Fixed Point Theory for Admissible Type Maps with Applications
© R. P. Agarwal and D. O’Regan 2009
Received: 8 December 2008
Accepted: 18 June 2009
Published: 20 July 2009
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.
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 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 .
Theorem 1.6 immediately yields the topological transversality theorem for Kakutani maps.
Also existence in Section 2 will be based on the following result of Petryshyn [8, Theorem 3].
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
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:
The class is due to Park  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  which we state here for the convenience of the reader.
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:
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 .
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:
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  and Theorem 1.10 can be found 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
(here we use the notation from , i.e., decreasing in the generalized sense) Let (or where is the generalized intersection ) denote the projective limit of (note for ) and note , so for convenience we write .
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).
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 ).
and the result above is again true.
Usually in our applications one has (so ). If is a pseudoopen subset of then for each one has (see ) 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.
Note that if for each then is essential in (see ).
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.
Condition (2.7) can be removed from the statement of Theorem 2.14.
Note that Remark 2.6 holds in this situation also.
One could also obtain a multivalued version of Theorem 2.18 by using the ideas in the proof below with the ideas in .
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.
Condition (2.36) can be removed from the statement of Theorem 2.20.
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.
Fix . We would like to apply Theorem 1.10. Note that we know from  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.
Note Remarks 2.4, 2.6, and 2.7 hold in this situation also.
Condition (2.45) can be removed from the statement of Theorem 2.27.
and the result above is again true.
Also one has the following result.
Condition (2.52) can be removed from the statement of Theorem 2.30.
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).
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).
Note Remarks 2.21 and 2.22 hold in this situation also.
To conclude the paper we apply Theorem 2.20 (or Theorem 2.36) to (2.17).
One could obtain a multivalued version of Theorem 2.37 by using the ideas in the proof below with the ideas in .
Thus (2.39) holds. Now essentially the same argument as in Theorem 2.18 guarantees that (2.41) and (2.42) hold.
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