- Research Article
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Fixed Point Theory for Admissible Type Maps with Applications
Fixed Point Theory and Applications volume 2009, Article number: 439176 (2009)
Abstract
We present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for multivalued maps between Fréchet spaces. As an application we show that our results are directly applicable to establish the existence of integral equations over infinite intervals.
1. Introduction
In this paper, assuming a natural sequentially compact condition we establish new fixed point theorems for Urysohn type maps between Fréchet spaces. In Section 2 we present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for admissible type maps. The proofs rely on fixed point theory in Banach spaces and viewing a Fréchet space as the projective limit of a sequence of Banach spaces. Our theory is partly motivated by a variety of authors in the literature (see [1–6] and the references therein).
Existence in Section 2 is based on a Leray-Schauder alternative for Kakutani maps (see [4, 5, 7] for the history of this result) which we state here for the convenience of the reader.
Theorem 1.1.
Let be a Banach space,
an open subset of
and
. Suppose
is an upper semicontinuous compact (or countably condensing) map (here
denotes the family of nonempty convex compact subsets of
). Then either
(A1) has a fixed point in
or
(A2) there exists (the boundary of
in
) and
with
.
Existence in Section 2 will also be based on the topological transversality theorem (see [5, 7] for the history of this result) which we now state here for the convenience of the reader. Let be a Banach space and
an open subset of
.
Definition 1.2.
We let denote the set of all upper semicontinuous compact (or countably condensing) maps
.
Definition 1.3.
We let if
with
for
.
Definition 1.4.
A map is essential in
if for every
with
there exists
with
. Otherwise
is inessential in
(i.e., there exists a fixed point free
with
).
Definition 1.5.
are homotopic in
, written
in
, if there exists an upper semicontinuous compact (or countably condensing) map
such that
belongs to
for each
and
with
.
Theorem 1.6.
Let and
be as above and let
. Then the following conditions are equivalent:
(i) is inessential in
;
(ii)there exists a map with
for
and
in
.
Theorem 1.6 immediately yields the topological transversality theorem for Kakutani maps.
Theorem 1.7.
Let and
be as above. Suppose that
and
are two maps in
with
in
. Then
is essential in
if and only if
is essential in
.
Also existence in Section 2 will be based on the following result of Petryshyn [8, Theorem 3].
Theorem 1.8.
Let be a Banach space and let
be a closed cone. Let
and
be bounded open subsets in
such that
and let
be an upper semicontinuous,
-set contractive (countably) map; here
,
and
denotes the closure of
in
. Assume that
(1) and
and
and
(here
and
denotes the boundary of
in
) or
(2) and
and
and
.
Then has a fixed point in
.
Also in Section 2 we consider a class of maps which contain the Kakutani maps.
Suppose that and
are Hausdorff topological spaces. Given a class
of maps,
denotes the set of maps
(nonempty subsets of
) belonging to
, and
the set of finite compositions of maps in
. A class
of maps is defined by the following properties:
(i) contains the class
of single-valued continuous functions;
(ii)each is upper semicontinuous and compact valued;
(iii)for any polytope ,
has a fixed point, where the intermediate spaces of composites are suitably chosen for each
.
Definition 1.9.
if for any compact subset
of
, there is a
with
for each
The class is due to Park [9] and his papers include many examples in this class. Examples of
maps are the Kakutani maps, the acyclic maps, the approximable maps, and the maps admissible in the sense of Gorniewicz.
Existence in Section 2 is based on a Leray-Schauder alternative [10] which we state here for the convenience of the reader.
Theorem 1.10.
Let be a Banach space,
an open convex subset of
and
. Suppose
is an upper semicontinuous countably condensing map with
for
and
. Then
has a fixed point in
.
Also existence in Section 2 will be based on some Lefschetz type fixed point theory. Let and
be Hausdorff topological spaces. A continuous single-valued map
is called a Vietoris map (written
) if the following two conditions are satisfied:
(i)for each , the set
is acyclic,
(ii) is a proper map, that is, for every compact
one has that
is compact.
Let be the set of all pairs
where
is a Vietoris map and
is continuous. We will denote every such diagram by
. Given two diagrams
and
, where
, we write
if there are maps
and
such that
,
,
and
. The equivalence class of a diagram
with respect to
is denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ1_HTML.gif)
or and is called a morphism from
to
. We let
be the set of all such morphisms. For any
a set
where
is called an image of
under a morphism
.
