FIXED POINT THEOREMS IN NON-ARCHIMEDEAN

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Asian-European Journal of Mathematics Vol. 5, No. 4 (2012) 1250050 (13 pages) c World Scientific Publishing Company
DOI: 10.1142/S1793557112500507

FIXED POINT THEOREMS IN NON-ARCHIMEDEAN MENGER PM-SPACES USING COMMON PROPERTY (E.A)

Sunny Chauhan∗ R. H. Government Postgraduate College Kashipur-244713, (U. S. Nagar), Uttarakhand, India [email protected] Suneel Kumar Government Higher Secondary School Sanyasiowala, PO-Jaspur, 244712, Uttarakhand, India ksuneel math@rediffmail.com

Communicated by N.-C. Wong Received May 12, 2012 Revised September 19, 2012 Published February 14, 2013 In this paper, we prove common fixed point theorems for weakly compatible mappings in Non-Archimedean Menger PM-spaces employing common property (E.A). Some examples are derived which demonstrate the validity of our results. As an application to our main result, we present a fixed point theorem for four finite families of self mappings. Our results improve and extend several existing results in the literature. Keywords: T-norm; Non-Archimedean Menger PM-space; weakly compatible mappings; property (E.A); common property (E.A); fixed point theorem. AMS Subject Classification: 54H25, 47H10

1. Introduction Istr˘atescu and Criv˘ at [12] introduced the notion of Non-Archimedean probabilistic metric space (N.A. PM-space) and established its topological preliminaries (see, also [11]). Further, Istr˘ atescu [9, 10] presented some fixed point theorems on N.A. Menger PM-spaces which generalized the results of Sehgal and Bharucha-Reid [20] and Sherwood [22]. The theory of probabilistic metric spaces is of fundamental importance in probabilistic functional analysis due to its extensive applications in random differential as well as random integral equations (see [3, 7]). ∗ Current

address: Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor-246701, Uttar Pradesh, India. 1250050-1

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Many mathematician studied the weaker forms of commutativity in N.A. Menger PM-spaces, e.g. Singh and Pant [25] studied the notion of weakly commuting mappings (introduced by Sessa [21] in metric space), Cho et al. [4] studied the notion of compatible mappings (introduced by Jungck [13] in metric space) and Rao and Ramudu [18] studied the notion of weakly compatible mappings (introduced by Jungck and Rhoades [14] in metric space) and proved several fixed point results in this direction. It is worth to mention that concept of weakly compatible mappings is the most general among all the commutativity concepts as each pair of weakly commuting self mappings is compatible and each pair of compatible self mappings is weakly compatible but the converse is not true. The study of common fixed points of non-compatible (as well as nontrivial compatible) mappings is also equally interesting which was initiated by Pant [16] in metric spaces. In 2009, Fang and Gao [6] extended the notion of property (E.A) (introduced by Aamri and Moutawakil [1] in metric space) to probabilistic metric space and proved common fixed point theorems under strict contractive conditions for a pair of weakly compatible mappings. Recently, Ali et al. [2] defined the notion of common (E.A) property (introduced by Liu et al. [15] in metric space) in framework of probabilistic settings which generalized several known results in Menger as well as metric spaces. Many authors proved extensively various results in N.A. Menger PM-spaces satisfying different contractive conditions (see [4, 5, 18, 23, 24]). In this paper, we prove common fixed point theorems for weakly compatible mappings in N.A. Menger PM-spaces satisfying common property (E.A) and give some examples to illustrate our results. We extend our main result to four finite families of self mappings. 2. Preliminaries Definition 2.1 ([19]). A mapping T : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (t-norm) if it satisfies the following conditions: for all a, b, c, d ∈ [0, 1] (1) (2) (3) (4)

T (a, 1) = a, T (0, 0) = 0; T (a, b) = T (b, a); T (a, b) ≤ T (c, d), whenever a ≤ c and b ≤ d; T (a, T (b, c)) = T (T (a, b), c).

