On the existence of a fixed point of the operator acting in the space of continuous functions

Nonlinear Analysis, Theory, Methods & Applications. Printed in Great Britain.

Vol. 20. No. 10, pp. 1257-1259, 1993. 0

0362-546X/93 $fl.oo+ .oil 1993 Pergamon Press Ltd

ON THE EXISTENCE OF A FIXED POINT OF THE OPERATOR ACTING IN THE SPACE OF CONTINUOUS FUNCTIONS A. POKROVSKII and D. RACHINSKII Institute for Information Transmission Problems, Emolovoy St. 19, 101447 Moscow, GSP-4, Russia (Received 1 October 1991; receivedforpublication

1 August 1992)

Key words and phrase: Existence of a fixed point.

1. INTRODUCTION

paracompact [l] topological space. Denote by C = C(3) the Banach space of continuous bounded functions x(t) (t E 3) with uniform norm. Denote by U the unit ball in C which has its center at the zero point and consider an operator A: U -+ U. The operator A is referred to as quasi-contracting if for every x(t), y(t) E CT there exists a natural n = n(x(t), y(t)), satisfying the inequality LET 3 BE A

NW

- A”Y(0ll <

llm - AOIl.

(1)

The property of quasi-contractness is much weaker than the property of general contractness [2]. Operator A is referred to as monotonous [2], if the inequality x(t) 2 y(t),

teu

implies the inequality Ax(t) 2 Ay(t), Now we can formulate

t E u.

the main result of this paper.

1. Let the operator A be quasi-contracting, AU 5 U, and let one of the operators and -A be monotonous. Then the operator A has at least one fixed point. THEOREM

+A

The proof of the theorem is presented at the end of this paper. Before that we discuss the formulation of theorem 1. (A) For many spaces Ei there exists a monotonous operator A which acts in U, satisfies inequality

IIMO - hwll 5 Ilw - Jwll and has no fixed points. The example is well known and it is very simple: let 3 = [0, l] and x(t) + (x(t) + 1)(2t - l), Ax(t)

if t I 0.5,

=

i x(t) + (1 - x(t))(2t

- l),

So the property of the operator being quasi-contracting 1257

if t > 0.5.

cannot be omitted.

A. POKROVSKII and D. RACHINSKII

1258

(B) Consider the space C, c C. This space includes functions which are equal to 0 in the point t, E 3. Denote by U, the unit ball in C,. If the point 1, is not isolated in 3, then as a rule there exists a monotonous operator without fixed points which acts in U0 and decreases the distance between each two noncoinciding elements. An example is 3 = [O, l] and Ax(t) = O.Q(l - x(t)). So theorem 1 demonstrates that the geometrical structure of the unit ball in the space C has advantages with respect to the geometrical structure of the unit ball in the space C, . It would be interesting to give a general description of such advantages. (C) As we know, the question about existence of a fixed point of every operator A which acts in U and’decreases distances between noncoincided points is open up to now. (D) Consider the space B of almost periodic scalar functions x(t) (t E IR’) with uniform norm. This space plays an important role in applications [3]. As a rule operators acting in B do not have the property of complete continuity. This fact makes all methods based on the theory of rotation of vector fields unapplicable [2]. For this reason modifications of the principle of contracted maps are very important in such type of spaces. The following statement is an immediate sequence of theorem 1 and of the famous Bore’s theorem about the existence of compactification of R’ after which almost periodic functions turn into continuous functions. THEOREM 2.

Let the operator A transform the ball of the space B into itself and one of the operators A, and -A is monotonous. Then op’erator A has at least one fixed point. These are true, of course, and analogous assertions about operators acting in the spaces of quasi-periodic functions.

Proof of theorem 1. Consider only the case of monotonous operator A. Introduce the space M = M(D), of all scalar functions x(t) bounded on 3 with the uniform norm. Extend operator A to the operator A, which is defined on the unit ball I’ C M(3) by the formula A*x(t) =

suPu_Y(t):

u(t)

E u, Y(f)

5 x(t)1

(x(t)

E V.

(2)

By the definition operator (2) is monotonous. The cone of nonnegative functions in the space M(3) is minihedral (i.e. any bonded set has the supremum). So by the Birkhoff-Tarsky theorem [2] operator (2) has a fixed point x,(l) E V. Theorem 1 will be proved if we establish the following assertion. LEMMA.

The function x*(r) is continuous.

Proof. It follows by the ordinal way from the Michael theorem that there exists a function z(t) E C which is the nearest to the function x*(t). So it remains to prove the equality z(t) = x*(t).

(3)

Let equality (3) not be true. Introduce the functions z+(t) = z(t) + p and z_(t) = z(t) - p, where p = I/z(t) - x*(t)ll. By the definition the functions z+(l) and z_(t) are continuous and satisfy the inequalities z_(t) 5 x*(t) 5 z+(t).

(4)

Existence

The relations

(4) and the monotonicity

of the operator

Az_(t) Besides the fact that the operator

A, implies

inequalities

5 z(t) I Az+(t).

A is quasi-contracting; IIAz-(f)

1259

of a fixed point

- -4z+Wll

(5) it implies

the inequality (6)

< 2P.

From (5) and (6) the inequality Ix*(t)

-

W)ll < PT

follows, where s(t) = O.S(Az_(t) + AZ+(~)). The continuity contradict the definition of the constant p. The equality (3) and simultaneously theorem 1 are proved.

(7) of s(t)

and

the

relation

(7)

REFERENCES Springer, Berlin (1955). 1. KELLY J., General Topology. 2. KRASNOSEL’SKII M. A. & ZABREJKO P. P., Geometrical 3. KRASNOSEL’SKII M. A.. BURD V. SH. & KOLESOV Yu. S.. York (1973).

Methods

of Nonlinear

Analysis.

Springer, Berlin (1984). Halsted Press, New

NonlinearAlmost Periodic Oscillations.