In this paper, first we prove a common fixed point theorem using weakly compatible mapping in 2- Menger space which generalize the well known results. Secondly, we prove a common fixed point theorem using (S-B) property along with weakly compatible maps. (S-B) property defined by Sharma and Bamoria [16] via implicit relation. Keywords: Common fixed points, Metric space, S-B property, 2-Menger space, weakly compatible mapping and implicit relation. AMS subject classification– 47H10, 54H25. DOI: 10.7176/MTM/9-5-01 Publication date:May 31st 201

Sharma, Geetanjali

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.5, 2019  1 Common Fixed Point Theorem in 2-Menger Space via (S-B) Property  Geetanjali Sharma1      Richa Gupta2      Pankaj Tiwari3      Akshay Sharma2 1.Research Scholar, Sarvepalli Radhakrishnan University, Bhopal (M.P.), India 2.Faculty of Science, Sarvepalli Radhakrishnan University, Bhopal (M.P.), India 3.Dept. of Mathematics, Govt. Naveen College, Lidhoura, Tikamgarh (M.P.), India  Abstract In this paper, first we prove a common fixed point theorem using weakly compatible mapping in 2- Menger space which generalize the well known results. Secondly, we prove a common fixed point theorem using (S-B) property along with weakly compatible maps. (S-B) property defined by Sharma and Bamoria [16] via implicit relation. Keywords: Common fixed points, Metric space, S-B property, 2-Menger space, weakly compatible mapping and implicit relation. AMS subject classification– 47H10, 54H25. DOI: 10.7176/MTM/9-5-01 Publication date:May 31st 2019  1. INTRODUCTION AND PRELIMINARIES In 1922, Banach proved the principal contraction result [4]. As we know, there have been published many works about fixed point theory for different kinds of contractions on some spaces such as quasi-metric spaces, cone metric spaces, convex metric spaces, partially ordered metric spaces, G-metric spaces, partial metric spaces, quasi-partial metric spaces, fuzzy metric spaces and Menger spaces.  The study of 2-metric spaces was initiated by Gahler[7] and some fixed point theorems in 2-metric spaces were proved in [8],[9], [10] and [15]. In 1987, Zeng [23] gave the generalization of 2-metric to Probabilistic 2-metric as follows; A  probabilistic metric space shortly PM-Space, is an ordered pair (X, F) consisting of a non empty set X and a mapping F  from X ×  X to  L, where L  is the collection of all distribution functions (a distribution function F is non decreasing and left continuous mapping of reals in to [0,1] with properties, inf F(x) = 0 and sup F(x) = 1).   1. The value of F at (x, y) ∈  X ×  X is represented by F,. The function F, are assumed satisfy the following conditions; 2. (FM-0) F, (t)  =  1, for all t > 0, iff x = y;  3. (FM-1) F, (0) =  0, if t = 0; 4. (FM-2) F,(t) =F,(t); 5. (FM-3) F, (t)  =  1 and  F, (s)  =  1 then F, (t +  s)  =  1. 6. A mapping T: [0,1] × [0,1] → [0,1] is a t-norm, if it satisfies the following conditions; 7. (FM-4) T(a, 1) = a for every a ∈ [0,1]; 8. (FM-5) T(0, 0) = 0, 9. (FM-6) T(a, b) = T(b, a) for every a, b ∈ [0,1]; 10. (FM-7) T(c, d)  ≥ T(a, b)for c ≥ a and d ≥ b 11. (FM-8) T(T(a, b), c) = T(a, T(b, c)) where a, b, c, d ∈ [0,1]. 