International Mathematical Forum, 3, 2008, no. 3, 135 - 145
Fixed Point Theorems for Weakly Compatible Mappings in ε-Chainable Probabilistic Metric Spaces Tamanna Kadian and Renu Chugh Department of Mathematics M.D. University, Rohtak - 124 001, India
[email protected] [email protected] Abstract The aim of this paper is to introduce the notion of ε-chainable probabilistic metric spaces and prove a common fixed point theorem for six weakly compatible mappings in this newly defined space.
Mathematics Subject Classification: 47H10, 54H25 Keywords: Fixed point, ε-chainable probabilistic metric space, probabilistic metric space, compatible mappings, weakly compatible mappings
1
Introduction
An essential feature of metric spaces is that for any two points in a metric space, a positive number called the distance between the points is defined. In fact, it is suitable to look upon the distance concept as a statistical or probabilistic rather than deterministic one, because the advantage of a probabilistic approach is that it permits from the initial formulation a greater flexibility rather than that offered by a deterministic approach, i.e., there have been a number of generalizations of metric spaces. One such generalization is initiated by Menger [10]. The idea thus appears that, instead of a single positive number, we should associate a distribution function with the point pairs. Thus, for any elements p, q in the space, we have a distribution function F (p, q; x) and interpret F (p, q; x) as the probability that the distance between p and q is less than x. Thus the concept of a probabilistic metric space (Menger space) corresponds to the situations when we do not know the distance between the points, i.e., the distance between the points is inexact rather than a single real
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number, we know only probabilities of possible values of this distance. Such a probabilistic generalization of a metric space appears to be well adapted for the investigation of physical quantities and physiological thresholds. Thus Menger [10] introduced the notion of probabilistic metric spaces or statistical metric spaces which is in fact, a generalization of metric space and the study of these spaces was expanded rapidly with the pioneering works of Schweizer-Sklar [2]. The theory of probabilistic metric spaces is of fundamental importance in probabilistic function analysis. Sehgal [17] initiated the study of fixed points in probabilistic metric spaces (PM-Spaces). For more details we refer the readers to [4-9, 11-13, 17]. First, recall that a real valued function defined on the set of real numbers is known as a distribution function if it is non-decreasing, left continuous and inf f (x) = 0, sup f (x) = 1. In what follows H(x) denotes the distribution function defined as follows: 0 , if x ≤ 0 H(x) = 1 , if x > 0 . The aim of this paper is to introduce the notion of ε-chainable probabilistic metric spaces and prove a common fixed point theorem for six weakly compatible mappings in this newly defined space.
2
Preliminary
In this section, we recall some definitions and known results in probabilistic metric space. Definition 2.1. A triangular norm Δ (shorty t-norm) is a binary operation on the unit interval [0, 1] such that for all a, b, c, d ∈ [0, 1], the following conditions are satisfied: (a) (b) (c) (d)
a Δ 1 = a; a Δ b = bΔa; a Δ b ≤ c Δ d whenever a ≤ c and b ≤ d; a Δ (b Δ c) = (aΔb) Δ c.
Examples. (i) Δ (a, b) = min(a, b). (ii) Δ (a, b) = ab. Definition 2.2 (Schweizer and Sklar [2]). The ordered pair (X, F ) is called a probabilistic metric space (shortly PM-space) if X is a nonempty set and F is a probabilistic distance satisfying the following conditions: for all x, y, z ∈ X and t, s > 0,
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(i) (ii) (iii) (iv)
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F (x, y; t) = 1 if and only if x = y ; F (x, y; 0) = 0; F (x, y; t) = F (y, x; t); if F (x, z; t) = 1, F (z, y; s) = 1, then F (x, y; t + s) = 1.
