COMMON FIXED POINTS OF COMPATIBLE MAPPINGS

Transcription

1 Iterat. J. Math. & Math. Sci. VOL. 13 NO. (1990) COMMON FIXED POINTS OF COMPATIBLE MAPPINGS S.M. KANG ad Y.J. CHO Departmet of Mathematics Gyeogsag Natioal Uiversity Jiju Korea G. JUNGCK Departmet of Mathematics Badley Uiversity Peoria, Illiois U.S.A. (Received October 19, 1988 ad i revised form September 27, 1988) ABSTRACT. I this paper, we preset a commo fixed poit theorem for compatible mappigs, which exteds the results of Dig, Divlccaro-Sessa ad the third author. KEY WORDS AND PHRASES. Commo fixed poits, commutig mappigs, weakly commutig mappigs ad compatible mappigs AMS MATHEMATICS SUBJECT CLASSIFICATION CODE. 54H INTRODUCTION. I [I], the cocept of compatible mappigs was itroduced as a geeralizatio of commutig mappigs. The utility of compatibility i the cotext of fixed poit theory was demostrated by extedig a theorem of Park-Bae [2]. I [3], the third author exteded a result of Sigh-Sigh [4] by employig compatible mappigs i lleu of commutig mappigs ad by usig four fuctios as opposed to three. O the other had, Diviccaro-Sessa [5] proved a commo fixed poit theorem for four mappigs, usig a well kow cotractive coditio of Meade-Sigh [6] ad the cocept of weak commutativity of Sessa [7]. Their theorems geeralize results of Chag [8], Imdad Kha [9], Meade-Sigh [6], Sessa-Fisher [I0] ad Sigh-Sigh [4]. I this paper, we exted the results of Dig [II], Diviccaro-Sessa [5] ad the third author [3]. The followig Defiitio 1.1 is give i [I]. DEFINITION I.I. Let A ad B be mappigs from a metric space (X,d) ito itself. The A ad B are said to be compatible if lira d(abx BAx 0 wheever Ix is a sequece i X such that llm Ax llm Bx z for some z i X. Thus, if d(abx BAx O as d(ax Bx 0, the A ad B are compatible.

2 62 S.M. KANG, Y.J. CHO AND G. JUNGCK,Mappigs which commute are clearly compatible, bu[ the coverse is false. S. Sessa [7] geeralized commutig mappigs by callig mappigs A ad B from a metric space (X,d) ito itself a weakly commutig pair if d(abx, BAx) d(ax, Bx) or all x i X. Ay weakly commutig pair are obviously compatible, but the coverse is false [3]. See [I] for other examples of the compatabile pairs whlch are ot weakly commutative ad hece ot commutig pairs. LEMMA I.! ([I]). Let A ad B be compatible mappigs from a metric space (X,d) ito itself. Suppose that llm Ax llm Bx z for some z i K. The llm BAx Az If A is cotiuous. 2. A FIXED POINT THEOREM. Throughout this paper, suppose that the fuctio : [0,) 5 [0, ) satisfies the followig coditios: (I) is odecreaslg ad upper semicotluous i each coordiate variable, (2) For each t > max [#(0,O,t,t,t), (t,t,t,2t,0), (t,t,t,0,2t)} < t. (2.1) LEMMA 2.1 ([12]). Suppose that : [0,) [0, ) is odecreasig ad upper semicotiuous from the right. If (t) < t for every t > 0, the lira (t) O, where? (t) deotes the compositio of (t) with itself -times. Now, we are ready to state our mai Theorem. THEOREM 2.2. Let A,B,S, ad T be mappigs from a complete metric space (X,d) ito itself. Suppose that oe of A,B,S ad T is cotiuous, the pairs A,S ad B, T are compatible ad that A(X) T(X) ad B(X) c S(X). If the iequality d(ax, By) (d(ax,sx), d(by,ty), d(sx,ty), d(ax,ty), d(by,sx)) (2.2) holds for all x ad y i X, where satisfies (I) ad (2), the A,B, S ad T have a uique commo fixed poit i X. PROOF. Let x X be give. Sice A(X)eT(X) ad B(X)c S(X), we ca choose x i o X such that Yl --TXl Ax ad, for this poit x I, there exists a poit x 2 i X such o that Y2 --Sx2 Bxl ad so o. Iductively, we ca defie a sequece {y i X such that Y2+l TX2+l AX2 ad (2.3) Y2 SX2 BX2-l" By (2.2) ad (2.3), we have d y2 + Y2+2 d(ax2, BX2+ i (d(ax2, SX2), d(bx2+l, TX2+l), d(sx2, TX2+l), d(ax2, TX2+I ) d(bx2+i, SX2 (d(y2+1, Y2 ), d(y2+2, Y2+l ), d(y2, Y2+l ), 0, d(y2+2, Y2 (d(y2 Y2+ d(y2+l Y2+2 ) d(y2 Y2+l ) O, d(y2, Y2+l + d(y2+l Y2+2 )"

