fix- gx x +0 ff x 0 and fgx- gfxl 2x" 0 if x 0 so that f and g are compatible on R with the usual metric.

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1 Internat. J. Math. & Math. Sci. VOL. Ii NO. 2 (1988) COMPATIBLE MAPPINGS AND COMMON FIXED POINTS (2) GERALD JUNGCK Department of Mathematics Bradley University Peoria, Illinois (Received November 17, 1986) ABSTRACT. A common fixed point theorem of S.L. S.P. Singh is generalized by weakening commutativity hypotheses by increasing the number of functions involved. KEY WORDS AND PHRASES. Common fixed points, commuting mappings, compatible mappings MATHEMATICS SUBJECT CLASSIFICATION CODE. 54H25 I. NTRODUCT ON. In 11] the concept of compatible mappings was introduced as a generalization of commuting maps (fg=gf) The utility of compatibility in the context of fixed point theory was demonstrated by extending a theorem of Park Bae [2] The purpose of this note is to further emulate the compatible map concept. We extend the following strong result of S.L. Singh S.P. Singh 3] by employing compatible maps in lieu of commuting maps, by using four functions as opposed to three. THEOREM. I.I. Let P,Q, T be self maps of a complete metric space (X,d) such that PT=TP, QT=TQ, P(X)LJQ(X)CT(X) If T is continuous there exists r(o,l) such that d(px,qy) < r max {d(tx,ty),d(px,tx), d(qy,ty), 1/2(d(Px,Ty) + d(qy,tx))} for all x,y in X (l.l) Then P, Q, T have a unique common fixed point. 2. PRELIMINARIES. The following definition was given in i] Definition 2.1. Self maps f g of a metric space (X,d) are compatible iff limnd(fgxn,gfxn) 0 whenever {x n is a sequence in X such that lim fx lim t for n some t in n ngxn X Thus, if d(fgx,gfx) +0 as d(fx,gx) -0 f g are compatible. For example, suppose that fx x 2 gx 2x for x in R the set of reals, f g are not commutative, but fix- gx x +0 ff x 0 fgx- gfxl 2x" 0 if x 0 so that f g are compatible on R with the usual metric.

2 286 G. JUNGCK Now maps which commute are clearly compatible, but the converse is false. In fact, compatible maps need not be weakly commutative. Sessa [4 defined self maps f g of a metric space (X,d) to be a weakly commuting pair iff d(gfx,fgx) d(fx,gx) for x in X If f g are weakly commutative they are obviously compatible, but the converse is false as the above example shows (e.g., let x=l.). See 1] for other examples of compatible pairs which are not weakly commutative hence not commuting pairs. 3. MAIN RESULTS. LEMMA 3.1 Let A,B,S, T be self maps of a metric space (X,d) such that A(X)cT(X) B(X)cS(X) let x X If r (0,I) such o that d(ax:by) r max(mxy) for xy cx where Mxy d(ax,sx),d(by,ty),d(sx,ty),1/2(d(ax,ty) + d(by,sx)) (3.1) then there is a Cauchy sequence in X beginning at x o defined by Y2n-I TX2n-i AX2n-2 Y2n SX2n BX2n-i for ncn the set of positive integers. PROOF. Since A(X)cT(X) B(X)cS(X) we can choose x x 2 in X such that Yl TXl AXo Y2 Sx2 BXl In general, we can choose X2n_l,X2n in X such that Y2n-1 TX2n-1 AX2n-2 Y2n SX2n BX2n-1 (3.2) Thus the indicated sequence exists. To see that is Cauchy, note that (3.1) (3.2) imply that d(tx2n+l,sx2n+2) d(ax2n,bx2n+l)_<r max(mn) where M n d(ax2n,sx2n) d(bx2n+ l,tx2n+ 1), d(sx2n,tx2n+l), 1/2(d(AX2n,TX2n+l) + d(bx2n+l,sx2n)) Then by (3.2) M n {d(tx2n+l,sx2n d(sx2n+2,tx2n+1 1/2d(SX2n+2,SX2n But 1/2d(SX2n+2,S2n)< 1/2(d(SX2n+2,TX2n+l + d(tx2n+l,sx2n))<_max d(sx2n+2,tx2n+l), d(tx2n+l,sx2n) since the larger of two numbers is greater or equal to their average. So we have max(mn) max {d(sx2n+2,tx2n+l), d(tx2n+l,sx2n)} with d(tx2n+l,sx2n+2) <_ r max(mn) But if d(tx2n+l,sx2n+2) <_ r d(tx2n/l,sx2n+2) d(tx2n/l,sx2n+2) 0 since r (0,1); thus max(mn) d(tx2n+l,sx2n) we conclude d(tx2n+l,sx2n/2)< r d(tx2n+l,sx2n) Similarly, d(tx2n+3,sx2n+2)! r d(sx2n+2,tx2n+l) Consequently, (3.2) implies that d(ym+l,ym)_< r d(ym,ym_ 1) for m even or odd. This last inequality implies that {Ym is Cauchy, as desired. / We shall also need the following simple result from [I (Proposition 2.2(2a)).