The ordered triple (X, F, Δ) is called Menger space if (X, F ) is a PM-space, Δ is a t-norm and the following inequality holds: for all x, y, z ∈ X and t, s > 0, (v) F (x, y; t + s) ≥ F (x, z; t)ΔF (z, y; s). Definition 2.3 (Mishra [15]). Let (X, F, Δ) be a Menger space and Δ be a continuous t-norm. A sequence {xn } in X is said to be (a) converge to a point x in X (written xn → x) if and only if for every ε > 0 and λ ∈ (0, 1), there exists an integer n0 = n0 (ε, λ) such that F (xn , x; ε) > 1 − λ for all n ≥ n0 (ε, λ). (b) Cauchy if for every ε > 0 and λ ∈ (0, 1), there exists an integer n0 = n0 (ε, λ) such that F (xn , xn+p ; ε) > 1 − λ for all n ≥ n0 and p > 0. (c) complete if every Cauchy sequence in X converges to x ∈ X . It may Δ is a continuous t-norm, it follows from (v) that the limit of sequence in Menger space is uniquely determined. Definition 2.4 (Singh and Jain [3]). Self maps A and B of a Menger space (X, F, Δ) are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e., if Ax = Bx for some x ∈ X then ABx = BAx. Definition 2.5 (Mishra [15]). Self maps A and B of a Menger space (X, F, Δ) are said to be compatible if F (ABxn , BAxn ; t)→1 for all t > 0, whenever {xn } is a sequence in X such that Axn , Bxn → x for some x in X as n → ∞. Remark 2.6. Weakly compatible maps need not be compatible. Example 2.7. Let (X, d) be a metric space where X = [0, 2] and (X, F, Δ) be the induced Menger space with F (x, y; t) = H(t − d(x, y)), for all x, y ∈ X and for all t > 0 with usual metric d. Define self maps A and B as follows: 2 − x , if 0 ≤ x < 1 Ax = 2, if 1 ≤ x ≤ 2, and
x , if 0 ≤ x < 1 Bx = 2 , if 1 ≤ x ≤ 2.
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Now for any x ∈ [1, 2], Ax = Bx = 2 and AB(x) = A(2) = 2 = B(2) = BA(x). Thus A and B are weakly compatible but not compatible. Consider the sequence {xn } = {1 − 1/n : n ∈ N}. Then F (Axn , 1; t) = H(t − (1/n)) and lim F (Axn , 1; t) = H(t) = 1. n→∞ Hence Axn → ∞ as n → ∞. Similarly, Bxn → ∞ as n → ∞. Also, F (ABxn , BAxn ; t) = H(t − (1 − 1/n)) and lim F (ABxn , BAxn ; t) = H(t − 1) = 1, for all t > 0.
n→∞
Hence the pair (A, B) is not compatible. Definition 2.8. Let (X, F, Δ) be a probabilistic metric space and ε > 0. A finite sequence x = x0 , x1 , · · · , xn = y is called ε-chain from x to y if F (xi , xi−1 , t) > 1 − ε for all t > 0 and i = 1, 2, · · · , n. A probabilistic metric space (X, F, Δ) is called ε-chainable if for any x, y ∈ X , there exists an ε-chain from x to y . Lemma 2.9. Let (X, F, Δ) be a probabilistic metric space. Then, for all x, y ∈ X, F (x, y, ·) is nondecreasing. Lemma 2.10 (Singh and Jain [3]). Let (X, F, Δ) be a probabilistic metric space. If there exists k ∈ (0, 1) such that F (x, y, kt) ≥ F (x, y, t) for all x, y ∈ X and t > 0, then x = y .
3
Main Results
Theorem 3.1. Let (X, F, Δ) be a complete ε-chainable Probabilistic metric space and let P, Q, A, B, S and T be self mappings of X satisfying the following conditions: (i) P (X) ⊂ ST (X) and Q(X) ⊂ AB(X), (ii) P and AB are continuous, (iii) AB = BA, ST = T S , P B = BP , T Q = QT , (iv) the pairs (P, AB) and (Q, ST ) are weakly compatible,
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(v) there exists q ∈ (0, 1) such that F (P x, Qy, qt) ≥ min{F (ABx, ST y, t), F (P x, ABx, t), F (Qy, ST y, t), F (P x, ST y, t)} for every x, y ∈ X and t > 0. Then P, Q, A, B, S and T have a unique common fixed point in X . Proof. Let x0 ∈ X . From condition (i) there exists x1 , x2 ∈ X such that P x0 = ST x1 = y0 and Qx1 = ABx2 = y1 . Inductively we can construct sequences {xn } and {yn } in X such that P x2n = ST x2n+1 = y2n and Qx2n+1 = ABx2n+2 = y2n+1 for n = 0, 1, 2, · · · . Now, we prove {yn } is a Cauchy sequence in X . Putting x = x2n , y = x2n+1 for x > 0 in (v) we get. From (v), F (y2n, y2n+1 , qt) = F (P x2n , Qx2n+1 , qt) ≥ min{F (ABx2n , ST x2n+1 , t), F (P x2n , ABx2n , t), F (Qx2n+1 , ST x2n+1 , t), F (P x2n, ST x2n+1 , t)} = min{F (y2n−1, y2n , t), F (y2n, y2n−1, t), F (y2n+1 , y2n, t), F (y2n, y2n , t)} ≥ min{F (y2n−1, y2n , t), F (y2n+1, y2n, t)}. From Lemma 2.9 and 2.10, we have that F (y2n, y2n+1 , qt) ≥ F (y2n−1, y2n , t)
(3.1.1)
Similarly, we have also F (y2n+1, y2n+2 , qt) ≥ F (y2n, y2n+1 , t)
(3.1.2)
From (3.1.1) and (3.1.2), we have F (yn+1 , yn+2 , qt) ≥ F (yn , yn+1 , t) From (3.1.3), F (yn , yn+1 , t) ≥ F (yn , yn−1 , t/q) ≥ F (yn−2 , yn−1 , t/q 2 ) ≥ · · · ≥ F (y1 , y2, t/q n ) → 1 as n → ∞. So F (yn , yn+1 , t) → 1 as n → ∞ for any t > 0. For each ε > 0 and each t > 0, we can choose n0 ∈ N such that F (yn , yn+1 , t) > 1 − ε for all n > n0 .
