In this paper, we define extend the results of many others. We prove common fixed point theorems on tangential property for a Gregus type on pair of fuzzy metric spaces. We also deal on some coupled coincidence and common fixed point theorems

Rajput, Anil

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      ISSN 2347-1921                                                            2608| P a g e                                                    N o v e m b e r  1 1 ,  2 0 1 4  COMMON FIXED POINT FOR TANGENTIAL MAPS OF GREGUS TYPE ON FUZZY METRIC SPACES Anil Rajput1 and Ritu Tiwari2 1Department of Mathematics, CSA Govt. PG College, Sehore MP India (dranilrajput@hotmail.com) 2 Department of Basic Science, JNCT Bhopal, India (deepuritutiwari@gmail.com)  Abstract: In this paper, we define extend the results of many others. We prove common fixed point theorems on tangential property for a Gregus type on pair of fuzzy metric spaces. We also deal on some coupled coincidence and common fixed point theorems.                              Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol. 9 , No. 5 www.cirjam.com , editorjam@gmail.com       ISSN 2347-1921                                                            2609| P a g e                                                    N o v e m b e r  1 1 ,  2 0 1 4  1 INTRODUCTION: The notion of fuzzy sets introduced by Zadeh (1965, [4]) proved a turning point in the development of Mathematics. The study of fixed points for multi-valued contraction mappings using the Hausdorff metric was initiated by Markin (1973, [2]) and Nadler (1972, [3] ).  Bhaskar and Lakshmikantham (2006, [1]) introduced the concepts of coupled fixed points and mixed monotone property and illustrated these results by proving the existence and uniqueness of the solution for a periodic boundary value problem.  2 Definitions and Preliminaries: To set up our results in the next section we recall some definitions and facts. 2.1 Definition: A fuzzy set A in X is a function with domain X and values in [0, 1]. 2.2 Definition: A binary operation *: [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if ([0, 1], *) is a topological abelian monoid with unit 1 s.t. a * b ≤ c * d whenever a  ≤  c and b ≤ d, ∀ a, b, c, d ∈ [0, 1]. Some examples are below: (i) *(a, b) = ab, (ii) *(a, b) = min{a, b}. 