Some Theorems for Real Increasing Functions in Elementary Fixed Point Theory ()
1. 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 where we consider the set F(x0) of selfmaps functions f of R (the reals) having a given point
as the fixed point of f, i.e.
. Furthermore, we consider the subset G(x0) of F(x0) formed from functions f such that
for any
(1.1)
and
for any
(1.2)
Moreover, let H(x0) be the subset of F(x0) of functions f “attracting” R, which means the sequence
as
for any
, being fn(x) the n-th iterate of f defined usually by
,
for any integer positive n. In general, this term is transferred to a contractive function in the sense of, historically speaking, a theorem of Banach-Caccioppoli [2] [3] and other famous authors (see, e.g., those cited in the good books [1] [4]), have a unique fixed point x0 “attracting the whole metric space”. Inspired from an old paper of Sgambati [5], we establish some results for a real increasing function (often considered as selfmap of a compact interval of R).
We start with the following theorem.
Theorem 1. Let
be increasing. If
for any x > x0, there exists a y > x such that
for
(1.3)
and
for any x < x0, there exists a y < x such that
for
, (1.4)
then
.
Proof. We limit to prove the theorem under (1.3) (the proof under (1.4) is similar).
The thesis is obvious if
for some n since if f is increasing and
, we have
for
. Then we can assume
for any integer positive n. By (1.1) and since f is increasing, we have that
for any integer positive n and for any
(1.5)
that is the sequence {fn(x)} is strictly decreasing and hence converges to the least upper bound r as
. Then for any
, there exists some integer positive s such that
for any
. We deduce
for
from (1.5). Assume that r > x0. By (1.3), there exists y > r such that
for
and choose an
. So there exists some integer positive m such that
for any
and then we should have, in particular,
which is a contradiction. Therefore r = x0.
Claim. The condition (1.3) or (1.4) is not necessary in order to have the belongness of a function
to the set H(x0). Indeed, let any
and consider a strictly decreasing sequence {hn} of rationals convergent to some rational x1 > x0. Hence it is enough to define the function
(irrational value) and
if
for any positive integer n in order to obtain a function f which proves such assertion.
Remark 1. Let
and f be increasing. Then it is immediate to verify (1.3) and (1.4) are equivalent, respectively, to the following:
x > x0 implies either
or there exists a y > x such that f(t) = x for
(1.6)
and
x < x0 implies either
or there exists a y < x such that f(t) = x for
(1.7)
Since an increasing function which has the property
(resp.,
) implies having the property f(x) < x (resp., f(x) > x), 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(x0) iff (x > x0 implies f(x) < x and the second member of (1.6) holds) and (x < x0 implies f(x) > x and the second member of (1.7) holds).
Corollary 3. A strictly increasing function
belongs to H(x0) iff
for any x > x0 and
for any x < x0.
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(x0), it is enough that f is right upper semicontinuous in every x > x0 and left lower semicontinuous in every x < x0.
Proof. Indeed a function
which is right upper (resp., left lower) semicontinuous verifies (1.3) (resp., (1.4)).
Corollary 5. In order to have the belongness of a continuous function
to the set H(x0), it is enough to satisfy (1.1) and (1.2).
2. Other Results
The following lemma is useful for the successive Theorems 7 and 8.
Lemma 6. Let a > 0 and f be an increasing selfmap of the compact interval [0, a] having zero as unique fixed point. Then
for any
.
Proof. We have that
. If
for some
, we should have that there striction g of f to the compact interval [x,a] is an increasing selfmap of [x, a] itself because
for any
holds. Then, e.g., by applying a famous theorem of Tarski [6] to an increasing selfmap g of the compact interval [x, a], minimum and maximum fixed points of g should exist in [x, a], which should be also fixed points of f in [0, a], a contradiction to the hypothesis.
Theorem 7. Let a > 0 and f be an increasing selfmap of the compact interval [0, a] having zero as unique fixed point and
implies either
or there exists an
such that
for
. Then we have that
as
for any
.
Proof. Lemma 6 assures that
for any
, so we can extend the function f from [0, a] to an increasing selfmap g of R in the following way:
if
,
if
,
if
which satisfies (1.1) and (1.2) for
. By invoking Theorem 2, we have
for any
.
Theorem 8. Let a > 0 and f be an increasing selfmap of the compact interval [0, a] having zero as unique fixed point and right continuous in any
. Then we have that
as
for any
.
Proof. By Lemma 6, we have that
for any
. Since f is right continuous, we also have that
for any
. 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 A0 is defined quasi compact if the sequence of measures of non-compactness {γ(An)} of the closed subsets of A0 recursively given by
for
converges to zero as
.
The following theorem holds:
Theorem 9. Let be a selfmap ψ of a bounded metric space A0. Suppose that
and there exists an f be an increasing selfmap of the compact interval [0, a] having zero as unique fixed point. Further
implies either
or there exists a y > x such that
for
. Then ψ is quasi compact provided that for any subset A of A0, we have that
(2.1)
Proof. From (2.1), it follows that
for any
,
. By Theorem 7, we have that
as
and hence the thesis.
