FIXED POINTS VIA "BIASED MAPS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 123, Number 7, July 1995 FIXED POINTS VIA "BIASED MAPS G. JUNGCK AND H. K. PATHAK (Commuicated by James E. West) Abstract. A geeralizatio of compatible maps called "biased maps" is itroduced ad used to prove fixed poit theorems for Meir-Keeler type cotractios ivolvig four maps. Extesios of kow results are thereby obtaied. I particular, a theorem by Kag ad Rhoades is geeralized. 1. Itroductio Self-maps A ad S of a metric space (X, d) are said to be compatible ([5]) iff d(sax, ASx) -> 0 wheever {x} is a sequece i X such that Ax, Sx t X. Compatible mappigs were itroduced i [5] as a geeralizatio of commutig mappigs ad have bee useful as a tool for obtaiig more comprehesive fixed poit theorems (see, e.g., [l]-[8], [10] [16]) ad i the study of periodic poits [9]. Now we itroduce the cocept of biased maps by softeig the restrictios imposed by compatibility. The result is a appreciable geeralizatio of compatible maps which, as we shall see, proves useful i the "fixed poit" area. Defiitio 1.1. Let A ad S be self-maps of a metric space (X, d). The pair {A, S} is S-biased iff wheever {x } is a sequece i X ad Ax, Sx -» t X, the (*) ad(sax, Sx) < ad(asx, Ax) if a = limif ad if a = lim sup. Of course, if the iequality i (*) holds with a = lim (which fact presupposes that the idicated limit exists), the lim if = lim sup = lim ad (*) is satisfied. We shall frequetly use this fact. The followig example shows why we could ot restrict a to "lim " if the bias cocept is to geeralize compatibility. (I this paper we shall use N, Q, Ir, ad / to deote the positive itegers, the ratioal umbers, the irratioal umbers, ad [0, 1], respectively.) Example 1.1. Let X = I, ad defie A, S : X > X by Ax Sx I - x for x [0, \], Ax - Sx = 0 for x Q (\, 1], ad Ax = Sx = 1 for x 7r(i, 1]. Let x2 = j ad x2 _i = ^ for N. The Sxk = \-xk -* 1 as k -* oo, SSx2 = 0, SSx2-X = 1, ad therefore lim*. d(ssxk, Sxk) does Received by the editors August 25, Mathematics Subject Classificatio. Primary 47H10, 54H25. Key words ad phrases. S-biased maps, compatible maps, ad (e, ^-cotractios America Mathematical Society 2049 Licese or copyright restrictios may apply to redistributio; see

2 2050 G. JUNGCK AND H. K. PATHAK ot exist although lim^. d(ssxk, SSxk) = 0 ; i fact, the pair {S, S} is trivially compatible for ay fuctio S. We remid the reader that lim if x = sup{x : «e N&x = iffc> xk} (for lim sup, switch sup ad if), ad that if a = limif or lim sup, ax < ay whe x <y + z for N ad z» 0 as «> oo. Also be assured that the "biased" map cocept arises aturally i the cotext of cotractive or relatively oexpasive ([7]) maps. See Propositio 2.1 below. Remark 1.1. If the pair {A, S} is compatible, the it is both S- ad ^-biased. For d(sax, Sx ) < d(sax, ASx) + d(asx, Ax) + d(ax, Sx ) for «N ; therefore, ad(sax, Sx) < 0 + ad(asx, Ax ) -I- 0 if Ax, Sx -> t X, {A, S} is a compatible pair, ad a is either lim if or lim sup. Thus {A, S} is S-biased. Similarly, by iterchagig A ad S i the above, we coclude that {A, S} is ^-biased if the pair is compatible. O the other had, cosider the followig. Example 1.2. Defie A, S : [0, 1] -> [0, 1] by Ax = I - 2x ad Sx = 2x for x [0, j], ad Ax = 0, Sx = 1 for x (, 1]. The, by