CUBO A Mathemati al Journal Vol.15, N o 03, (31�44). O tober 2013 Coin iden e and ommon �xed point theorems in Non-Ar himedean Menger PM-spa es Sunny Chauhan R.H. Government Postgraduate College, Kashipur-244713, (U.S. Nagar), Uttarakhand, India. sun.gkv�gmail. om B. D. Pant Government Degree College, Champawat-262523, Uttarakhand, India. badridatt.pant�gmail. om Mohammad Imdad Department of Mathemati s, Aligarh Muslim University, Aligarh 202 002, India. mhimdad�yahoo. o.in ABSTRACT The obje t of this work is to point out a falla y in the proof of Theorem 1 ontained in the re ent paper of Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137�150℄ proved in Non-Ar himedean Menger PM-spa e by using the notions of sub- ompatibility and sub-sequential ontinuity. We show that the results of Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137�150℄ an be re overed in two ways. Further, we establish some illustrative examples to show the validity of the main results. Our results improve a multitude of relevant �xed point theorems of the existing literature. RESUMEN El objetivo de este trabajo es señalar una fala ia en la demostra ión del Teorema 1 ontenido en un artí ulo re iente de Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137�150℄ probado en un espa io-PM No-Arquimedeano Menger usando no iones de ontinuidad sub ompatible y sub se uen ial. Mostramos que el resultado de Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137�150℄ puede re uperarse de dos maneras. Además, estable emos algunos ejemplos ilustrativos que muestran la validez de los resultados prin ipales. Nuestro resultado mejora una gran antidad de teoremas de punto �jo importantes existentes en la literatura. Keywords and Phrases: t-norm, ompatible mappings, re ipro al ontinuity, sub ompatible mappings, subsequential ontinuity. 2010 AMS Mathemati s Subje t Classi� ation: 47H10, 54H25. 32 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) 1 Introdu tion Istr tes u and Criv t [19℄ introdu ed the on ept of Non-Ar himedean probablisti metri spa es (brie�y, N.A. PM-spa es) in 1974. In this sequen e, Istr tes u [16,17℄ obtained some �xed point theorems on N.A. Menger PM-spa es and generalized the results of Sehgal and Bharu ha-Reid [32℄ (see [18,20℄). Further, Had�zi¢ [13℄ improved the results of Istr tes u [16,17℄. In 1987, Singh and Pant [33℄ introdu ed the notion of weakly ommuting mappings on N.A. Menger PM-spa es and proved some ommon �xed point theorems. Dimri and Pant [10℄ studied the appli ation of N.A. Menger PM-spa es to produ t spa es. In 1997, Cho et al. [8℄ introdu ed the on epts of ompatible mappings and ompatible mappings of type (A) in N.A. Menger PM- spa es and obtained some �xed point theorems for these mappings. Most of the ommon �xed point theorems for ontra tion mappings invariably require a ompatibility ondition besides assuming ontinuity of at least one of the mappings. Sin e then, Pant [27℄ noti ed these riteria for �xed points of ontra tion mappings and introdu ed a new ontinuity ondition, known as re ipro al ontinuity and obtained a ommon �xed point theorem by using the ompatibility in metri spa es. He also showed that in the setting of ommon �xed point theorems for ompatible mappings sat- isfying ontra tion onditions, the notion of re ipro al ontinuity is weaker than the ontinuity of one of the mappings. Jung k and Rhoades [21℄ weakened the notion