Definition 2.2 ([19]). A mapping F : R → R+ is said to be a distribution function if it is non-decreasing and left continuous with inf{F (t) : t ∈ R} = 0 and sup{F (t) : t ∈ R} = 1. We shall denote  by the set of all distribution functions. If X is a non-empty set, F : X × X →  is called a probabilistic distance on X and F (x, y) is usually denoted by Fx,y . Definition 2.3 ([19]). The ordered pair (X, F ) is said to be N.A. PM-space if X is a non-empty set and F is a probabilistic distance satisfying the following conditions: for all x, y, z ∈ X and t, t1 , t2 > 0, 1250050-2

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(1) Fx,y (t) = 1 ⇔ x = y; (2) Fx,y (t) = Fy,x (t); (3) if Fx,y (t1 ) = 1 and Fy,z (t2 ) = 1 then Fx,z (max{t1 , t2 }) = 1. The ordered triplet (X, F , T ) is called a N.A. Menger PM-space if (X, F ) is a N.A. PM-space, T is a t-norm and the following inequality holds: Fx,z (max{t1 , t2 }) ≥ T (Fx,y (t1 ), Fy,z (t2 )), for all x, y, z ∈ X and t1 , t2 > 0. Example 2.1. Let X be any non-empty set with at least two elements. If we define Fx,x (t) = 1 for all x ∈ X, t > 0 and
0 if t ≤ 1; Fx,y (t) = 1 if t > 1, where x, y ∈ X, x = y, then (X, F , T ) is a N.A. Menger PM-space with T (a, b) = min{a, b} or (ab). Example 2.2. Let X = R be the set of real numbers equipped with metric defined by d(x, y) = |x − y| and  t  if t > 0; Fx,y (t) = t + |x − y| 0 if t = 0. Then (X, F , T ) is a N.A. Menger PM-space with T as continuous t-norm satisfying T (r, s) = min{r, s} or (rs). Definition 2.4 ([5]). A N.A. Menger PM-space (X, F , T ) is said to be of type (C)g if there exists a g ∈ Ω such that g(Fx,z (t)) ≤ g(Fx,y (t)) + g(Fy,z (t)), for all x, y, z ∈ X, t ≥ 0, where Ω = {g | g : [0, 1] → [0, ∞) is continuous, strictly decreasing with g(1) = 0 and g(0) < ∞}. Definition 2.5 ([5]). A N.A. Menger PM-space (X, F , T ) is said to be of type (D)g if there exists a g ∈ Ω such that g(T (t1 , t2 )) ≤ g(t1 ) + g(t2 ), for all t1 , t2 ∈ [0, 1]. Remark 2.1 ([5]). (1) If a N.A. Menger PM-space (X, F , T ) is of type (D)g then it is of type (C)g . 1250050-3

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(2) If a N.A. Menger PM-space (X, F , T ) is of type (D)g , then it is metrizable, where the metric d on X is defined by  1 g(Fx,y (t))dt, d(x, y) = 0

for all x, y ∈ X. Throughout this paper, (X, F , T ) is a N.A. Menger PM-space with a continuous strictly increasing t-norm T . Let φ : [0, ∞) → [0, ∞) be a function satisfying the condition (Φ): φ is upper semi-continuous from the right and φ(t) < t for t > 0. Lemma 2.1 ([4]). If a function φ : [0, ∞) → [0, ∞) satisfies the condition (Φ). Then (1) for all t ≥ 0, limn→∞ φn (t) = 0, where φn (t) is the nth iteration of φ(t). (2) {tn } is a non-decreasing sequence of real numbers and tn+1 ≤ φ(tn ) where n = 1, 2, . . . then limn→∞ tn = 0. In particular, if t ≤ φ(t), for each t ≥ 0 then t = 0. Definition 2.6 ([5]). A pair (A, S) of self mappings of a N.A. Menger PM-space (X, F , T ) is said to be compatible if limn→∞ g(FASxn ,SAxn (t)) = 0, for all t > 0, whenever {xn } is a sequence in X such that limn→∞ Axn = limn→∞ Sxn = z, for some z ∈ X. Definition 2.7. A pair (A, S) of self mappings of a N.A. Menger PM-space (X, F , T ) is said to be non-compatible if there exists at least one sequence {xn } in X such that limn→∞ Axn = limn→∞ Sxn = z, for some z ∈ X, but for some t > 0, either limn→∞ g(FASxn ,SAxn (t)) = 0 or the limit does not exist. Definition 2.8. A pair (A, S) of self mappings of a N.A. Menger PM-space (X, F , T ) is said to satisfy the property (E.A) if there exists a sequence {xn } in X such that lim Axn = lim Sxn = z,