Remark 2. J. Einsenfeld and V. Lakshmikantham [7] assume all the hypothesis of Theorem 9 except that f is considered right continuous in any
. Thus our Theorem 9 is an extension of that result since a right continuous function satisfies the hypothesis contained in Theorem 9.
3. 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
such that
. Then the limit z of the sequence {fn(x0)} is the greatest fixed point of f in
.
Proof. z is a fixed point of f in S_(x0) since f is right continuous. If
is another fixed point of f in S_(x0), being f increasing, we have that
for any integer positive n and then
as
.
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
if for any decreasing sequence {xn} converging to x and such that is convergent the sequence {f(xn)} to the limit l, there exists a non-negative integer
for which we have that
. 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
.
Remark 3. Any right continuous selfmap f of an interval X of R is upper broad sequentially semicontinuous on the right on X because
for any
(we naturally understand that
).
Definition 12. Let f be a selfmap of interval X of R and let f be an upper broad sequentially semicontinuous on the right at the point
. We define upper broad sequentially semicontinuity on the right parameter of a decreasing sequence {xn} converging to x and such that is convergent the sequence {f(xn)} to the limit l, denoted by
for abuse of notation, the minimum of the set {p non-negative integer:
}.
Theorem 13. Let f be a selfmap of a compact interval X of R and f be an increasing and upper broad sequentially semicontinuous on the right over X. Moreover there exists in X a point x0 such that
and let
be the upper broad sequentially semicontinuity on the right parameter of the decreasing sequence {fn(x0)}. Then, if z is the limit of the sequence {fn(x0)}, fk(z) is the greatest fixed point of f in
.
Proof. For brevity, let
for any non-negative integer, we have
by definition 12. On the other hand,
because {xn} converges to its least upper bound, from which
because f is increasing and hence
, from it follows that
. Therefore
. If
is another fixed point of f in S_(x0), we have that
and then
as
. Since f is increasing, we have
.
By Remark 3, Theorem 13 is an effective extension of Theorem 10. In fact Theorem 13 holds in the following example where Theorem 10 is not applicable:
Example 14. Let f be a selfmap of [0, 1] be defined as
if
,
,
if
,
if
. It is easily seen that f is right continuous in [0, 1] - {1/2}, and hence upper broad sequentially semicontinuous on the right because of Remark 3 and also at 1/2 because
. Indeed f is not right continuous at 1/2 and hence Theorem 10 is not applicable but it can easily see that
is the greatest fixed point in S_(1/2) for any positive integer p with 1 ≤ p.
4. A Simple Result in Posets
The next result holds in a partially ordered set (poset) but it can hold in subsets of R as we show in a suitable example. This result can be deduced also from our previous works [8] [9] but we prefer to give a proof for making self-contained the present paper.
Theorem 15. Let (X, ≤) be a poset and f be an increasing selfmap of X. Assume the existence of a point x0 such that
and the following condition holds:
(C1) if C is a chain which has not a supremum in X, there exists a positive integer 1 ≤ n such that there exists the supremum of the chain fn(C), denoted by sup fn(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
by hypothesis. Let C be a chain of Y and assume that there no exists its supremum in X, then there exists
for some positive integer 1 ≤ n.
So, from
for any
, we deduce that
for any
since f is increasing, that is y is an upper bound of C in X. But we claim that y lives in Y. Indeed,
for any
, then
for any
due to the increasingness of f, i.e. f(y) is an upper bound of the chain fn(C), which means
, i.e.
and thus any chain C of Y has an upper bound in Y. By Zorn lemma, Y has a maximal element z such that
and therefore
, i.e. the thesis.
We illustrate Theorem 15 with a suitable example:
Example 16. Let
with usual ordering and f be defined as
if
and
if
. Then f is increasing and the unique chains of X not having supremum in X are the chains with supremum 1/2 which does not belong to X.
Note that
for any
and
for any integer n such that 1 ≤ n. Indeed, if C is a chain included in [0, 1/2), then f(C) is included in (1/2, 1] and
if included in [1, 2]. Note that z = 1 if
and
if
.
5. Conclusion
We have proved various theorems for real increasing functions f giving the new concept of upper broad sequentially semicontinuity on the right at a point x and related parameter k of a decreasing sequence {xn} convergent to x and such that also converges the sequence {f(xn)}. Naturally, a dual concept like lower broad sequentially semicontinuity on the left and related parameter k of an increasing sequence {xn} at a point x, such that also converges the sequence {f(xn)}, holds as well, however it should be interesting to establish geometrical and topological properties of the parameter k, here not investigated for brevity.
Acknowledgements
In memoriam of the masterly friend Prof. Antonio Zitarosa (Napoli).
Authors’ Contributions
The authors read and approved the final manuscript.