SOME FIXED POINT THEOREMS FOR COMPATIBLE MAPS

. A collection of fixed point theorems is generalized by replacing hypothesized commutativity or weak commutativity of functions involved by compatibility.

INTRODUCTION.The last two decades have produced a spate of articles which propose generalizations and/or extensions of the Banach Contraction Principle, which Principle states that a contraction f of a complete metric space (X,d) has a unique fixed point.Typical approaches have been either to vary the contraction requirement that d(fx, fy) < r d(x, y) for some r 6 (0,1) and all x,y 6 X, or to introduce more functions with conditions appended.For example, in 1976 the following result appeared: THEOREM 1.1.[1] Let f and g be commuting (g'f=fg) self maps of a complete metric space (X,d) such that f(X) C g(X) and g is continuous.If 3 r 6 (0,1) such that d(fx, fy) < r d(gx, gy) for x,y 6 X, then f and g have a unique common fixed point a 6 X (i.e., fa=ga=a).
The above theorem and article promoted commutative maps as a tool for generalizing.Subsequently, a variety of variations and generalizations of Theorem 1 which utilized the commuting map concept appeared See, e.g., [2, 3, 4, 5, 6, 7,]  In 1982, Sessa [8] introduced a generalization of the commuting map concept by saying that maps f,g:(X,d)---(X,d) axe weakly commutative iff d(fgx, gfx) < d(fx, gx) for x 6 X.In response, variations on Banach and Theorem 1. appeared in terms of "weakly commuting pairs f,g" see, e.g., [9], [10].Then, in 1986, the first author introduced the concept of compatibility.
Clearly, commuting mappings are weakly commuting and weakly comnmting pairs are compatible; exanaples in [8] and [11] show that neither converse is true.Articles already in print demonstrate that known results can be generalized by using compatibility in lieu of commutativity or weak commutativity.We refer the reader to [11, 12, 13, 14, 15,16, 17]; in particular we note [17] in which Rhoades, Park, and Moon obtain a very general fixed point theorem by using Meir-Keeler type contraction maps in conjuction with compatibility.
The purpose of this paper is to further demonstrate the effectiveness of the compatible map concept as a means of generalizing.We shall show that an appreciable number of fixed point and coincidence theorems can be improved by substituting compatibility for commutativity or weak commutativity.Such an effort seems to be in order, indeed, called for, since 2.as the reader will see-the method of attack for one theorem is typically very similar to that for another theorem.The approach becomes "standard" because the definition of compatibility and one proposition regarding compatibility are the only tools needed.The proposition we need is Proposition 2.2. in [11].
PROPOSITION 1.1.( [11]) Let f and g be compatible self maps of a metric space (X,d). 1.If f(t)=g(t), then fg(t)=gf(t). 2. Suppose that limnf(Xn) limng(xn)  for some E X and x n E X. (a) If f is continuous at t, limngf(xn) f(t).(b) If f and g are continous at t, then f(t) g(t) and fg(t) gf(t). 2.

GENERALIZATIONS VIA COMPATIBILITY.
We shall now state generalizations of published results, generalizations obtained in the main by replacing the hypothesised commutativity or weak commutativity with compatibility.Proofs of some of these results will be given in relative detail so as to demonstrate techniques involved.Of course, in most instances goodly portions of the proofs of results being generalized will pertain and will be appealed to so as to avoid repetition.
We have taken care to not to duplicate results already in the literature, such as the general theorem of Rhoades, Park, and Bae, and Moon.
The first theorem is a generalization of Theorem 1. in [18], a 1986 paper by Diviccara, Sessa and Fisher.We substitute compatibility for weak commutativity in the hypothesis.
Suppose S(X)U W(X) I(X).If either is continuous and compatible with one of S,T, or one of S or T is continuous and compatible with I, then I, S, and T have a unique common fixed point z.Further, z is the unique common fixed point of S and and of T and I.
But as n--, oo we obtain d(Iw,Tw)< 0, a contradiction.Thus, Tw=Iw.The argument given in the third paragraph of page 279 in [18] shows that, in fact, Iw=Sw=Tw.
The case in which is continuous and compatible with T follows from the above because of the svmmetric roles of S and T; i.e., Iw=Sw=Tw in this case also.
Next, suppose that S is continuous and compatible with I. Then (*) above and Propositio.n(1.1)2(a)imply that SIx2n SSx2n--* Sw and ISx2n Sw.Since S(X) c__ I(X), there exists w X such that IwP=Sw.In fact, the line of reasoning at the bottom of page 279 and top of page 280 is valid for us because the above sequences do converge to Sw, and we have Iw=Sw=Tw=Sw.As above, we can appeal to "symmetry" to conclude that Iz=Sz=Tz for sortie z when T is continuous and compatible with I.