n→∞

n→∞

for some z ∈ X. Remark 2.2. From Definition 2.8, it is easy to see that any two non-compatible self mappings of a N.A. Menger PM-space (X, F , T ) satisfy the property (E.A) but the converse need not be true (see [6, Example 1]). Definition 2.9. Two pairs (A, S) and (B, T ) of self mappings of a N.A. Menger PM-space (X, F , T ) are said to satisfy the common property (E.A) if there exist two sequences {xn }, {yn } in X such that lim Axn = lim Sxn = lim Byn = lim T yn = z,

n→∞

n→∞

n→∞

for some z ∈ X. 1250050-4

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Definition 2.10. A pair (A, S) of self mappings of a non-empty set X is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, that is, if Az = Sz for some z ∈ X, then ASz = SAz. Remark 2.3. If self mappings A and S of a N.A. Menger PM-space (X, F , T ) are compatible then they are weakly compatible but the reverse need not be true (see [18, Example 12]). It is also noticed that the notions of weak compatibility and property (E.A) are independent to each other (see [17, Example 2.2]). Definition 2.11 ([8]). Two families of self mappings {Ai } and {Sj } are said to be pairwise commuting if: (1) Ai Aj = Aj Ai , i, j ∈ {1, 2, . . . , m}, (2) Sk Sl = Sl Sk , k, l ∈ {1, 2, . . . , n}, (3) Ai Sk = Sk Ai , i ∈ {1, 2, . . . , m}, k ∈ {1, 2, . . . , n}. 3. Main Results We begin with the following observation. Lemma 3.1. Let A, B, S and T be self mappings of a N.A. Menger PM-space (X, F , T ) satisfying the following conditions: (1) the pair (A, S) (or (B, T )) satisfies property (E.A), (2) A(X) ⊂ T (X) (or B(X) ⊂ T (X)), (3) B(yn ) converges for every sequence {yn } in X whenever T (yn ) converges (or A(xn ) converges for every sequence {xn } in X whenever S(xn ) converges), (4)    g(FSx,T y (t)), g(FAx,Sx (t)), g(FBy,T y (t)),  (3.1) g(FAx,By (t)) ≤ φ max 1   (g(FSx,By (t)) + g(FAx,T y (t))) 2 for all x, y ∈ X, t > 0, where g ∈ Ω and φ satisfies the condition (Φ). Then (A, S) and (B, T ) satisfy the common property (E.A). Proof. Since (A, S) satisfies property (E.A), therefore there exists a sequence {xn } in X such that lim Axn = lim Sxn = z,

n→∞

n→∞

for some z ∈ X. Since A(X) ⊂ T (X), for each sequence {xn } ⊂ X there corresponds a sequence {yn } ⊂ X such that Axn = T yn . Hence, lim T yn = lim Axn = z,

n→∞

n→∞

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for some z ∈ X. Thus in all, we have Axn → z, Sxn → z and T yn → z. Now we assert that Byn → z. Putting x = xn and y = yn in inequality (3.1), we have   g(FSxn ,T yn (t)), g(FAxn ,Sxn (t)), g(FByn ,T yn (t)), . g(FAxn ,Byn (t)) ≤ φ max 1   (g(FSxn ,Byn (t)) + g(FAxn ,T yn (t))) 2 

Let Byn → l(= z) ∈ X for t > 0 as n → ∞. Then, passing to limit as n → ∞, we get   g(Fz,z (t)), g(Fz,z (t)), g(Fl,z (t)),  g(Fz,l (t)) ≤ φ max  1 (g(Fz,l (t)) + g(Fz,z (t)))  2 

 1 = φ max g(1), g(1), g(Fl,z (t)), (g(Fz,l (t)) + g(1)) 2

  1 = φ max 0, 0, g(Fl,z (t)), (g(Fz,l (t))) 2 

≤ φ(g(Fz,l (t))), for all t > 0, which implies that g(Fz,l (t)) = 0. By Lemma 2.1, we get z = l. Thus, we conclude that the pairs (A, S) and (B, T ) share the common property (E.A).

