COINCIDENCE AND FIXED POINTS FORCOMPATIBLE AND RELATIVELY NONEXPANSIVE MAPS

The concept of relatively nonexpansive maps is introduced. Fixed point and coincidence results for families of four self maps of metric spaces are obtained. Non-continuous compatible and relatively nonexpansive maps on star-shaped compact subsets of normed linear spaces are highlighted, and two theorems of Dotson are generalized.


INTRODUCTION
In [1] Dotson proved the following theorem.THEOREM 1.1 (Dotson).If T is a nonexpansive self-map of a compact star-shaped subset C of a Banach space (i.e., Tz Ty < z y for x,y e C), then T has a fixed point in C.
Our intent is to generalize this result.We do so by introducing the concept of relatively nonexpansive functions.Our first result is a fixed point theorem for four self maps of a general metric space, which has known results as corollaries, and is used to prove our main result which pertains to star-shaped compact subsets of a linear space.Of interest is the fact that none of the results in the body of this paper require that functions be continuous this is effected in part by using the concepts of compatibility and surjectivity.
Self maps I and g of a metric space (X,d) are compattble ( [2]) iff whenever {Zn} is a sequence in x such that .frn,gZn-t for some x, then d(J'gz n, gIrn)-,O.An immediate consequence of the definition (Proposition 2.2, [2]) is that if I and g are compatible and a e X is a coincidence point of y and g (ya ga), then 1'ga gla.(We shall write la for l(a) when convenient and confusion is not likely.)In fact, Cor. 2.3 in [3] asserts that whenever y and g are continuous and (X,d) is compact, then I and g are compatible iff laa la whenever la a.
We should also note that a subset C of a linear space X is star-shaped iff :q e C such that tz+(1-t)qC for te[0,1] and xeC.In this event, we shall say that C is star-shaped with respect to q.Of course, if C is convex, C is star-shaped with respect to any q C'.
We need the following definition.
DEFINITION 2.1.([2]) Self maps A and B of a metric space (X,d) are (e,$)-S,T- contractions relative to maps S,T:X-X iff A(X)C_T(X), B( X) C_ S( X), and there is a function & (0,oo)-.(0,o)such that 6(e) > for all e, and for all z,y E X we have: (i) <_ d(St, TU) < (e) implies d(At, By) < , and (ii) At By whenever St Ty.
THEOREM 2.1.Let S and T be self maps of a metric space (X,d), and let the pair A,B be (,)-S,T-contractions.If T(X) is complete, there exist u,v,p X such that Au Su p By Tv.
If furthermore, the pair A,S (B,T) is compatible, then Ap Sp I(Bp Tp p).And if both pairs A,S and B,T are compatible, p is the unique common fixed point of A,B,S, and T.
PROOF.Since A and B are (e,)-S,T-contractions, A(X)C_ T(X) and B(X)C_ S(X), and as a consequence of (i) and (ii) in the definition we know that St Ty implies At By, and d(At, By) < d(St, Ty) if St # Ty. (2.1) In particular, d(At, By) < d(St, Ty) for t,y X.Let t O q X.For n N, let Y2n-Tt2n-At2n-2 and Y2n St2n Bt2n-1" Since A(X) c_ T(X) and B(X) C_ S(X), the t can be so chosen.By Lemma 3.1 in [2], he sequence {Yn} thus inductively defined is a Cauchy sequence.But ghen he sequence {Y2n-1}, which is in T(X), is also Cauchy.Since T(X) is complete, {Y2n-1} converges o a point p Tv for some v e X.Therefore, Yn-'l-Now (2.1) implies hat for n E N: d(p, By) < d(p, At2n)+d(At2n, Bv < d(p, At2n)+d(St2n Tv). ( Since {Yn}, and hence any subsequence thereof, converges to 10, (2.2) and the definition of {Vn} imply that d(p, Bv)=O; i.e., lo By= Tv.But B(X) C_ S(X), so there exists u X such that St,= By= Tt, (2.1) therefore implies that Au By.We have: io By Tv $ A.
