A fixed point theorem in Menger space through weak compatibility

J. Math. Anal. Appl. 301 (2005) 439–448 www.elsevier.com/locate/jmaa

A fixed point theorem in Menger space through weak compatibility Bijendra Singh a , Shishir Jain b,∗ a S.S in Mathematics Vikram, University Ujjain (MP), India b Shri Vaishnav Institute of Technology and Science, Gram Baroli Post Alwasa, Indore (MP), India

Received 10 May 2004

Submitted by William F. Ames

Abstract This paper contains a fixed point theorem for six self maps in Menger space. Our result generalizes and extends many known results in Menger spaces and metric spaces.  2004 Elsevier Inc. All rights reserved. Keywords: Menger space; t-norm; Common fixed points; Compatible maps; Weak-compatible maps

1. Introduction There have been a number of generalizations of metric space. One such generalization is Menger space initiated by Menger [6]. It is a probabilistic generalization in which we assign to any two points x and y, a distribution function Fx,y . Schweizer and Sklar [9] studied this concept and gave some fundamental results on this space. It is observed by many authors that contraction condition in metric space may be exactly translated into PM-space endowed with min norms. Sehgal and Bharucha-Reid [10] obtained a generalization of Banach Contraction Principle on a complete Menger space which is a milestone in developing fixed-point theorems in Menger space. * Corresponding author.

E-mail address: [email protected] (S. Jain). 0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.07.036

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Sessa [8] initiated the tradition of improving commutativity in fixed-point theorems by introducing the notion of weakly commuting maps in metric spaces. Jungck [3] soon enlarged this concept to compatible maps. The notion of compatible mapping in a Menger space has been introduced by Mishra [7]. Recently, Jungck and Rhoades [5] (also Dhage [1]) termed a pair of self maps to be coincidentally commuting or equivalently weakly compatible if they commute at their coincidence points. This concept is most general among all the commutativity concepts in this field as every pair of R-weakly commuting self maps is compatible and each pair of compatible self maps is weakly compatible but the reverse is not true always. In this paper a fixed point theorem for six self maps has been proved using the concept of weak compatibility and compatibility of pair of self maps, which turns out to be a material generalization of the results of Mishra [7] and others. For the sake of completeness, following Mishra [7] and Sehgal and Bharucha Reid [10], we recall some definitions and known results in Menger space.

2. Preliminaries Definition 1. A mapping F : R → R + is called a distribution if it is non-decreasing left continuous with inf{F (t): t ∈ R} = 0 and sup{F (t): t ∈ R} = 1. We shall denote by L the set of all distribution functions while H will always denote the specific distribution function defined by
H (t) =

0, t
0 1, t > 0.

Definition 2 [7]. A probabilistic metric space (PM-space) is an ordered pair (X, F ) where X is an abstract set of elements and F : X × X → L is defined by (p, q) → Fp,q , where L is the set of all distribution functions, i.e., L = {Fp,q : p, q ∈ X}, where the functions Fp,q satisfy: (a) (b) (c) (d)

Fp,q (x) = 1 for all x > 0, if and only if p = q; Fp,q (0) = 0; Fp,q = Fq,p ; If Fp,q (x) = 1 and Fq,r (y) = 1 then Fp,r (x + y) = 1.

Definition 3. A mapping t : [0, 1] × [0, 1] → [0, 1] is called a t-norm if (e) (f) (g) (h)

t (a, 1) = a, t (0, 0) = 0; t (a, b) = t (b, a); t (c, d)  t (a, b) for c  a, d  b; t (t (a, b), c) = t (a, t (b, c)).