We have considered all possibilities to show that Iw=Sw=Tw for some w X when d n # 0. The case in which dn=0 for some n is covered in (ii) and (iii) on page 280 and holds for us.Thus, in any case, Iw=Sw=Tw fox some w e X.
As we now show, Iw is a common fixed point of I, S, and T. Note that the argument given depends on compatibility without any reference to continuity.If and S are compatible, then Tw=Iw=Sw and Proposition (1.1) 1. imply that SSw SIw ISw IIw.But then d(IIw, SIw) + d(Iw, Tw) 0, so that d(SIw, Tw) 0 by (b) of the hypothesis.Therefore, Iw=Tw=SIw=IIw, and z=Iw is a common fixed point of and S.
We have shown that, in any case, I, S, and T have a common fixed point.The uniqueness assertions follow immediately from (b) of the hypothesis.VI  The next theorem generalizes Theorem 1.
[19] of Imdad, Kahn, and Sessa by replacing the weakly commuting requirement of the hypothesis by compatibility.Note that our approach simplifies the argument give in [19] on pages 31-32.
THEOREM 2.2.Let X be a uniformly convex Banach space and K a nonempty closed subset of X.Let A, S, and T be self maps of K satisfying: (i) S and W are continuous, and A(K) C S(K)t3 T(K).
(ii) {A,S and {A,T} are compatible pairs on K.
We have, Au Su Tu.The remainder of the proof is the same as that in [19], beginning on the second line from the bottom of page 32 and continuing to middle of page 33, the end of the proof.E! Our next theorem generalizes Theorem 1. ( [20]) of Devi Prasad by relaxing the requirement that hf=fh and gh=hg by merely requiring that each of the pairs f,h and g,h be compatible.THEOREM 2.3.Let f, g, and h be self mappings of a complete metric space (X,d) which satisfy: f(X)LIg(X)_ h(X), f and h are compatible and g and h are compatible.Suppose further that (i) d(fx, gy) )2 < ( d(hx, fx) d(hy, gy), d(hx, gy) d(hy, fx), d(hx, fx) d(hx, gy), d(hy, fx) d(hy, gy)) for any x,y E X, where : +--.R+ is upper semi-continuous and nondecreasing in each coordinate variable and satisfies ( t, t, alt, a2t < for any t>0, where a E {0,1,2} with a + a 2 2. If h is continuous, then f, g, and h have a unique common fixed point. PROOF.Follow the proof of Prasad to the bottom of page 1074.Then we have {fX2n}, {gx2n+l}, and {hxn} converging to u.Since h is continuous, h2xn -, hu and hfx2n--hu, and since h and f axe also compatible, fhx2n-hu, by Proposition(1.1)2(a).
The remainder of the proof is the same as in the proof of Theorem 1. of Prasad. 13  The next theorem is a generalization of a Theorem 1. in [21] by S. L. Singh on L- spaces.L-spaces utilize semi-metrics d (See [21]).We extend our definition of compatibility to L-spaces by-saying that self maps P and Q of an L-space (X,--) are compatible relative to a semimetric d on X iff whenever {Xn} is a sequence in X such that Pxn--.and Qxn for some E X, then d(PQxn,QPxn) 0. Also note that in a separated L-space d is continuous.THEOREM 2.4.Let (X,-) be a separated L-space which is d-complete for a semimetric d.Let P, Q, T be continuous selfmaps of (X,) such that the pairs P.T and Q,T are each compatible relative to d and satisfy P(X)uQ(X)c_ T(X).If there exists h E (0,1) such that for all x,y G X: d(Px, Qy) _< h max d(Px, Tx), d(Qy, Ty), d(Wx, Ty),} then P, Q, and T have a unique common fixed point.
PROOF.The proof of Theorem 1. in [21] up to the bottom of page 92 is valid under our hypothesis.We thus have, Txn-z, Px2nz, and Qx2n+l z.The continuity of T, P, Q and of d, in conjunction with the compatibility of the T and P and of T and Q imply that Pz=Tz and Qz=Tz.Therefore, by compatibility let x n z for all n in the definition), PTz=TPz =TTz=TQz=QTz=PQz=Qpz=QQz.But then d(pQz, Qz)_< h max {d(pQz, TQz), d(Qz, Tz), d(TQz, Tz) h max {0, 0, d(PQz, Qz)}, so that PQz Qz.By the above equalities we Qz is a common fixed point of P, Q, and T.
