Generalised Common Fixed Point Theorems of A-Compatible and S-Compatible Mappings

Shahidur Rahman, Yumnam Rohen, M. Popeshwar Singh

  Open Access OPEN ACCESS  Peer Reviewed PEER-REVIEWED

Generalised Common Fixed Point Theorems of A-Compatible and S-Compatible Mappings

Shahidur Rahman1, Yumnam Rohen1,, M. Popeshwar Singh1

1National Institute of Technology Manipur, Imphal, India

Abstract

In this paper we prove a common fixed point theorem of four self mappings satisfying a generalized inequality using the concept of A-compatible and S-compatible mappings. Our result generalizes many earlier related results in the literature.

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Cite this article:

  • Rahman, Shahidur, Yumnam Rohen, and M. Popeshwar Singh. "Generalised Common Fixed Point Theorems of A-Compatible and S-Compatible Mappings." American Journal of Applied Mathematics and Statistics 1.2 (2013): 27-29.
  • Rahman, S. , Rohen, Y. , & Singh, M. P. (2013). Generalised Common Fixed Point Theorems of A-Compatible and S-Compatible Mappings. American Journal of Applied Mathematics and Statistics, 1(2), 27-29.
  • Rahman, Shahidur, Yumnam Rohen, and M. Popeshwar Singh. "Generalised Common Fixed Point Theorems of A-Compatible and S-Compatible Mappings." American Journal of Applied Mathematics and Statistics 1, no. 2 (2013): 27-29.

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1. Introduction

The first important result in the theory of fixed point of compatible mappings was obtained by Gerald Jungck in 1986 [2] as a generalization of commuting mappings. In 1993 Jungck, Murthy and Cho [3] introduced the concept of compatible mappings of type (A) by generalizing the definition of weakly uniformly contraction maps. Pathak and Khan [6] introduced the concept of A-compatible and S-compatible by splitting the definition of compatible mappings of type (A). Fixed point results of compatible mappings are found in [1-8][1].

Sharma and Sahu [8] proved the following theorem.

THEOREM 1.1 Let A, S and T be three continuous mappings of a complete metric space (X, d) into itself satisfying the following conditions:

(i)   A commutes with S and T respectively

(ii)  S (X) ⊆ A(X) and T(X) ⊆ A(X)

(iii) [d(Sx, Tx)]2a1d(Ax, Sx)d(Ay, Ty)+a2d(Ay, Sx)d(Ax, Ty)+a3d(Ax, Sx)d(Ax, Ty) +a4d(Ay, Ty)d(Ay, Sx)+a5d2(Ax, Ay)

For all x, yX, where ai ≥ 0, i = 1, 2, 3, 4, 5 and a1+a4+ a5< 1, 2a1+3a3+2a5<2.

Then A, S and T have a unique common fixed point in X.

Murthy [6] pointed out that the constraints taken by Sharma and Sahu in condition (iii) of theorem 1.1 is not true and suggested the corrected replacement as max {a1+2a3+ a5, a1+2a4+a5, a2+ a5} < 1 and proved a new fixed point theorem.

The aim of this paper is to prove a common fixed point theorem of S-compatible mappings in metric space by considering four self mappings. Further we give another common fixed point theorem of A-compatible mappings.

2. Preliminaries

Following are definitions of types of compatible mappings.

Definition 2.1 [2]: Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be compatible if d(ASxn, SAxn) = 0 whenever {xn} is a sequence in X such that Axn = Sxn = t for some tX.

Definition 2.2 [3]: Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be compatible of type (A) if d(ASxn, SSxn) = 0 and d(SAxn, AAxn) = 0 whenever {xn} is a sequence in X such that forAxn = Sxn = t for some tX.

Definition 2.3 [5]: Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be A-compatible if d(ASxn, SSxn) = 0 whenever {xn} is a sequence in X such that forAxn = Sxn = t for some tX.