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Definition 4. A Menger space is a triplet (X, F, t), where (X, F ) is PM-space and t is a t-norm such that for all p, q, r ∈ X and for all x, y  0   Fp,r (x + y)  t Fp,q (x), Fq,r (y) . Proposition 1 [10]. If (X, d) is a metric then the metric d induces a mapping X × X → L, defined by Fp,q (x) = H (x −d(p, q)), p, q ∈ X and x ∈ R. Further, if the t-norm t : [0, 1]× [0, 1] → [0, 1] is defined by t (a, b) = min{a, b}, then (X, F, t) is a Menger space. It is complete if (X, d) is complete. The space (X, F, t) so obtained is called the induced Menger space. Definition 5 [7]. A sequence {pn } in X is said to converge to a point p in X (written as pn → p) if for every ε > 0 and λ > 0, there is an integer M(ε, λ) such that Fpn ,p (ε) > 1 − λ, for all n  M(ε, λ). The sequence is said to be Cauchy if for each ε > 0 and λ > 0, there is an integer M(ε, λ) such that Fpn ,pm (ε)  1 − λ, ∀n, m  M(ε, λ). A Menger space (X, F, t) is said to be complete if every Cauchy sequence in it converges to a point of it. Definition 6. Self-maps A and S of a Menger space (X, F, t) are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Ap = Sp for some p ∈ X then ASp = SAp. Definition 7 [7]. Self mappings A and S of a Menger space (X, F, t) are called compatible if FASpn ,SApn (x) → 1 for all x > 0, whenever {pn } is a sequence in X such that Apn, Spn, → u, for some u in X, as n → ∞. Proposition 2. Self mappings A and S of a Menger space (X, F, t) are compatible then they are weakly compatible. Proof. Suppose Ap = Sp, for some p in X. Consider the constant sequence {pn } = p. Now, {Apn } → Ap and {Spn } → Sp (= Ap). As A and S are compatible we have FASp,SAp (x) = 1 for all x > 0. Thus, ASp = SAp and we get that (A, S) is weakly compatible. 2 The following is an example of pair of self maps in a Menger space which are weakly compatible but not compatible. Example 1. Let (X, d) be a metric space where X = [0, 2] and (X, F, t) be the induced Menger space with Fp,q (ε) = H (ε − d(p, q)), ∀p, q ∈ X and ∀ε > 0. Define self maps A and S as follows:

2 − x, if 0
x < 1, x, if 0
x < 1, Ax = and Sx = 2, if 1
x
2, 2, if 1
x
2. Take xn = 1 − 1/n. Now,   FAxn ,1 (ε) = H ε − (1/n)

∴

lim FAxn ,1 (ε) = H (ε) = 1.

n→∞

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Hence Axn → 1 as n → ∞. Similarly, Sxn → 1 as n → ∞. Also    1 FASxn ,SAxn (ε) = H ε − 1 − , lim FASxn ,SAxn (ε) = H (ε − 1) = 1, n→∞ n ∀ε > 0. Hence the pair (A, S) is not compatible. Also set of coincidence points of A and S is [1, 2]. Now for any x ∈ [1, 2], Ax = Sx = 2, and AS(x) = A(2) = 2 = S(2) = SA(x). Thus A and S are weakly compatible but not compatible. From the above example it is obvious that the concept of weak compatibility is more general than that of compatibility. Proposition 3. In a Menger space (X, F, t), if t (x, x)  x ∀x ∈ [0, 1], then t (a, b) = min{a, b} ∀a, b ∈ [0, 1]. Lemma 1 [11]. Let {pn } be a sequence in a Menger space (X, F, t) with continuous tnorm and t (x, x)  x. Suppose, for all x ∈ [0, 1], ∃k ∈ (0, 1) such that for all x > 0 and n ∈ N, Fpn ,pn+1 (kx)  Fpn−1 ,pn (x). Then {pn } is a Cauchy sequence in X. Lemma 2. Let (X, F, t) be a Menger space. If there exists k ∈ (0, 1) such that for p, q ∈ X, Fp,q (kx)  Fp,q (x). Then p = q. Proof. As Fp,q (kx)  Fp,q (x), we have:   Fp,q (x)  Fp,q k −1 x . By repeated application of above inequality, we get:       Fp,q (x)  Fp,q k −1 x  Fp,q k −2 x  · · ·  Fp,q k −m t  · · · ,

m∈N

which → 1 as m → ∞. Hence, Fp,q (x) = 1, ∀x > 0 and we get p = q.