2.3 Definition: The 3-tuple (X, M, *) is called a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on   X2 x [0, ∞) satisfying the following conditions:     (FM-1)  M(x, y, t) > 0 and M(x, y, 0) = 0 (FM-2)   M(x, y, t) = 1 iff x = y, (FM-3)  M(x, y, t) = M(y, x, t), (FM-4)  M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s), (FM-5)  M(x, y, .) : (0, ∞) → [0, 1) is continuous, for all x, y,   z ∈ X and s, t > 0. therefore, M(x, y, .) is non-decreasing for all x, y ∈ X 2.4 Definition: Let (X, M,) be a fuzzy metric space. (i) A sequence{xn}  is said to be convergent to a point x ∊ X if limn→∞ M(xn, x, t) = 1 for all t > 0. (ii) A subset A ⊆ X is said to be closed if each convergent sequence {xn} with xn ∊ X (A) and xn → x, we have x ∊ A. (iii) A subset A ⊆ X is said to be compact if each sequence in A has a convergent subsequence.  Throughout the paper X will represent the fuzzy metric space                     (X, M, ∗) and 𝜅(X), the set of compact subsets of X. For A, B ∊ 𝜅(X) and for every t > 0, denote MA, B, t) min{minaA(a, B, t), minbB A, b, t }, M(A, y, t) = max{M(x, y, t)}; x, y A} Remark: Obviously, MA, B, t) ≤ Ma, B, t) whenever a A and        MA, B, t) = 1 A = B. Also MA, y, t) = 1 if y A. 3 Main Results 3.1 Theorems: Let A, B: X  X and S, T: X x X  𝜅(X) be single and set-valued mappings satisfying the following conditions: (1) there exist contained coupled weak tangential points (z1, z2) to the mappings A and B. (2) (A, B) is tangential w. r .t (S, T) (3)   𝑠 𝑑𝑠 ≥    𝑠 𝑑𝑠𝑚(𝑥 ,   𝑦 ,   𝑢 ,   𝑣 ,   𝑡) 0𝑀∇(𝑆 𝑥 ,   𝑦 ,   𝑇 𝑢 ,   𝑣 ,   𝑡)0   where  m(x, y, u, v, t)        ISSN 2347-1921                                                            2610| P a g e                                                    N o v e m b e r  1 1 ,  2 0 1 4  = ∅       𝑎   𝑀∆ 𝑆 𝑥, 𝑦 ,𝐴𝑥, 𝑡 ∗  𝑀∆ 𝑇 𝑢,𝑣 ,𝐵𝑢, 𝑡 +𝑀∆  𝑆 𝑥, 𝑦 ,𝐵𝑢,𝑡2 ∗ 𝑀∆  𝑇 𝑢, 𝑣 ,𝐴𝑥,𝑡2  +  1 − 2𝑎  𝑚𝑎𝑥  𝑀∆ 𝑆 𝑥, 𝑦 ,𝐴𝑥, 𝑡 ,𝑀∆ 𝑇 𝑢, 𝑣 ,𝐵𝑢, 𝑡 ,𝑀∇ 𝑆 𝑥, 𝑦 ,𝐴𝑥, 𝑡 ,𝑀∇(𝑇 𝑢, 𝑣 ,𝐵𝑢, 𝑡)           (4) AAa = Aa, BBC = Bc, S(Aa, Ab) = T(Bc, Bd) and AAb = Ab, BBd = Bd, S(Ab, Aa) = T(Bd, Bc) for                 (a, b) 𝜖 C(A, S ) and (c, d) 𝜖 C(B,T), (5) the pair (A, S) is weakly compatible, for all x, y, u, v  𝜖 X,  0 ≤ a < 1 and  ∅ :  0, 1] → [0,1] be a non-decreasing