Uniqueness follows immediately from the contractive definition.VI   In the above proof we veritably showed that two compatibl self maps of a separated L-space commute at coincidence points of the maps.This fact is noted for metric spaces in Proposition (1.1) 1.However, Proposition (1.1)2.(b) says that if E and F are compatible and continuous self maps of a metric space and Exn, Fxnt then Et--Ft and EFt=FEt.The proof of the following theorem, which is a generalization of Theorem 2. in [22] by Yeh,   appeals to this fact.We again generalize by replacing the hypothesised commutativity of pairs of maps by hypothesising compatibility for the corresponding pairs.THEOREM 2.5.Let E, F, and T be continuous self maps of a complete metric space (X,d) such that E,T and F,T are compatible, and that E(X)U F(X) C_ T(X).Suppose that d(Ex, Fy) < a( d(Tx, Ty)) d(Tx,Ty) + b(d(Tx, Ty))[ d(Tx, Ex) + d(Ty, Fy)] + c(d(Tx, Ty)) d(Tx, Fy) + d(Ty, Ex)]) for all x,y X x y, where a. b, and c are mappings from N+ into [0, 1) satisfying the following: If A a + 2b + 2c where 0 < A(t) < for t[+, and {tn} is amonotone increasing sequence in N+ for which A(tn)-as n-oc then tn 0 as n-oc.Then E F, and T have a unique common fixed point.
PROOF.Proceed as in the proof of Theorem 2. of Yeh until line 5 of page 119.We have: Tx n, Ex2n, FX2n+l--x X.Since Tx2n,Ex2nX and the continuous functions E and T are compatible, Ex Tx and ETx=TEx by Proposition(1.1)2.(b.).Similarly, Fx=Tx and FTx=TFx.Thus, T(Tx) T(Ex)= E(Tx)= E(Ex)= T(Ex)= F(Tx)= F(Ex)= F(Fx).The remainder of the proof is as in [22]. [3  In [23], Diviccaro, Fisher, and Sessa prove a common fixed point theorem of the "Gregus" type.However, as was communicated to us by Sessa, a very recent paper (1991) by Davies ([24]) subsumes the "Gregus" type theorem in [23].We now appreciably generalize Davies' result Theorem 1. in [24] by replacing the nonexpansive requirement on the linear map by continuity, and the weakly commuting hypothesis by compatibility.THEOREM 2.6.Let and T be compatible self maps of C, a closed convex subset of a Banach space X, satisfying" Ix Ty -< c, Ix Iy +/ max Tx Ix II, Wy Iy + + 7 max {llIx-Iyll IITx-Ixll, IITy-Iyll forx,yC, where c,,,> 0 and a+/+'r 1.If is linear and continuous in C and T(C) C_ I(C), then T and have a unique common fixed point w and T is continuous at w. PROOF.Define K n xC :llTx-Ixll <l/n for allnN, the set of positive integers.The proof in [24] holds for our hypothesis through to (13), page 240, where we have {w}=A= f3 {cl(I(Kn)) n N} and we use cl to denote "closure".Since w A, for each nN =1 ynI(Kn) such that d(yn, w)<l/n.Then q v nKn such that yn Ivn; thus d(Ivn, w)<l/n and we infer that Ivn--w.But v nKn for nN, so that IlWvn-Ivnl[ < 1/n and we also have Tvnw.Since is continuous, ITvn---Iw and IIvn Iw.Moreover, TIvn-Iw by Proposition(1.1), since and T are compatible and is continuous.
That w is that unique common fixed point of and T follows from the fact that any common fixed point of and T is in A, and A is a singleton.However, Davies appeals to the nonexpansiveness of to prove T continuous at w. Since we are only asssuming that is continuous, we proceed as follows.
Let Xn w.Since is continuous, Ixn Iw Tw.Now by hypothesis, using for n (f N. Therefore, since Ixn-lw, Tx n-+ Tw as desired. The next Theorem is a generalization of Theorem 3. in [25], a paper published in 1986 by Fisher and Sessa.We generalize by substituting compatibility for weak commutativity.THEOREM 2.7.Let {S,I} and {T,J} be two pairs of compatible self maps of a complete metric space (X,d) such that d(Sx, Ty) < g( d(Ix, Jy), d(Ix, Sx), d(Jy, Ty) for any x,y E X, where g: 1.3{.I+, is continuous, and satisfies" (i) g(1,1,1)=h<l, and (ii) whenever u,v >0 and either u< g(u,v,v), u< g(v,u,v), or u< g(v,v,u), thenu < hr.If T(X) C_ I(X), S(X) C_ J(X), and if one of I, J, S, or T is continuous, then I, J, S, and T have a unique common fixed point z.Further, z is the unique common fixed point of and S and of J and T. PROOF.Follow the proof of Theorem 3. by Fisher and Sessa to line 6 on page 48.