2

In Mishra [7], the following result has been established. Theorem [7]. Let A, B, S and T be self maps of a Menger space (X, F, t), with continuous t-norm with t (x, x)  x, for all x ∈ [0, 1], satisfying conditions (a) A(X) ⊆ T (X), B(X) ⊆ S(X); (b) ∀p, q ∈ X, x > 0 and for some k ∈ (0, 1), FAp,Bq (kx)      ;  t FAp,Sp (x), t FBq,T q (x), t FSp,T q (x), t FAp,T q (αx), FBq,Sp (2 − α)x (c) the pairs (A, S) and (B, T ) are compatible;

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(d) S and T are continuous. Then A, B, S and T have a unique common fixed point.

3. Main result In the following, we extend this result to six self maps and generalize it in other respects too. Theorem 3.1. Let A, B, S, T , L and M are self maps on a complete Menger space, (X, F, t) with t (a, a)  a, for all a ∈ [0, 1], satisfying: (1) (2) (3) (4) (5)

L(X) ⊆ ST (X), M(X) ⊆ AB(X); AB = BA, ST = T S, LB = BL, MT = T M; either AB or L is continuous; (L, AB) is compatible and (M, ST ) is weakly compatible; there exists k ∈ (0, 1) such that  FLp,Mq (kx)  min FABp,Lp (x), FST q,Mq (x), FST q,Lp (βx),    FABp,Mq (2 − β)x , FABp,ST q (x) for all p, q ∈ X, β ∈ (0, 2) and x > 0.

Then A, B, S, T , L and M have a unique common fixed point in X. Proof. Let x0 ∈ X. From condition (1) there exists x1 , x2 ∈ X such that Lx0 = ST x1 = y0 and Mx1 = ABx2 = y1 . Inductively we can construct sequences {xn } and {yn } in X such that Lx2n = ST x2n+1 = y2n and Mx2n+1 = ABx2n+2 = y2n+1 for n = 0, 1, 2, . . . : Step 1. Putting p = x2n , q = x2n+1 for x > 0 and β = 1 − q with q ∈ (0, 1) in (5), we get:  FLx2n ,Mx2n+1 (kx)  min FABx2n ,Lx2n (x), FST x2n+1 ,Mx2n+1 (x),     FST x2n+1 ,Lx2n (1 − q)x , FABx2n ,Mx2n+1 (1 + q)x ,  FABx2n ,ST x2n+1 (x) , Fy2n, y2n+1 (kx)      min Fy2n−1, y2n (x), Fy2n, y2n+1 (x), 1, Fy2n−1, y2n+1 (1 + q)x , Fy2n−1, y2n (x)    min Fy2n−1, y2n (x), Fy2n, y2n+1 (x), Fy2n−1, y2n (x), Fy2n,, y2n+1 (qx),    min Fy2n−1, y2n (x), Fy2n, y2n+1 (x), Fy2n, y2n+1 (qx) . As t-norm is continuous, letting q → 1 we get:   Fy2n, y2n+1 (kx)  min Fy2n−1, y2n (x), Fy2n, y2n+1 (x), Fy2n, y2n+1 (x)   = min Fy2n−1, y2n (x), Fy2n, y2n+1 (x) .

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Hence, Fy2n, y2n+1 (kx)  min{Fy2n−1, y2n (x), Fy2n, y2n+1 (x)}. Similarly, Fy2n+1, y2n+2 (kx)  min{Fy2n, y2n+1 (x), Fy2n+1, y2n+2 (x)}. Therefore, for all n even or odd we have:   Fyn ,yn+1 (kx)  min Fyn−1 ,yn (x), Fyn ,yn+1 (x) . Consequently,

     Fyn ,yn+1 (x)  min Fyn−1 ,yn k −1 x , Fyn ,yn+1 k −1 x .

