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Internat. J. Math. & Math. Sci.

Vol. 9 No. 3 (1986) 621-622

621

ALMOST PERIODIC MOTION IN COMPLETE SPACE

ANANT KUMAR

Satgharwa, P.O. Buniad Ganj

GAYA (INDIA), Pin-823003 (Received December 17.

1985)

ABSTRACT. It is interesting that in a complete space an almost periodic motion is periodic if the null set is closed and if is not closed then every point of its trajectory is a limit point.

KEY

WORDS AND PHRASES.

Compact, periodic and

almost

periodic

motions.

1980 AMS

SUBJECT CLASSIFICATION CODE.

34C35.

i. INTRODUCTION.

We studied "Poisson stable distal dynamical

systems"

in

Ill

where the word

"compact"

is missing from the statement and proof of theorem 2.7. It should be stated as theorem

3.1(below).

Now by means of this theorem we shall establish an interesting relation (theorem

3.2)

between almost periodic and periodic motions in a complete space X.

2. DEFINITIONS AND NOTATIONS.

We shall follow definitions and notations of [i].

3. MAIN RESULTS.

3.1 THEOREM. Compact almost periodic motion is Poisson stable and distal.

3.2 THEOREM. In a complete space X an almost periodic motion is periodic if the null set is closed in X and if is not closed then

y(x)

which is perfect and

x compact set.

PROOF. An almost periodic motion

(x,t)

is recurrent

[2,

theo 8.02 P.384] and if a recurrent motion is situated in a complete space then

cly(x)

is compact minimal [2, theo 7.07 P.377]. Therefore the motion

(x,t)

is compact. Hence

(x,t)

is compact almost periodic motion therefore it is Poisson stable and distal (3.1 above). Thus y(x) is closed and perfect set [, theo 2.1].

Now by [theo. VI.3 of 3,

P.87],

if x is not periodic then cl(cly(x)

y(x))

cly(x)

x

cl((x) (x)) (x)

x

cl

y(x)

if is closed set in X.

x

Which is impossible, as

xey(x)

and in case of a compact motion is also non-empty x

[3,

theo. II.8 P.20]. Therefore x is periodic. But if is not closed then

cl x) //.

x

3.3 COROLLARY. If in theorem 3.2,T=R (the set of reals) then y(x) is connected.

(2)

622

A. KUMAR

REFERENCES

i. PRASAD, S.S. and

KUMAR,

A.

"Stable

P and Distal Dynamical

Systems."

Internat. J.

Math. & Math. Sci. Vol. 7 No. (1984) 181-185.

2. NEMYTSKII,

V.V.

and STEPANOV, V.V.

"Qualitative Theory

of Differential

Equations",

University Press, 1960.

3. SELL, G.R.

"Topological Dynamics

and

Ordinary

Differential

Equations",

V.N.R.

Mathematical Studies 33

(1971).

参照

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