Consider vector spaces over a field . Let
be a vector space and
an endomorphism. Now let
where
is the
th iterate of
, and let
. Since
one has the induced endomorphism
. We call
admissible if
; for such
we define the generalized trace
of
by putting
where tr stands for the ordinary trace.
Let be an endomorphism of degree zero of a graded vector space
. We call
a Leray endomorphism if (i) all
are admissible and (ii) almost all
are trivial. For such
we define the generalized Lefschetz number
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ2_HTML.gif)
Let be the
ech homology functor with compact carriers and coefficients in the field of rational numbers
from the category of Hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus
is a graded vector space, with
being the
-dimensional
ech homology group with compact carriers of
. For a continuous map
,
is the induced linear map
where
.
The ech homology functor can be extended to a category of morphisms (see [11, page 364]) and also note that the homology functor
extends over this category, that is, for a morphism
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ3_HTML.gif)
we define the induced map
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ4_HTML.gif)
by putting .
Let be a multivalued map (note for each
we assume
is a nonempty subset of
). A pair
of single valued continuous maps of the form
is called a selected pair of
(written
) if the following two conditions hold:
(i) is a Vietoris map,
(ii) for any
Definition 1.11.
An upper semicontinuous compact map is said to be admissible (and we write
) provided that there exists a selected pair
of
.
Definition 1.12.
An upper semicontinuous map is said to be admissible in the sense of Gorniewicz (and we write
) provided that there exists a selected pair
of
.
Definition 1.13.
A map is said to be a Lefschetz map if for each selected pair
the linear map
(the existence of
follows from the Vietoris theorem) is a Leray endomorphism.
If is a Lefschetz map, we define the Lefschetz set
(or
) by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ5_HTML.gif)
Definition 1.14.
A Hausdorff topological space is said to be a Lefschetz space provided that every
is a Lefschetz map and
that implies
has a fixed point.
Also we present Krasnoselskii compression and expansion theorems in Section 2 in the Fréchet space setting. Let be a normed linear space and
a closed cone. For
let
and it is well known that
where
. Our next result, Theorem 1.8, was established in [12] and Theorem 1.10 can be found in [13].
Theorem 1.15.
Let be a normed linear space,
a closed cone,
,
constants, and
. Suppose that
is compact with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ6_HTML.gif)
Then has a fixed point in
.
Theorem 1.16.
Let be a normed linear space,
a closed cone,
,
constants, and
. Suppose that
is completely continuous with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ7_HTML.gif)
Then has a fixed point in
.
Now let be a directed set with order
and let
be a family of locally convex spaces. For each
for which
let
be a continuous map. Then the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ8_HTML.gif)
is a closed subset of and is called the projective limit of
and is denoted by
(or
or the generalized intersection [14, page 439]
).
2. Fixed Point Theory in Fréchet Spaces
Let be a Fréchet space with the topology generated by a family of seminorms
; here
. We assume that the family of seminorms satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ9_HTML.gif)
A subset of
is bounded if for every
there exists
such that
for all
. For
and
we denote
. To
we associate a sequence of Banach spaces
described as follows. For every
we consider the equivalence relation
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ10_HTML.gif)
We denote by the quotient space, and by
the completion of
with respect to
(the norm on
induced by
and its extension to
is still denoted by
). This construction defines a continuous map
. Now since (2.1) is satisfied the seminorm
induces a seminorm on
for every
(again this seminorm is denoted by
). Also (2.2) defines an equivalence relation on
from which we obtain a continuous map
since
can be regarded as a subset of
. Now
if
and
if
. We now assume the following condition holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ11_HTML.gif)
Remark 2.1.
-
(i)
For convenience the norm on
is denoted by
.
-
(ii)
In our applications
for each
.
-
(iii)
Note if
(or
) then
. However if
then
is not necessaily in
and in fact
is easier to use in applications (even though
is isomorphic to
). For example if
, then
consists of the class of functions in
which coincide on the interval
and
.
Finally we assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ12_HTML.gif)
(here we use the notation from [14], i.e., decreasing in the generalized sense) Let (or
where
is the generalized intersection [14]) denote the projective limit of
(note
for
) and note
, so for convenience we write
.
For each and each
we set
, and we let
,
and
denote, respectively, the closure, the interior, and the boundary of
with respect to
in
. Also the pseudointerior of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ13_HTML.gif)
The set is pseudoopen if
. For
and
we denote
.
