Common Fixed Point Theorem for Weakly ... - Semantic Scholar

International Mathematical Forum, 2, 2007, no. 57, 2831 - 2839

Common Fixed Point Theorem for Weakly Compatible Mapping H. K. Pathak School of Studies in Mathematics Pt. Ravishankar Shukla University Raipur (C.G.) 492010, India [email protected] Prachi Singh1 Department of Mathematics Govt.V.Y.T.PG.Autonomous College Durg (C.G.), 491001, India [email protected] Abstract In this paper, we prove some fixed point theorems for weakly compatible mappings by improving the conditions of Som [8].

1. Introduction After introduction of fuzzy sets by Zadeh [5], many researchers have defined fuzzy metric space in different ways. In this paper we are using the definition of Kramosil and Michalek [4]. Sessa [7] proved some theorem of commutativity by weakening the condition to weakly commutativity and Jungck [3] enlarged it for compatibility of two mappings. Recently, Jungck and Rohades [2] defined weakly compatible mappings of two mappings at their coincident points. In this paper we are using weakly compatible mappings. 2. Preliminaries and Definitions In this section, we recall some notions and definitions in fuzzy metric spaces. Definition 2.1(Schweizer and Sklar [1]). ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions: (a) ∗ is associative and commutative, (b) ∗ is continuous, 1

Corresponding author

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H. K. Pathak and P. Singh

(c) a ∗ 1 = a, ∀ a ∈ [0, 1], (d) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, for each a, b, c, d ∈ [0, 1]. Definition 2.2(Kramosil and Michalek [4]). A triplet (X, M, ∗) is a fuzzy metric space if X is an arbitrary set, ∗ is continuous t-norm and M is a fuzzy set on X × X × [0, ∞) → [0, 1] satisfying, ∀ x, y ∈ X, the following conditions: (1)M (x, y, 0) = 0, (2)M (x, y, t) = 1, ∀ t > 0 iff x = y, (3)M (x, y, t) = M(y, x, t), (4)M (x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s), t, s ∈ [0, 1), (5)M (x, y, .) : [0, ∞) → [0, 1] is left continuous. Remark 2.3(Gragiec [6]). It is easy to prove that M(x, y, .) is non-decreasing for every x, y ∈ X. If (X, M, ∗) is a fuzzy metric space we can say that M is a fuzzy metric on X. Let (X, d) be a metric space. Let a ∗ b = ab for every a, b ∈ [0, 1] and let Md : X × X × [0, ∞) → [0, 1] be the function defined, for all x, y ∈ X by Md (x, y, 0) = 0 and for t > 0 by Md (x, y, t) =

t . t + d(x, y)

The triplet (X, Md , ∗) is a fuzzy metric space and Md is called the fuzzy metric induced by d. Definition 2.3 (Grabeic [6]). A sequence {xn }∞ n=1 in a fuzzy metric space (X, M, ∗) is G-Cauchy if M(xn+p , xn , t) > 1 −  for every t > 0 and for every p > 0. Definition 2.4. A sequence {xn } ∈ X in a fuzzy metric space (X, M, ∗) converges to x ∈ X if M(xn , x, t) > 1 − , ∀ t > 0. Definition 2.5. Let (X, M, ∗) be a fuzzy metric space. Then M is called bicontinuous if the sequences {xn } and {yn } in X are such that {xn } → x and {yn } → y, t > 0 implies M(xn , yn , t) → M(x, y, t). Definition 2.6. A fuzzy metric space (X, M, ∗) is complete if every G-Cauchy sequence is convergent. If (X, M, ∗) is complete fuzzy metric space, then M is fuzzy metric on X. Definition 2.7 (Jungck and Rhoades [2]). Two self mappings A and B are said to be weakly compatible if they commute at their coincidence points i.e., ABu = BAu whenever Au = Bu, u ∈ X. Note that compatible mappings are weakly compatible but weakly compatible mappings are not necessarily compatible. Theorem 2.8( Som [8]). Let S and T are two continuous self mappings of a complete metric space (X, M, ∗). Let A be a self mapping of X satisfying the following conditions (1) A(X) ⊂ S(X) ∩ T (X),

