COMPATIBLE MAPPINGS AND COMMON FIXED POINTS

A generalization of the commuting mapping concept is introduced. Properties of this "weakened commutativity" are derived and used to obtain results which generalize a theorem by Park and Bae, a theorem by Hadzic, and others.

Clearly, any weakly commuting pair are compatible.On the other hand, the functions f(x)=x 3 and g(x)=2x 3 are compatible since If(x)-g(x)l=Ixl3O iff Ifg(x)-gf(x) 61x190, but as noted above they are not a weakly commuting pair.
(In this instance X can be taken as R.) EXAMPLE 2.1.Let f(x)=x 3 and g(x):2-x with X=R.,f(Xmn)-g(Xn), i n iXn_llix n2 + Xn + 21 /0 iff x n/l and limnlfg(xn)-gf(xn)I=l 61Xn-l12=o if /I Thus f and g are compatible but are not a weakly commuting pair; e.g., x n let x=O.EXAMPLE 2.2.Let f(x)=2-x 2 and g(x)=x 2 with X=R.As in Example 2.1, it is easy to show that f and g are compatible but not weakly commutative.EXAMPLE 2.3.Let f(x)=cosh(x) and g(x)=sinh(x) with X=R.Then If(x)-g(x)l=e -x 0 iff x-.But f(x),g(x) /+ as x/ + i.e., f and g do not converge to an element t of X.Thus the condition of compatibility is satisfied vacuously, but f and g do not commute.
The following observation provides a criterion for identifying compatible functions.
PROPOSITION 2.1.Let f,g'(X,d) (X,d) be continuous, and let F={ x X" f(x)=g(x)=x }.Then f and g are compatible if any one of the following conditions is satisfied. (a) If f(Xn),g(xn) /t (X), then t F.
If (b) holds, the compatibility of f and g follow easily from (a) and (2.1) upon noting that F is closed since f and g are continuous.So to comDlete the proof, suppose that F is compact and that (2.1) and (c) hold.Then d(f(Xn), F) +0 Since F is compact there is a subsequence x k g(Xn)) 0 and D(x n, of x n which converges to some element c of F. Then f(Xkn) f(c)=c n by the continuity of f and the definition of F. Consequently, (2.1) implies that c=t F, so f and g are compatible by (a).EXAMPLE 2.4.Let f(x) x4+ax 2 (a> I) and g(x)=x 2 with X=R.Then If(Xn)-g(Xn)l= Xn21Xn 2 +(a-l) I0 iff x n 0( F), so that f and g are compatible by Proposition 2.1(c).But f and g are not weakly commutative; let a=2,x=l.COROLLARY 2.1.Suppose that f and g are continuous self maps of R such that f-g is strictly increasing.If f and g have a common fixed point, then f and g are compatible.
PROOF.Immediate, since f(Xn),g(x t implies that F={ t }.EXAMPLE 2.5.If f(x)=x3+ax and (x)=mx with X=R and a>m, then f(O)=g(O)=O and f-g is increasing, so f and g are compatible by Corollary 2.1.EXAMPLE 2.6.If f(x)=eX-I and g(x)=x 2 (X=R), f and g are compatible since f(O)=g(O)=O and f-g is increasing.
It is easy to show that if gi,fi:R/ R and the pairs fi,g are compatible for i=1,2 n, and if G(Xl'X2 Xn)=(gl(Xl) gn(Xn) and F(x Xn)= (fl(Xl) fn(Xn)), then F and G are compatible on (Rn,d) where d is the Euclidean metric.Thus the above examples show that G(x,y)=(eX-l,7y) and F(x,y)=(x2,y3+8y) are compatible on (R2,d).
The following result will be useful in section 3.
(b) If f and g are continuous at t, then f(t)=g(t) and fg(t)=gf(t).
We prove 2(b) by noting that gf(Xn)/f(t) by 2(a) and the continuity of f, whereas gf(Xn)/g(t) by the continuity of g.Thus f(t)=g(t) by uniqueness of limit, and therefore gf(t)=fg(t) by part I. / 3.
COMMON FIXED POINTS FOR ( a CONTRACTIONS.Definition 3.1.A pair of self maps A and B of a metric space (X,d) are ( , )-S,T-contractions relative to maps S,T:X X iff A(X)CT(X), B(X)CS(X), andthere is a function R + R + such that ()> for all and for x,y X-G.JUNGCK (i) < d(Sx,Ty) < a () implies d(Ax,By) < and (ii) Ax=By whenever Sx=Ty.Note that if A and B are (c ,6 )-S,T-contractions, then d(Ax,By)<d(Sx,Ty) for all x,y, strict inequality holding when Sx#Ty.Also observe that in the above definition the pairs A,S and B,T are evaluated at the same points so that order is significant.Consequently, even though A and B are ( ,6 )-S,Tcontractions, the pair B,A may not be as Example 3.1 will reveal.Definition 3.2.Let A,B,S,T be self mappings of a set X such that A(X)C T(X) and B(X)CS(X).For x o X, any sequence Yn defined by Y2n_l=TX2n_l AX2n_2 and Y2n=SX2n=BX2n_l for n N will be called an S,T-iteration of x o under A and B.
