ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEQO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X X (1978)

unpublished
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, but 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)
more » ... ric 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 + dfa, x) < d{b, «<) + У(®)-7(«<) (if for x0 e AV{xq) = a, then we obtain in a similar way d(b, x0) = 0) and then by the lower semicontinuity of V d{b,x) < V {x)-V (b). 13-Roczniki PTM Prace Mat. XX.2
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