Fixed Point Theorem on Weakly Compatible Mappings in Metric Space

1. INTRODUCTION

Gerald Jungck gave a common fixed point theorem for driving mappings, which sums up the Banach's fixed point theorem. This outcome was additionally summed up and reached out in different manners by numerous creators. S. Sessa5 characterized weak commutativity and demonstrated common fixed point theorem for weakly commuting maps. Further G. Jungck1 presented the idea of good guides which is more vulnerable than weakly commuting maps. A short time later, Jungck and Rhoades4 characterized a more vulnerable class of guides known as weakly compatible maps. The idea of occasionally weakly compatible mappings in metric space is presented by A1-Thagafi and Shahzad10 which is generally broad among all the commutativity ideas. The reason for this paper is to demonstrate a common fixed point theorem for four self-maps in which two sets are occasionally weakly compatible.

Definition 2.1. Let S and T be two self-mappings of a metric space (X, d) then S and T are said to be commuting on X if STx = TSx for all x in X . Definition 2.2.Two self-maps S and T of a metric space (X, d) are said to be compatible mappings if d (STxn, TSxn)=0, whenever is a sequence < xn > in X such that = Txn = t for some tX.

Definition 2.4.Two self-maps S and T of a metric space ( X , d ) are said to be occasionally weakly compatible if S and T are commuting at some coincident points. That means S and T are not commuting at all coincidence points. Weakly compatible mappings are occasionally weakly compatible mappings but converse is not true. P. C. Lohani and V. H. Badshah proved the following theorem. Theorem 2.5: Let P,Q, S and T be self-mappings from a complete metric space ( X , d ) into itself satisfying the following conditions for all x, y in X. then P, Q, S and T have a unique common fixed point in X. Now we generalize Theorem2.5 using occasionally weakly compatible mappings and associated sequence. Associated Sequence2.69: Suppose P, Q, S and T are self-maps of a metric space ( X , d ) satisfying the condition (2.5.1).Then for an arbitrary x0 X such that Sx0 = Qx1 and for x1, there exists a point x2 in X such that Tx1 and so on. Proceeding in the similar manner, we can define a sequence in X such that y2n and y2n+ 1 = Px2n+ 2 = Tx2n+ 1 for n. We shall call this sequence as an ―Associated sequence of x0, relative to the four self-maps P,Q,S and T. Now we prove a lemma which plays an important role in our main Theorem. Lemma2.7: Let P, Q, S and T be self-mappings from a complete metric space ( X , d ) into itself satisfying the conditions (2.5.1) and (2.5.2).Then the associated sequence{yn } relative to four self-maps is a Cauchy sequence in X. Theorem 2.8 ([7]). Let A Mm,m(R+), The followings are equivalent. (i) A is convergent towards zero; (ii) An as n ; (iii) the eigenvalues of A are in the open unit disc, that is ||< 1, for every C with det(A - I) = 0; (iv) the matrix I -A is nonsingular and (I – A)-1 = I + A +…….+ An + …… ;

(v) The matrix I -A is nonsingular and (I – A)-1 has nonnegative elements;

(v) for An and qAn as n for

each q Rm

Remark: Some examples of matrix convergent to zero are (a) any matrix A := where a, b R+ and a+b < 1; (b) any matrix A := where a, b R+ and a+b < 1; and max{a, c}<1; For other examples and considerations on matrices which converge to zero, see [8] and [11]. Theorem 2.9 ([2]). Let (X, d) be a complete vector valued generalized metric space and the mapping f : X X with the property that there exists a matrix A Mm,m(R+) such that d(f(x), f(y)) Ad(x, y) for all x, y X. If A is a matrix convergent towards zero, then (1) Fix(f) = {x}; (2) the sequence of successive approximations {xn} such that xn = f n(x0) is convergent and it has the limit x, for all x0 X. (3) One has the following estimation: d(xn, x) A(I-A)-1 d(x0, x1)

Theorem 3.1: Let P, Q, S and T are self-maps of a metric space ( X , d ) satisfying the conditions for all x, y in X. where ,  ≥ 0, the pairs (S,P) and (Q,T) both are occasionally weakly compatible . Further the associated sequence relative to four self-maps P, Q, S and T such that the sequence Sx0 ,Tx1, Sx2 ,Tx3......Sx2n ,Tx2n+1.. converges to zX. Then P, Q, S and T have a unique common fixed point in X.

