A COUNTER EXAMPLE ON COMMON PERIODIC POINTS OF FUNCTIONS

By a counter example we show that two continuous functions defined on a compact metric space satisfying a certain semi metric need not have a common periodic point.


INTRODUCTION
In [1] we defined the notion of a semi-metric and used it in a contractive type inequality to obtain some results regarding common fixed points of two functions.We proved Theorem 1.1 and gave a counter example illustrating that we cannot replace the contractive coefficient a with 1.However, it is natural to ask (see [2]) if it is possible to prove a version of Theorem 1.1 with (1.1) amended to read strict inequality, a replaced by 1, and with the additional requirement that x y, for the situation in which the functions are defined on a compact metric space X Theorem 1.2 provides a partial answer to this question Here we show that in general we can ot expect to prove such a result.We begin with Theorem 1.1 and Theorem 1.2 as well as some preliminaries from [1].
THEOREM 1.1.Let f and g be selfmaps of the unit interval and let h" I x I [0, oo) bca function having property P1.Suppose g is continuous on I and A is a nonempty closed g-znvariant subset of F(f).If there exists a real number a, 0 < a < such that for all x and y ,,, F(.f), f and g satisfy the following inequality: h(fz, fy) <_ a. max{h(gx, gy), h(gx, fx), h(gy, fy), h(gy, Ix), h(fx, gy) }, (1.1) then f and g have a unique common fixed point.
THEOREM 1.2.Suppose f and g are two selfmaps of a compact metric space X with continuous, and let h X X [0, c) be a function having property P1.If for all z 1 in X. ./"and # satisfy the following inequality: h(fx, fy) < max{h(gx,gy),h(gx, fx),h(gy, fy), h(gy, fz), h(fz, gy) }, ( then one of the following holds: (z) either f and g have a common fized point.(zz) o; every nonempty closed g-invariant subset of F(f) contains a perfect minimal set B .uch that the functions 1(x) h(gx, x) and 2(x) h(x,gx), do not attain their minimum or mazimum on B. Throughout g" denotes the n fold composition of g with itself and X is a compact metric space.The orbit of z under the homeomorphism g a one to one function g ), O(g, x) is the set {gk(x) -oo < k < oo}.A subset Y of X is called invariant under g if g(Y) C_ Y.A closed, invariant, nonempty subset of X is called minimal if it contains no proper subset that is also closed, invariant and nonempty.The sets P(f) and F(f) are the sets of periodic points and tlie fixed points of f, respectively.The space 2 {s (soss...) sj 0 or 1} is called the sequence space on the two symbols 0 and 1.For two sequences s (soslsz...) and (totlt2...), their distance is defined by dis, t] E,__ o z,-t, I/2'.It is clear that (E, d) is a compact metric spa.ce.
Let C be the Cantor Middle-Third set obtained as follows.Let Ao (1/3, 2/3) be the middle third of the unit interval I and 10 1 A0.Let A (1/9,2/9) U (7/9,8/9) be the middle third of the two intervals in lo and 1 10-A.Inductively, let A, denote the middle third of the intervals in 1,_ and let 1,, 1,_ A,, and C I%,>oI,.For each x q C, we attach a.n infinite sequence of O's and l's, S(z) (0ss2...), according to the'rule: so if x belongs to the left component of I0; so 0 if z belongs to the right component of 10.Since x belongs to some component of 1,_, and 1, is obtained by removing the middle third of this interval.