3 COMPATIBLE MAPPINGS AND COMMON FIXED POINTS 287 PROPOSITION 3.1. If f g are compatible self maps of a metric space (X,d) limnfxn limngx n t for some t in X then limngfx n ft if f is continuous. We can now state prove our generalization of Theorem I.I. THEOREM 3.1. Let A,B,S T be self maps of a complete metric space (X,d). Suppose that S T are continuous, the pairs A,S B,T are compatible, that A(X)(3 T(X) B(X) S(X). If there exists r (0,1) such that d(ax,by) < r max(mxy) for x,y in X where Mxy {d(ax,sx),d(by,ty),d(sx,ty), 1/2(d(ax,Ty) + d(by,sx))} (3.3) then there is a unique point z in X such that z Az Bz Sz Tz PROOF. By the Lemma 3.1. there is a sequence x n in X such that SX2n BX2n-I Y2n TX2n-I AX2n-2 Y2n-I such that the sequence Ym is Cauchy. Since (X,d) is complete {Ym converges to a point z in X Consequently, the subsequences AX2n SX2n,{ TX2n_l,{ BX2n_l converge to z (3.4) Since A S are compatible B T are compatible, the continuity of S T (3.4), Proposition 3.1. imply TTX2n_l,BTX2n_l Tz SSX2n,ASX2n Sz (3.5) Then (3.3) implies d(sz,tz) limnd(asx2n,btx2n_l)! r max(limm n) where M n {d(asx2n,ssx2n),d(btx2n_l,ttx2n_l),d(ssx2n,ttx2n_1), 1/2(d(ASX2n,TTX2n_ I) + d(ssx2n,btx2n_l)) }. By (3.5), lim n M {0,O,d(Sz,Tz),1/2(d(Sz,Tz) + n d(sz,tz)) so that d(sz,tz) r d(sz,tz) Since 0< r< Sz=Tz Also, d(az,tz) limnd(az,btx2n_l) r max(limnmn) where M n {d(az,sz) d(btx2n_l,ttx2n_l) d(sz,ttx2n_l), 1/2(d(Sz,BTX2n_1 + d(az,ttx2n_1)) Since Sz=Tz (3.5) yields" limnmn {d(az,tz),o,o,1/2(d(az,tz)) therefore, d(az,tz) < r d(az,tz) from which (as above) we infer Az=Tz(=Sz) But if we use this last stated equality in (3.3) with x=y=z we obtain" Az Bz Sz Tz (3.6) In fact, z is a common fixed point of A,B,S, T For (3.3) (3.4) yield" d(z,bz) limnd(ax2n,bz) r max(limnmn) with Mn d(ax2n,sx2n),d(bz,tz),d(sx2n,tz),1/2(d(ax2n,tz) + d(bz,sx2n))} Then limnmn {O,O,d(z,Tz),1/2(d(z,Bz) + d(bz,z))} by (3.4) (3.6). We thus

4 288 G. JUNGCK obtain d(z,bz) r d(z,bz) we conclude that z Bz Az Sz Tz That z is the only common fixed point of A,B,S, T follows easily from (3.3)./ We conclude with an example of four functions which satisfy the hypothesis of Theorem 3.1., no three of which satisfy the hypothesis of Theorem I.I.. EXAMPLE 3.1. Let X [I, ) d(x,y) x-y for x,y X Define Ax x Bx x Sx 2x 6 Tx 2x for x in X The functions are all continuous satisfy A(X)=B(X)-S(X)=T(X) X Moreover, Sx-Ax j2x + lj x 0 iff x since x ASx-SAx 6x(x-l) 0 iff x since x Thus, d(ax,sz) 0 only if x- in which instance d(asx,sax) 0 So A S are compatible; but they are not a weakly commuting hence not a commuting pair (Let x 2) Similarly, T B are compatible, since JTx-BxJ (2x+l)J x 0 iff x (x _>_ I) JTBx-BTxJ 2(x 1) 0 iff x (x >_ I) Finally, Sx-Tw 2J x y2 x: + y >_ 2J Ax-Bw 2 for x,y >_ I; therefore, Ax-ByJ J Sx-Tyj _< max(mxy) for x,y in X Hence (3.3), thus the hypothesis of Theorem 3.1., is satisfied. Observe also that no one of A,B,S, or T commutes with any two of the remaining three functions. Of course, common fixed point theorems other than Theorem I.I follow from Theorem See, for example, Corollary 3.2 of which in turn has Theorem I. of [5] as a corollary. REFERENCES I. Jungck, G. Compatible mappings common fixed points Internat. J_. Math. Math. Sci. _9(1936) 2. Park, S. Bae, Jong Sook Extensions of a fixed point theorem of Meir Keeler. Ark. Mat. 19(1981) Singh, S.L. Singh, S.P. A fixed point theorem. Indian J. Pure Ap_p. Math. I i(1980) Sessa, S. On a weak commutativity condition in fixed point considerations, Publ. Inst. Math. 3_2(46)(1982) Hadzic, Olga Common fixed point theorems for family of mapings in complete metric spaces. Math. Japonica 29(1984)

5 Journal of Applied Mathematics Decision Sciences Special Issue on Intelligent Computational Methods for Financial Engineering Call for Papers As a multidisciplinary field, financial engineering is becoming increasingly important in today s economic financial world, especially in areas such as portfolio management, asset valuation prediction, fraud detection, credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied applied to improve the quality of the financial decisions that need to be made. Until now, computational methods models are central to the analysis of economic financial decisions. However, more more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, fuzzy models) can be used to develop intelligent, easy-to-use, /or comprehensible computational systems (e.g., decision support systems, agent-based system, web-based systems) This special issue will include (but not be limited to) the following topics: Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Application fields: asset valuation prediction, asset allocation portfolio selection, bankruptcy prediction, fraud detection, credit risk management Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation Authors should follow the Journal of Applied Mathematics Decision Sciences manuscript format described at the journal site Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Guest Editors Lean Yu, AcademyofMathematicsSystemsScience, Chinese Academy of Sciences, Beijing , China; Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; yulean@amss.ac.cn Shouyang Wang, AcademyofMathematicsSystems Science, Chinese Academy of Sciences, Beijing , China; sywang@amss.ac.cn K. K. Lai, Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; mskklai@cityu.edu.hk Hindawi Publishing Corporation