(2.3) Thus, the first conclusion of the theorem is verified.If moreover, A and S are compatible, (2.3) implies that ASu SAt,, or Al0 S10.In fact, 10 A0.Otherwise, Tv # Sp, so (2.1) yields d(p, Ap) d(Bv, Ap) < d(Tv, Sp) d(p, Ap), & contradiction.We thus have, l0 Al0 S0.Similarly, 1o B0 Ti0, provided B and T are compatible.If both pairs A,S and B,T are compatible, the fact that 1o is the only common fixed point of A, B, S, and T follows easily from (2.1).COROLLARY 2.1.Let A, B, S, and T be self maps of a complete metric space (X,d), and suppose that S and T are surjective.If : r (0,1) such that for t,y x: d(At, By) <_ r d(Sz, Ty), (2.4) then :u,v,pX such that Au=Su=p=Bv=Tv.If moreover, the pairs A,S and B,T are each compatible, then A, B, S, and T have a unique common fixed point.PROOF.Define &(0,oo)-(0,oo) by (t)= t/r.To highlight the central role of compatibility, we observe COROLLARY 2.2.Let A and S be compatible self maps of a complete metric space (X,d).If S is surjective and if : rE (0,1) such that d(At, Ay)<r d(St, Sy) for t,y X, then A and $ have a unique common fixed point.
The major conclusion of Theorem 1 by Park  [4] follows from Theorem 2.1 with A B and S T. Since the identity map i(t)= t commutes with and is therefore compatible with any map f: xx, Theorem of Rhoades [5] is a consequence of Corollary 2.1 with A B i.

RELATIVELY NONEXPANSIVE MAPS.
Sehie Park [6] defined a self map g of a metric space (X,d) to be f-nonexpansive if 3 a continuous self map ! of (X,d) such that d(gz, gy)<_ d(fz, fy) for z,y X.Since we wish to extend the concept to four functions and drop the continuity requirement we shall say: DEFINITION 3.1.Let A,B,S, and T be self maps of a metric space (X,d).A and B are nonexpansive (n.e.) relative to S and Tiff d(Az, By) <_ d(Sz, Ty) for z,y X.Of course, if A B and S T, we shall say that A is nonexpansive relative to S.
Note that "order" is crucial in this definition.Thus, the pair A,B may be nonexpansive relative to the pair S,T, whereas the pair B,A may not be n.e.relative to the pair S,T (See [2], Remark 3.1).
We now state and prove our main result.
THEOREM 3.1.Let A,B,S,T be self maps of a compact subset C of a normed linear space x, and suppose that C is star-shaped with respect to q E C. If S and T are surjective and if for all z, y C: Ax By <-II Sz Ty II, (3.1) then there exist p,t,z E C such that Bz Tz , At St. If A and $ (B and T) are compatible, th,en A S (B, Tp).
PROOF.Let k n (0,1) such that karl, d for n N d C define: Anz nAz + (1 kn)q, d Bnz knBz + kn) q. (3.2) Since A,B:C d C is st-shad with rt to q, (3.2) s us that An, Bn:CC.
Morver, (3.2) d (3.1) imply that for z,yC d fixed nN:llAnz-BnYll k n 1Az By kn Sz Ty 11.But C S(C) T(C) is compact d therefore complete, d 0 < kn < 1; conquently, Corolly 2.1 implies that for eh n N there est z n, Yn C such that: Anzn SZn Pn BnYn TYn" Since C is compt, there is a subsequence {in} such that AinZin 8ZinP C.However, AZin Szin II Azin Ainin II II Azin (kinaZin + (1 kin)q ( kin) Azin + ( in) q (1-kin)M for some M > 0 by (3.2) d (3.3) since C is unded.Therefore, Ai-S=inll 0 kinl that AZin, SZinP n. (3.4) Now since C T(C), Tz l for some z C. By (3.1) we also have Since Tz p, (3.4) and the preceding inequalities imply that p Bz Tz.By a similar argument it follows that At St p for some C.
If moreover, A and S are compatible, At St p implies that SAt ASt; i.e., Sp Ap.In like fashion, if B and T are compatible, Tp Bp. / If A B in the statement of Theorem 3.1 and A is also injective, then Bz At implies that z, and we hve Az Sz Tz.So we can say the following.COROLLARY 3.1.Let A,S, and T be self maps of a compact subset C of a normed linear space x, and suppose that C is star-shaped with respect to q C. If 5" and T are surjective and [1Ax Ay <_ SZ Ty for z,y C, ( then ap C such that ap Sp Tp provided one of (a) and (b) below obtains: (a) A is one-to-one, (b) A,S and A,T are compatible pairs.With S T in Corollary 3.1, we obtain: COROLLARY 3.2.Let A and S be compatible self maps of a compact star-shaped subset C of a normed linear space.If S is surjective and A is nonexpansive relative to S, then A and S have a coincidence point.Again, with A i, the identity map, we have by Corollaxy 3.1" COROLLARY 3.3.If S and T are surjectve self maps of a star-shaped compact subset C of normed linear space X such that y _< s-Ty for ,y c, then p Sp Tp for some p C.