Theorem 3.1. Let A, B, S and T be self mappings of a N.A. Menger PM-space (X, F , T ) satisfying inequality (3.1) of Lemma 3.1. Suppose that (1) the pairs (A, S) and (B, T ) satisfy the common property (E.A), (2) S(X) and T (X) are closed subsets of X. Then the pairs (A, S) and (B, T ) have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point provided both the pairs (A, S) and (B, T ) are weakly compatible. Proof. Since the pairs (A, S) and (B, T ) share the common property (E.A), there exist two sequences {xn } and {yn } in X for some z ∈ X such that lim Axn = lim Sxn = lim Byn = lim T yn = z.

n→∞

n→∞

n→∞

n→∞

Since S(X) is a closed subset of X, hence limn→∞ Sxn = z ∈ S(X). Therefore there exists a point u ∈ X such that Su = z. Now we show that Au = Su. Putting 1250050-6

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x = u and y = yn in inequality (3.1), we get   g(FSu,T yn (t)), g(FAu,Su (t)), g(FByn ,T yn (t)),  , g(FAu,Byn (t)) ≤ φ max 1   (g(FSu,Byn (t)) + g(FAu,T yn (t))) 2 

passing to limit as n → ∞, we have   g(Fz,z (t)), g(FAu,z (t)), g(Fz,z (t)),   g(FAu,z (t)) ≤ φ max  1 (g(Fz,z (t)) + g(FAu,z (t)))  2 

 1 = φ max g(1), g(FAu,z (t)), g(1), (g(1) + g(FAu,z (t))) 2

  1 = φ max 0, g(FAu,z (t)), 0, (g(FAu,z (t))) 2 

≤ φ(g(FAu,z (t))), for all t > 0, which implies that g(FAu,z (t)) = 0. By Lemma 2.1, we get Au = z. Therefore Au = Su = z which shows that u is a coincidence point of the pair (A, S). Since T (X) is also a closed subset of X, therefore limn→∞ T yn = z ∈ T (X). Therefore, there exists a point v ∈ X such that T v = z. Now we assert that Bv = T v, by putting x = u and y = v in inequality (3.1), we have   g(FSu,T v (t)), g(FAu,Su (t)), g(FBv,T v (t)),  , g(FAu,Bv (t)) ≤ φ max 1   (g(FSu,Bv (t)) + g(FAu,T v (t))) 2 

and so   g(Fz,z (t)), g(Fz,z (t)), g(FBv,z (t)),   g(Fz,Bv (t)) ≤ φ max  1 (g(Fz,Bv (t)) + g(Fz,z (t)))  2 

 1 = φ max g(1), g(1), g(FBv,z (t)), (g(Fz,Bv (t)) + g(1)) 2

  1 = φ max 0, 0, g(FBv,z (t)), (g(Fz,Bv (t))) 2 

≤ φ(g(Fz,Bv (t))), for all t > 0, which implies that g(Fz,Bv (t)) = 0. By Lemma 2.1, we get Bv = z. Therefore Bv = T v = z which shows that v is a coincidence point of the pair (B, T ). 1250050-7

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Since the pair (A, S) is weakly compatible, therefore Az = ASu = SAu = Sz. Putting x = z and y = v in inequality (3.1), we have    g(FSz,T v (t)), g(FAz,Sz (t)), g(FBv,T v (t)),  , g(FAz,Bv (t)) ≤ φ max 1   (g(FSz,Bv (t)) + g(FAz,T v (t))) 2 and so

  g(FAz,z (t)), g(FAz,Az (t)), g(Fz,z (t)),   g(FAz,z (t)) ≤ φ max 1   (g(FAz,z (t)) + g(FAz,z (t))) 2 