of ompatible mappings by introdu ing weakly ompatible mappings and proved ommon �xed point theorems without any requirement of ontinuity of the involved mappings. In 2009, Kutuk u and Sharma [26℄ introdu ed the on ept of ompatible mappings of type (A-1) and type (A-2) in N.A. Menger PM-spa es and showed that they are equivalent to ompatible mappings under ertain onditions. Many math- emati ians proved several ommon �xed point theorems in Non-Ar himedean Menger PM-spa es using di�erent ontra tive onditions (see [4,6,9,22�24,34℄). In 2008, Al-Thaga� and Shahzad [1℄ introdu ed the on ept of o asionally weakly ompatible (shortly, ow ) mappings in metri spa es. Bouhadjera and Godet-Thobie [2℄ weakened the on ept of o asionally weak ompatibility and re- ipro al ontinuity in the form of sub- ompatibility and sub-sequential ontinuity respe tively and proved some interesting results with these on epts in metri spa es. Re ently, Imdad et al. [14℄ showed that the results ontained in [2℄ an easily re overed by repla ing sub- ompatibility with ompatibility or sub-sequential ontinuity with re ipro ally ontinuity (also see [3,5,12℄). In this paper, we prove ommon �xed point theorems for two pairs of self mappings by using the notions of ompatibility and sub-sequentially ontinuity (alternately sub- ompatibility and re ipro ally ontinuity) in N.A. Menger PM-spa es. Some examples are also derived to support our results. 2 Preliminaries De�nition 2.1. [31℄ A triangular norm T (brie�y, t-norm) is a binary operation on the unit interval [0,1℄ su h that for all a, b, c, d ∈ [0, 1] and the following onditions are satis�ed: CUBO 15, 3 (2013) Coin iden e and ommon �xed point theorems in N.A. Menger . . . 33 (1) T (a, 1) = a; (2) T (a, b) = T (b, a); (3) T (a, b) ≤ T (c, d), whenever a ≤ c and b ≤ d; (4) T (a, T (b, c)) = T (T (a, b), c). De�nition 2.2. [31℄ A mapping F : R → R+ is said to be a distribution fun tion if it is non- de reasing and left ontinuous with inf{F(t) : t ∈ R} = 0 and sup{F(t) : t ∈ R} = 1. We shall denote by Im, the set of all distribution fun tions whereas H stands for spe i� distribution fun tion (also known as Heaviside fun tion) de�ned as H(t) = { 0, if t ≤ 0; 1, if t > 0. If X is a non-empty set, F : X × X → Im is alled a probabilisti distan e on X and F(x, y) is usually denoted by Fx,y. De�nition 2.3. [17,19℄ The ordered pair (X, F) is said to be an N.A. PM-spa e if X is a non- empty set and F is a probabilisti distan e satisfying the following onditions: for all x, y, z ∈ X and t, t1, t2 > 0, (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 alled an N.A. Menger PM-spa e if (X, F) is an N.A. PM- spa e, 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 2.4. Let X be any set with at least two elements. If we de�ne Fx,x(t) = 1 for all x ∈ X, t > 0 and Fx,y(t) = { 0, if t ≤ 1; 1, if t > 1, where x, y ∈ X, x 6= y, then (X, F, T ) is an N.A. Menger PM-spa e with T (a, b) = min{a, b} or (ab) for all a, b ∈ [0, 1]. 