Since (S,P)and (Q,T) both are occasionally weakly compatible. So there are points x, y in X such that

using the conditions Sx = Px and Qy = Ty , we get

which is a contradiction. Hence Therefore

Similarly there exists a common fixed point of Q and

Suppose w ≠ v put x = w and y = v in the condition (3.1.2), we get

This is a contradiction. Therefore w v .Hence w is a common fixed point of P, Q, S and T Theorem (3.2):- Let . Let C Lp be nonempty, -closed and -bounded. Let T: C C be a - contraction. Then, T has a unique fixed point f C. Moreover, for any f C., (Tn (f) f) 0 as n where Tn is the nth iterate of T. Let us fix f0 C. Since C is bounded, hence Observe that for any n, k 1. Since < 1 and we conclude that is-Cauchy. The - completeness of implies the existence of Such that Because C is -closed, we get . Since Therefore, = , which means that is a fixed point of T. To prove the uniqueness part observe that if , then Since < 1 and the right-hand side is finite, equality (3.3) can hold only if . Theorem (3.3):- Let us assume that . Let C be non empty, -closed, and -bounded. Let T: C C be a point wise contraction or asymptotic point wise -contraction. Then T has at most one fixed point f0 C. moreover, if f0 is a fixed point of T, then the orbit {Tn (f)} -converges to f0 for any f C. Proof: Since every point wise -contraction is an asymptotic point wise -contraction, we can assume that T is an asymptotic point wise -contraction, i.e., there exist a sequence of mappings : C [0, } such that where {} converges pointwise to  : C [0, 1). Let , C be two fixed points of T. Then we have for all n 1. If we let n , we will get Since and C is -bounded, we conclude that , i.e. . This proves that T has at most one fixed point. To prove the convergence, assume that f0 is the fixed of T. Fix an arbitrary C. Let us prove that {Tn (f)} -converges to f0. Indeed we have for any n, m 1. Hence for any n 1. If we let n , we obtain Since we get which implies the desired conclusion of x0 converges, the metric space ( X , d ) need not be complete.

Theorem3.1 is speculation of Theorem2.5 by prudence of the weaker conditions such as occasionally weakly compatibility of the sets (S,P) and (Q,T) instead of compatibility of the sets (S,P) and (Q,T). The continuity of any of the mappings is being dropped and the convergence of related sequence comparative with four self-maps S, P, Q and T is utilized set up the complete metric space.

1. G. Jungck (1986). Compatible mappings and common fixed points, Internat. J.Math. & Math. Sci., 9, pp. 771-778. 3. Jungck (1988). Compatible mappings and common fixed points (2), Internat. J. Math. & Math. Sci., 11, pp. 285-288. 4. G. Jungck and B.E. Rhoades (1998). Fixed point for set valued functions without continuity, Indian J. Pure. Appl. Math, 29(3), pp. 227-238. 5. S. Sessa (1980) on weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math, 32(46), pp. 149-153. 6. P. C. Lohani and V. H. Badshah (1998). Compatible mappings and common fixed point for four mappings, Bull. Cal. Math. Soc., 90, pp. 301-308. 7. B. Fisher (1983). Common fixed points of four mappings, Bull. Inst. Math. Acd. Sinica, 11, pp. 103-108. 8. Ravi Sriramula and V. Srinivas (2017). A fixed point theorem using using weakly semi compatible mappings in metric space, International Journal of Mathematical Analysis, vol. 11(15), pp. 515-524. https://doi.org/10.12988/ijma.2017.7688 9. Ravi Sriramula and V. Srinivas (2017). A Result on fixed point theorem using compatible mappings of type(K), Annals of Pure and Applied Mathematics, 13(1), pp. 41-47. DOI: http://dx.doi.org/10.22457/apam.v13n1a5 10. M.A.A-Thagafi, N. A. Shahzad (2009). A note on occasionally weakly compatible maps, Int. J. Math. Anal.3, pp. 55-58. 11. Popa, V. (2001). A general common fixed point theorem for weakly compatible mappings in compact metric spaces, Turk. J. Math. 25, pp. 465-474.

Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana -124001, India sangeetadhaniamaths@gmail.com