12. A Menger space is a triplet (X, F, T), where (X, F) is a PM-Space, X is a non-empty set and a t − norm satisfying instead of (FM-8) a stronger requirement. 13. (FM-9)  F, (t +  s) ≥  T *F,(t), F,(s)+ for all x ≥ 0, y ≥ 0. 14. For a given metric space (X, d) with usual metric d, one can put F, (t)  =  H (t − d(x, y)) for all x, y ∈ X and t >  0. where H is defined as:                       H(t)  =  - 1  if s > 0, 0  if s ≤ 0.         and t-norm T is defined as T(a, b) =  min 0a, b1.  For the proof of our result we required the following definitions. Definition 1.1 :-A triangular norm ∗(shortly 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: (1) a ∗ 1 = a, (2) a ∗ b = b ∗ a, (3) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.5, 2019  2 (4) a ∗ (b ∗ c) = (a ∗ b) ∗ c. Examples of t-norms are a ∗ b = min0a, b1 , a ∗ b = ab and a ∗ b = max0a + b − 1,01. Definition 1.2 :- Let (X, F,∗) be a Menger space and  be a continuous t-norm. (a) A sequence {x8} in X is said to be converge to a point x in X (written x8→x) iff for every ε > 0 and λ ∈ (0, 1), there exists an integer n; =  n;(ε, λ) such that F<,(ε) > 1 − λ for all n ≥  n;. (b) A sequence {x8} in X is said to be Cauchy if for every ε > 0 and λ ∈  (0, 1), there exists an integer n; = n;(ε, λ) such       that F<,<=> (ε) > 1 − λ for all n ≥  n; and p >  0. (c) A Menger space in which every Cauchy sequence is convergent is said to be complete. Remark 1.3:- If  is a continuous t-norm, it follows from (FM − 4) that the limit of sequence in Menger space is uniquely determined. Definition 1.4:- Self maps A and B  of a Menger space (X, F,∗)  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. Weakly Compatible Maps In 1982, Sessa [17], weakened the concept of commutativity to weakly commuting mappings. Afterwards, Jungck [4] enlarged the concept of weakly commuting mappings by adding the notion of compatible mappings. In 1991, Mishra [16] introduced the notion of compatible mappings in the setting of probabilistic metric space. Definition 1.5 :- Self maps A and B of a Menger space (X, F,∗) are said to be compatible if FBCD,CB<,(t)  →  1 for all t >  0, whenever 0x81 is a sequence in X such that Ax8 →  x, Bx8 →  x for some x in X as n → ∞. . Definition 1.6:- Let S and T be weakly compatible of a Menger space (X, M,∗) and Su =  Tu for some u in X then STu =  TSu =  SSu =  TTu. Definition 1.7:- (Implicit Relation) Let 4 be the set of real and continuous function from  4R R   so that (i)   is non-increasing in 2 ,3nd rd argument and (ii) For  , 0 , , , 0u v u v v v u v    Example 1.8:- Let X =  [0, 3] be equipped with the usual metric d(x, y) = |x − y| Define f, g: [0,3] → [0,3] by f(x)  =  J x  if x ∈ [0,1), 3  if x ∈ [1,3]. And                                                            g(x) = J3 − x  if x ∈ [0,1), 3       if x ∈ [1,3].  