 1] and supx∈X A(x) = 1. We denote by W (X ), the family of all approximate quantities in X . Let A, B ∈ W (X ), then A is said to be more accurate than B, denoted by A ⊂ B, if and only if A(x) ≤ B(x) for each x in X , where B denotes the membership function of B. For x ∈ X , we write {x} the characteristic function of the ordinary subset {x} of X . We denote W 0 (X ) = {{x} : x ∈ X }. For α ∈ (0, 1], the fuzzy point (x)α of X is the fuzzy set of X given by xα (x) = α and α ̸= x. Let I X be the collection of all fuzzy subsets in X and W (X ) be a sub collection of all approximate quantities. For A, B ∈ W (X ), α ∈ [0, 1], define pα (A, B) =
inf
x∈Aα , y∈Bα
d(x, y),
p(A, B) = sup Pα (A, B), α
∗
Corresponding author. Fax: +381 11 3370364. E-mail addresses: [email protected] (L. Ćirić), [email protected] (M. Abbas), [email protected] (B. Damjanović), [email protected] (R. Saadati).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.12.050
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L. Ćirić et al. / Mathematical and Computer Modelling 53 (2011) 1737–1741
Dα (A, B) = H (Aα , Bα ), D(A, B) = sup Dα (A, B), α
where H is the Hausdorff metric induced by the metric d. We note that Pα is a non-decreasing function of α and D is a metric on W (X ). Let α ∈ [0, 1], then the family Wα (X ) is given by {A ∈ I X : Aα is non empty convex and compact }. Let X be an arbitrary set, Y be a metric linear space. A mapping T is called fuzzy mapping if T is a mapping from X into W (Y ), that is, Tx ∈ W (Y ) for each x in X . Thus if we characterize a fuzzy set Tx in a metric linear space Y by a member ship function Tx, then Tx(y) is the grade of member ship of y in Tx. Therefore a fuzzy mapping T is a fuzzy subset on X × Y with membership function Tx(y). A fuzzy point xα in X is called a fixed fuzzy point of the fuzzy mapping T if xα ⊂ Tx [1]. If {x} ⊂ Tx, then x is a fixed point of T . Definition 1. Let X be a nonempty set. Then (X , d, ≤) is called an ordered metric space iff. (i) (X , d) is a metric space, (ii) (X , ≤) is partial ordered. Definition 2. Let (X , ≤) be a partial ordered set. x, y ∈ X are called comparable if x ≤ y or y ≤ x holds. Following lemmas are needed in the sequel. Lemma 3 (Heilpern [2]). Let (X , d) be a metric space, x, y ∈ X and A, B ∈ W (X ): (1) if pα (x, A) = 0, then xα ⊂ A (2) pα (x, A) ≤ d(x, y)+ pα (y, B) (3) if xα ⊂ A, then pα (x, B) ≤ Dα (A, B). Lemma 4 (Lee and Cho [3]). Let (X , d) be a complete metric space, T be a fuzzy mapping from X into W (X ) and x0 ∈ X . Then there exists a x1 ∈ X such that {x1 } ⊂ Tx0 . Zadeh [4] introduced the concept of a fuzzy set which motivated a lot of mathematical activity on the generalization of the notion of a fuzzy set. Boričić in [5] considered fuzzification of propositional logics. Heilpern [2] introduced the concept of a fuzzy mappings in a metric linear space and proved a fixed point theorem for fuzzy contraction mapping which is the generalization of a fixed point theorem for multi-valued mappings of Nadler [6]. Estruch and Vidal [1] proved a fixed point theorem for fuzzy contraction mappings in a complete metric spaces which in turn generalized Heilpern fixed point theorem. Further generalization of the result given in [1] was proved in [7,8]. Recently Dutta and Choudhury [9] gave a generalization of Banach contraction principle, which in turn generalize Theorem 1 of [10] and corresponding result of [11]. Very recently Altun et al. [12] proved fixed point theorems in the frame work of ordered cone metric spaces. Bose and Shani [13] extended the result of Heilpern to pair of mappings. Kamran [14] and Sahin [15] also obtained some common fixed point theorems for fuzzy mappings in metric spaces. Recently Ðorić [16], Abbas and Ðorić [17] and Azam and Beg [18] proved common fixed point theorem for mappings which satisfy Alber and Guerr-Delabriere type contractive condition. Very recently Azam [19] established common fixed point theorems for fuzzy mappings under a ϕ -contraction condition on a metric space with the d∞ -metric. The aim of this paper is to establish the existence of a common fuzzy fixed point of generalized contractive mappings without employing any commutativity condition. Our result generalize, improve and extend many known results in the comparable literature [18,20,7]. 2. Main results We begin with the following result. Theorem 5. Let X be a complete ordered space. Suppose that T1 , T2 : X −→ Wα (X ) are two fuzzy mapping on X satisfying
ϕ(Dα (T1 x, T2 y)) ≤ ϕ(d(x, y)) − φ(d(x, y))
(1)
for all comparable elements x, y ∈ X , where, ϕ : [0, ∞) → (0, ∞) is a continuous and monotone nondecreasing functions with ϕ(t ) = 0 if and only if t = 0 and φ : [0, ∞) → (0, ∞) is lower semi continuous with ϕ(t ) = 0 if and only if t = 0. Suppose that if {y} ⊂ T1 (x0 ), then y, x0 ∈ X are comparable. Further, if x, y ∈ X are comparable, then every u ∈ (T1 x)α and every v ∈ (T2 y)α are comparable. Also suppose that if a sequence {xn } → x and its consecutive terms are comparable, then xn , x ∈ X are comparable for all n. Then there exists a point x in X such that xα ⊂ T1 x and xα ⊂ T2 x.