The above definition ensures us that for nonempty sets X S,T-iterations will exist since A(X) CT(X) and B(X)CS(X), although the sequences Yn certainly need not be unique.
Yn is a Cauchy sequence.
To see that (b) is true, remember that d(Ax,By) < d(Sx,Ty) for all x,y by the hypothesis on A,B,S and T. So if m is even, say m=2n, d(Ym,Ym+I) d(Y2n,Y2n+l d(BX2n_l,AX2n < d(TX2n_l,SX2n d(Y2n_l,Y2n) d(Ym_l,Ym).Similarly, d(Ym,Ym+I) < d(Ym_ l,ym if m is odd.Thus the sequence d(Ym,Ym+I) is nonin- creasing and converges to the greatest lower bound of its range which we denote by r.Now r_> O; in fact, r O. Otherwise, part (a) implies d(Ym+l,Ym+2) < r whenever r < d(Ym,Ym+I) < 6 (r) since m and m+l are certainly of opposite parity.But since d(Ym,Ym+I) converges to r, there is a k such that d(Yk,Yk+l) < (r) so that d(Yk+l,Yk+2) < r-which contradicts the designation of r.
To prove part (c) of the Lemma, let 2 be given.With r (E) E part (a) of the Lemma asserts that d(yp+l,Yq+l) whenever d(yp,yq) < + r and p and q are of (3.2)   opposite parity.In order to show that (3.4) yields a contradiction, we first want an m p such that + r/3 < d(yp,ym) < + r with p and m of opposite parity.In this instance let m=k+l.
LEMMA 3.2 Let S and T be self maps of a metric space (X,d) and let A and B be (,6)-S,T-contractions such that the pairs A,S and B,T are compatible.
If there exists z X such that Az=Sz and Bz=Tz, then c=Tz is the unique common fixed point of A,B,S and T.
PROOF.The definition of ( ,6 )-S,T-contractions implies d(Ax,By) < d(Sx,Ty) if SxTy.Thus if SzTz, the hypothesis yields the contradiction d(Az,Bz) d(Sz,Tz) d(Az,Bz).We conclude that Sz=Tz=Az=Bz.Now let c=Tz and suppose that cTc.Since T and B are compatible and Tz=Bz, TBz=BTz by Proposition 2. We can now state and prove our first main result.THEOREM 3.1.Let S and T be self maps of a metric space (X,d) and let A and B be (,)-S,Tcontractions such that the pairs A,S and B,T are compatible.Let x oE X and let Yn by any S,T-iteration of x under A and B. If Yn has a cluster point z in X, then Yn converges to z, and Tz is the unique common fixed point of A,B,S and T provided these functions are continuous at z. PROOF.By Lemma 3.1, Yn is Cauchy and therefore converges to the cluster point z.Then AX2n,SX2n,BX2n_l,TX2n_l +z (See Definition 3.2).Since A and S are compatible and also continuous at z, Az=Sz by Proposition 2.2.2.b.. Similarly, Bz=Tz, and the conclusion follows from Lemma 3.2../ The next result follows immediately upon noting that under the given hypothesis, any S,T-iteration of any point under A and B converges to a cluster point since (X,d) is complete.COROLLARY 3.1.Let S and T be continuous self maps of a complete metric space (X,d) and let A and B be (E , )-S,T-contractions such that the pairs A,S and B,T are compatible.If A and B are continuous, then A,B,S and T have a unique common fixed point.
Observe that Theorem 1.2.by Park and Bae is a consequence of Corollary 3.1.with f=S=T and g=A=B.This follows upon noting that since g is an ( ,5 )-f- contraction, d(g(x),g(y)) < d(f(x),f(y)) for all x,y so that the stated continuity of f ensures the continuity of g.
The following example of ( , )-contractions pertains to most of the preceding and as such should be instructive.EXAMPLE 3.1.Let X [I,) and d(x,y) Ix-yl.Let Sx--(x4+l)/2, Ax=x 2 Bx=x and Tx=(x2+l)/2 for x in X.Then A,B,S, and T are continuous self maps of the complete metric space (X,d).Since A1 Sl and (S-A)x (x2-I)2/2 is increasing on X, A and S are compatible on X by Corollary 2.1.which is clearly validon any connected subset of R.However, A and S are not even weakly commutative; consider x=2 for example.On the other hand, B and T are compatible since BT=TB.
To see that the hypothesis of Corollary 3.1.is satisfied, we have yet to show that A and B are ( , )-S,T-contractions.To this end let >0o We want IAx-Byl x -Yl < -Remembering that x,y__> I, it is easy to show that i(x4-y2)/21 __> implies that x2+y (I + C2 + I).But then l(x4-y2)/21< 6 () (x2+y) -I implies that Ix2-yl < 2 (E so that 2 ()(I + /2 +-I) -I Ix2-yl if () (I + v'2 + I)/2.Now so defined for all 0 is a continuous mapping from R + into R + such that (E) so that property (i) of Definition 3.1.is satisfied.To confirm property (ii) note that ISx Ty I(x 4 y2)/21 IAx Byll(x2+ y)/2 I, (3.11) which requires that Ax=By when Sx=Ty since x,y I.In Example 3.1.the function 6 is continuous.Our next result tells us that if we merely require that be lower semicontinuous, A and B in Corollary 3.1.