map such that ∅(t) > t, t ≥ 0. Then A, B, S and T have a common coupled fixed point in X. Proof: Since z1, z2 𝜖 A (x) ∩ B (X) so there exist points w1, w2, w1/, w2/ 𝜖 X such that z1 = Aw1 = Bw1/, z2 = Aw2 = Bw2 /. Again (A, B) is tangential w.r.t (S, T) so there exist sequences {xn} and {yn} in X such that    limn→∞ Axn = z1 = limn→∞ Byn 𝜖 C 𝜖 Xlimn→∞S xn, yn) ∩ limn→∞ T(yn, xn)  limn→∞ Ayn = z2 = limn→∞ Bxn 𝜖 D 𝜖 Xlimn→∞S yn, xn) ∩ limn→∞ T(xn, yn) Now, we shall prove that  Aw1 𝜖 S(w1, w2), Aw2 𝜖 S(w2, w1), Bw1/ 𝜖 T(w1/, w2/)  and Bw2/ ϵ T(w2/, w1/) If not, putting x = xn, y = yn, u = w1/ and v = w2 / in (3), we get  m(xn, yn, 𝑤1/,  𝑤2/, t)  = ∅       𝑎   𝑀∆ 𝑆 𝑥𝑛 ,  𝑦𝑛 , A𝑥𝑛 , t ∗  𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡 +𝑀∆  𝑆 𝑥𝑛 ,  𝑦𝑛 ,𝐵𝑤1/,𝑡2 ∗ 𝑀∆  𝑇 𝑤1/,  𝑤2/ ,𝐴𝑥𝑛 ,𝑡2  +  1 − 2𝑎  𝑚𝑎𝑥  𝑀∆ 𝑆 𝑥𝑛 ,  𝑦𝑛 ,𝐴𝑥𝑛 , 𝑡 ,𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡 ,𝑀∇ 𝑆 𝑥𝑛 ,  𝑦𝑛 ,𝐴𝑥𝑛 , 𝑡 ,𝑀∇(𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡)             𝑠 𝑑𝑠 ≥    𝑠 𝑑𝑠𝑚 (𝑥𝑛 , 𝑦𝑛 ,𝑤1/, 𝑤2/  𝑡) 0𝑀∇(𝑆 𝑥𝑛 , 𝑦𝑛  ,   𝑇 𝑤1/, 𝑤2/ ,   𝑡)0     Letting   n → ∞, we have    𝑠 𝑑𝑠 ≥    𝑠 𝑑𝑠limn→∞𝑚(𝑥𝑛 , 𝑦𝑛 ,𝑤1/, 𝑤2/  𝑡) 0limn→∞𝑀∇(𝑆 𝑥𝑛 , 𝑦𝑛  ,   𝑇 𝑤1/, 𝑤2/ ,   𝑡)0, but    limn→∞ m(xn, yn, 𝑤1/,  𝑤2/, t)  = ∅       𝑎  1 ∗  𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡 +𝑀∆  𝑧1 ,𝐵𝑤1/,𝑡2 ∗ 𝑀∆  𝑇 𝑤1/,  𝑤2/ , 𝑧1,𝑡2  +  1 − 2𝑎  𝑚𝑎𝑥  1,𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡 ,1,𝑀∇(𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡)           = ∅     𝑎   𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡 +𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡  +  1 − 2𝑎  𝑚𝑎𝑥  𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡          = ∅ [2𝑎 + (1 – 2𝑎)] 𝑀∆ 𝑇 𝑤1/,  𝑤2/ ,𝐵𝑤1/, 𝑡    𝑠 𝑑𝑠 𝑀∇(𝐶,T(w1/,   w2/),𝑡) 0 ≥    𝑠 𝑑𝑠∅(𝑀∆(T(w1/,   w2/),   𝐵w1/,   𝑡)) 0        ISSN 2347-1921                                                            2611| P a g e                                                    N o v e m b e r  1 1 ,  2 0 1 4  Since, z1 = Bw1/ 𝜖 C, we have   𝑠 𝑑𝑠𝑀∆(𝐵w1/,   T( w1/, w2/),   𝑡)) 0 ≥   𝑠 𝑑𝑠 𝑀∇(𝐶,   T(w1/,   w2/),   𝑘𝑡) 0       ≥   𝑠 𝑑𝑠∅(𝑀∆(T(w1/,   w2/),   𝐵w1/,   𝑡)) 0 ⇒  𝑠 𝑑𝑠𝑀∆(𝐵w1/,   T( w1/, w2/),   𝑡)) 0  ≥   𝑠 𝑑𝑠∅(𝑀∆(𝐵w1/,T(w1/,   w2/),   𝑡)) 0       ≥   𝑠 𝑑𝑠(𝑀∆( 𝐵w1/,   T(w1/,   w2/),   𝑡)) 