34 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) Example 2.5. Let X = R be the set of real numbers equipped with metri de�ned by d(x, y) =| x−y | and Fx,y(t) = { t t+|x−y| , if t > 0; 0, if t = 0. Then (X, F, T ) is an N.A. Menger PM-spa e with T as ontinuous t-norm satisfying T (a, b) = min{a, b} or ab for all a, b ∈ [0, 1]. De�nition 2.6. [8℄ An N.A. Menger PM-spa e (X, F, T ) is said to be of type (C)g if there exists a g ∈ Ω su h 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 ontinuous with g(1) = 0 i� t = 1}. De�nition 2.7. [8℄ An N.A. Menger PM-spa e (X, F, T ) is said to be of type (D)g if there exists a g ∈ Ω su h that g(T (t1, t2)) ≤ g(t1) + g(t2), for all t1, t2 ∈ [0, 1]. Remark 2.8. [8℄ If an N.A. Menger PM-spa e (X, F, T ) is of type (D)g, then (X, F, T ) is of type (C)g. On the other hand, (X, F, T ) is an N.A. Menger PM-spa e su h that T (a, b) ≥ max{a+b−1, 0} for all a, b ∈ [0, 1], then (X, F, T ) is of type (D)g for g ∈ Ω de�ned by g(t) = 1−t, t ∈ [0, 1]. Throughout this paper (X, F, T ) is an N.A. Menger PM-spa e with a ontinuous stri tly in reasing t-norm T . Let φ : [0, ∞) → [0, ∞) be a fun tion satisfying the ondition (Φ): φ is upper semi- ontinuous from the right and φ(t) < t for t > 0. Lemma 2.9. [8℄ If a fun tion φ : [0, ∞) → [0, ∞) satis�es the ondition (Φ) then we have: (1) for all t ≥ 0, limn→∞ φ n(t) = 0, where φn(t) is the nth iteration of φ(t). (2) If {tn} is a non-de reasing sequen e of real numbers and tn+1 ≤ φ(tn) where n = 1, 2, . . . then limn→∞ tn = 0. In parti ular, if t ≤ φ(t), for ea h t ≥ 0 then t = 0. De�nition 2.10. [8℄ A pair (A, S) of self mappings de�ned on an N.A. Menger PM-spa e (X, F, T ) is said to be ompatible if and only if FASxn,SAxn(t) → 1 for all t > 0, whenever {xn} is a sequen e in X su h that Axn, Sxn → z for some z ∈ X as n → ∞. CUBO 15, 3 (2013) Coin iden e and ommon �xed point theorems in N.A. Menger . . . 35 De�nition 2.11. A pair (A, S) of self mappings de�ned on an N.A. Menger PM-spa e (X, F, T ) satis�es the (E.A) property, if there exists a sequen e {xn} su h that lim n→∞ Axn = lim n→∞ Sxn = z, for some z ∈ X. De�nition 2.12. [29℄ A pair (A, S) of self mappings of a non-empty set X is said to be weakly ompatible (or oin identally ommuting) if they ommute at their oin iden e points, i.e. if Az = Sz for some z ∈ X, then ASz = SAz. It is easy to see that two ompatible mappings are weakly ompatible but onverse is not true. De�nition 2.13. [21℄ A pair (A, S) of self mappings of a non-empty set X is ow i� there is a point x ∈ X whi h is a oin iden e point of A and S at whi h A and S ommute. In an interesting note, �ori et al. [11℄ showed that the notion of ow redu es to weak om- patibility in the presen e of a unique point of oin iden e (or a unique ommon �xed point) of the given pair of single valued mappings. Thus, no generalization an be obtained by repla ing weak ompatibility with ow . Inspired by Bouhadjera and Godet-Thobie [2℄, we de�ne the notion of sub- ompatible map- pings in N.A. Menger PM-spa e as follows: De�nition 2.14. A pair (A, S) of self mappings de�ned on an N.A. Menger PM-spa e (X, F, T ) is said to be sub ompatible i� there exists a sequen e {xn} su h that lim n→∞ Axn = lim n→∞ Sxn = z, for some z ∈ X and limn→∞ FASxn,SAxn(t) = 1, for all t > 0. Remark 2.15. Two ow mappings are sub- ompatible, however the onverse is not true in general (see [3, Example 1.2℄). Remark 2.16. A pair of sub- ompatible mapping satis�es the (E.A) property. Obviously, ompat- ible mappings whi h satisfy the (E.A) property are sub- ompatible but the onverse statement does not hold in general (see [30, Example 2.3℄). De�nition 2.17. A pair (A, S) of self mappings de�ned on an N.A. Menger PM-spa e (X, F, T ) is alled re ipro ally ontinuous if for a sequen e {xn} in X, limn→∞ ASxn = Az and limn→∞ SAxn = Sz, whenever lim n→∞ Axn = lim n→∞ Sxn = z, for some z ∈ X. 36 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) Remark 2.18. If two self mappings A and B are ontinuous, then they are obviously re ipro ally ontinuous but onverse is not true. Moreover, in the setting of ommon �xed point theorems for ompatible pair of self mappings satisfying ontra tive onditions, ontinuity of one of the mappings implies their re ipro al ontinuity but not onversely (see [27℄). Now we de�ne the notion of sub-sequentially ontinuous mappings in N.A. Menger PM-spa e due to Bouhadjera and Godet-Thobie [2℄: De�nition 2.19. A pair of self mappings (A, S) de�ned on an N.A. Menger PM-spa e (X, F, T ) is alled sub-sequentially ontinuous i� there exists a sequen e {xn} in X su h that, lim n→∞ Axn = lim n→∞ Sxn = z, for some z ∈ X and limn→∞ ASxn = Az and limn→∞ SAxn = Sz. Remark 2.20. One an easily he k that if two self-mappings A and S are both ontinuous, hen e also re ipro ally ontinuous mappings but A and S are not sub-sequentially ontinuous (see [28, Example 1℄). 3 Results In 2012, Khan et al. [25℄ proved the following ommon �xed point theorem for two pairs of sub- ompatible as well as sub-sequentially ontinuous mappings in N.A. Menger PM-spa e. Theorem 3.1. [25, Theorem 1℄ Let A, B, S and T be four self mappings of an N.A. Menger PM- spa e (X, F, T ). If the pairs (A, S) and (B, T) are sub- ompatible and sub-sequentially ontinuous, then (1) A and S have a oin iden e point, (2) B and T have a oin iden e point. Further, if g(FAx,By(t)) ≤ φ ( max { g(FSx,Ty(t)), g(FSx,Ax(t)), g(FTy,By(t)), g(FSx,By(t)), g(FTy,Ax(t)) }) , (1) holds for all x, y ∈ X, t > 0, φ ∈ Φ and g : [0, 1] → [0, ∞) is ontinuous and stri tly de reasing with g(1) = 0 and g(0) < ∞. Then A, B, S and T have a unique ommon �xed point in X. Unfortunately, Theorem 3.1 is not true in its present form. To substantiate this viewpoint, we refer to Imdad et al. [15, Example 0.1℄ wherein it an be easily seen that involved mappings do not have a oin iden e or ommon �xed point in the underlying spa e. CUBO 15, 3 (2013) Coin iden e and ommon �xed point theorems in N.A. Menger . . . 37 Motivated by a re ent note of Imdad et al. [14℄, the on lusions of Theorem 3.1 remain valid if we repla e ompatibility with sub- ompatibility and sub-sequential ontinuity with re ipro al ontinuity. However, Theorem 3.1 an be orre ted in two ways under more general onditions as follows: Theorem 3.2. Let A, B, S and T be self mappings of an N.A. Menger PM-spa e (X, F, T ). If the pairs (A, S) and (B, T) are ompatible and sub-sequentially ontinuous, then (1) the pair (A, S) has a oin iden e point, (2) the pair (B, T) has a oin iden e point. (3) There exists φ ∈ Φ su h that g(FAx,By(t)) ≤ φ ( max { g(FSx,Ty(t)), g(FSx,Ax(t)), g(FTy,By(t)), 1 2 (g(FSx,By(t)) + g(FTy,Ax(t))) }) , (2) holds for all x, y ∈ X, t > 0 and g ∈ Ω. Then A, B, S and T have a unique ommon �xed point in X. Proof. Sin e the pair (A, S) (also (B, T)) is sub-sequentially ontinuous and ompatible mappings, therefore there exists a sequen e {xn} in X su h that lim n→∞ Axn = lim n→∞ Sxn = z, (3) for some z ∈ X, and lim n→∞ FASxn,SAxn(t) = FAz,Sz(t) = 1, for all t > 0 then Az = Sz, whereas in respe t of the pair (B, T), there exists a sequen e {yn} in X su h that lim n→∞ Byn = lim n→∞ Tyn = w, (4) for some w ∈ X, and lim n→∞ FBTyn,TByn(t) = FBw,Tw(t) = 1, for all t > 0 