ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEQO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE X X (1978)
L.
P a s i c k i(Lublin)
A short proof of the Caristi theorem
In 1975 Caristi [2] proved a fixed point theorem which was next used in proving some other results [2], [4]. The original proof used the transfinite induction method and was rather complicated. Then several other proofs лГеге found [1], [3], [4], but all of them are more compli
cated than the new proof presented below, which uses Zorn’s lemma, bu t in a simple way.
T h e o r e m
1 (Wong [4]). Let f be a self map on a non-empty complete metric space (X , d) and V : _X-> [
0, oo) a lower semicontinuous function.
Let us presume that the following condition holds:
(
1) Lor any x
eX , x Ф f(x) there exists у
eX
—{x} such that : d{x, y) < V { x ) - V ( y ) .
By these assumptions f has a fixed point in X .
P ro o f. In view of Zorn’s lemma there exists a maximal set А <=z X , a
eA such th a t for all points x, у
eA
d{x, y) < \V {x )-V { y )\.
For a = inf(F(a?): x e A) there exists a sequence of points' zi e A such th a t ( V {z{))ieN is non-increasing and lim V (£*•) = a.
I t follows from г_>0°
<*(«<, % ) < \ У ( Ъ ) - У { * j ) \
th a t there exists b
eX , b = lim zt .
i-*CQ
For any x
eA , if V(x) Ф a, then for sufficiently large i we have d(b, x) < d(b, z j + d fa , x) < d{b, «<) + У(®) —7(«<)
(if for x 0
eA V {
xq) = a, then we obtain in a similar way d(b, x 0) =
0) and then by the lower semicontinuity of V
d { b ,x ) < V { x ) - V ( b ) .
13 — Roczniki PTM Prace Mat. XX.2
428 L. P a sick i
This means th a t b e A and th a t there is no point y e X such th a t b Ф у and
d{b, y) < V ( b ) - V ( y )
because such у would belong to A. Then it must be so th a t:
d{b,f(b)) = 0 Q.E.D.
The theorem which follows, is a consequence of Theorem 1.
Theorem 2
(Caristi
[2]).Let all assumptions of Theorem 1 except
(1)be satisfied. Let us assume that for x e X
M ) < V{x) — V (f(x )).
Then f has a fixed point.
We can modify Wong’s theorem as follows:
Theorem 3.
Let (X , d) be a non-empty complete metric space and V: X ->[0, oo) a lower semicontinuous function. I f a sentence formula g satisfies the following condition on X :
from ~g(x) it follows that there exists y e X such that:
d(x, y) < V ( x ) - V ( y ) , then there exists b e X such that g(b).
R e f e r e n c e s
[1] F .E . B r o w d e r , On theorem of Caristi and KirTc (to appear).
[2] J. C a r is ti, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. (to appear).
[3] W. A. K irk , CarisWs fixed point theorem and metric convexity (to appear).
[4] C. S. W on g, On a fixed point theorem of contractive type, preprint.