= φ(max{g(FAz,z (t)), g(1), g(1), g(FAz,z (t))}) = φ(max{g(FAz,z (t)), 0, 0, g(FAz,z (t))}) ≤ φ(g(FAz,z (t))), for all t > 0, which implies that g(FAz,z (t)) = 0. By Lemma 2.1, we get Az = z. Therefore Az = Sz = z which shows that z is a common fixed point of the pair (A, S). Also the pair (B, T ) is weakly compatible, therefore Bz = BT v = T Bv = T z. Putting x = u and y = z in inequality (3.1), we have    g(FSu,T z (t)), g(FAu,Su (t)), g(FBz,T z (t)),  , g(FAu,Bz (t)) ≤ φ max 1   (g(FSu,Bz (t)) + g(FAu,T z (t))) 2 and so

  g(Fz,Bz (t)), g(Fz,z (t)), g(FBz,Bz (t)),   g(Fz,Bz (t)) ≤ φ max 1   (g(Fz,Bz (t)) + g(Fz,Bz (t))) 2 

= φ(max{g(Fz,Bz (t)), g(1), g(1), g(Fz,Bz (t))}) = φ(max{g(Fz,Bz (t)), 0, 0, g(Fz,Bz (t))}) ≤ φ(g(Fz,Bz (t))), for all t > 0, which implies that g(Fz,Bz (t)) = 0. By Lemma 2.1, we get Bz = z. Therefore Bz = T z = z which shows that z is a common fixed point of the pair (B, T ). Therefore z is a common fixed point of both the pairs (A, S) and (B, T ). Uniqueness of common fixed point is an easy consequence of inequality (3.1). Remark 3.1. Theorem 3.1 improves the results of Cho et al. [4], Singh et al. [23, Theorem 3.1, Theorem 3.2] and Singh et al. [24, Theorem 3.1, Corollary 3.3] as one never requires any conditions on containment of ranges amongst the involved 1250050-8

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mappings and continuity of one or more mappings while completeness of the underlying space is replaced by closedness of subspaces. Theorem 3.1 also generalizes the results of Rao and Ramudu [18, Theorem 14]. The following example illustrates Theorem 3.1. Example 3.1. Let (X, F , T ) be a N.A. Menger PM-space, where X = [1, 15) and metric d is defined as condition (2) of Remark 2.1. Define the self mappings A, B, S and T by

1 if x ∈ {1} ∪ (3, 15); 1 if x ∈ {1} ∪ (3, 15); A(x) = B(x) = 14 if x ∈ (1, 3]; 5 if x ∈ (1, 3];   1 if x = 1;  if x = 1;   1    4 if x ∈ (1, 3]; T (x) = 10 + x if x ∈ (1, 3]; S(x) =     x + 1    x − 2 if x ∈ (3, 15). if x ∈ (3, 15); 4 Taking {xn } = {3 + n1 }, {yn } = {1} or {xn } = {1}, {yn } = {3 + n1 }, it is clear that both the pairs (A, S) and (B, T ) satisfy the common property (E.A). lim Axn = lim Sxn = lim Byn = lim T yn = 1 ∈ X.

n→∞

n→∞

n→∞

n→∞

It is noted that A(X) = {1, 14}
[1, 13] = T (X) and B(X) = {1, 5}
[1, 4] = S(X). On the other hand, S(X) and T (X) are closed subsets of X. Thus, all the conditions of Theorem 3.1 are satisfied and 1 is a unique common fixed point of the pairs (A, S) and (B, T ) which also remains a point of coincidence as well. Also, all the involved mappings are even discontinuous at their unique common fixed point 1. Theorem 3.2. The conclusion of Theorem 3.1 remains true if condition (2) of Theorem 3.1 is replaced by the following: (2) A(X) ⊂ T (X) and B(X) ⊂ S(X), where A(X) is the closure range of A and B(X) is the closure range of B. Proof. Since the pairs (A, S) and (B, T ) satisfy the common property (E.A), there exist two sequences {xn }, {yn } in X such that lim Axn = lim Sxn = lim Byn = lim T yn = z,

n→∞

n→∞

n→∞

n→∞

for some z ∈ X. Since z ∈ A(X) and A(X) ⊂ T (X), there exists a point v ∈ X such that z = T v. By the proof of Theorem 3.1, we can show that the pair (B, T ) has a coincidence point, call it v, that is, Bv = T v. Since z ∈ B(X) and B(X) ⊂ S(X), there exists a point u ∈ X such that z = Su. Similarly we can also prove that the pair (A, S) has a coincidence point, call it u, that is, Au = Su. The rest of the proof is on the lines of the proof of Theorem 3.1, hence it is omitted. 1250050-9