then Bw = Tw. Hen e z is a oin iden e point of the pair (A, S) whereas w is a oin iden e point of the pair (B, T). Now we show that z = w. On using inequality (2) with x = xn, y = yn, we get g(FAxn,Byn(t)) ≤ φ ( max { g(FSxn,Tyn(t)), g(FSxn,Axn(t)), g(FTyn,Byn(t)), 1 2 (g(FSxn,Byn(t)) + g(FTyn,Axn(t))) }) , 38 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) passing to limit as n → ∞, we get g(Fz,w(t)) ≤ φ ( max { g(Fz,w(t)), g(Fz,z(t)), g(Fw,w(t)), 1 2 (g(Fz,w(t)) + g(Fw,z(t))) }) , = φ ( max { g(Fz,w(t)), g(1), g(1), 1 2 (g(Fz,w(t)) + g(Fz,w(t))) }) = φ (max{g(Fz,w(t)), 0, 0, g(Fz,w(t))}) = φ (g(Fz,w(t))) . Owing Lemma 2.9, we have z = w. We assert that Az = z. On using (2) with x = z and y = yn, we get g(FAz,Byn(t)) ≤ φ ( max { g(FSz,Tyn(t)), g(FSz,Az(t)), g(FTyn,Byn(t)), 1 2 (g(FSz,Byn(t)) + g(FTyn,Az(t))) }) , passing to limit as n → ∞, we get g(FAz,z(t)) ≤ φ ( max { g(FAz,z(t)), g(FAz,Az(t)), g(Fz,z(t)), 1 2 (g(FAz,z(t)) + g(Fz,Az(t))) }) , = φ ( max { g(FAz,z(t)), g(1), g(1), 1 2 (g(FAz,z(t)) + g(Fz,Az(t))) }) = φ (max {g(FAz,z(t)), 0, 0, g(FAz,z(t))}) = φ (g(FAz,z(t))) . On employing Lemma 2.9, we have z = Az. Therefore Az = z = Sz and hen e z is a ommon �xed point of (A, S). Now we show that z is a ommon �xed point of (B, T). On using (2) with x = xn and y = z, we get g(FAxn,Bz(t)) ≤ φ ( max { g(FSxn,Tz(t)), g(FSxn,Axn(t)), g(FTz,Bz(t)), 1 2 (g(FSxn,Bz(t)) + g(FTz,Axn(t))) }) , passing to limit as n → ∞, we get g(Fz,Bz(t)) ≤ φ ( max { g(Fz,Bz(t)), g(Fz,z(t)), g(FBz,Bz(t)), 1 2 (g(Fz,Bz(t)) + g(FBz,z(t))) }) , = φ ( max { g(Fz,Bz(t)), g(1), g(1), 1 2 (g(Fz,Bz(t)) + g(FBz,z(t))) }) = φ (max{g(Fz,Bz(t)), 0, 0, g(Fz,Bz(t))}) = φ (g(Fz,Bz(t))) . In view of Lemma 2.9, we have z = Bz. Therefore Bz = z = Tz. Thus we on lude that z is a ommon �xed point of A, B, S and T. The uniqueness of ommon �xed point is an easy onsequen e of inequality (2). CUBO 15, 3 (2013) Coin iden e and ommon �xed point theorems in N.A. Menger . . . 39 Theorem 3.3. Let A, B, S and T be self mappings of an N.A. Menger PM-spa e (X, F, T ). If the pairs (A, S) and (B, T) are sub- ompatible and re ipro ally ontinuous, then (1) the pair (A, S) has a oin iden e point, (2) the pair (B, T) has a oin iden e point. (3) Further, the mappings A, B, S and T have a unique ommon �xed point in X provided the involved mappings satisfy the inequality (2) of Theorem 3.2. Proof. Sin e the pair (A, S) (also (B, T)) is sub- ompatible and re ipro ally ontinuous, therefore there exists a sequen es {xn} in X su h that lim n→∞ Axn = lim n→∞ Sxn = z, for some z ∈ X, and lim n→∞ FASxn,SAxn(t) = lim n→∞ FAz,Sz(t) = 1, for all t > 0, whereas in respe t of the pair (B, T), there exists a sequen e {yn} in X with lim n→∞ Byn = lim n→∞ Tyn = w, for some w ∈ X, and lim n→∞ FBTxn,TBxn(t) = lim n→∞ FBz,Tz(t) = 1, for all t > 0. Therefore, Az = Sz and Bw = Tw, i.e., z is a oin iden e point of the pair (A, S) whereas w is a oin iden e point of the pair (B, T). The rest of the proof an be ompleted on the lines of Theorem 3.2. Remark 3.4. The on lusions of Theorem 3.2 and Theorem 3.3 remain true if we repla e the inequality (2) by one of the following: g(FAx,By(t)) ≤ φ (maxg(FSx,Ty(t)), g(FSx,Ax(t)), g(FTy,By(t)), g(FSx,By(t))) , (5) for all x, y ∈ X, t > 0, where g ∈ Ω and φ satis�es the ondition (Φ). Or, g(FAx,By(t)) ≤ φ (max g(FSx,Ty(t)), g(FSx,Ax(t)), g(FTy,By(t))) , (6) for all x, y ∈ X, t > 0, where g ∈ Ω and φ satis�es the ondition (Φ). 