ON A FIXED POINT THEOREM OF FISHER AND SESSA

A fixed point theorem of Fisher and Sessa is generalized by replacing the requirements of commutativity and nonexpansiveness by compatibility and continuity respectively.

In [1], Fisher and Sessa proved the following generalization of a theorem of Gregus [2].
Let T and I be two weakly commuting mappings of a closed convex subset C of a Banach Space X into itself satisfying the inequality for all x,y in C, where a (0,1).If I is linear and nonexpansive in C and if T(C) E I(C), then T and I have a unique common fixed point.
Sessa defined (see [I]) self maps I and T of a metric space (X,d) to be weakly commuting iff d(ITx,TIx) < d(Ix,Tx) for x X.Subsequently, Jungck [3] defined two such self maps to be compatible iff whenever {x n} is a sequence in X such that TXn, Ix t for some t E X, then d(ITx ,TIx 0.
Clearly, commuting maps are n n n weakly commuting, and weakly commuting maps are compatible.
[I] and [3] give examples which show that neither implication is reversible.
The purpose of this note is to show that Theorem 1.1 can be appreciably generalized by substituting compatibility for weak commutativity and continuity for the nonexpansive requirement.
2. Suppose that IImnf(Xn limng(Xn t for some t in X. (a) If f is continuous at t, llmngf(xn) f(t).
(b) If f and g are continuous at t, then f(t) g(t) and fg(t) gf(t).REMARK 2.1.We shall use N to denote the set of positive Integers and cl(S) to denote the closure of a set S. PROPOSITION 2.1.Let T and I be compatible self maps of a metric space (X,d) with I continuous.Suppose there exist real numbers r > 0 and a (0,I) such that for all x,y X, d(Tx,Ty) rd(Ix,ly) + a max {d(Tx,lx), d(Ty,ly)}. (2.1) Then Twffilw for some w , where n K, {x X: d(Tx,lx) I/n}.n PROOF.Suppose that Twffilw for some w X.Then w K for all n and thus n Tw E T(K n) c__ cl(Tn(Kn)) for all n; i.e, Tw A.

Conversely, if w
A, for each n there exists Yn T(Kn such that d(w,y n) < I/n.
But then the assumption that d(Tw,lw) > 0 demands that a )I, which contradicts the choice of a.
We conclude that d(Tw,lw) 0, and thus lw Tw.
THEOREM 2.1.Let I and T be compatible self maps of a closed convex subset C of a Banach space X.Suppose that I is continuous, linear, and that T(CI I(CI.If there exists a (0,11 such that for all x,y C then I and T have a unique common fixed point in C. pages n 24-26 without appeal to the weak commutatlvlty of I and T or the nonexpanslveness of I that by appeal to the convexity of C, the llnearity of I, and property (2.1), we can infer that A N {cI(T(K )):n E N} is a si, lton and therefore nonempty.n Consequently, Proposition (2.1) tien assures us that lw=Tw for some w e C. We now show that Tw is a common fixed point of I and T.
Since I and T are compatible and lw=Tw, Lemma 2.1.implies that ITwTlw.We then have T2w Tlw ITw, so that by (2.1) thus, TT,W=-Tw since a E (0,11.By the above we then have Tw TTw ITw; i.e., Tw is a common fixed point of T and I. That Tw is the only common fixed point of I and T follows Immediately from (2.1).
In conclusion, we wish to present an example which shows that our Theorem 2.1 Is indeed a generalization of Theorem I.I.To effect this, the following result from [4] is helpful.
LEMMA 2.2.(Corollary 2.6, [4]).Suppose that f and g are continuous self maps of a metric space and that f is proper.
If fx-gx implies x-fx, then f and g are compatible.
-I We remind the reader that a map f: for all x,y E C so that condition (2.11 is satisfied. To see that I and T are compatible, note that both are continuous and that I is clearly proper, so we can appeal to Lemma 2.2.Now Tx Ix implies that x(3x 2 + I) 0, so x 0. But I(0) 0

EXAMPLE 2 .
1. Let X be the reals with the usual norm, C[0,-I, IX 5x/2 and Tx x + x(x 2 + 11 -I for x C. Now C is convex and closed, and I,T: C C where T(CII(C)C and I is linear and continuous.Moreover, it is easy to see that the maximum value of dT/dt on C is 2, so by the Mean Value Theorem we can