By repeated application of above inequality, we get:      Fyn ,yn+1 (x)  min Fyn−1 ,yn k −1 x , Fyn ,yn+1 k −m x . Since Fyn ,yn+1 (k −m x) → 1 as m → ∞, it follows that Fyn ,yn+1 (kx)  Fyn−1 ,yn (x) ∀n ∈ N and ∀x > 0. Therefore, by Lemma 1, {yn } is a Cauchy sequence in X, which is complete. Hence {yn } → z ∈ X. Also its subsequences converge as follows: {Mx2n+1 } → z and {ST x2n+1 } → z,

(3.1)

{Lx2n } → z

(3.2)

and {ABx2n } → z.

Case I. AB is continuous. As AB is continuous, (AB)2 x2n → ABz and (AB)Lx2n → ABz. As (L, AB) is compatible, we have L(AB)x2n → ABz. Step 2. Putting p = ABx2n , q = x2n+1 with β = 1 in condition (5), we get:  FLABx2n ,Mx2n+1 (kx)  min FABABx2n ,LABx2n (x), FST x2n+1 ,Mx2n+1 (x), FST x2n+1 ,LABx2n (x), FABABx2n ,Mx2n+1 (x),  FABABx2n ,ST x2n+1 (x) . Letting n → ∞, we get:

  FABz,z (kx)  min FABz,ABz (x), Fz,z (x), Fz,ABz (x), FABz,z (x), FABz,z (x) ,

i.e. FABz,z (kx)  FABz,z (x). Therefore, by Lemma 2, we get ABz = z. Step 3. Putting p = z, q = x2n+1 with β = 1 in condition (5), we get:  FLz,Mx2n+1 (kx)  min FABz,Lz (x), FST x2n+1 ,Mx2n+1 (x), FST x2n+1 ,Lz (x),  FABz,Mx2n+1 (x), FABz,ST x2n+1 (x) . Letting n → ∞, we get:   FLz,z (kx)  min Fz,Lz (x), Fz,z (x), Fz,Lz (x), FLz,z (x), FLz,z (x) , FLz,z (kx)  FLz,z (x), which gives Lz = z. Therefore ABz = Lz = z.

(3.3)

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Step 4. Putting p = Bz, q = x2n+1 with β = 1 in condition (5), we get:  FLBz,Mx2n+1 (kx)  min FABBz,LBz (x), FST x2n+1 ,Mx2n+1 (x), FST x2n+1 ,LBz (x),  FABBz,Mx2n+1 (x), FABBz,ST x2n+1 (x) . As BL = LB, AB = BA, so we have L(Bz) = B(Lz) = Bz and AB(Bz) = B(ABz) = Bz. Letting n → ∞, we get:   FBz,z (kx)  min FBz,Bz (x), Fz,z (x), Fz,Bz (x), FBz,z (x), FBz,z (x) , i.e. FBz,z (kx)  FBz,z (x); which gives Bz = z and ABz = z implies Az = z. Therefore Az = Bz = Lz = z.

(3.4)