usig Propositio 1.1 below, it is easy to show that {A, S} is both A- ad '-biased but ot compatible. (Note that both A ad S are cotiuous ad [0, 1] is compact, so that both A ad S are proper maps; i.e., A~X(M) is compact if M is.) The ext result is the aalogue to Theorem 2.2 i [8] for compatible maps. Propositio 1.1. Let A ad S be self-maps of a metric space (X, d). (a) If the pair {A, S} is S-biased ad Ap = Sp, the d(sap, Sp) < d(asp, Ap). (b) If A ad S are cotiuous ad oe of A or S is proper, the {A, S} is S-biased iff Ap = Sp implies that d(sap, Sp) < d(asp, Ap). Proof. To see that (a) holds, suppose that Ap = Sp. Let x = p for «e N, so Ax = Sx» Ap = Sp. The d(sap, Sp) = lim d(sax, Sx ) < lim d(asx, Ax ) = d(asp, Ap) as desired, sice {A, S} is 5-biased. Of course, the ecessity portio of (b) follows from (a). To see that the coditio give i (b) is sufficiet to esure that {A, S} is -biased, suppose that {x } is a sequece such that Ax, Sx t X ad that 5 is proper. The M = {Sx, «e N} U {/} is compact ad therefore S~X(M) is compact. But the the sequece {x } i S~X(M) has a subsequece {xkj which coverges to a poit p, ad therefore {Axk}, {Sxk} coverge to Ap ad Sp, respectively, sice A ad 5" are cotiuous. The Ap = Sp t by "uiqueess of limits", so that d(sap, Sp) < d(asp, Ap) by hypothesis. But the, sice Ax > Ap = t ad Sx -* Sp, SAx -» SAp ad ASx ASp because A ad S are cotiuous. We thus have lim d(sax, Sx) < lim d(asx, Ax ), as desired. D I Example 1.2 the pair {A, S} was both yl-biased ad S-biased. Of course, this eed ot be the case. Cosider: Licese or copyright restrictios may apply to redistributio; see

3 FIXED POINTS VIA "BIASED MAPS" 2051 Example 1.3. Let I = [0, 1] with the absolute value metric. Defie A, S : I -* I by A(x) = (x - \)2 ad S(x) = 2A(x) for x I. The A ad S are certaily proper sice both are cotiuous ad I is compact. We thus appeal to Propositio 1.1. Now Ax = Sx iff x = \. Sice A(\) = S(^) - 0, SA(\) = 5(0) = \ ad AS(\) =.4(0) = ±. Thus \AS(\) - A(\)\ = \ ad \SA(\) -S(\)\ = \, so by Propositio 1.1 the pair {A, S} is,4-biased ad ot S-biased. Cosequetly, Remark 1.1 tells us that {A, S} is ot compatible. For future referece, ote that \Ax - Ay\ = j\sx - Sy\ forx,y l. 2. (e, ^-CONTRACTIONS FOR FOUR MAPS Meir-Keeler cotractios for four maps were itroduced i [5] ad called (e, (J)-cotractios. To expedite the esuig discussio of theory ad results, we exted the (e, ô) cocept as follows. Defiitio 2.1. A pair of self-maps A ad 77 of a metric space (X, d) are (e, S)-S, r(/?)-cotractios relative to maps S, T : X -* X iff A(X) ç T(X), B(X) ç S(X), ad there exist fuctios p : XxX -> [0, oo) ad ö : (0, oo) -> (0, oo) such that ô(e) > e for all e, ad for x, y X : (i) e < P(x, y) < ô(e) implies that d(ax, By) < e. We shall refer to (e, ô)(p)-cotractios as (m) cotractios if p(x, y) = m(x,y) = max I d(sx, Ty), -(d(sx, By) + d(ax, Ty)) >, ad as (M) cotractios if p(x, y) = M(x, v) = max I d(sx, Ty), d(ax, Sx), d(by, Ty), X-(d(Sx,By) + d(ax,ty))y Remark 2.1. Thus, suppose A ad B are (e,ô)-s, T(p) cotractios with p(x, y) = m(x, y) or M(x,y). The Ax = By whe p(x, y) 0. Cosequetly, if Ax ^ By, p(x, y) ^ 0 ad (i) i Defiitio 2.1 therefore implies that d(ax, By) < p(x, y). So i geeral, d(ax, By) < p(x, y) for all x, y X. The followig