ON SEQUENCES OF CONTRACTIVE MAPPINGS AND THEIR FIXED POINTS

By using a condition of Reich, we establish two fixed point theorems concerning sequences of contractive mappings and their fixed points. A suitable example is also given.

where for any t>O, a(t)+2b(t) i.

rt +
Evidently (A) implies (B) and (C), (B) and (C) imply (D), (D) imply (E) and (F), (E) and (F) imply (G).Suitable examples can be found in Rhoades [6] to illustrate some of the above implications.In the sequel, N stands for the set of natural numbers.
The following result was established in [5] and [6].
THEOREM I. Let T heN, be mappings of (X,d) into itself satisfying condition n (F) with the same functions a,b,c and with fixed points z Suppose that a mapping n T of X into itself can be defined pointwise by T(x)=limn Tn(X) for any x in X.Then T has a unique fixed point z and z=limn z N.
The proof of Theorem consists essentially in the fact that the sequence {Zn} is regular, i.e. it possesses a limit z(say).It appears that a result corresponding to Theorem for mappings satisfying condition (G) does not exist in print in the literature of fixed point theory.In this note, we wish to study this problem.
Other results and open problems related to the stability of fixed point of mappings can be found in Rhoades [6], Nadler [i0], Rus [II] and Singh [12].
We first state the following result more general than Theorem I.
THEOREM 2. Let T ,neN, be mappings of (X,d) into itself satisfying condition n (G) with the same functions a,b,c and with fixed points z Suppose that a mapping n T of X into itself can be defined pointwlse by T(x)= lim T (x) for any x in X n n and the sequence {Zn is regular.Then T has a unique fixed point z and z=limn_=Zn.
PROOF.Since the metric d: X X [0,) is continuous, the limit mapping T satisfies the inequality (1.3).By condition (G), T has a unique fixed point z.We claim that inf d(Zn,Z) 0, neN otherwise assume Z > 0. By observing that d(z ,z)>-Z>0 and hence z z for any neN, for any neN.If we denote with {Zk(n) a subsequence of {Zn} such that we obtain that (2.2) n-Following Reich [4], we oserve that the assumptions about the functions a,b,c of condition (G) imply the existence of two functions h,k: (0,)(0,) for which, given t>0, there exists an h(t)>0 such that 0r-t<h(t) implies a(r)+2b(r)+2c(r)k(t)<l.
for any n>.p.From (2.3), it follows that a(d(Zk(n),Z))+2c(d(Zk(n),Z))<.k(Z)< 1 for any n>p.On the other hand, since the sequence {Tk(n)(Z) converges to z=T(z), we can find an integer qeN such that d(Tz,Tk(n)(Z))< (l-k(Z))/2 for any n>q.By (2.I), then we have for any n>max {p,q}, 2d(Tz,Tk(n a contradiction.Thus Z=0 and therefore the sequence {Zk(n) converges to z.Since {z is regular, it has limit z and this concludes the proof.n THEOREM 3. Let T be mappings of (X,d) into itself with at least one fixed point n Zn.If {T n} converges uniformly to a mapping T of X into itself satisfying condition (G) and if the sequence {z is regular, then z=limnZ where z is the unique fixed n n point of T.
PROOF.We have that d(z n z)=d(Zn,TZ)<=d( z n TZn)+d(TZn,TZ).for any neN and proceeding as in the proof of Theorem 2, we get the thesis.
REMARK i.It is evident that there certainly exists a subsequence of {z n converging to z, even if {z is not regular.It is not yet known if the regularity n of {Zn} is a necessary condition in Theorems 2 and 3.
REMARK 3. Following Ray [13] and Fraser and Nadler [14], one can establish a result analogous to Theorem i0 of Reich [3] by using condition (G).
3. AN EXAMPLE.In order to illustrate the degree of generality of Theorem 2 over Theorem i, we furnish an example which shows that there exist mappings T of X into itself satls- n fying condition (G) but no condition (F).
EXAMPLE.Let X=[0,1] be equipped with metric d defined as follows, Ix-yl f x,y [0,1], d(x,y) x+y if one of x,ygN {I} Then (X,d) is a complete metric space because it is isometric to a closed subspace of the space of absolutely sunnable sequences.For further details, see Boyd and Wong [15].Now we define T XX nN by setting Tl(X)=T2(x)=0 for any x in X and for n nZ3, x-I if x e N {I}.
This implies, since a and c are monotonically decreasing functions, that q-i < a(1) + 2c(i). q As q+, we obtain Ia(i)+2c(1)<i, a contradiction which shows that the condition (F) is not satisfied by T for n3.
n On the other hand, for any neN the condition (G) holds if we choose b(t) c(t) 0 for any t>0 and a(t)=l-t/2 if 0<tl, a(t)=l-i/t if t>l.The condition (G) is obviously satisfied by T and T 2. For n3 and x,y in [0,I], xy, we get d(TnX,TnY)=Ix-y [I n   (x+y)] Furthermore, if one of x,y lies in N {I} with xy and n3, then we have d(TnX,TnY) TnX+Tn y "< x+y (x+y)- 1  1/(x+y) + c(d(x,y))-d(x,y).
We now define T(x)=x x2/2 if 0<x<=l, T(x)--x-I if x is in N {i}.
z =z=O are the unique fixed points of T and T respectively and we have n n Of course, lim T n(X) T(x)   nfor any x in X.Thus the conclusion of Theorem 2 holds good since the sequence {z n} converges to z.The idea of this example appears in [15].
ACKNOWLEDGEMENT.The authors thank an anonymous referee for some useful suggestions.

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.