We now show how easily one can extend fixed point theory in Banach spaces to applicable fixed point theory in Fréchet spaces. In this case the map will be related to
by the closure property (2.11).
Theorem 2.2.
Let and
be as described above,
a subset of
and
where
for each
. Also for each
assume that there exists
and suppose the following conditions are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ19_HTML.gif)
Then has a fixed point in
.
Remark 2.3.
Notice that to check (2.10) we need to show that for each the sequence
is sequentially compact.
Proof.
From Theorem 1.1 for each there exists
with
(we apply Theorem 1.1 with
and note
). Let us look at
. Notice
and
for
from (2.7). Now (2.10) with
guarantees that there exists a subsequence
and a
with
in
as
in
. Look at
. Now
for
. Now (2.10) with
guarantees that there exists a subsequence
of
and a
with
in
as
in
. Note from (2.4) and the uniqueness of limits that
in
since
(note
for
). Proceed inductively to obtain subsequences of integers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ20_HTML.gif)
and with
in
as
in
Note
in
for
.
Fix . Note
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ21_HTML.gif)
for every . We can do this for each
. As a result
and also note
since
for each
. Also since
in
for
and
in
as
in
one has from (2.11) that
in
.
Remark 2.4.
From the proof we see that condition (2.7) can be removed from the statement of Theorem 2.2. We include it only to explain condition (2.10) (see Remark 2.3).
Remark 2.5.
Note that we could replace above with
a subset of the closure of
in
if
is a closed subset of
(so in this case we can take
if
is a closed subset of
). To see this note
,
and
in
as
and we can conclude that
(note that
if and only if for every
there exists
,
for
with
in
as
).
Remark 2.6.
Suppose in Theorem 2.2 we replace (2.10) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ22_HTML.gif)
In addition we assume with
for each
is replaced by
and suppose (2.11) is true with
replaced by
. Then the result in Theorem 2.2 is again true.
The proof follows the reasoning in Theorem 2.2 except in this case and
.
Remark 2.7.
In fact we could replace (in fact we can remove it as mentioned in Remark 2.4) (2.7) in Theorem 2.2 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ23_HTML.gif)
and the result above is again true.
Remark 2.8.
Usually in our applications one has (so
). If
is a pseudoopen subset of
then for each
one has (see [15]) that
is a open subset of
so
.
Essentially the same reasoning as in Theorem 2.2 (now using Theorem 1.7) establishes the following result. We will need the following definitions.
Let and
be as described in Section 2. For the definitions below
and
with
for each
(or
a subset of the closure of
in
if
is a closed subset of
). In addition assume for each
that
Definition 2.9.
We say if for each
one has
(i.e., for each
,
is an upper semicontinuous countably condensing map).
Definition 2.10.
if
and for each
one has
for
.
Definition 2.11.
is essential in
if for each
one has that
is essential in
(i.e., for each
, every map
with
has a fixed point in
).
Remark 2.12.
Note that if for each
then
is essential in
(see [7]).
Definition 2.13.
(We assume for
.)
are homotopic in
, written
in
, if for each
one has
in
.
Theorem 2.14.
Let and
be as described above,
a subset of
and
where
for each
or
a subset of the closure of
in
(if
is a closed subset of
). Also for each
assume that there exists
and suppose
, (2.6), (2.7), and the following condition holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ24_HTML.gif)
Also assume (2.10) and (2.11) hold. Then has a fixed point in
.
Proof.
Fix . Now Remark 2.12 guarantees that the zero map (i.e.,
) is essential in
for each
. Now Theorem 1.7 guarantees that
is essential in
so in particular there exists
with
. Essentially the same reasoning as in Theorem 2.2 (with Remark 2.5) establishes the result.
Remark 2.15.
Notice that (2.6) and (2.17) could be replaced by in
(of course we assume
and we must specify
for
here).
Remark 2.16.
Condition (2.7) can be removed from the statement of Theorem 2.14.
Remark 2.17.
Note that Remark 2.6 holds in this situation also.
As an application of Theorem 2.2 we discuss the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ25_HTML.gif)
Theorem 2.18.
Let be a constant and
the conjugate to
. Suppose the following conditions are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ29_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ31_HTML.gif)
Then (2.17) has at least one solution in .
Remark 2.19.
One could also obtain a multivalued version of Theorem 2.18 by using the ideas in the proof below with the ideas in [16].
Proof.