L. Ćirić et al. / Mathematical and Computer Modelling 53 (2011) 1737–1741
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Proof. Let x0 be in X . By Lemma 2, there exists x1 in X such that {x1 } ⊂ T1 (x0 ) which implies that p α ( x1 , T 1 x0 ) = 0
for each α in [0, 1],
which is possible if and only if x1 ∈ (T1 x0 )α . By assumption, x0 and x1 are comparable. Since (T2 x1 )α is nonempty compact subset of X , there exists x2 ∈ (T2 x1 )α such that d(x1 , x2 ) = pα (x1 , T2 x1 ) ≤ Dα (T1 x0 , T2 x1 ). Moreover, x1 and x2 are comparable. Continuing this process, we can construct a sequence {xn } in X such that x2n+1 ∈ (T1 (x2n ))α and x2n+2 ∈ (T2 (x2n+1 ))α for all n ≥ 0, x2n and x2n+1 are comparable and d(x2n+1 , x2n+2 ) ≤ Dα (T1 x2n , T2 x2n+1 ). Since ϕ is nondecreasing, ϕ(d(x2n+1 , x2n+2 )) ≤ ϕ(Dα (T1 x2n , T2 x2n+1 )). Since x2n and x2n+1 are comparable. Thus by taking x2n for x and x2n+1 for y in the inequality (1), it follows that
ϕ(d(x2n+1 , x2n+2 )) ≤ ϕ(Dα (T1 x2n , T2 x2n+1 )) ≤ ϕ(d(x2n , x2n+1 )) − φ(d(x2n , x2n+1 )). Similarly
ϕ(d(x2n+3 , x2n+2 )) ≤ ϕ(d(x2n+2 , x2n+1 )) − φ(d(x2n+2 , x2n+1 )). Therefore, for all n
ϕ(d(xn , xn+1 )) ≤ ϕ(Dα (T1 xn−1 , T2 xn )) ≤ ϕ(d(xn−1 , xn )) − φ(d(xn−1 , xn )). Hence ϕ(d(xn , xn+1 )) ≤ ϕ(d(xn−1 , xn )). Thus, we have d(xn , xn+1 ) ≤ d(xn−1 , xn ), which shows that {d(xn , xn+1 )} is non-increasing sequence of positive real numbers which is bounded below by 0. Therefore there is a real number r ≥ 0 such that lim d(xn , xn+1 ) = r .
n→∞
Suppose that r > 0, then 0 < ϕ(r ) ≤ ϕ(d(xn , xn+1 )) ≤ ϕ(Dα (T1 xn−1 , T2 xn ))
≤ ϕ(d(xn , xn−1 )) − φ(d(xn , xn−1 )). Now by continuity of ϕ and lower semicontinuity of φ we get
ϕ(r ) ≤ lim sup ϕ(d(xn , xn−1 )) − lim inf φ(d(xn−1 , xn )) n→∞
n→∞
and hence ϕ(r ) ≤ ϕ(r ) − φ(r ) < ϕ(r ), a contradiction. Therefore r = 0 and so lim d(xn , xn+1 ) = 0.
n→∞
Following the similar arguments to those given in [21], it can be shown that {xn } is a Cauchy sequence in X . It follows from the completeness of X that xn −→ x ∈ X . Since consecutive terms of {xn } are comparable and xn ≤ x. Now, we claim that pα (x, T2 x) = 0 for each α ∈ [0, 1]. From
|pα (x, T2 x) − d(x, x2n+1 )| ≤ pα (x2n+1 , T2 x) ≤ Dα (T1 x2n , T2 x) and (1) we get
ϕ(|pα (x, T2 x) − d(x, x2n+1 )|) ≤ ϕ(Dα (T1 x2n , T2 x)) ≤ ϕ(d(x2n , x)) − φ(d(x2n , x)). Hence we obtain, as ϕ is continuous and φ is lower semicontinuous,
ϕ(pα (x, T2 x)) ≤ ϕ(0) − φ(0) = 0, that is, ϕ(pα (x, T2 x)) = 0. Hence pα (x, T2 x) = 0. Therefore xα ⊂ T2 x. Similarly, xα ⊂ T1 x.