(3.1.3)
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For m, n ∈ N , we suppose m ≥ n. Then we have that F (yn , ym , t) ≥ min{F (yn , yn+1 , t/m − n), F (yn+1 , yn+2 , t/m − n), · · · , F (ym−1 , ym , t/m − n)} > min {(1 − ε), (1 − ε), · · · , (1 − ε)} ≥ 1 − ε, and hence {yn } is a Cauchy sequence in X which is complete. Hence {yn } → z ∈ X and the subsequences {Qx2n+1 }, {ST x2n+1 }, {P x2n} and {ABx2n+2 } of {yn } also converge to z . Since X is ε-chainable, there exists ε-chain from xn to xn+1 , that is, there exists a finite sequencexn = y1 , y2 , · · · , yl = xn+1 such that F (yi, yi−1 , t) > 1 − ε for all t > 0 and i = 1, 2, · · · , l . Thus, we have F (xn , xn+1 , qt) ≥ min{F (y1, y2 , t), F (y2, y3 , t), · · · , F (yl−1, yl , t)} > min{(1 − ε), (1 − ε), · · · , (1 − ε)} ≥ 1 − ε. For m > n, F (xn , xm , t) ≥ min{F (xn , xn+1 , t/m − n), F (xn+1 , xn+2 , t/m − n), · · · , F (xm−1 , xm , t/m − n)} > min{(1 − ε), (1 − ε), · · · , (1 − ε)} > 1 − ε, and so {x n }is a Cauchy sequence in X and hence there exists x ∈ X such that xn → x. From (ii), P x2n−2 → P x and ABx2n → ABx. Since X is Hausdorff, P x = z = ABx. Because (P, AB) is weakly compatible, P ABx = ABP x and so P z = ABz . From(ii), P ABx2n → P ABx and hence P ABx2n → ABz. Also, from continuity of AB , ABABx2n → ABz. Step (i): By taking x = ABx2n and y = x2n−1 in (v), we have F (P ABx2n , Qx2n−1 , qt) ≥ min{F (ABABx2n , ST x2n−1 , t), F (P ABx2n , ABABx2n , t), F (Qx2n−1 , ST x2n−1 , t), F (P ABx2n , ST x2n−1 , t)}.
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Taking limit as n → ∞, F (ABz, z, qt) ≥ min{F (ABz, z, t), F (ABz, ABz, t), F (z, z, t), F (ABz, z, t)} ≥ F (ABz, z, t). From lemma 2.10, we get ABz = z . But ABz = P z . Therefore P z = z = ABz . Step (ii): By taking x = Bz and y = x2n+1 in (v), we have F (P Bz, Qx2n+1 , qt) ≥ min{F (ABBz, ST x2n+1 , t), F (P Bz, ABBz, t), F (Qx2n+1 , ST x2n+1 , t), F (P Bz, ST x2n+1, t)}. Since AB = BA and BP = P B , we have P (Bz) = B(P z) = Bz
and AB(Bz) = B(ABz) = Bz.
Letting n → ∞, we have F (Bz, z, qt) ≥ min{F (Bz, z, t), F (Bz, Bz, t), F (z, z, t), F (Bz, z, t)} F (Bz, z, qt) ≥ F (Bz, z, t). From Lemma 2.10, we get Bz = z. Since z = ABz , we have z = Az. Therefore, z = Az = Bz = P z . Step (iii): Since P (X) ⊂ ST (X), there exists v ∈ X such that ST v = P z = z. By taking x = x2n and y = v in (v), we have F (P x2n, Qv, qt) ≥ min{F (ABx2n , ST v, t), F (P x2n, ABx2n , t), F (Qv, ST v, t), F (P x2n, ST v, t)}. Letting n → ∞, we have F (z, Qv, qt) ≥ min{F (z, ST v, t), F (z, z, t), F (Qv, ST v, t), F (z, ST v, t)} = min{F (z, z, t), F (z, z, t), F (Qv, z, t), F (z, z, t)} ≥ F (Qv, z, t) and so Qv = z and hence ST v = Qv = z . Since (Q, ST ) is weakly compatible, ST Qv = QST v and hence ST z = Qz.