Here ,
consists of the class of functions in
which coincide on the interval
,
with of course
defined by
. We will apply Theorem 2.2 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ32_HTML.gif)
here . Fix
and note
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ33_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ34_HTML.gif)
Let be given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ35_HTML.gif)
Also let (we will use Remark 2.5) and let
be given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ36_HTML.gif)
Clearly (2.6) and (2.7) hold, and a standard argument in the literature guarantees that is continuous and compact so (2.8) holds. To show (2.9) fix
and suppose that there exists
(so
) and
with
. Then for
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ37_HTML.gif)
so , that is,
. This contradicts (2.23), so (2.9) holds. To show (2.10) consider a sequence
with
,
on
and
. Now to show (2.10) we will show for a fixed
that
is sequentially compact for any subsequence
of
. Note for
that
so
is uniformly bounded since
for
implies
for
. Also
is equicontinuous on
since for
and
(note there exists
with
for a.e.
) one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ38_HTML.gif)
The Arzela-Ascoli theorem guarantees that is sequentially compact. Finally we show (2.11). Suppose there exists
and a sequence
with
and
in
such that for every
there exists a subsequence
of
with
in
as
in
. If we show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ39_HTML.gif)
then (2.11) holds. To see (2.31) fix . Consider
and
(as described above). Then
for
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ40_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ41_HTML.gif)
(here (2.21) guarantees that there exists with
for a.e.
) Let
through
and use the Lebesgue Dominated Convergence theorem to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ42_HTML.gif)
since in
. Finally let
(note (2.19)) to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ43_HTML.gif)
Thus (2.11) holds. Our result now follows from Theorem 2.2 (with Remark 2.5).
Essentially the same reasoning as in Theorem 2.2 (now using Theorem 1.8) establishes the following result.
Theorem 2.20.
Let and
be as described in the beginning of Section 2,
a closed cone in
,
, and
are bounded pseudoopen subsets of
with
, and
. Also assume either
for each
(here
) or
a subset of the closure of
in
(if
is a closed subset of
). Also for each
assume
and suppose that the following conditions hold (here
):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ45_HTML.gif)
Also for each assume either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ46_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ47_HTML.gif)
hold. Finally suppose that the following hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ50_HTML.gif)
Then has a fixed point in
.
Proof.
Fix . We would like to apply Theorem 1.8. Note that we know from [15] that
is a cone and
and
are open and bounded with
. Theorem 1.8 guarantees that there exists
with
in
. As in Theorem 2.2 there exists a subsequence
and a
with
in
as
in
. Also
together with (2.40) yields
for
and so
. Proceed inductively to obtain subsequences of integers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ51_HTML.gif)
and with
in
as
in
. Note
in
for
and
. Now essentially the same reasoning as in Theorem 2.2 (with Remark 2.5) guarantees the result.
Remark 2.21.
Condition (2.36) can be removed from the statement of Theorem 2.20.
Remark 2.22.
Note (2.40) is only needed to guarantee that the fixed point satisfies
for
. If we assume all the conditions in Theorem 2.20 except (2.40) then again
has a fixed point in
but the above property is not guaranteed.
Essentially the same reasoning as in Theorem 2.2 (just apply Theorem 1.10 in this case) establishes the following result.
Theorem 2.23.
Let and
be as described above,
a convex subset of
and
where
for each
or
a subset of the closure of
in
(if
is a closed subset of
). Also for each
assume that there exists
and suppose that (2.6), (2.7), (2.9), (2.10), (2.11) and the following condition hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ52_HTML.gif)
Then has a fixed point in
.
Proof.
Fix . We would like to apply Theorem 1.10. Note that we know from [15] that
is convex. From Theorem 1.10 for each
there exists
with
in
. Now essentially the same reasoning as in Theorem 2.2 (with Remark 2.5) guarantees the result.
Remark 2.24.
Note Remarks 2.4, 2.6, and 2.7 hold in this situation also.
Now we present some Lefschetz type theorems in Fréchet spaces. Let and
be as described above.
Definition 2.25.
A set is said to be PRLS if for each
,
is a Lefschetz space.
Definition 2.26.
A set is said to be CPRLS if for each
,
is a Lefschetz space.
Theorem 2.27.
Let and
be as described above,
is an PRLS, and
. Also for each
assume that there exists
and suppose that the following conditions are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ54_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ55_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ56_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ57_HTML.gif)
Then has a fixed point in
.
Proof.
For each there exists
Now the same reasoning as in Theorem 2.2 guarantees the result.
Remark 2.28.
Condition (2.45) can be removed from the statement of Theorem 2.27.