0 which is a contradiction. Hence Bw1/ 𝜖 T(w1/, w2/). Similarly, by putting x = yn, y = xn, u = w2/ and v = w1/ in (3), we get Bw2/ 𝜖 T(w2/, w1/). Again by taking x = w1, y = w2, and u = yn and v = xn in (3), we get m(w1, w2, yn, xn, t) = ∅       𝑎   𝑀∆ 𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 , 𝑡 ∗  𝑀∆ 𝑇 𝑦𝑛 , 𝑥𝑛 ,𝐵𝑦𝑛 , 𝑡 +𝑀∆  𝑆 𝑤1 ,𝑤2 ,𝐵𝑦𝑛 ,𝑡2 ∗ 𝑀∆  𝑇 𝑦𝑛 , 𝑥𝑛 ,𝐴𝑤1 ,𝑡2  +  1 − 2𝑎  𝑚𝑎𝑥  𝑀∆ 𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 , 𝑡 ,𝑀∆ 𝑇 𝑦𝑛 , 𝑥𝑛 ,𝐵𝑦𝑛 , 𝑡 ,𝑀∇ 𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 , 𝑡 ,𝑀∇(𝑇 𝑦𝑛 , 𝑥𝑛 ,𝐵𝑦𝑛 , 𝑡)             𝑠 𝑑𝑠 ≥    𝑠 𝑑𝑠𝑚 𝑤1 , 𝑤2 ,𝑦𝑛 ,𝑥𝑛 ,𝑡 0 𝑀∇(𝑆 𝑤1 ,𝑤2 ,   𝑇 𝑦𝑛 ,𝑥𝑛  ,𝑡)0, hence   𝑠 𝑑𝑠limn→∞ (𝑀∇(𝑆 𝑤1 ,𝑤2 ,𝐶,𝑡)0     ≥   𝑠 𝑑𝑠limn→∞ 𝑚 𝑤1 , 𝑤2 ,𝑦𝑛 ,𝑥𝑛 ,𝑡 0                                       =   𝑠 𝑑𝑠  ∅(𝑀∆𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 ,𝑡))0  As z1 = Aw1 C, we have          𝑠 𝑑𝑠(𝑀∆𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 ,𝑡))0 ≥   𝑠 𝑑𝑠𝑀∇(𝑆 𝑤1 ,𝑤2 ,𝐶,𝑡0           ≥   𝑠 𝑑𝑠∅(𝑀∆𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 ,𝑡))0           ≥   𝑠 𝑑𝑠𝑀∆𝑆 𝑤1 ,𝑤2 ,𝐴𝑤1 ,𝑡0 Which is a contradiction. Hence Aw1  S(w1, w2). Similarly, by putting x =w2, y = w1, u = xn and v = yn in (3), we get z2 = Aw2 S(w2, w1).  Hence (w1, w2) C(A, S) and (w1/, w2/) C(B, T). Now (4), gives AAw1 = Aw1, BBw1/ = Bw1/ and S(Aw1, Aw2) = T(Bw1/, Bw2/) AAw2 = Aw2, BBw2/ = Bw2/ and S(Aw2/, Aw1/) = T(Bw2/, Bw1/). But we have z1 = Aw1 = Bw1/, z2 = Aw2 = B w2/. This gives,  Az1 = z1 = Bz1 and S(z1, z2) = T(z1, z2) Az2 = z2 = Bz2 and S(z2, z1) = T(z2, z1) Also, weak compatibility of (A, S) gives AS(w1, w2)  S(Aw1, Aw2)  z1 = Bz1 = Az1 AS(w1, w2)  S(Aw1, Aw2) = S(z1, z2) = T(z1, z2). Similarly, we can have z2 = Bz2 = Az2 S(z2, z1) = T(z2, z1). Hence    (z1, z2) is a common coupled fixed point of the mappings A, B, S and T. Example:Let X = R and a * b = ab and M (x, y, t) = 𝑡𝑡+𝑑  (𝑥 ,𝑦)  then           (X, M,) is a fuzzy metric space. Define A, B: X → X and                      S, T: X  X → 𝜅(X) by setting       ISSN 2347-1921                                                            2612| P a g e                                                    N o v e m b e r  1 1 ,  2 0 1 4   Ax =  2𝑥 − 1,𝑥 ≤ 1 3 ,𝑥 > 1   ,    Bx =  2 − 𝑥, 1 ≤  𝑥 ≤ 2 3 ,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒      and S(x, y) =   𝑥 + 𝑦 − 4, 𝑥 + 𝑦 + 2 , 𝑖𝑓 𝑥, 𝑦 0, 3] x − 1, y − 1 ,   otherwise . T(x, y) =   2𝑥 − 𝑦 + 1, 3𝑥 + 𝑦 , 𝑥 < 𝑦 𝑥 − 2𝑦 − 1,   𝑥 + 3 , x ≥ y  Consider the sequences, {xn} = 1 - 1𝑛 and {yn} = 1 + 1𝑛 , then  limn→∞ Axn = limn→∞ Byn → 1 𝜖 limn→∞S(xn, yn) ∩ limn→∞T(yn, xn)  limn→∞Ayn = limn→∞Bxn → 3 𝜖 limn→∞S(yn, xn) ∩ limn→∞T(xn, yn) This shows that (A, B) is tangential w.r.t (S, T) A1 = B1 = 1 𝜖 [0, 6] = S(1, 3) ∩ T(3, 1)  Also,  A3 = B3 = 3 ϵ [0, 6] = S(3, 1) ∩ T(1, 3) Hence all the conditions of above theorems are satisfied and (1, 3) is a coupled fixed point of the maps A, B, S and T. 3.2 Theorem: Let A, B : X → X and S, T : X X X) be single and set-valued mappings satisfying the following conditions: (1) there exist contained coupled weak tangential points (z1, z2) to the mappings A and B. (2) (A, B) is tangential with respect to (S, T).  3    𝑠 𝑑𝑠 ≥    𝑠 𝑑𝑠𝑚(𝑥 ,   𝑦 ,   𝑢 ,   𝑣 ,   𝑡) 0𝑀∇(𝑆 𝑥 ,   𝑦 ,   𝑇 𝑢 ,   𝑣 ,   𝑡)0    where m(x, y, u, v, t)  = ∅       𝑎  𝑀∆ 𝑆 𝑥, 𝑦 ,𝐴𝑥, 𝑡 , 𝑀∆ 𝑇 𝑢, 𝑣 ,𝐵𝑢, 𝑡  +𝑎  𝑀∆  𝑆 𝑥, 𝑦 ,𝐵𝑢,𝑡2 , 𝑀∆  𝑇 𝑢, 𝑣 ,𝐴𝑥,𝑡2  +  1 − 2𝑎  𝑚𝑎𝑥  𝑀∆ 𝑆 𝑥, 𝑦 ,𝐴𝑥, 𝑡 ∗ 𝑀∆ 𝑇 𝑢, 𝑣 ,𝐵𝑢, 𝑡 ,𝑀∇ 𝑆 𝑥, 𝑦 ,𝐴𝑥, 𝑡 ,𝑀∇(𝑇 𝑢, 𝑣 ,𝐵𝑢, 𝑡)           a. AAa = Aa = BBc, S(Aa, Ab) = T(Bc, Bd) and AAb = Ab = BBd,     S(Ab, Aa) = T(Bd, Bc) for (a, b) ϵ C(A, S) and (c, d) ϵ C(B, T) b. the pair (A,S) is weakly compatible, for all x, y, u, v ϵ X, 0 a < 1 and ∅: [0,1] be a non-decreasing map such that ∅(t) > t, t Then A, B, S and T have a common coupled fixed point in X. Proof The result follows directly from theorem (3.1). References: [1]. Bhaskar T. G. and Lakshmikantham V.: Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis: Theory, Methods and Applications, Vol.65, no.7, pp. 1379-1393, 2006.  [2]. Markin J.T.: Continuous dependence of fixed point sets , Proc. Am. Math. Soc. 38(1973)-0313897-4. [3]. Nadler S.: Multi-valued contraction mappings, Pacific J. Math. 30 (1972), MR 40#8035.Zbl 187.45002. [4]. Zadeh L.A.: Fuzzy Sets, Information and Control, Vol. 89, pp. 338-353, 1965. 

COMMON FIXED POINT FOR TANGENTIAL MAPS OF GREGUS TYPE ON FUZZY METRIC SPACES

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COMMON FIXED POINT FOR TANGENTIAL MAPS OF GREGUS TYPE ON FUZZY METRIC SPACES

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