RESULTS
We first show that A is a homeomorphism on E or in another word C) and the orbit of every point of C under A is dense in C E Since E does not have a nonempty proper closed invarint subset under A, it is a perfect minimal set.LEMMA 2.1 A is a homeomorphism from C to itself.PROOF.To see this we show that A is continuous, one one, onto on C with Aalso continuous.
To see that A is continuous, let x be an arbitrary point of C and > 0. Let N be a positive integer such that 1/2 N < e. Choose 8 1/2/v.If d(,x) < , the sequences x nd y have identical first N elements, hence A(z) and A(y) have also identical first N terms.Thus d(A(x),A(y)) < 1/2 N < e, implying the continuity of A at x.
To see that A is one one on C, let x (xox...),y (yoy...) be two points of C with z y, then there exists a least nonnegative integer N such that XN YlV.Obviously the corresponding elements of the sequences A(x) and A(y) are different, hence A is one one.
To see that A is onto, let y (yoyy...) and N be the smallest nonnegative integer such that YN 1. Then for x (ll...0yN+yN+...) we have A(x) y.Since (Ez, d) is a compact metric space and A is continuous on C E, the image of every closed subset of C under A is a closed set, implying the continuity of A-.
LEMMA 2.2.The orbit of every point of C under A is dense in C. PROOF.Let x (xoxx...) and y (yoyy...) be two rbitrary points of C E. For > 0, choose a positive integer N so that 1/2 iv < e.Let N be the least positive integer such that the sequences x and y have identical first N elements.Then for k 2 N'-the two sequences A"(x) and y have at least N identical first elements.Similarly suppose N is the lea,st positive integer such that the first N elements of the two sequences Aa(x) and y are identical.Then N2 > N + and for k 2 N-the two sequences Ak'+(x) and y have identical first N elements.By repeating this process we obtain a positive integer m k + k + + k sch that A(x) and y have identical first N elements, implying d(A'(x),y) < 1/2 g < e.Since :r and y were arbitrary we may interchange the role of x with y.Thus the result is established.DEFINITION 2.1.Let X be a compact metric space.The function h X x X [0, oo) is said to have property P if it satisfies the following conditions: (i)" h(z,y) 0 if and only if x y, (ii)" if lim-oo x xo, lirn,,_.ooy yo, and lirr_.ooh(x,,, y) O, the, Xo y0.
The following theorem is based on an example which illustrates that assertion (ii) of Theorem 1.2 ma.y occur.THEOREM 2.1.There exist two continuous functions f and g selfmaps of a compact metric space (X,d), a g-minimal perfect set B C_ F(f) and a function h X x Z --, [0, oo) having property P, such that for all x # y both in B, f and g satisfy the following: h(fx, fy) < max{h(gx,gy),h(gx, fx),h(gy, fy), h(gy, fx), h(fx, gy) }, (2.1) yet .fand g do not have a common periodic point.
PROOF.Consider the compact metric space (E,d).For each z 6 Ez, let f(z) x and g(:r) A(z), where A is the adding machine.It is clear that B E2 is a g-invariant perfect subset of F(f).Suppose g {O(g,x): x B}.By the axiom of choice there is a set E such tha.tE has exactly one element from each element of H. Choose an arbitrary point xo E.
Since g is one to one on B, the function h is well defined on B x B. The function h satisfies tle property P1 since, for x y, h(z, y) 0 and for each (x, y) B x B, h(x, y) > 1.It remains to show that for every # s in B the inequality (2.1) is satisfied.To show this let t, s B, and # s.We distinguish several different cases.
We see that inequality (2.1) is satisfied for every t,s E B, yet f and g do not have any common periodic point.In fact the set of fixed points of f is identical to B, while g does not have any periodic point in B.
Remark 2.1.We may choose the functions f and g to be continuous selfmaps of the unit interval.For example let (am, b,) },= be the complementary intervals of the middle third Cantor set C. Define x xEC, f(x) a,, a,, < x <_ (a, + b,)/2, 2(x b,) + b, (a, + b,)/2 < x <_ b,, ( for n 1,2, 3,..., g a continuous extension of A, and the semi metric h on C as in the above theorem.It is clear that F(f) C and the orbit of each point outside C under .f(z.e:.{x,f(x),f(x),f3(x),...})is attracted to a point in C, also f and g satisfy the inequality (2.1) on F(f), yet they do not have any common periodic points.
ACKNOWLEDGMENT.would like to thank professor B.E. Rhoades for his comments (reference [3]) which led to this work.Also wish to thank professor A. M. Bruckner for his valuable sggestions.
AUTHOR'S PRESENT ADDRESS.Department of Mathematics, The University of Ten- nessee at Chattanooga, Chattanooga, TN 37403-2598.

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.
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