Since S(X) C J(X), =! z' such that Jz' z.As in [25], line 9, page 49, to line 12, page 49, we have Tz z.But Jzt= Tz implies that T and J commute at zt, by Proposition (1.1)1.This implies Tz TJz JTz Jz.That Tz Jz z follows from the last five lines of page 49, [24].Therefore, I, S, T, and J have a common fixed point z if is continuous.
The proof for the case in which J is continuous is analogous to the preceding proof.In fact, the remainder of the proof in [25] beginning with line 6, page 50, holds if the phrase, Since and are compatible" is substituted for every appearance of Since and weakly commute", with one exception.Beginning with the fifth line from the bottom of page 51, we would say, Since S and are compatible, the fact tht Sz z Iz implies Iz=ISz=SIz=Sz.We thus have Iz=Szand z=Tz=Jz from above.But then, d(Sz, z d(Sz, Wz) < g( d(Iz, Jz), d(Iz, Sz), d(Jz, Tz) =g(d(Sz, z),0,0) < hd(Sz, z), and this implies that Sz z.Thus, z is a common fixed point of I, J, S, and T." 13 The following theorem generalizes Theorem 3.1 of M. S. Kahn and M. Swaleh in [26].The only change in the statement of theorem is to require {A,S} and {A,T} to be compatible pairs as opposed to weakly commuting pairs.THEOREM 2.8.Let A, S, and T be self maps of a complete metric space (X, d).
Then A, S, and T have a unique common fixed point.
PROOF.The proof is the same as the proof of Theorem 3.1 in [26] down to ten lines from the bottom of page 986.Now since Axn--* z and SXn z, A 2 Xn Az and ASxn Az since A is continuous.But then Proposition (1.1)), SAxnAz since {A, S} is a compatible pair.Similarly, we conclude that ATxn--* Az and TAxn Az.The remainder of the proof is as in [6] gl We now consider compatibility and/or generalizations thereof in the context of multi- valued maps.3.

MULTI-VALUED FUNCTIONS AND COMPATIBILITY.
We shall consider three papers involving multi-valued functions.The first two let B(X) denote the set of bounded subsets of a complete metric space (X,d) and define a function g: B(X)xB(X)-[0,o) by g(A,B) sup d(a, b): a A and b B }. See [27] or [28] for a discussion and listing of properties of g.We do note that 0 < g(A, B) _< g(A, C) + g(C, B) for A,B,C B(X), and g(A, B)=0 iff A=S={a}.If x X, we write g(x, A) for g({x}, A) when convenient and confusion is not likely.
If {An} is a sequence in B(X), we say that {An} converges to A C_ X, and write An--oA, iff (i) a A implies that a=-xmooa n for some sequence {an} with an An for n N, and   (ii) for anye>0 =lmNsuchthat An_CAe= {xX: d(x,a) <e for someaA forn> m.
We need the following lemmas.LEMMA 3.1 ([271) Suppose {An} and {Bn} are sequences in B(X) and (X, d) is a complete metric space.If An-A B(X) and Bn-, B B(X), then g(An, Bn) -g(A, B).LEMMA 3.2 ([28]) If {An} is a sequence of nonempty bounded sets in the complete metric space (X,d) and ifnli__moog(An,{y}) 0 for some y X, then 'An {y}.
To define "compatibility" in this context, we say the following.DEFINITION 3.1.Let (X, d) be a metric space.Let I: X--o X and F: X B(X).F and are &compatible iff IFx B(X) for x X and g(IFxn, FIxn) 0 whenever {Xn} is a sequence in X such that Ixn-.and Fxn--{t for some X.
Observe that even though the conditions of the above definition are satisfied non- vacuously, F need not be single valued.Consider, e.g., I:R---.R and F: R-B(R) defined by Ix x/3 and Fx [0, x/2], where R denotes the reals with the usual topology.
The following result regarding g-compatibility will prove useful.Note that by definition, a function F:XB(X)is continuous iff Xn-.z in (X,d)implies Fxn Fz in B(X).PROPOSITION 3.1.Let (X,d) be a complete metric space.F:XB(X), and and F are &compatible.