3 COMMON FIXED POINTS OF COMPATIBLE MAPPINGS 63 If d(v2+i Y2+2 > d(y2. Y2+l) d(y2+ I, Y2+2 (d(y2+l Y2+2 ), d(y2+ I, Y2+2 ), d(y2+l), Y2+2 ), O, 2d(Y2+l Y2+2 )) --< (d(y2+l Y2+2 )) d(y2+l Y2+2 ) which is a cotradlcito. Thus,.Similarly, we have i the above Iequality, the we have d(y2+l Y2+2 (d(y2 Y2+ ) d(y2 V2+! 0, 2d(Y2 Y2+?(d(Y2, Y2+l ))" d(y2 Y2+I ) (2.4) d(y2+2 Y2+3 (d(y2+l Y2+2)). (2.5) It follows from (2.4) ad (2.5) that d d(y y (d(y y))?-l(d(y y2)) (2 6) + - By (2.6) ad Lemma 2.1, we obtai ltm d O. (2.7) I order to show that {y is a Cauchy sequece, it is sufficiet to show that {Y2 [s a Cauchy sequece. Suppose that [Y2 is ot a Cauchy sequece. The there is a a > 0 such that, for each eve Iteger 2k, there exist eve itegers 2re(k) ad 2(k) such that d(y2m(k), Y2(k)) > for 2re(k) > 2(k) 2k. For each eve iteger 2k, let 2re(k) be the least eve itege exceedig (2.8) 2(k) satisfyig (2.8), that is, d(y2(k), Y2m(k)_2 ad d(y2(k), Y2m(k) The, for each eve iteger 2k, < d(y2(k) Y2m(k) d(y2(k) Y2m(k)-2 It follows from (2.7) ad (2.9) that lira d(y2(k), Y2m(k)). k+ By the triagle iequality, >. (2.9) + d2m(k)_2 + d2m(k)_ ad d(y2(k), Y2m(k)_l) d(y2(k), Y2m(k)) d2m(k)_ + d(y2(k)+l Y2m(k)-I d(y2(k) Y2m(k))l d2m(k)- d From (2.7) ad (2.10), as k, d(y2(k), Y2m(k)_[) e ad d(y2(k)+i, Y2m(k)_l e. 2(k) (2. I0)

4 64 S.M. FANG, Y.J. CHO AND G. JUNGCK By (2.2) ad (2.3), we have d(y2(k) d Y2m(k) + d(ax 2 (k) 2 (k) Bx2m (k)-i < d2(k + #(d2(k), d2m(k)- I d(y2(k) Y2m(k)-I ) d(y2(k) + l Y2m(k)- ) d(y2m(k) Y2(k) )" is upper semlcotluous, (0, O,,,c) eas k/, which is a cotradictio. Hece {y is a Cauchy sequece ad it coverges to some poit z i X. Cosequetly the subsequeces {AX2} {SX2} {BX2-l} ad {TX2_I coverge to z. Suppose that S is cotiuous. Sice A ad S are compatible, Lemma I. 2 implies that SSX2 ad ASX2 By (2.2), we obtai d( ASX2 Bx 2 -I Lettig, we have Sz. #(d(asx2, SSX2), d(bx2_ I, TX2_l), d(ssx2, TX2_l), d(asx2, TX2_l) d(bx2_i, SSX2 )" d(sz, z) (0, O, d(sz, z), d(sz, z), d(z, Sz)), so that z Sz. By (2.2), we also obtai d(az, Bx2_l) l(d(az, Sz), d(bx2_l, TX2_l), d(sz, TX2_l), d(az, TX2_i), d(bx2_i, Sz)). Lettig +, we have d(az, z) #(d(az, Sz), O, d(sz, z), d(az, z), d(z, Sz)), so that z kz. Sice A(X)C T(X), z T(X) ad hece there exists a poit w i X such that z Az Tw. d(z, Bw) d(az, Bw) t(3, d(bw, Tw), d(sz, rw), d(az, Tw), d(bw,z)), which implies that z Bw. Sice B ad T are compatible ad Tw Bw z, d(tbw, BTw) 0 ad hece Tz TBw BTw Bz. Moreover, by (2.2), d(z, Tz) d(az, d(bz, Tz), d(z, Tz), d(z, Tz), d(bz, z)), so that z Tz. Therefore, z is a commo fixed poit of A,B,S ad T. Similarly, we ca complete the proof i the case of the cotiuity of T. Now, suppose that k is cotiuous. Sice A ad S are compatible, Lamia 1.2 implies that By (2.2), we have AAX2 ad SAX2 Az. d(aax 2 (d( SAX2), d(bxo TX2 ), BX2-I AAx2 -I d(sax2, TX2_ I), d(aax2, TX2_ I), d(bx2_ [, SAX2))- Lettig, we obtai d(az, z) (0, O, d(az, z), d(az, z) d(z, Az)), so that z Az. Hece, there exists a poit v i X such that z Az Tv.