propositio tells us that biased maps arise quite aturally. I particular, relatively oexpasive maps [7] ad thus (e, ^-cotractios iduce "bias". Propositio 2.1. Let A, B, S, ad T be self-maps of a metric space (X, d) such that A(X) ç T(X) ad d(ax, By) < d(sx, Ty) for x, y X. If S is cotiuous, the pair {A, S} is A-biased. Proof. For suppose Ax, Sx > t( X). Sice A(X) ç T(X), for each «e A^ 3y X such that Ty = Ax. The d(ty, By) = d(ax, By) < d(sx, Ty) -» 0, so By -> t. We thus have Ax, By, Sx, Ty -* t. Now d(asx, Ax) < d(asx, By) + d(by, Ax) for N, Licese or copyright restrictios may apply to redistributio; see

4 2052 G. JUNGCK. AND H. K. PATHAK so the cotiuity of S implies ad(asx, Ax) < ad(asx, By) < lim d(ssx, Ty) = limß?(5^x, Sx ), whether a = lim if or lim sup; i.e., {A,S} is ^-biased. D O the other had, Example 1.3 tells us that eve though A = B ad 5 = T, ad both A ad 5 are cotiuous i Propositio 2.1, the pair {A, 5} eed ot be S-biased. The followig result o (e, ô)-cotractios will prove useful. Propositio 2.2. Let S ad T be self-maps of a metric space (X, d), ad let A ad B be (e, S)-S, T(p)-cotractios of (X, d) with S lower-semicotiuous. If {x} ad {y} are sequeces i X such that limp(x, y) = e > 0 ad lim sup d(ax, By) - r R, the r < e. Proof. Sice ô(e) > e ad 5 is a lower-semicotiuous fuctio, there is a eighborhood Ns of e such that ô(t) > e for t Ne. We ca therefore choose to NE such that 0 < t0 < e < S(t0). Sice p(x,y )->e, there exists m N such that p(x, y ) (to, S(to)) for «> m. The, by (i) i Defiitio 2.1, d(ax, By) < t0 for «> m ; i.e., limsupd(ax, 77y ) = r < to < e. 3. Mai results Propositio 1.1 prompts the followig coveiet defiitio. Defiitio 3.1. Let A ad S be self-maps of a metric space (X, d). The pair {A, S} is weakly S-biased iff Ap = Sp implies d(sap, Sp) < d(asp, Ap). Of course, if {A, S} is S-biased, it is weakly S-biased by Propositio 1.1(a). Lemma 3.1. Let A, B, S, ad T be self-maps of a metric space (X, d). Suppose that (*) Ax / By implies d(ax,by) < m(x, y) = max jd(sx, Ty), ^(d(sx, By) + d(ax, Ty))\. If there exist u, v, p e X such that p = Au = Su = Bv = Tv ad {A, S} is weakly S-biased ({B, T} is weakly T-biased), the p = Ap = Sp (p = Bp = Tp). Proof. Suppose that {A, S} is weakly S-biased. Sice p = Au - Su, we have d(sau,su) < d(asu,au); i.e., (1) d(sp,p) < d(ap,p). We assert that Ap = p, ad hece p = Sp by (1). For if Ap ^ p, the Ap ^ Bv by hypothesis, ad (*) therefore implies that d(ap, p) = d(ap, Bv) < m(p, v) = max{d(sp, Tv), x2(d(sp, Bv) + d(ap, Tv))} = max{d(sp, p), x2(d(sp, p) + d(ap, p))} < d(ap, p) by (1). But we the have the cotradictio, d(ap, p) < d(ap, p). The proof that p = Bp = Tp whe {B, T} is weakly biased is aalogous. D The proof of the followig result uses the fact that ay (m) cotractio is a (M) cotractio. Licese or copyright restrictios may apply to redistributio; see

