On Some Weak Conditions of Commutativity in Common Fixed Point Theorems

We generalize common fixed point theorems of Fisher and Sessa in complete metric spaces, using some conditions of weak commutativity between a set-valued mapping and a single-valued mapping. Suitable examples prove that these conditions do not imply each of the others.

for all A,B,C in B(X).We recall some definitions and a basic lemma of Fisher [I].
Let {A n=l,2,...} be a sequence of ncnempty subsets of X.We say that the n sequence {A converges to a subset A of X if each point a in A is the limit n of a convergent sequence {a with a in A for n=l,2 and if for any n n n e>O, there exists an integer N such that A A for n>N, A being the union of n e g all open spheres with centers in A and radius g.The following lemma holds.
LEMMA I.If {A and {B are sequences of bounded subsets of (X,d) which n n converge to the bounded subsets A and B respectively, then the sequence {6(An,Bn)} converges to 6(A,B).
A set-valued mapping F of X into B(X) is continuous at the point x in X if whenever {x is a sequence of points of X converging to x the sequence {Fx n n in B(X) converges to Fx. F is said to be continuous in X if it is continuous at each point x in X.We say that z is a fixed point of F if z is in Fz.
Following the notations of our foregoing paper [2], we denote by the set of all real functions of [0,+) into [0,+) such that @ is nondecreasing, right continuous and @(t) < t for all t 0.
Let I be a mapping of X into itself such that F(X) c_ I(X).inductively the sequence {Xn such that IXn+ Zn FXn for n=O,l,2... Using this iterative process, Fisher [3] proved the following result.THEOREM I. Let F be a mapping of X into B(X) and let I be a continuous mapping of X into itself satisfying the inequality for all x,y in X, where 0 c < I.If Flx= IFx for all x in X and if (2.1) holds, then F and I have a unique common fixed point z and further Fz {z}.
Fisher [I] also proved that THEOREM 2. Theorem holds if one assumes the continuity of F in X instead of the continuity of I.
The proof of both these theorems are based on the fact that the sequence {6(Fx n,Fxl): n=0,1,2,...} is bounded for any Xo in X.As in [2], from now on we assume that sup {6(FXn,FXI) n=O,l,2...} < + (2.3) for some point x in X.Further in [2], we defined two mappings F and I to be o weakly commuting if IFx e B(X) and 6(FIx,IFx) max {6(Ix,Fx), diam IFx} (2.4) for all x in X. Clearly two commuting mappings F and I are weakly commuting, but in general two weakly commuting mappings do not commute as it is shown in the Example of [2].
COMMON FIXED POINT THEOREMS 291 Using this concept, the first part of Theorem 3.1 of [2], that generalizes Theorem I, runs as follows.
If there exists a point x in X satisfying o condition (2.3), if F and I weakly commute and if (2.1) holds, then F and I have a unique common fixed point and further Fz {z}.
Note that if we assume (t) c t for all t 0 and 0 I, (2.5) be- comes (2.2).Following Itoh and Takahashi [4], we also consider two mappings F and I such that IFx Fix for all x in X.In this case, we say that F and I quasi commute and it is evident that if F and I commute, they also quasi commute.
When we wrote the paper [2], we were unaware about the result of [1] and, under the same assumptions of Theorem 3, in the second part of Theorem 3.1 of [2], we assumed the continuity of F instead of the continuity of I.In [2], we established the following inequality (see inequality (3.6) of [2]), d(Zm,Zn) <= 6(Zm,FXn) <= 6(FXm,FXn for any m,n p,p being a suitable nonnegative integer.This inequality implies that {Zn} is a Cauchy sequence, which converges to a point z in X since X is complete Unfortunately, the second part of the proof of Theorem 3.1 of [2] was not correctly established.Strictly speaking, the gap consists in the fact that (2.6) does not imply the following inequality of [2], 6(z,Fz n) =< d(Z,Zn + 6(Zn'FZn < d(Z,Zn + e for n > p, from which, as n + one should deduce that Fz {z}.
Here we point out that the second part of Theorem 3.1 of [2] can be substituted by the following result, which is a direct generalization of Theorem 2.
THEOREM 4. Let F be a continuous mapping of X into B(X) and let I be a mapping of X into itself satisfying the inequality (2.5) for all x,y in X, where e .If there exists a point x in x satisfying condition (2.3), if F o and I quasi commute and if (2.1) holds, then F and I have a unique common fixed point z and further Fz {z}.
PROOF.It is a minor variant of the proof of Theorem of [I] and we omit it for brevity.
Note that if F and I commute and (t) c't for all t 0, 0 C < 1, we deduce Theorem 2 from Theorem 4.
The following example proves the greater generality of Theorem 4 over Theorem 2.