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Common fixed point theorem

(2){A, S} and {A, T } are R-weakly commuting, (3) M(Ax, Ay, t) ≥ r[min{M(Sx, T y, t), M(Sx, Ax, t), M(Sx, Ay, t), M(T y, Ay, t)}] for all x, y ∈ X, where r : [0, 1] → [0, 1] is a continuous function such that (4) r(t) > t, for each t < 1 and r(t) = 1 for t = 1. Let the sequences {xn } and {yn } in X be such that {xn } → x and {yn } → y, t > 0 implies M(xn , yn , t) → M(x, y, t). Then A, S, T have a common fixed point in X. Theorem 2.9( Som [8]). Let S and T be two continuous self mappings of a complete fuzzy metric space (X, M, ∗). Let A and B be two self mappings of X satisfying the following conditions (1 ) A(X) ∪ B(X) ⊂ S(X) ∩ T (X), (2 ) {A, T } and {B, S} are R-weakly commuting pairs, (3 ) aM(T x, Sy, t) + bM(T x, Ax, t) + cM(Sy, By, t) + max{M(Ax, Sy, t), M(By, T x, t)} ≤ qM(Ax, By, t) for all x, y ∈ X, where a, b, c ≥ 0, q > 0 with q < a + b + c < 1. Then A, B, S and T have a unique common fixed point. 3. Main Results Now we state and prove our main theorem for weakly compatible mappings. Theorem 3.1. Let S and T be two continuous self mappings of a complete F M− space (X, M, ∗) such that a ∗ b = min(a, b) for all a, b in X. Let A be a self mapping of X satisfying the following conditions: (1) A(X) ⊂ S(X) ∩ T (X), (2) {A, S} and {A, T } are weakly compatible, (3) M(Ax, Ay, t) ≥ r(min{M(Sx, T y, t), M(Sx, Ax, t), M(Sx, Ay, t), M(T y, Ay, t)}) for all x, y ∈ X and t > 0, where r : [0, 1] → [0, 1] is some continuous function such that (4) r(t) > t, for each t < 1. Then A, S, T have a common fixed point in X. Proof. Let x0 ∈ X be any arbitrary point. Since A(X) ⊂ S(X) then there must exists a point x1 ∈ X such that Ax0 = Sx1 . Also since A(X) ⊂ T (X), there exists another point x2 ∈ X such that Ax1 = T x2 . In general, we get a sequence {yn } recursively as y2n = Sx2n+1 = Ax2n and y2n+1 = T x2n+2 = Ax2n+1 ∀n ∈ N ∪ {0}. Let M2n = M(y2n+1 , y2n, t) = M(Ax2n+1 , Ax2n , t). Then M(Ax2n+2 , Ax2n+1 , t) = M2n+1 .

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Using inequality (3), we get M2n+1 ≥ r(min{M(Sx2n+2 , T x2n+1 , t), M(Sx2n+2 , Ax2n+2 , t), M(Sx2n+2 , Ax2n+1 , t), M(T x2n+1 , Ax2n+1 , t)}) = r(min{M(Ax2n+1 , Ax2n , t), M(Ax2n+1 , Ax2n+2 , t), M(Ax2n+1 , Ax2n+1 , t), M(Ax2n , Ax2n+1 , t)}) = r(min{M2n , M2n+1 , 1, M2n })

(3.1)

If M2n > M2n+1 , then by definition of r we have M2n+1 ≥ r(M2n+1 ) > M2n+1 , a contradiction. So M2n+1 ≥ M2n . Thus, from (3.1) we get M2n+1 ≥ r(M2n ) > M2n

(3.2)

Hence {M2n }∞ n=0 is an increasing sequence of positive numbers in [0,1] and, therefore, tends to a limit L ≤ 1. We claim that L = 1. If L < 1, then on taking limit n → ∞ in (3.2) we get L ≥ r(L) ≥ L; i.e., r(L) = L, which contradicts the fact that L < 1. Hence, L = 1. Now for any positive integer p, t t M(Axn , Axn+p , t) ≥ M(Axn , Axn+1 , ) ∗ M(Axn+1 , Axn+2 , ) ∗ ........ p p .... ∗ M(Axn+p−1 , Axn+p , pt ) > (1 − ) ∗ (1 − ) ∗ (1 − ) ∗ · · · p − times = 1 − . Thus, M(Axn , Axn+p , t) > 1 − , ∀ t > 0. Hence {Axn } is a Cauchy sequence. Since X is complete , {Axn } tends to a limit, say z ∈ X. Hence the subsequences {Sxn } and {T xn } of {Axn } also tends to the same limit. Since A(X) ⊂ S(X), there must exists a point u ∈ X, such that z = Su. Then by (3) we have M(Au, Axn , t) ≥ r(min{M(Su, T xn , t), M(Su, Au, t), M(Su, Axn, t), M(T xn , Axn , t)}). Taking limit as n → ∞ in the above inequality we get M(Au, z, t) ≥ r(min{M(z, z, t), M(z, Au, t), M(z, z, t), M(z, z, t)}) ≥ r(min{1, M(z, Au, t), 1, 1}) ≥ r(M(z, Au, t)) > M(z, Au, t),