Of course, the fixed point 10 of Corollary 3.3 need not be unique; e.g., let S T i.
Note that Dotson's result, Theorem 1.1, which states that any nonexpansive self map of a star- shaped compact subset of a normed linear space has a fixed point, follows from Corollary 3.1(b) with S T i, the identity map.
Our next theorem extends Dotson's Theorem 2. [1] for weakly compact sets.We shall use the symbol _,w to denote weak convergence.We sketch the proof since it is similar to that of Theorem 3.1 and it incorporates ideas used in the proof of Dotson's Theorem 2. We remind the reader' that a mapping S:C-,X of a subset C of a Banach space X is demi-closed provided whenever {zn} c_ c, Zn--,x C, .andSzn-y .X, then Sz --' y.THEOREM 3.2.Let A,B,S, and T be self maps of a weakly compact subset 6' of a Banach space X, and suppose that C is star-shaped with respect to q e 6".If S and T are surjective and the pair A,B is nonexpansive relative to the pair S,T, there exists (u)e C such that provided S-A(T-B) is demi-closed.PROOF.Define kn, An, and B n as in the proof of Theorem 3.1.Since 6' T(C) is weakly compact and therefore norm-closed, C is complete.Consequently, the argument given in the proof of Theorem 3.1 up to (3.3) pertains and ] z n, un 6' such that (3.3) holds.Since 6" is weakly compact and Zn, Anzn C for n N, there exists a subsequence {in} such that w Z in--,x and to for some , p C. And as in the proof of Theorem 3.1, we'have for all n: A inZin SZ in'-" P ai n Si n <_ (1 in)II azi n / (1 in)II q II.(3.if)Since weakly convergent sequences axe norm-bounded and since kinl, (3.6) implies that (A S)Xin.-.O.But w Zin--,x and A-S is demi-closed, so that (A-S)z =0; i.e., Az Sz.Similarly, TV By for some V C if B-T is demi-closed, so that (A-S)r 0; i.e., Ar Sz.Similarly, TV for some V C if B T is demi-closed./ COROLLARY 3.4.Let A and S be self maps of a weakly compact subset C of a Banach space X and suppose that C is star-shaped with respect to q C. If S is surjective, A is nonexpansive relative to S, and if A-S is demi-closed, then A, S for some C. It is known-as a consequence of Theorem 3.2 in [3J-that if f and are continuous and compatible self maps of a compact metric space (X,d) such that f(x) c_ (X) and d(fr, fv) < d(r,v) when gz gy, then :l a unique z E x such that fx gx z.In view of Corollary 3.2 above, we are prompted to ask under what circumstances we could relax the preceding strict inequality by merely requiring that I be nonexpansive relative to g, and still obtain a common fixed point for f and .
The next theorem provides a possible beginning point for such inquiry.We prove this theorem by appeal to the following lemma, which lemma is actual proved in [7].
LEMMA 4.1 ([8]) Let f,g:l--.lbe continuous and compatible.If I and g have a coincidence point but no common fixed points, there exists a,b E I [0,1] such that (i) f(a) g(a) >_ b < a >_ f(b) g(b), and (ii) f(z) < g(t) for z (a,b).THEOREM 4.1.Let f and g be continuous, compatible self maps of the interval I [0,1] such that f(l)c_ g(l).If f is nonexpansive relative to g, then fz gz z for some z I.
PROOF.Since f and g are continuous and l(1)c_ g(l), fz gz for some z I. Suppose that f and g have no common fixed point.Then a,b I such that (i) and (ii) of Lemma 4.1 hold.Since g(a)>a and g(b)<b, c.(a,b) such that g(c)=c.
EXAMPLE 4.3.Let l(z)= I-= and g(z)= (1-z)2 for z e I.Note that I is not n.e.relative to g, and conversely.Consider successively: I/2,y l, and z 0, tt 1/2.On the other hand, the remainder of the hypothesis is satisfied (e.g., ! and g commute at z=0,1, their point of "coincidence" ).QUESTION.What compacta can be substituted for I in Theorem 4.1 and still yield a true result?