need not be continuous.THEOREM 3.2.Let S and T be continuous self maps of a complete metric space (X,d) and let A and B be ( ,6 )-S,T-contractions such that the pairs A,S and B,T are compatible.If a is lower semicontinuous, then A,B,S and T have a unique common fixed point.
PROOF.Let x o X and let Yn be an S,T-iteration of x o under A and B. Since (X,d) is complete and since Yn is Cauchy by Lemma 3.1.,Yn converges to some element z of X.Then AX2n, SX2n,BX2n_l,TX2n_l /z (Definition 3.2.).
Since S and T are continuous and the pairs A,S and B,T are compatible, Proposition 2.2. and the indicated continuities yield: T2X2n_l BTX2n_l Tz and S2X2n ASX2n Sz (3.12)We assert that Sz=Tz.For suppose that d(Sz,Tz) z for some > O.
Since 6:R + R + is lower semicontinuous and 6 () > by definition, there is a neighborhood N( of such that (t) for t N( ).We can thus choose t o such that 0 < t E < ,6 (t) By (3.12), d(S2X2n,T X2n-l ' and we can ,T 2 ) (to,6 (t o)) for n > n o Then therefore choose n o such that d(S2X2n X2n_l the definition of 6 (Definition 3.1) implies that d(ASX2n,BTX2n_l < t o for n_>n o But then (3.12) implies that d(Sz,Tz) < t o < d(Sz,Tz), the anti- cipated contradiction.Now Sz=Tz implies that Az=Bz by Definition 3.1..Moreover, Az=Tz since limnd(Sz,T2X2n_l d(Sz,Tz) O.By the above we d(Az,Tz) imnd(Az,BTX2n_ have Az=Bz=Sz=Tz, so that Tz is the unique common fixed point of A,B,S, and T by Lemma 3.2.. / COROLLARY 3.2.Let S and T be self maps of a complete metric space (X,d), and let A,B'X S(X)i')T(X).Suppose that S and T are continuous and that the pairs A,S and B,T are compatible.If there exists r (O,l) such that d(Ax,By) < r d(Sx,Ty) for x,y X, (3.13) Then A,B,S, and T have a unique common fixed point.
PROOF.Define 6 by 6 () =/r.Then 6 is a continuous self map of R + such that 6 () > Iso, d(Sx,Ty) < 6 () implies d(Ax,By) < r(/r) = Thus A and B are , 6)-S,T-contractions and the hypothesis of Theorem 3.2. is satisfied./ Corollary 3.2.clearly generalizes Fisher's Theorem 2. in [15].Moreover, the corollary below generalizes Theorem I. of [20] by substituting compatibility for commu ta t v ty.COROLLARY 3.3.Let S and T be continuous self maps of a complete metric space (X,d).Let A i" be a family of maps Ai.X+ S(X)-.T(X) compatible with both S and T and let be any indexing set.If there exists r (0,I) such that d(Aix,Ajy) !r d(Sx,Ty) for all x,y X and ij, (3.14) then there is a unique point c X such that c Sc Tc A for all I.
PROOF.Let i,j (ij).By Corollary 3.2 there is a unique point c X such that c Aic Ajc Sc Tc.Now if k z (i#k), there is a unique point d X such that d Aid Akd Sd Td.Then (3.14) implies" d(c,d) d(AiC,Akd) < r d(Sc,Td) d(c,d).
Since r < I, c must equal d and the conclusion follows./ REMARK 3.2..The functions in Example 3.1.show that the concept of ( , -S,T-contractions does indeed generalize the relation (3.13) of Corollary 3.2., since in (3.11)we have !SI-TwI IAI-BylI(I+y)/2)I where (l+y)/2 converges to as y approaches from the right; i.e., there exists no r c (0,I) such that IAx-By <_ r ISx-Tyl for all x,y>_ I.
The preceding results suggest theobvious general question, "To what extent can other fixed point theorems involving commuting maps be strengthened by sub- stituting "compatibility" hypotheses for "commutativity" ?".We however close with more specific QUESTION 4.1.To what extent is the hypothesis that 6 be lower semi-con- tinuous necessary in Theorem 3.2.?

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

LEMIIA 3 .
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 x o X and Yn is an S,T-iteration of x o under A and B, then (a) for each O, < d(yp,yq) < () implies d(yp+l,Yq+l) when p and q are of opposite parity,(b) d(Yn,Yn+I)O, and Assume without loss of generality that r e By part (b) of the Lemma we can choose n e N such that o d(Ym,Ym+l) r/6 for m __> n o (3.3)We now let q p __> n o and show that d(yp,yq) thereby proving that Yn is indeed Cauchy.So suppose that d(yp,yq) __> 2 (3.4)

First
Round of Reviews May 1, 2009