**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 *

^{e}*X , x Ф f(x) there exists у *

^{e }*X *

^{— }*{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 *

**e***A such th a t for all points x, у *

^{e}*A*

*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 *

*e*

* X , b = lim zt .*

**i-*CQ**

*For any x *

*e*

* A , 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 *

*e*

*A V {*

*x*

*q) = 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.**