5 FIXED POINTS VIA "BIASED MAPS" 2053 Theorem 3.1. Let S ad T be self-maps of a complete metric space (X, d). Suppose that A ad B are (e, 6)-S, T(m)-cotractios ad that the pair {A, S} is S-biased ad {B, T} is T-biased. If oe of A, B, S, or T is cotiuous ad ö is lower semicotiuous, the there is a uique poit p X such that p = Ap - Bp = Sp = Tp. Proof. Let xq X, ad let {y } be defied iductively by y2-\ = Tx2-X = Ax2-2 ad y2 = Sx2 = Bx2-X for N. Sice A(X) ç T(X) ad B(X) ç S(X), the Xj ca be so chose. As is kow (see, e.g., [16], [11]) ad ot difficult to prove, the sequece {y } thus defied is Cauchy. Sice X is complete, 3p X such that y > p. I particular, (3.1) Ax2, Sx2, Bx2-X, Tx2-X -* p. We first use (3.1) to show that for ay sequece {v} i X ad N, (3.2) (i) d(av, Bx2-X) < d(sv, Tx2-X) + ß, (ii) d(ax2, Bv) < d(sx2, Tv) + y where ß, y -* 0 as «-» oo. To prove (i), ote that by defiitio of (m) cotractios, d(av, Bx2-X) < m(v, x2-i) = max I d(sv, Tx2 -i), -(d(av, Tx2-X) + d(sv, Bx2-X))\, so d(av,bx2-x)<d(sv,tx2-x) or d(av, Bx2-X) < \d(av, Tx2 -X) + jd(sv, Bx2-X) for «N. The first iequality satisfies (i) with ß = 0, so we eed cosider oly the secod iequality. But the secod iequality ad the triagle iequality imply: 2d(Av, Bx2-X) < d(av, Bx^-i) + d(bx2-x, Tx2-X) + d(sv, Tx2-X) + d(tx2_x, Bx2_x), which yields: d(av, Bx2 -X) < d(sv, Tx2-X) + 2d(Bx2~x, Tx2-X). This last iequality produces (i), sice (3.1) implies ß = 2d(Bx2-X, Tx2-X)» 0. The proof of (3.2)(ii) follows similarly with y = 2(Ax2, Sx2). Now assume that oe of S or 7, say S, is cotiuous. The SSx2, SAx2 Sp by (3.1). We assert that Sp = p. For suppose that d(sp, p) = e > 0. The (3.1) implies e = d(sp, p) = limd(ssx2, Tx2-X) (3.3) " = limd(ssx2, /7x2 _i) = limd(sax2, Sx2). Sice {A, S} is S-biased (see Defiitio 1.1), e = limd(sax2, Sx2) = ad(sax2, Sx2) < ad(asx2, Ax2). Now d(asx2, Ax2) < d(asx2, Bx2-i) + d(bx2-x, Ax2) for N. Therefore, (3.4) e < ad(asx2, Ax2) < ad(asx2, Bx2-X), sice d(bx2-.x, Ax2) -» 0. Licese or copyright restrictios may apply to redistributio; see

6 2054 G. JUNGCK AND H. K. PATHAK But d(asx2, Bx2 -\) < d(ssx2, Tx2-X) + ß by (3.2)(i), so lim sup d(asx2, Bx2_x) < limsup(7(ss.x:2, Tx2-i) = limd(ssx2, Tx2-X). The (3.3), (3.4), ad the precedig iequality imply that 0 < ß = limd(asx2, Bx2-X) = limd(ssx2, Tx2-X) = limd(ssx2, 5x2 _i) = limd(asx2, Tx2-X), the last equality followig from (3.1). But so m(sx2, x2-x) = max{d(ssx2, Tx2-i), 2(d(ASx2, Tx2-X) + d(ssx2, Bx2 x))}, 0 < e limm(sx2, x2 -X) = limd(asx2, Bx2-X), cotradictig Propositio 2.2. Thus, Sp p. But the Remark 2.1 ad (3.2) imply that d(ap, p) lim d(ap, Bx2-X) < lim (d(sp, Tx2-X)+ß ) = d(sp, p) = 0. We therefore have Sp = Ap - p. But A(X) ç T(X) by hypothesis, so Hu X such that Tu Ap = Sp. Therefore, by Remark 2.1, d(ap, Bu) < m(p, u) = max{d(sp, Tu), \(d(sp, Bu) + d(ap, Tu))} = \d(ap,bu). We coclude that Ap = Bu, ad we have Bu = Tu = p = Ap = Sp ; cosequetly, Bp = Tp = p = Ap Sp by Lemma 3.1. By symmetry, the argumet above applied to B ad T yields a commo fixed poit if T is cotiuous. Assume ext that A is cotiuous. The (3.1) implies that AAx2, ASx2 Ap. Suppose that d(ap, p) e > 0. The (3.5) 0 < e = limd(aax2, Bx2-X) = lim<7(^lsa:2, Ax2). Sice {A, S} is S-biased, limúí(^sx2, Ax2) = limsupúí(/4sx2, Ax2) > limsupi/(sax2, Sx2). But (3.1) implies limsupúf(s^x2, Sx2) = lim sup d(sax2, Tx2-X), so by (3.5) (3.6) 0 < e = lim<7(^l^x2, Bx2-X) > limsup<7(s^x2, Tx2-X). Moreover, by Remark 2.1 ad (3.2), limifú?(s^x2, Tx2_x) > limifi/(/l^x2, Bx2-X) = limd(aax2, Bx2-X). Licese or copyright restrictios may apply to redistributio; see