40 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) Or, g(FAx,By(t)) ≤ φ     max    g(FSx,Ty(t)) + g(FSx,Ax(t)) + g(FTy,By(t)), g(FSx,Ax(t)) + g(FSx,By(t)), g(FAx,Ty(t)) + +g(FTy,By(t))        , (7) for all x, y ∈ X, t > 0, where g ∈ Ω and φ satis�es the ondition (Φ). Remark 3.5. Theorem 3.2 and Theorem 3.3 (also in view of Remark 3.4) improve the results of Rao and Ramudu [29, Theorem 14℄, Khan and Sumitra [23, Theorem 2℄ and Kutuk u and Sharma [26, Theorem 1℄. By hoosing A, B, S and T suitably, we an drive a multitude of ommon �xed point theorems for a pair or triod of mappings. As a sample, we outline the following natural result for a pair of self mappings. Corollary 3.6. Let A and S be self mappings of an N.A. Menger PM-spa e (X, F, T ). If the pair (A, S) is ompatible and sub-sequentially ontinuous (alternately sub- ompatible and re ipro ally ontinuous), then (1) the pair (A, S) has a oin iden e point. (2) There exists φ ∈ Φ su h that, g(FAx,Ay(t)) ≤ φ ( max { g(FSx,Sy(t)), g(FSx,Ax(t)), g(FSy,Ay(t)), 1 2 (g(FSx,Ay(t)) + g(FSy,Ax(t))) }) , (8) holds for all x, y ∈ X, t > 0 and g ∈ Ω. Then A and S have a unique ommon �xed point in X. Remark 3.7. The results similar to Corollary 3.6 an also be outlined in respe t of inequalities (5)-(7). Now we give some illustrative examples. Example 3.8. Let (X, d) be a metri spa e with the usual metri d where X = [0, ∞) and (X, F, T ) be the indu ed N.A. Menger PM-spa e with g(t) = 1−t for all t ∈ [0, 1], and Fx,y(t) = H(t−d(x, y)) for all x, y ∈ X and all t > 0 and T (a, b) = min{a, b} for all a, b ∈ [0, 1]. Set A = B and S = T. De�ne the self mappings A and S on X by A(x) = { x 4 , if x ∈ [0, 1]; 5x − 4, if x ∈ (1, ∞). S(x) = { x 5 , if x ∈ [0, 1]; 4x − 3, if x ∈ (1, ∞). CUBO 15, 3 (2013) Coin iden e and ommon �xed point theorems in N.A. Menger . . . 41 Consider a sequen e {xn} = { 1 n } n∈N in X. Then lim n→∞ A(xn) = lim n→∞ ( 1 4n ) = 0 = lim n→∞ ( 1 5n ) = lim n→∞ S(xn). Next, lim n→∞ AS(xn) = lim n→∞ A ( 1 5n ) = lim n→∞ ( 1 20n ) = 0 = A(0), lim n→∞ SA(xn) = lim n→∞ S ( 1 4n ) = lim n→∞ ( 1 20n ) = 0 = S(0), and lim n→∞ FASxn,SAxn(t) = 1, for all t > 0. Consider another sequen e {xn} = { 1 + 1 n } n∈N in X. Then lim n→∞ A(xn) = lim n→∞ ( 5 + 5 n − 4 ) = 1 = lim n→∞ ( 4 + 4 n − 3 ) = lim n→∞ S(xn). Also, lim n→∞ AS(xn) = lim n→∞ A ( 1 + 4 n ) = lim n→∞ ( 5 + 20 n − 4 ) = 1 6= A(1), lim n→∞ SA(xn) = lim n→∞ S ( 1 + 5 n ) = lim n→∞ ( 4 + 20 n − 3 ) = 1 6= S(1), but limn→∞ FASxn,SAxn(t) = 1. Thus, the pair (A, S) is ompatible as well as sub-sequentially ontinuous but not re ipro ally ontinuous. Therefore all the onditions of Corollary 3.6 are sat- is�ed. Here, 0 is a oin iden e as well as unique ommon �xed point of the pair (A, S). It is noted that this example annot be overed by those �xed point theorems whi h involve ompatibil- ity and re ipro al ontinuity both or by involving onditions on ompleteness (or losedness) of underlying spa e (or subspa es). Also, in this example neither X is omplete nor any subspa e A(X) = [ 0, 1 4 ] ∪ (1, ∞) and S(X) = [ 0, 1 5 ] ∪ (1, ∞) are losed. It is noted that this example annot be overed by those �xed point theorems whi h involve ompatibility and re ipro al ontinuity both. Example 3.9. In the setting of Example 3.8, de�ne X = R (set of real numbers) and the self mappings A and S on X by A(x) = { x 4 , if x ∈ (−∞, 1); 5x − 4, if x ∈ [1, ∞). S(x) = { x + 3, if x ∈ (−∞, 1); 4x − 3, if x ∈ [1, ∞). Consider a sequen e {xn} = { 1 + 1 n } n∈N in X. Then lim n→∞ A(xn) = lim n→∞ ( 5 + 5 n − 4 ) = 1 = lim n→∞ ( 4 + 4 n − 3 ) = lim n→∞ S(xn). 