# Fixed Point Theory for Admissible Type Maps with Applications

- Ravi P. Agarwal
^{1}Email author and - Donal O'Regan
^{2}

**2009**:439176

**DOI: **10.1155/2009/439176

© R. P. Agarwal and D. O’Regan 2009

**Received: **8 December 2008

**Accepted: **18 June 2009

**Published: **20 July 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.

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 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 .

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.

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.

Then has a fixed point in .

Theorem 1.16.

Then has a fixed point in .

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

- (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 .

(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 .

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.

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.

and with in as in Note in for .

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.

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.

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.

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.

Theorem 2.18.

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.

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.

Then has a fixed point in .

Proof.

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.

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.

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.

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 ).

and the result above is again true.

Also one has the following result.

Theorem 2.30.

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.

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.

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.

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.

Proof.

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.

## Authors’ Affiliations

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