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Corollary 3.1. The conclusions of above proved results remain true if condition (2) of Theorem 3.1 and condition (2) of Theorem 3.2 are replaced by the following: (2) A(X) and B(X) are closed subsets of X provided A(X) ⊂ T (X) and B(X) ⊂ S(X). Theorem 3.3. Let A, B, S and T be self mappings of a N.A. Menger PM-space (X, F , T ) satisfying the conditions (1) − (4) of Lemma 3.1. Suppose that (1) S(X) (or T (X)) is a closed subset of X. Then the pairs (A, S) and (B, T ) have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point provided both the pairs (A, S) and (B, T ) are weakly compatible. Proof. By Lemma 3.1, both the pairs (A, S) and (B, T ) satisfy the common property (E.A), therefore there exist two sequences {xn }, {yn } in X such that lim Axn = lim Sxn = lim Byn = lim T yn = z,

n→∞

n→∞

n→∞

n→∞

for some z ∈ X. If S(X) is a closed subset of X then from the proof of Theorem 3.1, we can easily show that the pair (A, S) has a point of coincidence u ∈ X, that is, Au = Su = z. Since A(X) ⊂ T (X) and Au = z, there exists a point v ∈ X such that Au = T v. The rest of the proof is on the lines of the proof of Theorem 3.1, hence the details are avoided. Example 3.2. In the setting of Example 3.1, replace the self mappings A, B, S and T by the following, besides retaining the rest:

1 if x ∈ {1} ∪ (3, 15); 1 if x ∈ {1} ∪ (3, 15); A(x) = B(x) = 4 if x ∈ (1, 3]; 5 if x ∈ (1, 3];   1 if x = 1;  1 if x = 1;       if x ∈ (1, 3]; T (x) = 11 + x if x ∈ (1, 3]; S(x) = 5     x + 1    x − 2 if x ∈ (3, 15). if x ∈ (3, 15); 4 Here, it is noted that A(X) = {1, 4} ⊂ [1, 14] = T (X) and B(X) = {1, 5} ⊂ [1, 4) ∪ {5} = S(X), that is, T (X) is a closed subset of X. Thus, all the conditions of Theorems 3.2 and 3.3 and Corollary 3.1 are satisfied and 1 is a unique common fixed point of the pairs (A, S) and (B, T ) which also remains a point of coincidence as well. It may be pointed out that Theorem 3.1 is not applicable to this example as S(X) is not a closed subset of X. Also, notice that all the mappings in this example are even discontinuous at their unique common fixed point 1. On taking A = B and S = T in Theorem 3.1, we get the following result. Corollary 3.2. Let A and S be self mappings of a N.A. Menger PM-space (X, F , T ). Suppose that 1250050-10

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(1) the pair (A, S) satisfies the property (E.A), (2) S(X) is a closed subset of X, (3)    g(FSx,Sy (t)), g(FAx,Sx (t)), g(FAy,Sy (t)),  g(FAx,Ay (t)) ≤ φ max 1   (g(FSx,Ay (t)) + g(FAx,Sy (t))) 2

(3.2)