7 FIXED POINTS VIA "BIASED MAPS" 2055 We therefore obtai by (3.6), the precedig iequality, ad (3.1) which implies 0 < e = limd(aax2, Bx2-X) = limd(sax2, Tx2 _x) = limd(aax2, Tx2-X) = limd(sax2, Bx2 -X), limm(ax2, x2-\) = e = limd(aax2bx2-x), cotradictig Propositio 2.2. We coclude that Ap = p. But A(X) ç T(X) implies that Tv = Ap = p for some v X, so d(bv, p) = lim d(bv, Ax2) < lim(d(tv, Sx2 ) + y) - d(p,p) = 0, by (3.1) ad (3.2). Therefore Bv - p = Tv = Ap. But B(X) ç S(X), so that Su = Bv = Tv for some u e X ad we obtai as above: d(au, Bv) < m(u, v) = \d(au, Bv). Cosequetly, Au = Bv ; therefore Tv = Bv = p = Su = Au, which implies Tp = Bp = p = Ap = Sp by Lemma 3.1. Of coarse, a completely aalogous argumet yields a commo fixed poit if B is assumed to be cotiuous. We have show that if oe of A, B, S, or T is cotiuous, the A, B, S, ad T have a commo fixed poit. The uiqueess of the commo fixed poit follows immediately from the defiitio of (e, ô)-s, r(m)-cotractios. D Corollary 3.1. Let A, B, S, T be self-maps of a complete metric space (X, d) such that A(X) ç T(X) ad B(X) ç S(X). Suppose there exists r (0, 1) such that d(ax, By) < rm(x, y) for x, y X. If {A, S} is S-biased ad {B, T} is T-biased, the A, B, S, ad T have a uique commo fixed poit, provided oe of A, B, S, or T is cotiuous. Proof. Let ô(e) - s/r for e (0, oo). The a : (0, oo) (0, oo), ô is cotiuous ad therefore certaily lower-semicotiuous, ad ô(e) > e sice r < 1. Moreover, m(x, y) < 3(e) = e/r implies that d(ax, By) < rm(x, y) < r(e/r) = e, so that A ad B are S, r-(m)-cotractios. G Of course, Corollary 3.1 holds if we replace m(x, y) by d(sx, Ty). However, Example 1.3 shows that eve though we were to make that substitutio, require that A - B, S - T, ad demad that both A ad S be cotiuous, the coclusio to Corollary 3.1 eed ot hold if the pair {A, S} is ot S-biased. The role of "biased" maps i producig fixed poits is demostrated eve more dramatically by the ext result. If we drop all cotiuity requiremets ad the demad that ô be lower semicotiuous i Theorem 3.1, we ca still secure a c.f.p. by merely requirig that oe of A(X), B(X), S(X), or T(X) be complete istead of X. Theorem 3.2. Let S ad T be self-maps of a metric space (X, d), ad let A,B be (e, S)S, T(m)-cotractios. If oe of A(X), B(X), S(X), or T(X) is complete, ad the pairs {A, S} ad {B, T} are weakly S-biased ad weakly T-biased respectively, the A, B, S, ad T have a uique commo fixed poit. Proof. As i the proof of Theorem 3.1, there exists a Cauchy sequece {y } defied by: y2«-i = Tx2_x = Ax2-2 ad y2 = Sx2 = Bx2-X for N. Licese or copyright restrictios may apply to redistributio; see

8 2056 G. JUNGCK AND H. K. PATHAK Suppose T(X) is complete. Sice {y} is Cauchy, the subsequece {y2 _i} (ç T(X)) is Cauchy ad therefore coverges to a poit p = T(v) for some v X. The the Cauchy sequece {y} also coverges to p, ad we have (3.7) Ax2, Sx2, Bx2 -X, Tx2-X > p. Sice A ad B are (e,s)-s, r(m)-cotractios, Remark 2.1 ad the triagle iequality yield d{p, Bv) < d(p, Ax2) + d(ax2, Bv) < d(p, Ax2) + m(x2, v) ; so for «N, d(p,bv) <d(p, Ax2) + max i d(sx2, Tv), -(d(sx2, Bv) + d(ax2, Tv)) The (3.7) implies that d(p, Bv) < \d(p, Bv) as -» oo, ad we ifer that p = Bv = Tv. But B(X) ç S(X), so there exists u X such that Su = Bv = Tv. Therefore, d(au, Bv) < max{d(su, Tv), {(d(su, Bv) + d(au, Tv))} = \d(au, Bv). Hece, Au - Bv, ad