5 COMMON FIXED POINTS OF COMPATIBLE MAPPINGS 65 d(aaz2 Bv) $(d(aax2 SAX2), d(bv, Tv), d(sax2, Tv), d(aax2, Tv) d(bv, SAX2)), Lettig, we have d(z, Bv) < (0, d(bv, Tv), d(z, Tv), d(az, Tv), d(bv,z)), which implies that z By. S[ce B ad T are compatble ad Tv Bv z, d(tbv, BTv) 0 ad hece Tz TBv BTv Bz. Moreover, by (2.2), we have d(ax2 Bz) (d(ax2 SX2), d(bz, Tz), d(sx2 Tz), d(ax2, Tz3, d(bz, SX2)). Lettig, d(z, Bz) (0, d(bz, Tz), d(z, Tz), d(z, Tz), d(bz, z)), so that z Bz. Sice B(X) c S(X), there exists a poiit w i X such that z Bz Sw. d(aw, z) d(aw, Bz) (d(aw, Sw), O, d(sw, z), d(aw, z), d(z, Sw)), so that Aw z. Sice A ad S are compatible ad Aw Sw z, d(sbw, BSw) 0 ad hece Sz SAw ASw Az. Themefore z is a commo fixed poit of A,B,S ad T. Similarly, we ca complete the proof i the case of the cotiuity of B. It follows easily from (2.2) that z is a uique commo fixed poit of A,B,S ad T. COROLLARY 2.3. Let A,B,S ad T be mappigs from a complete metmic space (X,d) ito itself. Suppose that oe of A,B,S ad T is cotiuous, the pairs A,S ad B,T are compatible ad that A(X) c T(X) ad B(X) S(X). I the iequality (2.2) holds for all x ad y i X, where satisfies (I) ad (2.[)" t) max[$(t,t,t,t,t), $(t,t,t,2t,o), $(t,t,t,o,2t)}<t (2.11) for each t > 0, the A,B,S ad T have a uique commo fixed poit i X. REMARK 2.4. From Theorem 2.2 ad Corollary 2.3, we exted the results of Dig [II] ad Diviccaro-Sessa [5] by employig compatibility i lieu of commutig ad weakly commutig mappigs, respectively. Further our theorem exteds also a result of Dig [II] by usig oe cotiuous fuctio as opposed to two. 5 REMARK 2.5. From Theorem 2.2 defiig : [0, ) [0, ) by $(tl,t2,t3,t4,t5) h max[tl,t2,t3, (t4+ ts)} for all tl,t2,t3,t4,t 5 [0,) ad h [0, I), we obtai a result of the third author [3] eve if oe fuctio is cotiuous as opposed to two. REFERENCES I. JUNGCK, G., Compatible mappigs ad commo fixed poits. Iterat. J. Math. & Math. Sci., 9 (1986), PARK, S. ad BAE, J.S., Extesios of a commo fixed poit theorem of Mier ad Keeler, Ark. Math., 19 (1981),

6 66 S.M. KANG, Y.J. CHO AND G. JUNGCK 3. JUNGCK, G., Compatible mappigs ad commo fixed poits (2), Iterat. J. Math. ad Math. Scl., 11 (1988), SINGH, S.L. ad SINGH, S.P., A fixed poit theorem, Idia J. Pure & Appll.e.d Math. II (1980), DIVICCARO, M.L. ad SESSA,S., Some remark o commo fixed poits of four mappigs, Jaabha I_5 (1985), MEADE, B.A. ad SINGH, S.P., O commo fixed poit theorems, Bull. Austral. Math. Soc., 16 (1977), SESSA, S., O a weak commtatlbity coditio i fixed poit cosideratios, Publ. Ist. Math.., 32(46) (1982), CHANG, C.C., O a fixed poit theorem of cotractive type, Comm. Math. Uiv. St. Paul 32 (1983), IMDAD, M. ad KHAN, M.S., Fixed poit theorems for a class of mappigs, Idia J. Pure ad Applied Math.., I4 (1983), I0. SESSA, S. ad FISHER, B., Commo fixed poits of weakly commutig mappigs, Bull. Polish. Acad. Scl. Math., 35, (1987), I. DING, X.P., Some commo fixed poit theorems of commutig mappigs II, Mat hh. Semiar Note I.(1983), MARKOWSKI, J., Fixed poit theorems for mappigs with cotractive iterate at a poit, Proc. Amer. math. Soc., 62 (1977),

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