42 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) Also, lim n→∞ AS(xn) = lim n→∞ A ( 1 + 4 n ) = lim n→∞ ( 5 + 20 n − 4 ) = 1 = A(1), lim n→∞ SA(xn) = lim n→∞ S ( 1 + 5 n ) = lim n→∞ ( 4 + 20 n − 3 ) = 1 = S(1), and lim n→∞ FASxn,SAxn(t) = 1, for all t > 0. Consider another sequen e {xn} = { 1 n − 4 } n∈N in X. Then lim n→∞ A(xn) = lim n→∞ ( 1 4n − 1 ) = −1 = lim n→∞ ( 1 n − 4 + 3 ) = lim n→∞ S(xn). Next, lim n→∞ AS(xn) = lim n→∞ A ( 1 n − 1 ) = lim n→∞ ( 1 4n − 1 4 ) = − 1 4 = A(−1), lim n→∞ SA(xn) = lim n→∞ S ( 1 4n − 1 ) = lim n→∞ ( 1 4n − 1 + 3 ) = 2 = S(−1), and limn→∞ FASxn,SAxn(t) 6= 1. Thus, the pair (A, S) is re ipro ally ontinuous as well as sub- ompatible but not ompatible. Therefore all the onditions of Corollary 3.6 are satis�ed. Thus 1 is a oin iden e as well as unique ommon �xed point of the pair (A, S). It is also noted that this example too annot be overed by those �xed point theorems whi h involve ompatibility and re ipro al ontinuity both. Re eived: June 2012. A epted: September 2013. Referen es [1℄ M.A. Al-Thaga� and N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approx- imations, A ta Math. Sin. (Engl. Ser.), 24(5) (2008), 867�876. [2℄ H. Bouhadjera and C. Godet-Thobie, Common �xed theorems for pairs of sub ompatible maps, arXiv:0906.3159v1 [math.FA℄ 17 June (2009) [Old version℄. [3℄ H. Bouhadjera and C. Godet-Thobie, Common �xed theorems for pairs of sub ompatible maps, arXiv:0906.3159v2 [math.FA℄ 23 May (2011) [New version℄. [4℄ S.S. Chang, Fixed point theorems for single-valued and multi-valued mappings in Non- Ar himedean Menger probabilisti metri spa es, Math. Japoni a 35(5) (1990), 875�885. CUBO 15, 3 (2013) Coin iden e and ommon �xed point theorems in N.A. Menger . . . 43 [5℄ S. Chauhan, Z. Kadelburg and S. Dalal, A ommon �xed point theorem in metri spa e under general ontra tive ondition, J. Appl. Math. 2013, vol. 2013, Arti le ID 510691, 7 pages. [6℄ S. Chauhan, B.D. Pant, S. Kumar and A. Tomar, A ommon �xed point theorem in Non-Ar himedean Menger PM-spa e, Analele Universit�aµii Oradea Fas . Matemati a XX(2) (2013), in printing. [7℄ S. Chauhan, S. Radenovi¢, M. Imdad and C. Vetro, Some integral type �xed point theorems in Non-Ar himedean Menger PM-Spa es with ommon property (E.A) and appli ation of fun - tional equations in dynami programming, Revista de la Real A ademia de Cien ias Exa tas, Fisi as y Naturales. Serie A. Matemati as (2013), in press. [8℄ Y.J. Cho, K.S. Ha and S.S. Chang, Common �xed point theorems for ompatible mappings of type (A) in Non-Ar himedean Menger PM-spa es, Math. Japon. 46(1) (1997), 169�179. MR1466131 [9℄ B.S. Choudhury, S. Kutuk u and K. Das, On �xed points in Non-Ar himedean Menger PM- spa es, Ko hi J. Math. 7 (2012), 41�50. [10℄ R.C. Dimri and B.D. Pant, Fixed point theorems in Non-Ar himedean Menger spa es, Kyung- pook Math. J. 31(1) (1991), 89�95. [11℄ D. �ori , Z. Kadelburg and S. Radenovi¢, A note on o asionally weakly ompatible mappings and ommon �xed point, Fixed Point Theory, 13(2) (2012), 475�479. [12℄ D. Gopal and M. Imdad, Some new ommon �xed point theorems in fuzzy metri spa es, Ann. Univ. Ferrara Sez. VII S i. Mat. 57(2) (2011), 303�316. [13℄ O. Had�zi¢, A note on Istr tes u�s �xed point theorem in Non-Ar himedean Menger spa es, Bull. Math. So . S i. Math. Rep. So . Roum. 24(72) (1980), 277�280. [14℄ M. Imdad, J. Ali and M. Tanveer, Remarks on some re ent metri al �xed point theorems, Appl. Math. Lett. 24(7) (2011), 1165�1169. [15℄ M. Imdad, D. Gopal and C. Vetro, An addendum to: A ommon �xed point theorem in intu- itionisti fuzzy metri spa e using sub ompatible maps, Bull. Math. Anal. Appl. 4(1) (2012), 168�173. [16℄ I. Istr t . es u, On some �xed point theorems with appli ations to the non-Ar himedean Menger spa es, Atti A ad. Naz. Lin ei Rend. Cl. S i. Fis. Mat. Natur. (8)58(3) (1975), 374�379. [17℄ I. Istr t . es u, Fixed point theorems for some lasses of ontra tion mappings on Non- Ar himedean probablisti metri spa e, Publ. Math. Debre en 25(1-2) (1978), 29�34. [18℄ I. Istr t . es u and G. Babes u, On the ompletion on Non-Ar himedean probabilisti metri spa es, Seminar de spatii metri e probabiliste, Universitatea Timisoara, Nr. 17, 1979. 44 Sunny Chauhan, B. D. Pant & Mohammad Imdad CUBO 15, 3 (2013) [19℄ I. Istr t . es u and N. Crivat, On some lasses of Non-Ar himedean probabilisti metri spa es, Seminar de spatii metri e probabiliste, Universitatea Timisoara, Nr. 12, 1974. [20℄ I. Istr t . es u and G. Palea, On Non-Ar himedean probabilisti metri spa es, An. Univ. Tim- i³oara Ser. �ti. Mat. 12(2) (1974), 115�118 (1977). [21℄ G. Jung k and B.E. Rhoades, Fixed points for set valued fun tions without ontinuity, Indian J. Pure Appl. Math. 29(3) (1998), 227�238. [22℄ M.A. Khan, Common �xed point theorems in Non-Ar himedean Menger PM-spa es, Int. Math. Forum 6(40) (2011), 1993�2000. [23℄ M.A. Khan and Sumitra, A ommon �xed point theorem in Non-Ar himedean Menger PM- spa e, Novi Sad J. Math. 39(1) (2009), 81�87. [24℄ M.A. Khan and Sumitra, Common �xed point theorems in Non-Ar himedean Menger PM- spa e, JP J. Fixed Point Theory Appl. 5(1) (2010), 1�13. [25℄ M.A. Khan, Sumitra and R. Kumar, Sub- ompatible and and sub-sequential ontinuous maps in Non-Ar himedean Menger PM-spa e, Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137�150. [26℄ S. Kutuk u and S. Sharma, A ommon �xed point theorem in Non-Ar himedean Menger PM- spa es, Demonstratio Math. 42(4) (2009), 837�849. [27℄ R.P. Pant, Common �xed points of four mappings, Bull. Cal. Math. So . 90(4) (1998), 281� 286. [28℄ R.P. Pant and R.K. Bisht, Common �xed point theorems under a new ontinuity ondition, Ann. Univ. Ferrara Sez. VII S i. Mat. 58(1) (2012), 127�141. [29℄ K.P.R. Rao and E.T. Ramudu, Common �xed point theorem for four mappings in Non- Ar himedean Menger PM-spa es, Filomat 20(2) (2006), 107�113. [30℄ F. Rouzkard, M. Imdad and H.K. Nashine, New ommon �xed point theorems and invariant approximation in onvex metri spa es, Bull. Belg. Math. So . Simon Stevin 19 (2012), 311� 328. [31℄ B. S hweizer and A. Sklar, Statisti al metri spa es, Pa i� J. Math. 10 (1960), 313�334. [32℄ V.M. Sehgal and A.T. Bharu ha-Reid, Fixed points of ontra tion mappings on probabilisti metri spa es, Math. Systems Theory 6 (1972), 97�102. [33℄ S.L. Singh and B.D. Pant, Common �xed points of weakly ommuting mappings on Non- Ar himedean Menger PM-spa es, Vikram J. Math. 6 (1985/86), 27�31. [34℄ S.L. Singh, B.D. Pant and S. Chauhan, Fixed point theorems in Non-Ar himedean Menger PM-spa es, J. Nonlinear Anal. Optim. Theory Appl. 3(2) (2012), 153�160.