we have p = Bv = Tv = Au = Su. Cosequetly, our hypothesis, Remark 2.1, ad Lemma 3.1 demad that p = Ap = Bp = Sp = Tp. That p is the oly commo fixed poit follows from the defiitio of (m) cotractios ad Remark 2.1. I the above we assumed that T(X) was complete. A comparable argumet yields (3.7) ad hece the coclusio if S(X) is complete. If o the other had, for example, A(X) is complete, we obtai (3.7) ad have p A(X). But A(X) C T(X), so that p T(X) ad the above argumet pertais. D The followig corollary to Theorem 3.2 geeralizes the mai theorem, Theorem 2.3, of Kag ad Rhoades i [13] by elimiatig cotiuity requiremets completely ad by replacig "compatibility" with "weak bias" ad d(sx, Ty) with m(x, y). (Note that the roles of the pairs A, B ad S, T are reversed i [13].) Corollary 3.2. Let A, B, S, ad T be self-maps of a complete metric space (X, d) with S ad T surjective. Suppose that the pair {A, S} is weakly S-biased ad {B, 71 is weakly T-biased. If there is a odecreasig upper semicotiuous fuctio tp : [0, oo) *[0, oo) such that <p(t) < t for all t > 0 ad (3.8) d(ax, By) < tp(m(x, y)) for x, y X, A, B, S, ad T have a uique commo fixed poit. Proof. We first show that the pair A, B is a (e,ô)-s, T-cotractio. Now A(X) ç T(X) ad B(X) c S(X) sice S ad T are surjectios. Sice <p is u.s.c. ad <p(e) < e whe e > 0, for each such e 3re > 0 such that <p(t) < e for t (e - re, e + re). We ca therefore defie ô : (0, oo)» (0, oo) by ô(e) = sup{t (e, e + 1) : cp(t) < e}. Clearly, ô(e) > e for e > 0. Moreover, by the above we ifer that if 0 < e < t < S(e), the defiitio of ô yields to Licese or copyright restrictios may apply to redistributio; see

9 FIXED POINTS VIA "BIASED MAPS" 2057 [t, 3(e)) such that <p(to) < e, ad hece <p(t) < e, sice tp is odecreasig. We coclude that for ay e, 0 < e < t < 3(e) implies q>(t) < e. Therefore, if e < m(x,y) < ô(e), d(ax, By) < q>(m(x,y)) < e by (3.8). Thus, property (i) i Defiitio 2.1 is satisfied. We have show that the pair A, B is a (e, ô)-s, T-cotractio by Defiitio 2.1. Moreover, sice T(X) = X, T(X) is complete. The hypothesis of Theorem 3.2 has bee show to be satisfied, ad the uique commo fixed poit is thereby assured. D I our cosideratio of Theorems 3.1 ad 3.2, we should ask whether or ot these results hold for the more geeral (M) cotractios, where by Defiitio 2.1, M(x, y) = max ld(sx, Ty), d(ax, Sx), d(by, Ty), \(d(ax,ty) + d(sx,by))y This questio merits a reply, sice results aalogous to Theorem 3.1 for compatible pairs {A,S} ad {77, T} which use M(x,y) istead of m(x,y) are i prit e.g., Theorems 8 ad 12 i [2], Theorem 3.2 i [3], Theorem 3.1 i [10], or the very geeral theorem by Rhoades, Park, ad Moo i [11] ad [16]. The followig example shows that although we replace m(x, y) i Theorems 3.1 ad 3.2 by p(x, y) - max{d(ax, Sx), d(by, Ty)} istead of M(x, y) ad permit all four fuctios to be cotiuous, we eed ot obtai a commo fixed poit if we require oly biased pairs of maps. Note that m(x, y) is obtaied from M(x, y) by deletig the p(x, y) terms. Example 3.1. Let X = I - [0, 1] ad d the absolute value metric. Defie A,B,S, T:I -+7 by ad \ ifxe[0,i], Ax = Bx= { l-2x ifx {\,\], 0 ifxe[i,l] s, = r* = i2x if*e[0^ 1 ifxe[i,l]. Clearly, A ad S are cotiuous ad A(X) = [0, j] Ç S(X) - X, so that both A(X) ad S(X) are