Remark 2.29.
Suppose in Theorem 2.27, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ58_HTML.gif)
instead of (2.48) and is replaced by
with
and
for each
and suppose that (2.49) is true with
replaced by
. Then the result in Theorem 2.27 is again true.
In fact we could replace above with
a subset of the closure of
in
if
is a closed subset of
(so in this case we can take
if
is a closed subset of
).
In fact in this remark we could replace (in fact we can remove it as mentioned in Remark 2.4) (2.45) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ59_HTML.gif)
and the result above is again true.
Also one has the following result.
Theorem 2.30.
Let and
be as described above,
is an CPRLS and
. Also assume
is a closed subset of
and for each
that
and suppose that the following conditions are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ60_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ61_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ62_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ63_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ64_HTML.gif)
Then has a fixed point in
.
Remark 2.31.
Condition (2.52) can be removed from the statement of Theorem 2.30.
Remark 2.32.
Note that we can remove the assumption in Theorem 2.30 that is a closed subset of
if we assume
with
and
(or
a subset of the closure of
in
if
is a closed subset of
) for each
with of course
replaced by
in (2.56).
Remark 2.33.
Of course there are analogue results for compact morphisms (see the ideas here and in [17]) and for compact permissible maps (see [18]).
Next we present some Krasnoselskii results in the Fréchet space setting (in the first we use Theorem 1.15 and the second Theorem 1.16).
Theorem 2.34.
Let and
be as described in the beginning of Section 2,
a closed cone in
,
, and
are constants with
, and
with
, and
(or
is a subset of the closure of
in
if
is a closed subset of
) for each
; here
where
. Also for each
assume
and suppose that (2.36) and the following conditions are satisfied (here
with
):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ65_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ66_HTML.gif)
Also assume (2.40), (2.41), and (2.42) hold. Then has a fixed point in
.
Remark 2.35.
Note Remarks 2.21 and 2.22 hold in this situation also.
Theorem 2.36.
Let and
be as described in the beginning of Section 2,
a closed cone in
,
and
are constants with
, and
with
,
and
(or
is a subset of the closure of
in
if
is a closed subset of
) for each
. Also for each
assume
and suppose that (2.36) and the following conditions are satisfied (here
where
and
with
):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ67_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ68_HTML.gif)
In addition assume (2.40), (2.41), and (2.42) hold. Then has a fixed point in
.
To conclude the paper we apply Theorem 2.20 (or Theorem 2.36) to (2.17).
Theorem 2.37.
Let be a constant and
the conjugate to
and suppose that (2.18), (2.19), (2.20), (2.21), (2.22), and (2.23) hold. In addition assume the following conditions are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ69_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ70_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ71_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ72_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ73_HTML.gif)
Then (2.17) has at least one solution in with
for
.
Remark 2.38.
One could obtain a multivalued version of Theorem 2.37 by using the ideas in the proof below with the ideas in [16].
Remark 2.39.
In (2.63) we picked for convenience (i.e., so we could take
; otherwise we would take
). Also if there exists a
,
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ74_HTML.gif)
then one could replace (2.64) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ75_HTML.gif)
Proof.
Here let ,
,
,
, and
be as in Theorem 2.18. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ76_HTML.gif)
and note that for each that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ77_HTML.gif)
Also let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ78_HTML.gif)
and note that for each that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ79_HTML.gif)
Finally we could take . As in Theorem 2.18 clearly (2.36) and (2.37) hold; we need only to note that if
then from (2.62) and (2.63) one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ80_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ81_HTML.gif)
Next we show that (2.39) is satisfied. Let . Then
and this together with (2.22) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ82_HTML.gif)
for , and so (2.23) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ83_HTML.gif)
Now let . Then
and
for
(in particular
for
). Now (2.64) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ84_HTML.gif)
so (2.65) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F439176/MediaObjects/13663_2008_Article_1142_Equ85_HTML.gif)
Thus (2.39) holds. Now essentially the same argument as in Theorem 2.18 guarantees that (2.41) and (2.42) hold.
Notice (2.40) is satisfied with . To see this fix
and take a subsequence
and let
be such that
(i.e.,
) for some
. Then
, so as a result
. The result now follows from Theorem 2.20.
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Agarwal, R.P., O'Regan, D. Fixed Point Theory for Admissible Type Maps with Applications. Fixed Point Theory Appl 2009, 439176 (2009). https://doi.org/10.1155/2009/439176
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DOI: https://doi.org/10.1155/2009/439176