Step 5. As L(X) ⊆ ST (X), there exists v ∈ X such that z = Lz = ST v. Putting p = x2n , q = v with β = 1 in condition (5), we get:  min FLx2n ,Mv (kx)  min FABx2n ,Lx2n (x), FST v,Mv (x), FST v,Lx2n (x),  FABx2n ,Mv (x), FABx2n ,ST v (x) . Letting n → ∞ using Eq. (3.2), we get:   Fz,Mv (kx)  min Fz,z (x), Fz,Mv (x), Fz,z (x), ∗Fz,Mz (x), Fz,z (x)  Fz,Mv (x). Therefore by Lemma 2, Mv = z. Hence, ST v = z = Mv. As (M, ST ) is weakly compatible, we have ST Mv = MST v. Thus, ST z = Mz. Step 6. Putting p = x2n , q = z with β = 1 in condition (5), we get:  FLx2n ,Mz (kx)  min FABx2n ,Lx2n (x), FST z,Mz (x), FST z,Lx2n (x),  FABx2n ,Mz (x), FABx2n ,ST z (x) . Letting n → ∞ and using Eqs. (3.1) and Step 5, we get:   Fz,Mz (kx)  min Fz,z (x), FMz,Mz (x), FMz,z (x), Fz,Mz (x), Fz,Mz (x) , i.e. Fz,Mz (kx)  Fz,Mz (x). Hence z = Mz. Step 7. Putting p = x2n , q = T z with β = 1 in condition (5), we get:  FLx2n ,MT z (kx)  min FABx2n ,Lx2n (x), FST T z,MT z (x), FST T z,Lx2n (x),  FABx2n ,MT z (x), FABx2n ,ST T z (x) . As MT = T M and ST = T S we have MT z = T Mz = T z and ST (T z) = T (ST z) = T z. Letting n → ∞, we get:   Fz,T z (kx)  min Fz,z (x), FT z,T z (x), FT z,z (x), Fz,T z (x), Fz,T z (x)  Fz,T z (x). Therefore, by Lemma 2, T z = z. Now ST z = T z = z implies Sz = z. Hence Sz = T z = Mz = z.

(3.5)

Combining (3) and (4), we get: Az = Bz = Lz = Mz = T z = Sz = z. Hence, the six self maps have a common fixed point in this case.

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Case II. L is continuous. As L is continuous, L2 x2n → Lz and L(ABx2n ) → Lz. As (L, AB) is compatible, we have (AB)Lx2n → Lz. Step 8. Putting p = Lx2n , q = x2n+1 with β = 1 in condition (5), we get:  FLLx2n ,Mx2n+1 (kx)  min FABLx2n ,LLx2n (x), FST x2n+1 ,Mx2n+1 (x), FST x2n+1 ,LLx2n (x), FABLx2n ,Mx2n+1 (x),  FABLx2n ,ST x2n+1 (x) . Letting n → ∞, we get:   FLz,z (kx)  min FLz,Lz (x), Fz,z (x), Fz,Lz (x), FLz,z (x), FLz,z (x) , which gives Lz = z. Now, using Steps 5–7 gives us Mz = ST z = Sz = T z = z. Step 9. As M(X) ⊆ AB(X) there exists w ∈ X such that z = Mz = ABw. Putting p = w, q = x2n+1 with β = 1 in condition (5), we get:  FLw,Mx2n+1 (kx)  min FABw,Lw (x), FST x2n+1 ,Mx2n+1 (x), FST x2n+1 ,Lw (x),  FABw,Mx2n+1 (x), FABw,ST x2n+1 (x) . Letting n → ∞, we get:   FLw,z (kx)  min Fz,Lw (x), Fz,z (x), Fz,Lw (x), Fz,z (x), Fz,z (x) , i.e. FLw,z (kx)  Fz,Lw (x), which gives Lw = z = ABw. As (L, AB) is weakly compatible, we have Lz = ABz. Also Bz = z follows from Step 4. Thus, Az = Bz = Lz = z and we obtain that z is the common fixed point of the six maps in this case also. Step 10 (Uniqueness). Let u be another common fixed point of A, B, P , Q, S and T ; then Au = Bu = P u = Su = T u = Qu = u. Putting p = z, q = u in condition with β = 1 in (5), we get:  FLz,Mu (kx)  min FABz,Lz (x), FST u,Mu (x), FST u,Lz (x), FABz,Mu (x),  FABz,ST u (x) , i.e. Fz,u (kx)  Fz,u (x), which gives z = w. Therefore, z is a unique common fixed point of A, B, L, M, S and T . 2 Remark 1. In view of Proposition 2, t (a, b) = min{a, b}; then the condition there exists k ∈ (0, 1) such that for all p, q ∈ X and x > 0    FLp,Mq (kx)  t FABp,Lp (x), t FST q,Mq (x), t FST q,Lp (βx),     t FABp,Mq (2 − β)x , FABp,ST q (x) becomes (5). Again if we take B = T = I , the identity map on X in Theorem 3.1, then the condition (2) is satisfied trivially and we get:

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Corollary 3.2. Let A, S, L and M are self maps on a complete Menger space (X, F, t) with t (x, x)  x for all x ∈ [0, 1], satisfying: (1) (2) (3) (4)

L(X) ⊆ S(X), M(X) ⊆ A(X). Either A or L is continuous. (L, A) is compatible and (M, S) is weakly compatible. There exists k ∈ (0, 1) such that    FLp,Mq (kx)  min FAp,Lp (x), FSq,Mq (x), FSq,Lp (βx), FAp,Mq (2 − β)x ,  FAp,Sq (x) for all p, q ∈ X, β ∈ (0, 2) and x > 0.

Then A, S, L and M have a unique common fixed point in X. Remark 2. In view of Remark 1, Corollary 3.2 is a generalization of the above result of Mishra [7] in the sense that the condition of compatibility of the second pair of self maps has been restricted to weak-compatibility and only one map of the first pair is needed to be continuous. In the following we prove the projection of Theorem 3.1 from complete Menger space to complete metric space. Theorem 3.3. Let A, B, S, T , L and M are self maps on a complete metric space (X, d) satisfying (1)–(4) of Theorem 3.1; then there exists k ∈ (0, 1) such that for all p, q ∈ X  d(Ap, Bq)
k max d(Lp, ABp), d(Mq, ST q), d(ABp, ST q),   1 2 d(Lp, ST q) + d(Mq, ABp) . Then A, B, S, T , L and M have a unique common fixed point in X. Proof. The proof follows from Theorem 3.1 and by considering the induced Menger space (X, F, t), where t (a, b) = min{a, b} and Fp,q (x) = H (x − d(p, q)), H being the distribution function as given in Definition 1. 2 This theorem also generalizes Theorem 2 of Mishra [7] for a complete metric space in the aforesaid sense. Remark 3. In view of Theorem 3.3, it is clear that Theorem 2 of Mishra [7], Theorem 3.1 of Jungck [4] and its follow-up corollaries by Hadgic [2] are special cases of our main result. Our result also improves the results of Xieping [12, Theorems 1 and 2] in the sense that the requirement of weak compatibility is more general than of compatibility and commutativity. Also the number of required continuities of maps has been reduced to one only in our results.

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References [1] B.C. Dhage, On common fixed point of coincidentally commuting mappings in D-metric spaces, Indian J. Pure Appl. Math. 30 (1999) 395–406. [2] O. Hadgic, Common fixed point theorems for families of mapping in complete metric space, Math. Japon. 29 (1984) 127–134. [3] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. (1986) 771–779. [4] G. Jungck, Compatible mappings and common fixed points (2), Internat. J. Math. Math. Sci. (1988) 285– 288. [5] G. Jungck, B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (1998) 227–238. [6] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942) 535–537. [7] S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon. 36 (1991) 283–289. [8] S. Sessa, On a weak commutative condition in fixed point consideration, Publ. Inst. Math. (Beograd) 32 (1982) 146–153. [9] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960) 313–334. [10] V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings in PM-spaces, Math. System Theory 6 (1972) 97–102. [11] S.L. Singh, B.D. Pant, Common fixed point theorems in PM spaces and extension to uniform spaces, Honam Math. J. 6 (1984) 1–12. [12] D. Xieping, A common fixed point theorem of commuting mappings in PM-spaces, Kexue Tongbao 29 (1984) 147–150.

Further reading [1] R. Dedeic, N. Sarapa, A common fixed point theorems for three mappings on Menger spaces, Math. Japon. 34 (1989) 919–923. [2] T.L. Hicks, Fixed point theory in probabilistic metric spaces II, Math. Japon. 44 (1996) 487–493.