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Common fixed point theorem

which gives Au = z. Therefore, Au = z = Su. Similarly since A(X) ⊂ T (X), there must exists a point v ∈ X, such that z = T v. Then by (3) we have Av = z = T v. Hence Au = z = Su = Av = T v. Since the pair of mappings (A, S) is weakly compatible so ASu = SAu gives Az = Sz. Now we show that z is a fixed point of A. Suppose Az = z, then by (3) we get M(Az, Av, t) ≥ r(min{M(Sz, T v, t), M(Sz, Az, t), M(Sz, Av, t), M(T v, Av, t)}) ≥ r(min{M(Az, z, t), M(Az, Az, t), M(Az, z, t), M(z, z, t)}) ≥ r(min{M(Az, z, t), 1, M(Az, z, t), 1}) ≥ r(M(Az, z, t)) > M(Az, z, t), which gives Az = z = Sz. Similarly, when the pair (A, T ) is weakly compatible we obtain Az = z = T z. Thus, we have Az = z = Sz = T z. (3.3) Hence z is a common fixed point of A, S and T .   Now for the uniqueness of z, suppose z and z , z = z are two common fixed     points of A, S and T ; i.e., Az = z = Sz = T z and Az = z = Sz = T z . Then, by (3) we have    M(Az, Az , t) ≥ r(min{M(Sz, T z , t), M(Sz, Az, t), M(Sz, Az , t),   M(T z , Az , t)}) 













i.e., M(z, z , t) ≥ r(min{M(z, z , t), M(z, z, t), M(z, z , t), M(z , z , t)}) 



i.e., M(z, z , t) ≥ r(min{M(z, z , t), 1, M(z, z , t), 1}) ≥ M(z, z , t), 

which gives z = z . This completes the proof. Example 3.2. Let X = [−1, 2], d(x, y) = |x − y|, ∀x, y ∈ X, A, S : X → X defined by A(x) =

⎧ ⎪ ⎨

1

⎪ ⎩

1+

⎧ ⎪ ⎪ ⎪ ⎨

3 4

1 2 x 32

1 + 14 x2 1 S(x) = ⎪ 2 ⎪ ⎪ ⎩ 1 − 18 x2 For −1 ≤ x < 1, we have

if − 1 ≤ x ≤ 1, if 1 < x < 54 , 5 if ≤ x ≤ 2, 4 if − 1 ≤ x < 1, if x = 1, if 1 < x < 54 , 5 if ≤ x ≤ 2. 4

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A(x) = 1 , S(x) = 1 + 14 x2 , SA(x) = 1 and AS(x) = 34 , then M(ASx, SAx, t) = t+t 1 and M(Ax, Sx, Rt ) = t+ 1tx2 R . 4 4 To test R-weakly commuting, we observe that M(ASx, SAx, t) ≥ M(Ax, Sx, Rt ) which gives R ≥ x12 , but there exists no R for x = 0 ∈ [−1, 1). Hence A and S are not R-weakly commuting. However, for x = 1, we have Sx = Ax = 1, SAx = ASx = 1. Hence A and S are weakly compatible at x = 1. Example 3.3. Let r(t) : [0, 1] → [0, 1] be defined by r(t) = Then it is clear that r(t) > t ∀t < 1 and r(1) = 1. Define A, S, T : [0, 1] → [0, 1] by ∀ x ∈ [0, ∞)

Ax = 1 

Sx = 

Tx =

√ t ∀t ∈ [0, 1].

