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Then for any x 0 ? X, (T n x 0 ) n?N is a forward Cauchy sequence. Hence by Lemma 3.3, there exists ? ? AX that is fixed under T * . Now Theorem 3.1 applies and we get a fixed point of T . Since T is a contraction ,
It is well-known that that any (·)-complete X is a complete lattice in the intrinsic order, and hence (·) is admissible. It is actually a metric version of Knaster-Tarski's theorem that we are going to prove ,
s fixed point theorems It is known from [6] that any complete metric space X can be embedded onto the set of maximal elements of a continuous dcpo BX := {{x, r | x ? X, ? > r 0} ? X, where x, r(z) := X(z, x) + r, and x, r BX y, s iff x, r X y, s iff r X(x, y) + s. In [9] it has been shown that X is an A-complete gms iff (BX, BX ) is a dcpo We will use this knowledge in proving a generalized version of Caristi's fixed point theorem ,
Let Z be a subgms of BX consisting of elements of the form x, ?(x) for x ? X. Hence x, ?(x) Z y, ?(y) iff ?(x) X(x, y) + ?(y) Moreover ? is lower semicontinuous iff it preserves directed suprema with respect to the order Z . All this means that ,
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