complete. Moreover, A ad S are proper sice I is compact, so we may use Propositio 1.1(b) to show that the pair {A, 5} is biased. To this ed ote that At = St iff t = \. Ad A(\) = \ = S(\), so SA(\) = 1, AS(\) = 0. Therefore, \AS(±)-A(±)\ = \ = \SA(±)-S(\)\, which implies {A,S} is both ^-biased ad S-biased. But sice SA(\) ^ AS(\), {A, S} is ot compatible ([8, Theorem 2.2]). We ow show \Ax - Ay\ < \Ay-Sy\ if x <y,so d(ax, Ay) < \max{d(ax, Sx), d(ay, Sy)} certaily Licese or copyright restrictios may apply to redistributio; see

10 2058 G. JUNGCK AND H. K. PATHAK holds. (Note: Because of symmetry i x ad y i this last iequality, we lose o geerality by assumig x <y.) Now if 0 < x, y < \, \Ax- Ay\ = 0 < \\Ay -Sy\. If 0 < x < \ < y < %, \Ax-Ay\ = lhl-2y) = Îl4y-l ad \Ay-Sy\ = (l-2y)-2y = l-4y = 2\Ax - Ay\. O the other had, if\<x<y<\, \Ax - Ay\ = 2(y - x), whereas.4y - Sy = 4y - 1 > 4y - 4x = 4(y - x), sice 4x > 1 ; thus, \Ax - Ay\ < \\Ay - Sy\. Fially, if y > \, \Ay - Sy\ = I, but for ay x, y : \Ax - Ay\ < \. We have show that i ay evet, x < y implies \Ax - Ay\ < j\ay - Sy\. But A ad S do ot have a commo fixed poit; either do 77 ad T, sice A = B ad S = T. The above example clearly demostrates that the potetially ill-behaved terms i M(x,y) are d(ax,sx) ad d(by, Ty) whe the pairs {A,S} ad {77, T} are ot compatible. Cosequetly, improvemets or geeralizatios of Theorems 3.1 ad 3.2 may be difficult to come by i the cotext of (M) cotractios ad biased maps. But whe the pairs {A, S} ad {B, T} are compatible, Propositio 2.2 i [5] guaratees the desired respose from d(ax, Sx) ad d(by, Ty). The iterested reader ca cofirm this by checkig the proofs of theorems i [2], [12], ad [16], for example. 4. Retrospect By the above, if we require that the pairs {A, S} ad {B, T} be compatible istead of beig S ad T biased, respectively, Theorem 3.1 is valid for (M) cotractios as well as (m) cotractios. Therefore, the followig "suggests" that Theorem 3.1 may ot be a ew result. Propositio 4.1. Let S ad T be self-maps of a metric space (X, d), ad let A ad B be (e,3)-s, T-(m)-cotractios with 8 lower semicotiuous. Suppose that the pair {A, S} is S-biased. If oe of A or S is cotiuous, the the pair {A, S} is compatible. Proof. Suppose {x} is a sequece i X ad t X such that Ax, Sx» t. We ca the appeal to the proof of Propositio 2.1 to obtai a sequece {y } such that 77y, Ty -»!. If we substitute x for x2 ad y for x2-x i that portio of the proof of Theorem 3.1 which verifies that Sp = p whe S is cotiuous, we obtai lim d(asx, By) = lim d(sax, Sx) = 0. But d(asx, SAx) < d(asx, By)+d(By, Sx)+d(Sx, SAx), for N, so d(asx, SAx) -> 0 ; i.e., {A, S} is compatible. The argumet i the istace i which A is cotiuous is comparable. D The followig example assures us that, i spite of Propositio 4.1, Theorem 3.1 pertais to situatios ot icluded by Theorem 2.1 of [6]. We agai refer to Remark 2.1 ad remid the reader that Propositio 4.1 certaily holds for (e, 3)-S, T-cotractios. Example 4.1. Let X = [0, 1]. We defie maps A, 77, S, T : X X such that A{X) = {i} ç T(X) = {0}U[i, l],b(x) = {±, } = S(X), ad oly A is cotiuous. These facts will be immediately apparet, ad we leave them for Licese or copyright restrictios may apply to redistributio; see