0 1

if x ∈ [0, 1), if x ∈ [1, ∞),

2 1

if x ∈ [0, 1), if x ∈ [1, ∞).

Now we see that A(X) = {1}, S(X) = {0, 1}, T (X) = {1, 2} and so  A(X) ⊂ S(X) T (X). Then, for d(x, y) = |x − y| we have M(T (1), 1, t) = 1, M(S(1), 1, t) = 1 and M(A(1), 1, t) = 1. Thus, 1 is the common fixed point of A, S and T. Also M(AS(1), SA(1), t) = 1 and M(AT (1), T A(1), t) = 1. Hence, the pairs of mappings (A, S) and (A, T ) are weakly compatible. 1 If xn = ( 2n ), then, for n → ∞ we have

M(SAxn , ASxn , t) →

t t+1

and M(T Axn , AT xn , t) →

t t+1

so SAxn = ASxn and T Axn = AT xn . Hence, the pairs of mappings (A, S) and (A, T ) are not commutative. Case 1. If x, y ∈ [0, 1), then we have M(Ax, Ay, t) = 1, M(Sx, T y, t) = t t t , M(Sx, Ay, t) = t+1 , M(T y, Ay, t) = t+1 and so M(Sx, Ax, t) = t+1 

t , t+2

t t = r( t+2 ) M(Ax, Ay, t) = 1 ≥ t+2 ≥ r(min{M(Sx, T y, t), M(Sx, Ax, t), M(Sx, Ay, t), M(T y, Ay, t)}).

Common fixed point theorem

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Case 2. If x ∈ [0, 1), y ∈ [1, ∞), then we have M(Ax, Ay, t) = 1, M(Sx, T y, t) = t t t , M(Sx, Ax, t) = t+1 , M(Sx, Ay, t) = t+1 , M(T y, Ay, t) = 1 and so t+1 

t t = r( t+1 ) M(Ax, Ay, t) = 1 ≥ t+1 ≥ r(min{M(Sx, T y, t), M(Sx, Ax, t), M(Sx, Ay, t), M(T y, Ay, t)}).

Case 3. It is same as Case 2 as M is a symmetric function. Case 4. If x, y ∈ [1, ∞), then we have M(Ax, Ay, t) = 1, M(Sx, T y, t) = 1, M(Sx, Ax, t) = 1, M(Sx, Ay, t) = 1, M(T y, Ay, t) = 1 and so M(Ax, Ay, t) = 1 = r(1) = r(min{M(Sx, T y, t), M(Sx, Ax, t), M(Sx, Ay, t), M(T y, Ay, t)}). Hence in all cases, we see that all the conditions of Theorem 3.1 are satisfied. Theorem 3.4. Let S and T be two continuous self mappings of a complete fuzzy metric space (X, M, ∗). Let A and B be two self mappings of X satisfying (1) A(X) ∪ B(X) ⊂ S(X) ∩ T (X), (2) {A, T } and {B, S} are weakly compatible pairs, and (3) aM(T x, Sy, t) + bM(T x, Ax, t) + cM(Sy, By, t) + max{M(Ax, Sy, t), M(By, T x, t)} ≤ qM(Ax, By, t) for all x, y ∈ X and t > 0, where a, b, c ≥ 0 with 0 < q < a + b + c < 1. Then A, B, S and T have a unique common fixed point. Proof. Let x0 ∈ X be any arbitrary point. Since A(X) ⊂ S(X), there must exists a point x1 ∈ X such that Ax0 = Sx1 . Also since A(X) ⊂ T (X), then there exists another point x2 ∈ X such that Ax1 = T x2 . In general, we get a sequence {yn } recursively as y2n = Sx2n+1 = Ax2n and y2n+1 = T x2n+2 = Ax2n+1 for all n ∈ N ∪ {0}. > 1 a Cauchy Using inequality (2), we get similar as Som [8] that for a+b q−c sequence in X. Hence the sequences {Ax2n }, {Bx2n+1 }, {Sx2n+1 } and {T x2n+2 } are Cauchy and converge to same limit, say z. Since A(X) ⊂ S(X) and A(X) ⊂ T (X) so there must exists a point u, v ∈ X, such that z = Su and z = T v. Then by (2) we have aM(T xn , Su, t) + bM(T xn , Axn , t) + cM(Su, Bu, t)+ max{M(Axn , Su, t), M(Bu, T xn , t)} ≤ qM(Axn , Bu, t) aM(z, z, t)+bM(z, z, t)+cM(z, Bu, t)+max{M(z, z, t), M(Bu, z, t)} ≤ qM(z, Bu, t) i.e.,

a+b+cM(z, Bu, t)+max{1, M(Bu, z, t)} ≤ qM(z, Bu, t)