Definition 2.4 [5]: Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be S-compatible if d(SAxn, AAxn) = 0 whenever {xn} is a sequence in X such that forAxn = Sxn = t for some tX.

Proposition 2.5 [6]: Let A and S be mappings from a complete metric space (X, d) into itself. If a pair (A, S) is A-compatible on X and St = At for t X, then ASt = SSt.

Proposition 2.6 [6]: Let A and S be mappings from a complete metric space (X, d) into itself. If a pair (A, S) is S-compatible on X and St = At for t X, then SAt = AAt.

Proposition 2.7 [6]: Let A and S be mappings from a complete metric space (X, d) into itself. If a pair (A, S) is A-compatible on X and Axn =Sxn = t for t X, then SSxnAt if A is continuous at t.

Proposition 2.8 [6]: Let A and S be mappings from a complete metric space (X, d) into itself. If a pair (A, S) is S-compatible on X and Axn =Sxn = t for t X, then AAxnSt if S is continuous at t.

Now we prove the following theorem.

LEMMA 2.9 Let A, B, S and T be mapping from a metric space (X, d) into itself satisfying the following conditions:

(1) A(X) ⊆ T(X) and B(X) ⊆ S(X)

(2) [d(Ax, Bx)]2a1d(Ax, Sx)d(By, Ty)+a2d(By, Sx)d(Ax, Ty)+a3d(Ax, Sx)d(Ax, Ty)+a4d(By, Ty)d(By, Sx) +a5d2(Sx, Ty)

where a1+ a2 +2a3 +a4+ a5< 1 and a1, a2, a3, a4, a5 ≥ 0

(3) Let x0X then by (1) there exists x1X such that Tx1 = Ax0 and for x1 there exists x2X such that Sx2 = Bx1 and so on. Continuing this process we can define a sequence {yn} in X such that

then the sequence {yn} is Cauchy sequence in X.

Proof. By condition (2) and (3), we have

Where

Since a1+ a2 +2a3 +a4+ a5< 1 and a1, a2, a3, a4, a5 ≥ 0.

In order to satisfy the inequation, one value of λ will be positive and the other will be negative. We also note that the sum and product of the two values of λ is less than 1 and -1 respectively. Neglecting the negative value, we have where 0<p<1.

Hence {yn} is Cauchy sequence.

3. Main Results

We prove the following theorem.

THEOREM 3.1: Let A, B, S and T be self maps of a complete metric space (X, d) satisfying the following conditions:

(1) A (X) ⊆ T(X) and B(X) ⊆ S(X)

(2) [d(Ax, Bx)]2 ≤ a1d(Ax, Sx)d(By, Ty)+a2d(By, Sx)d(Ax, Ty)+a3d(Ax, Sx)d(Ax, Ty)+a4d(By, Ty)d(By, Sx) +a5d2(Sx, Ty)

where a1+ a2 +2a3 +a4+ a5< 1 and a1, a2, a3, a4, a5 ≥ 0

(3) Let x0 ∈ X then by (1) there exists x1X such that Tx1 = Ax0 and for x1 there exists x2X such that Sx2 = Bx1 and so on. Continuing this process we can define a sequence {yn} in X such that

then the sequence {yn} is Cauchy sequence in X.

(4) One of A, B, S or T is continuous.

(5) [A, S] and [B, T] are S-compatible mappings on X.

Then A, B, S and T have a unique common fixed point in X.

Proof: By lemma 2.9, {yn} is Cauchy sequence. Since X is complete, there exists a point zX such that lim yn = z as n → ∞. Consequently subsequences Ax2n, Sx2n, Bx2n-1 and Tx2n+1 converges to z.

Let S be a continuous mapping. Since A and S are S-compatible mappings on X, then by proposition 2.8., we have AAx2nSz and SAx2nSz as n → ∞.

Now by condition (2) of lemma 2.9, we have

As n→∞, we have

which is a contradiction. Hence Sz = z,

Now

Letting n→∞, we have [d(Az, z)]2a3[d(Az, z)]2. Hence Az = z.