for all x, y ∈ X, t > 0, where g ∈ Ω and φ satisfies the condition (Φ). Then the pair (A, S) has a coincidence point. Moreover, A and S have a unique common fixed point provided the pair (A, S) is weakly compatible. Now we utilize the notion of commuting pairwise due to Imdad et al. [8] and extend Theorem 3.1 to four finite families of self mappings in N.A. Menger PM-spaces. Corollary 3.3. Let {A1 , A2 , . . . , Am }, {B1 , B2 , . . . , Be }, {S1 , S2 , . . . , Sn } and {T1 , T2 , . . . , Tf } be four finite families of self mappings of a N.A. Menger PMspace (X, F , T ) such that A = A1 A2 . . . Am , B = B1 B2 . . . Be , S = S1 S2 . . . Sn and T = T1 T2 . . . Tf which also satisfy inequality (3.1) of Lemma 3.1 and conditions (1) and (2) of Theorem 3.1. Then the pairs (A, S) and (B, T ) have a point of coincidence each. n Moreover, if the family {Ai }m i=1 commutes pairwise with the family {Si }j=1 f e whereas the family {Br }r=1 commutes pairwise with the family {Tw }w=1 , then (for all i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , n}, r ∈ {1, 2, . . . , e} and w ∈ {1, 2, . . . , f }) Ai , Bj , Sr and Tw have a common fixed point. Remark 3.2. Corollary 3.3 improves and extends the result of Singh et al. [24]. By setting A1 = A2 = · · · = Am = A, B1 = B2 = · · · = Be = B, S1 = S2 = · · · = Sn = S and T1 = T2 = · · · = Tf = T in Corollary 3.3, we deduce the following corollary. Corollary 3.4. Let A, B, S and T be self mappings of a N.A. Menger PM-space (X, F , T ). Suppose that the pairs (Am , S n ) and (B e , T f ) share the common property (E.A) along with closedness of S n (X) and T f (X) such that    g(FS n x,T f y (t)), g(FAm x,S n x (t)),            g(F e y,T f y (t)), B m e g(FA x,B y (t)) ≤ φ max  (3.3)       1    (g(FS n x,B e y (t)) + g(FAm x,T f y (t))) 2 for all x, y ∈ X, t > 0, g ∈ Ω where φ satisfies the condition (Φ) and m, n, e, f are fixed positive integers. Then A, B, S and T have a unique common fixed point provided both the pairs (Am , S n ) and (B e , T f ) commute pairwise. 1250050-11

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Remark 3.3. The conclusions of Lemma 3.1 and Theorems 3.1 and 3.3 remain true if we replace the inequality (3.1) by one of the following inequalities, for all x, y ∈ X, t > 0, g ∈ Ω and φ satisfies the condition (Φ): g(FAx,By (t)) ≤ φ(max{g(FSx,T y (t)), g(FAx,Sx (t)), g(FBy,T y (t)), g(FSx,By (t))}), g(FAx,By (t)) ≤ φ(max{g(FSx,T y (t)), g(FAx,Sx (t)), g(FBy,T y (t))}).

(3.4) (3.5)

Remark 3.4. Results similar to Corollary 3.3 can also be outlined in respect of Remark 3.3. The listing of the possible corollaries are not included. Remark 3.5. In view of Remark 3.3, the results improve the results of Singh et al. [23, Corollary 3.3, Corollary 3.4]. Acknowledgments The authors are thankful to Professor J¨ org Koppitz and the anonymous referee for their appreciation, valuable comments and suggestions. References 1. M. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270(1) (2002) 181–188. 2. J. Ali, M. Imdad and D. Bahuguna, Common fixed point theorems in Menger spaces with common property (E.A), Comput. Math. Appl. 60(12) (2010) 3152–3159. 3. S. S. Chang, Y. J. Cho and S. M. Kang, Nonlinear Operator Theory in Probabilistic Metric Space (Nova Science Publishers, New York, 2001). 4. Y. J. Cho, K. S. Ha and S. S. Chang, Common fixed point theorems for compatible mappings of type (A) in Non-Archimedean Menger PM-spaces, Math. Jap. 46(1) (1997) 169–179. 5. Y. J. Cho, S. M. Kang and S. S. Chang, Coincidence point theorems for nonlinear hybrid contractions in Non-Archimedean Menger probabilistic metric spaces, Demonstratio Math. 28(1) (1995) 19–32. 6. J. X. Fang and Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Anal. 70(1) (2009) 184–193. 7. M. Grabiec, Y. J. Cho and V. Radu, On Nonsymmetric Topological and Probabilistic Structures (Nova Science Publishers, New York, 2006). 8. M. Imdad, J. Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces, Chaos Solitons Fractals 42(5) (2009) 3121–3129. 9. I. Istr˘ at.escu, On some fixed point theorems with applications to the non-Archimedean Menger spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(3) (1975) 374–379. 10. I. Istr˘ at.escu, Fixed point theorems for some classes of contraction mappings on nonArchimedean probablistic metric space, Publ. Math. Debrecen 25(1–2) (1978) 29–34. 11. I. Istr˘ at.escu and G. Babescu, On the Completion on Non-Archimedean Probabilistic Metric Spaces, Seminar de spatii metrice probabiliste, Vol. 17 (Universitatea Timisoara, 1979). 1250050-12

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