Then for any x ∈  [1,3], x is a coincidence point and fgx =  gfx, showing that f, g are weakly compatible maps on [0, 3]. Lemma 1.9:-  Let (X, M,∗) be a Menger space. Then for all x, y ∈  X, M(x, y, . ) is a non-decreasing function. Lemma 1.10:- Let (X, M,∗) be a Menger space. If there exists k ∈  (0, 1) such that  for all x, y ∈  X M,(t) ≥ M,(t)   ∀ t > 0 then x =  y. Lemma 1.11:- Let 0xn1 be a sequence in a Menger space (X, M,∗). If there exists a number k ∈ (0, 1) such that M<=M,<=N(kt) ≥ M<=N,<(t) ∀ t > 0 and n ∈ N. Then 0xn1 is a Cauchy sequence in X. Lemma 1.12:- The only t-norm ∗ satisfying r ∗  r ≥  r for all r ∈  [0, 1] is the minimum t-norm, that is a ∗  b =  min 0a, b} for all a, b ∈  [0, 1]. Lemma 1.13:- Let  , ,X M   be a Menger space and  , , 0x y X t   and if for a number  0,1k  ,    , , , ,M x y kt M x y t  then x y . Example 1.14:- Let (X, d) be a metric space. Define a ∗  b =  min 0a, b1 and  M,(t) = PPQR(,), for all x, y ∈  X.and all t >  0. Then (X, M,∗) is a Menger space. It is called the Menger space induced by d. Remark 1.15:- If self maps A and B of a Menger space (X, F,∗) are compatible then they are weakly compatible.   2. MAIN RESULT Now we prove the following results: Theorem 2.1: Let  , ,X M   be a common fixed point theorem in 2- Menger space with compatible maps. Let Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.5, 2019  3 , ,A B S  and T be mappings of X into itself satisfying following conditions: (2.1) AX TX  and BX SX  (2.2)  ,A S  or  ,B T  satisfy the (S-B) property (2.3) there exists a constant  0,1q  such that x, y, a ∈ X and t > 0, α TMB,C,U(qt) ∗ MW,X,U(t) + MB,W,U(t)2 ∗MC,X,U(t) + MB,X,U(t)2 Z ≥ 0                                                                                                                                         (2.1.1) (2.4) If the pairs  ,A S  or  ,B T  are weakly compatible (2.5) One of A(X), B(X), S(X)or T(X)  is closed subset of X . Indeed, , ,A B S  and T  have a unique common fixed point in X . Proof. Suppose that  ,B T  satisfies the (S-B) property. Then there exists a sequence  nx  in X  such that lim limn nn nBx Tx z    for some z X . Since BX SX , there exists in X  a sequence  ny  such that n nBx Sy .  Hence lim nnSx z .  Let us show that lim nnAy z . Now by equation (2.1.1), we have  α TMB<,C<,U(qt) ∗ MW<,X<,U(t) + MB<,W<,U(t)2 ∗MC<,X<,U(t) + MB<,X<,U(t)2 Z ≥ 0 α TMB<,C<,U(qt) ∗ MC<,X<,U(t) + MB<,C<,U(t)2 ∗MC<,X<,U(t) + MB<,X<,U(t)2 Z ≥ 0 Since lim limn nn nBx Tx    , , 1n nM Bx Tx t   So taking limn α TMB<,C<,U(qt) ∗ 1 + MB<,C<,U(t)2 ∗1 + MB<,C<,U(t)2 Z ≥ 0   is non-increasing in 2 ,3nd rd  argument  α *MB<,C<,U(qt) ∗ MB<,C<,U(t) ∗ MB<,C<,U(t)+ ≥ 0 By the definition (1.7) MB<,C<,U(qt) ≥ MB<,C<,U(t) Since M  is continuous function lim8→] MB<,C<,U(qt) ≥ lim8→] MB<,C<,U(t) By lemma (1.13) lim limn nn nAy Bx   and we deduce that lim nnAy z  Suppose SX  is a closed subset of X .  Then z Su  for some u X .  Subsequently we have,  lim lim lim limn n n nn n n nAy Bx Tx Sy Su       . By  (2.3), we have α TMB^,C<,U(qt) ∗ MW^,X<,U(t) + MB^,W^,U(t)2 ∗MC<,X<,U(t) + MB^,X<,U(t)2 Z ≥ 0 Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.5, 2019  4 α TMB^,C<,U(qt) ∗ MW^,X<,U(t) + MB^,W^,U(t)2 ∗MC<,X<,U(t) + MB^,X<,U(t)2 Z ≥ 0 Taking limn , we have  α TMB^,W^,U(qt) ∗ MW^,W^,U(t) + MB^,W^,U(t)2 ∗MW^,W^,U(t) + MB^,W^,U(t)2 Z ≥ 0 α TMB^,W^,U(qt) ∗ 1 + MB^,W^,U(t)2 ∗1 + MB^,W^,U(t)2 Z ≥ 0   is non-increasing in 2 ,3nd rd  argument  α *MB^,W^,U(qt) ∗ MB^,W^,U(t) ∗ MB^,W^,U(t)+ ≥ 0 By the definition (1.7) MB^,W^,U(qt) ≥ MB^,W^,U(t) Thus by lemma (1.13) We have Au Su . The weak compatibility of A and S implies that ASu SAu  and then AAu ASu SAu SSu   .  