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Step (iv): By taking x = x2n and y = z in (v), we have F (P x2n, Qz, qt) ≥ min{F (ABx2n , ST z, t), F (P x2n , ABx2n , t), F (Qz, ST z, t), F (P x2n , ST z, t)} which implies that taking limit as n → ∞, F (z, Qz, qt) ≥ min{F (z, ST z, t), F (z, z, t), F (Qz, ST z, t), F (z, ST z, t)} = min{F (z, Qz, t), F (z, z, t), F (Qz, Qz, t), F (z, Qz, t)} ≥ F (z, Qz, t) From Lemma 2.10, we have Qz = z. Therefore, z = Az = Bz = P z = Qz = ST z . Step (v): By taking x = x2n and y = T z in (v), we have F (P x2n , QT z, qt) ≥ min{F (ABx2n , ST T z, t), F (P x2n, ABx2n , t), F (QT z, ST T z, t), F (P x2n , ST T z, t)}. Since QT = T Q and ST = T S , we have QT z = T Qz = T z
and ST (T z) = T (ST z) = T z.
Letting n → ∞, we have F (z, T z, qt) ≥ min{F (z, T z, t), F (z, z, t), F (T z, T z, t), F (z, T z, t)}. From Lemma 2.10, we have z = T z. Since T z = ST z , we also have z = Sz. Therefore, z = Az = Bz = P z = Qz = Sz = T z . That is, z is the common fixed point of six maps. For uniqueness, let w be another common fixed point of P, Q, A, B, S and T . Then, F (z, w, qt) = F (P z, Qw, qt) ≥ min{F (ABz, ST w, t), F (P z, ABz, t), F (Qw, ST w, t), F (P z, ST w, t)} ≥ F (z, w, t). From Lemma 2.10, z = w .
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Corollary 3.2. Let (X, F, Δ) be a complete ε-chainable probabilistic metric space and let P, Q, A, B, S and T be self mappings of X satisfying (i)-(iv) of theorem 3.1 and there exists q ∈ (0, 1) such that F (P x, Qy, qt) = min{F (ABx, ST y, t), F (P x, ABx, t), F (ABx, Qy, 2t), F (Qy, ST y, t), F (ST y, P x, t)}, for every x, y ∈ X and t > 0. Then P, Q, A, B, S and T have a unique common fixed point in X . Proof. From definition, we have that min{F (ABx, ST y, t), F (P x, ABx, t), F (Qy, ST y, t), F (Qy, ABx, 2t), F (P x, ST y, t)} ≥ min{F (ABx, ST y, t), F (P x, ABx, t), F (Qy, ST y, t), F (ABx, ST y, t), F (ST y, Qy, t), F (P x, ST y, t)} ≥ min{F (ABx, ST y, t), F (P x, ABx, t), F (Qy, ST y, t), F (P x, ST y, t)} and hence, from theorem 3.1, P, Q, A, B, S and T have a unique fixed point in X . Corollary 3.3. Let (X, F, Δ) be a complete ε-chainable probabilistic metric space and let P, Q, A, B, S and T be self mappings of X satisfying (i)-(iv) of theorem 3.1 and there exists q ∈ (0, 1) such that F (P x, Qy, qt) ≥ F (ABx, ST y, t),
for every x, y ∈ X and t > 0.
Then P, Q, A, B, S and T have a unique common fixed point in X . Proof. We have that F (ABx, ST y, t) = min{F (ABx, ST y, t), 1} = min{F (ABx, ST y, t), F (P x, P x, 5t)} ≥ min{F (ABx, ST y, t), F (P x, ABx, t), F (ABx, Qy, 2t), F (Qy, ST y, t), F (ST y, P x, t)} and hence, from Corollary 3.2, P, Q, A, B, S and T have a unique fixed point in X . Let AB and ST be the identity mapping on X in Corollary 3.3. Then we get the next result. Corollary 3.4. Let (X, F, Δ) be a complete ε-chainable probabilistic metric space and let P and Q be self mappings of X satisfying the following condition: there exists q ∈ (0, 1) such that F (P x, Qy, qt) ≥ F (x, y, t),
for every x, y ∈ X and t > 0.
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Then P and Q have a unique common fixed point in X . In Corollary 3.4, if we take P = Q, then this result becomes to Banach contraction theorem. Corollary 3.5. Let (X, F, Δ) be a complete ε-chainable probabilistic metric space and let P be self mapping of X satisfying the following condition: there exists q ∈ (0, 1) such that F (P x, P y, qt) ≥ F (x, y, t)
for every x, y ∈ X and t > 0.
Then P has a unique fixed point in X .
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