.3) holds since X is a bounded metric space.So all the assumptions of Theorem 4 hold but Theorem 2 is not applicable since F and I do not commute.Note that F and I also weakly commute The Example 2 of [2] proves that the weak commutativity is a necessary condition in Theorem 3. Now we prove that the quasi commutativity is a necessary condition in Theorem 4.
EXAMPLE 2. Let X {x,y,z} be a finite set with metric d defined as d(x,y)= d(x,z)=l,d(y,z)=2.Define F and I as Fx=Fz=x}, Fy={x,z} and Ix=y, ly=x, Iz=z.
Of course F is continuous in X and the conditions (2.1) (2.5) is satisfied if (t)=t/2 for all tO and then all the assumptions of Theorem 4 hold, except the quasi commutativity since IFy= I{y,z} {x,z} {x} Fx =FIy but F and I do not have common fixed points.
The Example 3 of [2] proves that the condition (2.1) is necessary in Theorem 3.
The next example proves that the same condition is necessary in Theorem 4, even if F and I are single-valued mappings.Thus the inequality (2.5)Is satisfied if (t)=t/2 for all t>0 and hence all the assumptions of Theorem 4 hold except the condition (2.1) since F(X)=X'#_X-{0} I(X), but I does not have fixed points.
The Example 4 of [2] proves that the continuity of I is a necessary condition in Theorem 3. Now we show that the continuity of F is necessary in Theorem (2.1) is satlsfied since F(X) (0, ---] c_ (0,11 I(X).
(Fx,Fy) .6(Ix,Fy) Then the inequality (2.5) is satisfied if (t)=t/2 for all t>0 and all the assumptions of Theorem 4 hold except the continuity of F, but F and I do not have common fixed points. 4. ANOTHER FIXED POINT THEOREM.
In this Section we establish another result by using a weaker condition than the commutativity between two single-valued mappings of X into itself, but, fol- lowing the ideas of this work, we give this condition between a set-valued mapping F of X into B(X) and a single-valued mapping I of X into itself.Precisely, we say that F and I slightly commute if IFx B(X) and 6(FIx,IFx) _-< max {6(Ix,Fx), diam Fx} (4.1) for all x in X.Note that if F is a single-valued mapping, then diam Fx diam IFx 0 for all x in X and hence (2.4) and (4.1) become d(FIx,IFx) < d(Ix,Fx) for all x in X, that is the condition given in [5].
In the sequel we use the following lemma of [6].
LEMMA 2. Let {An} be a sequence of nonempty bounded subsets of (X,d) and y be a point of X such that lim 6(A ,y) 0.
n n Then the sequence IAn} converges to the set {y}.
Now we give the following result.
THEOREM 5. Let F be a mapping of X into B(X) and I be a mapping of X into itself satisfying the inequality (2.5) for all x,y in X, where e.If there exists a point x in X satisfying condition (2.3), if F and I slightly commute, if (2.1) o holds and if F or I is continuous in X, then F and I have a unique common fixed point z and further Fz {z}.
PROOF.We omit the first part of this proof since it is identical to the first part of the proof of Theorem 3.1 of [2].
As in [2] we can prove that the sequence {Ix converges to a point z in X and n the sequence of sets {FXn} converges to the set {z}.Since F and I slightly commute, we have that Now we assume that F is continuous in X.Then the sequence of subsets {FIXn} converges to the set {Fz} and using inequality (2.5), we have that 6(FIXn+l,Fxn)--< (max{d(I2Xn+l,IXn), =< (max 6(IFXn,FlXn) + 6(FIXn' "Fx.n) + 6(FXn'IXn)' is in IFx and is nondecreasing.
It follows that {z}=Fz={Iz} and thus z is also a fixed point of I.
Now we assume the continuity of I instead of the continuity of F. Then the sequence {I2x converges to the point Iz and the sequence of sets {IFx converges n n to the set {Iz}.
Then Fz={z} and hence z is also a fixed point of F. In any case, z is a common fixed point of F and I suppose that F and I have another common fixed point z'.Using inequality (2.5), we have that d(z,z') 6(Fz,Fz') _-< @(d(z,z')).
This means that z=z and therefore z is the unique common fixed point of F and I.
This completes the proof.
Therefore the slight commutativity is a necessary condition in Theorem 5.
REMARK 2. The Example 3 proves that the condition (2.1) is also necessary in Theorem 5.
REMARK 3. In Example 4, we note that F slightly commutes with I since 6(FIO,IFO)=I/2=d(IO,FO) and 6(FIx,IFx)=x/2=diam Fx if x > 0. Since all the assumptions of Theorem 5 are satisfied except the continuity of F and I in X, we can say that the continuity of F or I in X is a necessary condition in Theorem 5.

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: EXAMPLE i.Let X [0,I] with the function 6 induced by the Euclidean metric d.Define the mappings F and I as Fx [0,x/(x+4)] for all x in X 7] c__ [0, 7] U {i} I(X) and IFx [0,x/

REMARK 4 .
It is not yet hno if (2.3) is a necessary condition in Theorems 3,4 and 5.

First
Round of Reviews March 1, 2009 4. EXAMPLE4.Let X= [0,I with the function 6 induced by the Euclidean metric d and define F and I as Fx i/2 if x=0 and Fx=(0,x/2] if x>0, Ix=1 if x=O and Ix=x if x>0.Of course, condition (2.3) holds since X is a bounded metric space and and since 12 Xn+ is in IFx we get that