On the other hand,  Since AX TX , there exists a point v X such that Au Tv . We claim that Au Bv  using (2.3); we have α TMB^,C_,U(qt) ∗ MW^,X_,U(t) + MB^,W^,U(t)2 ∗MC_,X_,U(t) + MB^,X_,U(t)2 Z ≥ 0  α TMB^,C_,U(qt) ∗ MW^,B^,U(t) + MB^,W^,U(t)2 ∗MC_,B^,U(t) + MB^,B^,U(t)2 Z ≥ 0 α TMB^,C_,U(qt) ∗ 1 ∗ 1 + MB^,C_,U(t)2 Z ≥ 0    is non-increasing in 2 ,3nd rd  argument  α *MB^,C_,U(qt) ∗ MB^,C_,U(t)  ∗ MB^,C_,U(t)+ ≥ 0 By the definition (1.7) MB^,C_,U(t) ≥ MB^,C_,U(t) Therefore by lemma, we have Au Bv  Thus .Au Su Tv Bv     The weak compatibility of B and T implies that BTv TBv and .TTv TBv BTv BBv     Let us show that Au  is a common fixed point of , ,A B S  and T .  In view of (2.3) we have  α TMBB^,C_,U(qt) ∗ MWB^,X_,U(t) + MBB^,WB^,U(t)2 ∗MC_,X_,U(t) + MBB^,X_,U(t)2 Z ≥ 0  α TMBB^,B^,U(qt) ∗ MBB^,B^,U(t) + MBB^,BB^,U(t)2 ∗MB^,B^,U(t) + MBB^,B^,U(t)2 Z ≥ 0 α TMBB^,B^,U(qt) ∗ 1 + MBB^,B^,U(t)2 ∗1 + MBB^,B^,U(t)2 Z ≥ 0   is non-increasing in 2 ,3nd rd  argument  α *MBB^,B^,U(qt) ∗ MBB^,B^,U(t) ∗ MBB^,B^,U(t)+ ≥ 0 By the definition (1.7) MBB^,B^,U(qt) ≥ MBB^,B^,U(t) Therefore by lemma, we have Au AAu SAu  and Au  is a common fixed point of A and S .  Similarly, we can validate that Bv  is a common fixed point of B  and T .  Since Au Bv , we achieve that Au is point of , ,A B S  and T , Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.5, 2019  5 which is called common fixed point.. If Au Bu Su Tu u    and Av Bv Sv Tv v    .  Then by (2.3), we have α TMB^,C_,U(qt) ∗ MW^,X_,U(t) + MB^,W^,U(t)2 ∗MC_,X_,U(t) + MB^,X_,U(t)2 Z ≥ 0  α TM^,_,U(qt) ∗ M^,_,U(t) + M^,^,U(t)2 ∗M_,_,U(t) + M^,_,U(t)2 Z ≥ 0  α TM^,_,U(qt) ∗ 1 + M^,_,U(t)2 ∗1 + M^,_,U(t)2 Z ≥ 0   is non-increasing in 2 ,3nd rd  argument  α *M^,_,U(qt) ∗ M^,_,U(t) ∗ M^,_,U(t)+ ≥ 0 By  the definition (1.7) M^,_,U(t) ≥ M^,_,U(t) Therefore by lemma, we have u v  and the common fixed point is a unique.   This explanation is verified the theorem. Hence , ,A B S  and T  have a unique common fixed point in X .  References 1. Aydi H., Karapınar, E., Postolache, M: Tripled coincidence point theorems for weak ϕ-contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012, 44 (2012). 2. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for (ψ,φ)-weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl. 63(1), 298-309 (2012). 3. 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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.5, 2019  6 23. Zeng Wenzhi” Probabilistic 2- metric Spaces” J. Math. Research Expo. 2, 241-245, (1987).   

Common Fixed Point Theorem in 2-Menger Space via (S-B) Property

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Common Fixed Point Theorem in 2-Menger Space via (S-B) Property

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