Now since Az = z, by condition (1), zT(X). Also T is self map of X so there exists a point uX such that z = Az = Tu. More over by condition (2), we obtain,

i.e., [d(z, Bu)]2a4[d(z, Bu)]2.

Hence Bu = z i.e., z = Tu = Bu.

By condition (5), we have

Hence d(Tz, Bz) = 0 i.e., Tz = Bz.

Now,

i.e., [d(z, Tz)]2a2[d(z, Tz)]2 which is a contradiction. Hence z = Tz i.e, z = Tz = Bz.

Therefore z is common fixed point of A, B, S and T. Similarly we can prove that z is a common fixed point of A, B, S and T if any one of A, B or T is continuous.

Finally, in order to prove the uniqueness of z, suppose w be another common fixed point of A, B, S and T Then we have,

which gives [d(z, Tw)]2 ≤ a2 [d(z, Tw)]2. Hence z = w.

This completes the proof.

THEOREM 3.2: Let A, B, S and T be self maps of a complete metric space (X, d) satisfying the following conditions:

(1) A (X) ⊆ T(X) and B(X) S(X).

(2) [d(Ax, Bx)]2a1d(Ax, Sx)d(By, Ty)+a2d(By, Sx)d(Ax, Ty)+a3d(Ax, Sx)d(Ax, Ty) +a4d(By, Ty)d(By, Sx) +a5d2(Sx, Ty)

where a1+ a2 +2a3 +a4+ a5< 1 and a1, a2, a3, a4, a5 ≥ 0.

(3) Let x0X then by (1) there exists x1X such that Tx1 = Ax0 and for x1 there exists x2X such that Sx2 = Bx1 and so on. Continuing this process we can define a sequence {yn} in X such that

then the sequence {yn} is Cauchy sequence in X.

(4) One of A, B, S or T is continuous.

(5) [A, S] and [B, T] are A-compatible mappings on X.

Then A, B, S and T have a unique common fixed point in X.

Proof: Similar to theorem 3.1.

Remark:

(i) By taking a1= a2 =k1 and a3= a4 =k2 and a5=0 and (A, S) and (B, T) as compatible mappings theorem 3.1 reduces to theorem 1 of Bijendra and Chouhan [1].

(ii) By taking S = T and (A, S) and (A, T) as commuting mappings or compatible mappings of type (A) theorem 3.1 reduce to results of Murthy [6] and Sharma and Sahu [8] under certain conditions.

References

[1]  Bijendra Singh and M.S. Chauhan, On common fixed points of four mappings, Bull. Cal. Math. Soc. 88, 1996 451-456.
In article      
 
[2]  Jungck G., Compatible maps and common fixed points, Inter .J. Math. and Math. Sci. 9, 1986 771-779.
In article      CrossRef
 
[3]  Jungck G., Murthy P.P. and Cho Y.J., Compatible mappings of type (A) and common fixed points, Math. Japonica 38 1993, 381-390.
In article      
 
[4]  Koireng M., Leenthoi N. and Rohen Y., Common fixed points of compatible mappings of type (R) General Mathematics Notes, 10(1) May 2012, 58-62.
In article      
 
[5]  P. P. Murthy, Remarks on fixed point theorem of Sharma and Sahu, Bull. of Pure and Appl. Sc., 12 E(1-2), 1993 7-10.
In article      
 
[6]  H.K. Pathak and M.S. Khan, A comparison of various types of compatible maps and common fixed points, Indian J. pure appl. Math., 28(4): April 1997. 477-485.
In article      
 
[7]  H.K. Pathak, M.S. Khan and Reny George, Compatible mappings of type (A-1) and type (A-2) and common fixed points in fuzzy metric spaces, International Math. Forum, 2(11): 2007 515-524.
In article      
 
[8]  Sharma B. K. and Sahu N. K., Common fixed point of three continuous mappings, The Math. Student, 59 (1) 1991 77-80.
In article      
 
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