COMMON FIXED POINTS OF COMPATIBLE MAPPINGS

In this paper, we present a common fixed point theorem for compatible mappings, which extends the results of Ding, Divlccaro-Sessa and the third author.


INTRODUCTION.
In [I], the concept of compatible mappings was introduced as a generalization of commuting mappings.The utility of compatibility in the context of fixed point theory was demonstrated by extending a theorem of Park-Bae [2].
In [3], the third author extended a result of Singh-Singh [4] by employing compatible mappings in lleu of commuting mappings and by using four functions as opposed to three.On the other hand, Diviccaro-Sessa [5] proved a common fixed point theorem for four mappings, using a well known contractive condition of Meade-Singh [6] and the concept of weak commutativity of Sessa [7].
In this paper, we extend the results of Ding [II], Diviccaro-Sessa [5] and the third author [3].
The following Definition 1.1 is given in [I].DEFINITION I.I.Let A and B be mappings from a metric space (X,d) into itself.
Then A and B are said to be compatible if lira d(ABx BAx 0 whenever Ix is a n n n sequence in X such that llm Ax llm Bx z for some z in X. ,Mappings which commute are clearly compatible, bu[ the converse is false.S. Sessa [7] generalized commuting mappings by calling mappings A and B from a metric space (X,d) into itself a weakly commuting pair if d(ABx, BAx) d(Ax, Bx) or all x in X.Any weakly commuting pair are obviously compatible, but the converse is false [3].
See [I] for other examples of the compatabile pairs whlch are not weakly commutative and hence not commuting pairs.LEMMA I.! ([I]).Let A and B be compatible mappings from a metric space (X,d) into itself.Suppose that llm Ax llm Bx z for some z in K. Throughout this paper, suppose that the function : [0,) 5 [0, ) satisfies the following conditions: (I) is nondecreaslng and upper semicontlnuous in each coordinate variable, (2) For each t > O, @(t) max [#(0,O,t,t,t), (t,t,t,2t,0), '(t,t,t,0,2t)} < t.
Suppose that : [0, ) [0, ) is nondecreasing and upper semicontinuous from the right.If (t) < t for every t > 0, then lira n(t) O, ?nwhere (t) denotes the composition of (t) with itself n-times.Now, we are ready to state our main Theorem.THEOREM 2.2.Let A,B,S, and T be mappings from a complete metric space (X,d) into itself.
Suppose that one of A,B,S and T is continuous, the pairs A,S and B, T are compatible and that A(X) T(X) and B(X) c S(X).If the inequality d(Ax, By) (d(Ax,Sx), d(By,Ty), d(Sx,Ty), d(Ax,Ty), d(By,Sx)) (2.2) holds for all x and y in X, where satisfies (I) and (2), then A,B, S and T have a unique common fixed point in X.
PROOF.Let x X be given.Since A(X)eT(X) and B(X)c S(X), we can choose x in o X such that Yl --TXl Ax and, for this point x I, there exists a point x 2 in X such o that Y2 --Sx2 Bxl and so on.Inductively, we can define a sequence {yn in X such that Y2n+l TX2n+l AX2n and Y2n SX2n BX2n-l" (2.3) 2) and (2.3), we have If d(V2n+I' Y2n+2 > d(Y2n.Y2n+l) in the above Inequality, then we have d( which is a contradlciton.O. ( n In order to show that {yn is a Cauchy sequence, it is sufficient to show that {Y2n [s a Cauchy sequence.Suppose that [Y2n is not a Cauchy sequence.Then there is an a > 0 such that, for each even Integer 2k, there exist even integers 2re(k) and 2n(k) such that d(Y2m(k), Y2n(k)) > for 2re(k) > 2n(k) 2k.
Hence {yn is a Cauchy sequence and it converges to some point z in X. Consequently the subsequences {AX2n}' {SX2n}' {BX2n-l} and {TX2n_I converge to z. Suppose that S is continuous.Since A and S are compatible, Lemma I.'2 implies that Letting n + , we have d(Az, z) #(d(Az, Sz), O, d(Sz, z), d(Az, z), d(z, Sz)), so that z kz.
Since A(X)C T(X), z T(X) and hence there exists a point w in X such that z Az Tw.
S[nce'B and T are compatble and Tv Bv z, d(TBv, BTv) Themefore z is a common fixed point of A,B,S and T.
Similarly, we can complete the proof in the case of the continuity of B. It follows easily from (2.2) that z is a unique common fixed point of A,B,S and T.
Let A,B,S and T be mappings from a complete metmic space (X,d) into itself.
Suppose that one of A,B,S and T is continuous, the pairs A,S and B,T are compatible and that A(X) c T(X) and B(X) S(X).I the inequality (2. for each t > 0, then A,B,S and T have a unique common fixed point in X. REMARK 2.4.From Theorem 2.2 and Corollary 2.3, we extend the results of Ding [II] and Diviccaro-Sessa [5] by employing compatibility in lieu of commuting and weakly commuting mappings, respectively.Further our theorem extends also a result of Ding [II] by using one continuous function as opposed to two. 5 REMARK 2.5.From Theorem 2.2 defining : [0, ) [0, ) by $(tl,t2,t3,t4,t5) h max[tl,t2,t3, (t4+ ts)} for all tl,t2,t3,t4,t 5 [0,) and h [0, I), we obtain a result of the third author [3] even if one function is continuous as opposed to two.