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H. K. Pathak and P. Singh

a+b+cM(z, Bu, t)+1 ≤ qM(z, Bu, t)

a + b + 1 ≤ (q − c)M(z, Bu, t) a+b+1 > 1, i.e., M(z, Bu, t) ≥ q−c which yields Bu = z, hence Bu = z = Su and aM(T v, Sxn , t) + bM(T v, Av, t) + cM(Sxn , Bxn , t)+ max{M(Av, Sxn , t), M(Bxn , T v, t)} ≤ qM(Av, Bxn , t)

i.e.,

aM(z, z, t) + bM(z, Av, t) + cM(z, z, t) + max{M(Av, z, t), M(z, z, t)} ≤ qM(Av, z, t) i.e.,

a+bM(z, Av, t)+c+max{M(Av, z, t), 1} ≤ qM(Av, z, t)

a + c + 1 ≤ (q − b)M(Av, z, t) a+c+1 i.e., M(Av, z, t) ≥ > 1, q−b which yields Av = z. Hence Av = z = T v. Since {A, T } and {B, S} are weakly compatible, AT v = T Av gives Az = T z and BSu = SBu gives Bz = Sz. Now we show that z is a fixed point of A. Suppose Az = z, then by (2) we have aM(T z, Su, t) + bM(T z, Az, t) + cM(Su, Bu, t)+ max{M(Az, Su, t), M(Bu, T z, t)} ≤ qM(Az, Bu, t) i.e.,

i.e., i.e.,

aM(Az, z, t) + bM(Az, Az, t) + cM(z, z, t) + max{M(Az, z, t), M(z, Az, t)} ≤ qM(Az, z, t)

(a + b + 1)M(Az, z, t) + c ≤ qM(Az, z, t)

c > 1, q−a−b−1 which yields Az = z. Hence Az = z = T z. Similarly, z is the fixed point of B, so Bz = Sz = z. Thus, Az = Bz = Sz = z = T z; i.e., z is the common fixed point of A, B, S and T.   Now for the uniqueness of z, suppose z , z = z , is another common fixed point     of A,B, S and T ; i.e. Az = Bz = z = Sz = T z and Az = Bz = z = Sz =  T z . Then by (2) we have    aM(T z, Sz , t) + bM(T z, Az, t) + cM(Sz , Bz , t)    + max{M(Az, Sz , t), M(Bz , T z, t)} ≤ qM(Az, Bz , t)    i.e., aM(z, z , t) + bM(z, z, t) + cM(z , z , t)    + max{M(z, z , t), M(z , z, t)} ≤ qM(z, z , t) i.e.,

M(Az, z, t) ≥

i.e.,

aM(z, z , t) + b + c + M(z , z, t) ≤ qM(z, z , t)







Common fixed point theorem

i.e.,



M(z, z , t) ≥

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b+c >1 q−1−a



which yields z = z . Hence z is the unique common fixed point of A, B, S and T.

References [1] A.Sklar and B.Schweizer, Statistical metric spaces, Pacefic J.Math., 10 (1960), 314-334. [2] B.E. Rhoades and G.Jungck , Fixed points for set valued functions without continuity, Indian J.pure and Appl.Math. 29(3)(1998), 227-238. [3] G.Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9(1986), 771-779. [4] I. Kramosil and J. Michalek , Fuzzy metric and statistical metric spaces, Kybernetica, 11(1975), 326-334. [5] L.A.Zadeh, Fuzzy Sets,Inform. Control 8(1965), 338-353. [6] M.Grabeic,Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27(1988), 385-389. [7] S.Sessa , On a weak commutativity condition in a fixed point considaration, Publ. Inst. Math. 32(46)(1986), 149-153. [8] T.Som , Some fixed point theorems on metric and Banach spaces, Indian J. Pure and Appl. Math., 16(6)(1985), 575-585. Received: April 5, 2007