FIXED POINT THEOREM OF NONLINEAR CONTRACTION IN METRIC SPACE

In the present paper, I present a shorter proof by generalizing the main results of Sayyed et al. by replacing the containment condition in the context of metric space.


Introduction
Aamri et al. [1] introduced the concept of property (E.A.) which was perhaps inspired by the condition of compatibility introduced by Jungck [11] and further Imdad et al. [10] extended this result.Babu et al. [7,8,9] proved common fixed point theorem for occasionally weakly compatible maps satisfying property (E.A.) using an inequality involving quadratic terms.Aliouche [4] proved a common fixed point theorem of Gregus type weakly compatible mappings satisfying generalized contractive conditions.
Abbas [2] established a common fixed point for Lipschitzian mapping satisfying rational contractive conditions.Murty et.al.[15] proved fixed points of nonlinear contraction in metric space.

Preliminaries
Throughout this paper (X, d) is a metric space which is denoted by X. Definition 2.1: Jungck and Rhoades [13].Let A and S be selfmaps of a set X.If Au = Su =  (say),   X, for some u in X, then u is called a coincidence point of A and S and the set of coincidence points of A and S is denoted by C (A, S), and  is called a point of coincidence of A and S. (III) Weakly compatible [12], if they commute at their coincidence point.

Remark 2.4
(I) [12] Every compatible pair is weakly compatible but its converse need not be true.FIXED POINT THEOREM OF NONLINEAR CONTRACTION IN METRIC SPACE (II) [16] Weak compatibility and property (E.A.) are independent of each other.
(III) [11] Every weakly compatible pair is occasionally weakly compatible but its converse need not be true.
(IV) [8] Occasionally weakly compatible and property (E.A.) are independent of each other.
Definition 2.5: [14] Let (X, d) be a metric space and A, B, S and T be four selfmaps on X.The pairs ( , ) AS and ( , ) BT are said to satisfy common property (E.A.) if there exists two sequences Proof: To be given Ax = Sx = {z} (say) for any  ( , ) x C A S . ( . Further, assume that the pairs ( , ) AS and ( , ) BT are compatible on X.If one of the mappings ,, A B S and T is continuous then ,, A B S and T have a unique common fixed point in X.

Main results
Proposition 3.1.Let A, B, S and T be self maps of a metric space (X, d) and satisfying the inequality.
, d(Sx,Ty) , for all  , x y X , where k ≥ 0 and k < 1.  .By proposition (2.9), z is the unique common fixed point of ,, A B S and T in X. Remark 3.4: Proposition (2.5) of [9] and theorem (2.6) of [9] remain true, if we replace completeness of S(X) and T(X) by the completeness of ( ) ( ) S X T X  in X.For this we have given an example 2.7 in the following manner without proof.Now we rewriting the proposition (2.5) and theorem 2.6 of [9].

Remark 2 . 6 :Example 2 . 7 :Remark 2 . 8 :Preposition 2 . 9 :
Let ,, A B S and T be self maps of a set X.If the pairs (A, S) and (B, T) have common point of coincidence in X then C(A, S)   and C (B, T)  .But converse is not true.Let X = [0, ) with usual metric and A, B, S and T self maps on x and defined by but the pairs (A, S) and (B, T) not having common point of coincidence.The converse of the remark 2.6 is true, provided it satisfies inequality(3.1).This is given as proposition (3.1).[2] Let A and S be two self maps of a set X and the pair (A, S) satisfies occasionally weakly compatible (owc) condition.If the pairs (A, S) have unique point of coincidence Ax = Sx = z then z is the unique common fixed point of A and S.