5
Define a class of functions G = {g : R+ → R+ } satisfying the following conditions: (g1 ) g is nondecreasing in the first and 5th variables. (g2 ) If u, v, ∈ R+ are such that g (u, v, v, u, u + v) ≤ 0, or g (u, v, u, v, u + v) ≤ 0, then u ≤ hv , where 0 < h < 1 is a constant. (g3 ) If u ∈ R+ is such that g (u, 0, 0, u, u) ≤ 0, or g (u, 0, u, 0, u) ≤ 0, then u = 0.
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Theorem 6. Let X be a complete ordered space. Suppose that T1 , T2 : X −→ Wα (X ) are two fuzzy mapping on X satisfying g (Dα (T1 x, T2 y), d(x, y), pα (x, T1 x), pα (y, T2 y), pα (x, T2 y) + pα (y, T1 x)) ≤ 0
(2)
for all comparable elements x, y ∈ X and for some g ∈ G. Suppose that for any y in X with {y} ⊂ T1 (x0 ) implies that y, x0 ∈ X are comparable and for comparable x, y ∈ X with u ∈ (T1 x)α and v ∈ (T2 y)α imply u, v ∈ X are comparable. Further, suppose that if a sequence {xn } → x and its consecutive terms are comparable, then xn , x ∈ X are comparable for all n. Then there exists a point x in X such that xα ⊂ T1 x and xα ⊂ T2 x. Proof. Let x0 be in X . By Lemma 2, there exists x1 in X such that {x1 } ⊂ T1 (x0 ) which implies that pα (x1 , T1 x0 ) = 0
for each α in [0, 1],
which is possible if and only if x1 ∈ (T1 x0 )α . By given assumption x0 and x1 are comparable. Since (T2 x1 )α is nonempty compact subset of X , therefore there exists x2 ∈ (T2 x1 )α such that d(x1 , x2 ) = pα (x1 , T2 x1 ) ≤ Dα (T1 x0 , T2 x1 ). Also, x1 and x2 are comaparable. Since x0 and x1 are comparable, then g (Dα (T1 x0 , T2 x1 ), d(x0 , x1 ), pα (x0 , T1 x0 ), pα (x1 , T2 x1 ), pα (x0 , T2 x1 ) + pα (x1 , T1 x0 )) ≤ 0. Since pα (x1 , T1 x0 ) = 0 and pα (x0 , T2 x1 ) ≤ d(x0 , x1 )+ pα (x1 , T2 x1 ), then g (d(x1 , x2 ), d(x0 , x1 ), d(x0 , x1 ), d(x1 , x2 ), d(x1 , x2 ) + d(x0 , x1 )) ≤ 0. Hence, as g ∈ G, d(x1 , x2 ) ≤ hd(x0 , x1 ). Similarly, we obtain that x2 and x3 are comparable and d(x2 , x3 ) ≤ hd(x1 , x2 ) ≤ h2 d(x0 , x1 ). Continuing this process, we can construct a sequence {xn } in X such that x2n+1 ∈ (T1 (x2n ))α and x2n+2 ∈ (T2 (x2n+1 ))α for all n ≥ 0, x2n and x2n+1 are comparable and d(x2n+1 , x2n+2 ) ≤ hd(x2n , x2n+1 ). Thus, by induction we have d(xn , xn+1 ) ≤ hn d(x0 , x1 ). From the proceeding inequality we conclude that {xn } is a Cauchy sequence in X . It follows from the completeness of X that xn −→ x ∈ X . Since consecutive terms of {xn } are comparable, then xn , x ∈ X are comparable for all n. Also, note that x ∈ limn→∞ (T1 x2n )α and x ∈ limn→∞ (T2 x2n+1 )α . Now, we claim that pα (x, T2 x) = 0 for each α ∈ [0, 1]. From
|pα (x, T2 x) − d(x, x2n+1 )| ≤ pα (x2n+1 , T2 x) ≤ Dα (T1 x2n , T2 x) and (2) we get, as g ∈ G, g (pα (x2n+1 , T2 x), d(x2n , x), pα (x2n , T1 x2n ), pα (x, T2 x), pα (x2n , T2 x) + pα (x, T1 x2n )) ≤ 0. Hence we obtain pα (x, T2 x) = 0. Therefore xα ⊂ T2 x. Similarly, xα ⊂ T1 x.
Acknowledgements The first and third authors accomplished research results on the project IO 174025, and resources for its implementation have been provided by Ministry of Science and Technological Development of Republic Serbia. The fourth author is grateful to the Young research Club, Islamic Azad University—Ayatollah Amoli Branch, Amol, Iran. References [1] [2] [3] [4] [5] [6] [7] [8]
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