Some Theorems for Real Increasing Functions in Elementary Fixed Point Theory

We obtain some theorems for real increasing functions showing that elementary fixed point theory can bring to astonishing results by assuming only a few properties, some of which intrinsically possessed from these functions. An application is given for a theorem of quasi-compactness and a known result in posets is also recalled and applied to real intervals.


Introduction
Notwithstanding some words of the title, Elementary Fixed Point Theory [1] does not mean to establish "elementarily" fixed point theorems in the context of "structurally simple spaces" such as metric spaces or Banach spaces, but to deduce fixed point theorems from some intrinsic properties of the selfmaps involved and thus we arrive at astonishing results, like in the case of this paper as n → ∞ for any x ∈ R , being f n (x) the n-th iterate of f defined usually by to the set H(x 0 ). Indeed, let any 0 x ∈ R and consider a strictly decreasing sequence {h n } of rationals convergent to some rational x 1 > x 0 . Hence it is enough to define the function ( ) ( ) for any positive integer n in order to obtain a function f which proves such assertion. Since an increasing function which has the property ( ) then we can enunciate the following theorem and its corollary due to the author of [5] without proof: Theorem 2. An increasing function belongs to H(x 0 ) iff (x > x 0 implies f(x) < x and the second member of (1.6) holds) and (x < x 0 implies f(x) > x and the second member of (1.7) holds). The following theorem and its corollary, due to the author of [5], are also easily proved: Theorem 4. In order to have the belongness of a function to the set H(x 0 ), it is enough that f is right upper semicontinuous in every x > x 0 and left lower semicontinuous in every x < x 0 .
Corollary 5. In order to have the belongness of a continuous function to the set H(x 0 ), it is enough to satisfy (1.1) and (1.2).

Other Results
The following lemma is useful for the successive Theorems 7 and 8.
Then we have that Proof. Lemma 6 assures that , so we can extend the function f from [0, a] to an increasing selfmap g of R in the following way: . So the thesis comes from Theorem 7.
As application of above Theorem 7, we give a generalization of a theorem of "quasi compactness" due to J. Einsenfeld and V. Lakshmikantham [7]. We recall that a selfmap ψ of a bounded metric space A 0 is defined quasi compact if the sequence of measures of non-compactness {γ(A n )} of the closed subsets of A 0 recursively given by . Thus our Theorem 9 is an extension of that result since a right continuous function satisfies the hypothesis contained in Theorem 9.

Upper Broad Sequentially Semicontinuity on the Right
The following theorem is widely known (e.g., [6]. We give a short proof.
Theorem 10. Let f be an increasing and right continuous selfmap of a compact interval X of R and there exists a point 0 x X ∈ such that ( ) Then the limit z of the sequence {f n (x 0 )} is the greatest fixed point of f in Proof. z is a fixed point of f in S_(x 0 ) since f is right continuous. If We provide a generalization of Theorem 10 starting with the following definition: Definition 11. Let f be a selfmap of subset X of R. We say that f is upper broad sequentially semicontinuous on the right at the point x X ∈ if for any decreasing sequence {x n } converging to x and such that is convergent the sequence {f(x n )} to the limit l, there exists a non-negative integer . We say that f is upper broad sequentially semicontinuous on the right on X if f is upper broad sequentially semicontinuous on the right at any point x X ∈ . teger 1 ≤ n such that there exists the supremum of the chain f n (C), denoted by sup f n (C). Then f has a fixed point z in Proof. Consider the subset of X defined by ( ) . Y is no empty because there at least 0 x Y ∈ by hypothesis. Let C be a chain of Y and assume that there no exists its supremum in X, then there exists ( ) sup n y f = C for some positive integer 1 ≤ n.
So, from