11 the reader to cofirm. Now defie FIXED POINTS VIA "BIASED MAPS" 2059 Ax = Bx = Sx -r ad Tx-l-x if x ad 1 3 Ax = -, Bx = Sx =, ad Tx = 0 if x ( ^, 1. First ote that {A, B} is a (e, 8)-S, T-cotractio sice it satisfies \Ax - By\ < ^\Sx - Ty\ for x,y X. To see this, observe that \Ax - By\ 0 oly whe y > \. The \Ax - By\ = \ ; whereas Sx > for all x, so that \Sx - Ty\ = \Sx - 0 >. Thus, i ay evet, 3\Ax - By\ < \Sx - Ty\. To see that {A, S} is compatible, suppose that Ax, Sx -» / e X. Clearly, t = j ad x < j for large sice \Ax - Sx\ = for x > \. The S.4x = S(\) = \ ad ASx = x2. Thus \SAx - ASx\ -> 0. O the other had, cosider 77 ad T. If Bx, Tx -> / X, the t = j, xh -», ad ^«< 2 f r ^arëe So r* {j, 1 - x }, 77x = 3, ad T5x = \ for all large. The r/7x - Tx -> ^ - j = 0, ad {/7, T} is therefore T-biased. O the other had, if x = \, e.g., Tx -> ' ad therefore 77Tx - 77x -> 11 - j =. Cosequetly, {B, T} is ot /7-biased ad thus ot compatible. We coclude by otig that Theorem 3.2 elimiated all cotiuity requiremet o A, B, S, ad T, ad the l.s.c. requiremets o 3 imposed i Theorem 3.1, ad merely required that oe of the rage spaces be complete i lieu of X beig complete. This prompts the Questio. To what extet ca the lower semicotiuity hypothesis o 3 be muted i Theorem 3.1? Refereces 1. N. A. Assad ad S. Sessa, Commo fixed poits for oseif-maps o compacta, SEA Bull. Math. 16(1992), Tog-Huei Chag, Fixed poit theorems for cotractive type set-valued mappigs, Math. Japo. 38(1993), Y. J. Cho, P. P. Murthy, ad G. Jugck, A commo fixed poit theorem of Meir ad Keeler type, Iterat. J. Math. Math. Sei. 16 (1993), R. O. Davies ad S. Sessa, A commo fixed poit of Gregus type, Facts Uiversitii (NIS) Ser. Math. Iform. 16 (1992), G. Jugck, Compatible mappigs ad commo fixed poits, Iterat. J. Math. Math. Sei. 9 (1986), _, Compatible mappigs ad commo fixed poits "revisited", Iterat. J. Math. Math. Sei. 17(1994), _, Coicidece ad fixed poits for compatible ad relatively oexpasive maps, Iterat. J. Math. Math. Sei. 16 (1993), _, Commo fixed poits for commutig ad compatible maps o compacta, Proc. Amer. Math. Soc. 103 (1988), , Commo fixed poits for compatible maps o the uit iterval, Proc. Amer. Math. Soc. 115(19921, Licese or copyright restrictios may apply to redistributio; see

12 2060 G. JUNGCK AND H. K. PATHAK 10. _, Compatible mappigs ad commo fixed poits (2), Iterat. J. Math. Math. Sei. 11 (1988), G. Jugck, K. B. Moo, S. Park, ad B. E. Rhoades, O geeralizatios of the Meir-Keeler type cotractio maps: Correctios, J. Math. Aal. Appl. 180 (1993), G. Jugck ad B. E. Rhoades, Some fixed poit theorems for compatible maps, Iterat. J. Math. Math. Sei. 16 (1993), S. M. Kag ad B. E. Rhoades, Fixed poits for four mappigs, Math. Japo. 37 (1992), H. K. Pathak, Commo fixed poit theorems without cotiuity for weak compatible mappigs, J. Austral. Math. Soc. Ser. A (1993) (to appear). 15. K. P. R. Rao, Some coicidece ad commo fixed poits for self-maps, Bull. Ist. Math. Acad. Siica 19 (1991), B. E. Rhoades, S. Park, ad K. B. Moo, O geeralizatios of the Meir-Keeler type cotractio maps, J. Math. Aal. Appl. 146 (1990), Departmet of Mathematics, Bradley Uiversity, Peoría, Illiois Departmet of Mathematics, Kalya Mahavidyalaya, Bhilai Nagar (M.P.), Idia Licese or copyright restrictios may apply to redistributio; see