Open Access

Fixed Point Theory for Admissible Type Maps with Applications

Fixed Point Theory and Applications20092009:439176

https://doi.org/10.1155/2009/439176

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 [16] 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
(1.1)

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
(1.2)

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
(1.3)
we define the induced map
(1.4)

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
(1.5)

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
(1.6)

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
(1.7)

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
(1.8)

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
(2.1)
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
(2.2)
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:
(2.3)
Remark 2.1.
  1. (i)

    For convenience the norm on is denoted by .

     
  2. (ii)

    In our applications for each .

     
  3. (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
(2.4)

(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
(2.5)

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:
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)

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
(2.12)

and with in as in Note in for .

Fix . Note
(2.13)

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
(2.14)

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
(2.15)

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:
(2.16)

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
(2.17)

Theorem 2.18.

Let be a constant and the conjugate to . Suppose the following conditions are satisfied:
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)

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
(2.24)
here . Fix and note
(2.25)
with
(2.26)
Let be given by
(2.27)
Also let (we will use Remark 2.5) and let be given by
(2.28)
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
(2.29)
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
(2.30)
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
(2.31)
then (2.11) holds. To see (2.31) fix . Consider and (as described above). Then for and so
(2.32)
so
(2.33)
(here (2.21) guarantees that there exists with for a.e. ) Let through and use the Lebesgue Dominated Convergence theorem to obtain
(2.34)
since in . Finally let (note (2.19)) to obtain
(2.35)

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 ):
(2.36)
(2.37)
Also for each assume either
(2.38)
or
(2.39)
hold. Finally suppose that the following hold:
(2.40)
(2.41)
(2.42)

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
(2.43)

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:
(2.44)

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:
(2.45)
(2.46)
(2.47)
(2.48)
(2.49)

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
(2.50)

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
(2.51)

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:
(2.52)
(2.53)
(2.54)
(2.55)
(2.56)

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 ):
(2.57)
(2.58)

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 ):
(2.59)
(2.60)

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:
(2.61)
(2.62)
(2.63)
(2.64)
(2.65)

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
(2.66)
then one could replace (2.64) with
(2.67)

Proof.

Here let , , , , and be as in Theorem 2.18. Let
(2.68)
and note that for each that
(2.69)
Also let
(2.70)
and note that for each that
(2.71)
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
(2.72)
so
(2.73)
Next we show that (2.39) is satisfied. Let . Then and this together with (2.22) yields
(2.74)
for , and so (2.23) yields
(2.75)
Now let . Then and for (in particular for ). Now (2.64) implies
(2.76)
so (2.65) yields
(2.77)

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

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Department of Mathematics, National University of Ireland

References

  1. Agarwal RP, Dshalalow JH, O'Regan D: Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach. International Journal of Mathematics and Mathematical Sciences 2005, (17):2775–2782.MathSciNetView ArticleMATHGoogle Scholar
  2. Agarwal RP, Frigon M, O'Regan D: A survey of recent fixed point theory in Fréchet spaces. In Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday. Vol. 1, 2. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:75–88.View ArticleGoogle Scholar
  3. Agarwal RP, O'Regan D: A Lefschetz fixed point theorem for admissible maps in Fréchet spaces. Dynamic Systems and Applications 2007,16(1):1–12.MathSciNetMATHGoogle Scholar
  4. Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications. Volume 495. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:x+399.View ArticleMATHGoogle Scholar
  5. Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003:xvi+690.View ArticleMATHGoogle Scholar
  6. O'Regan D: An essential map approach for multimaps defined on closed subsets of Fréchet spaces. Applicable Analysis 2006,85(5):503–513. 10.1080/00036810500474861MathSciNetView ArticleMATHGoogle Scholar
  7. Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.View ArticleMATHGoogle Scholar
  8. Petryshyn WV: Existence of fixed points of positive -set-contractive maps as consequences of suitable boundary conditions. Journal of the London Mathematical Society 1988,38(3):503–512.MathSciNetMATHGoogle Scholar
  9. Park S: A unified fixed point theory of multimaps on topological vector spaces. Journal of the Korean Mathematical Society 1998,35(4):803–829.MathSciNetMATHGoogle Scholar
  10. Agarwal RP, O'Regan D: Homotopy and Leray-Schauder principles for multi maps. Nonlinear Analysis Forum 2002,7(1):103–111.MathSciNetMATHGoogle Scholar
  11. Górniewicz L, Granas A: Some general theorems in coincidence theory. I. Journal de Mathématiques Pures et Appliquées 1981,60(4):361–373.MathSciNetMATHGoogle Scholar
  12. O'Regan D: A Krasnoselskii cone compression theorem for maps. Mathematical Proceedings of the Royal Irish Academy 2003,103A(1):55–59.View ArticleMathSciNetMATHGoogle Scholar
  13. Izydorek M, Kucharski Z: The Krasnosielski theorem for permissible multivalued maps. Bulletin of the Polish Academy of Sciences. Mathematics 1989,37(1–6):145–149.MathSciNetMATHGoogle Scholar
  14. Kantorovich LV, Akilov GP: Functional Analysis in Normed Spaces. The Macmillan, New York, NY, USA; 1964:xiii+771.MATHGoogle Scholar
  15. O'Regan D: Leray-Schauder and Krasnoselskii fixed point theory in Fréchet spaces for Urysohn type operators,. In Legacy of the Legend. V. Lakshmikantham, Cambridge Scientific;Google Scholar
  16. Agarwal RP, O'Regan D: Multivalued nonlinear equations on the half line: a fixed point approach. The Korean Journal of Computational & Applied Mathematics 2002,9(2):509–524.MathSciNetMATHGoogle Scholar
  17. Agarwal RP, O'Regan D: Lefschetz fixed point theorems for compact morphisms. Fixed Point Theory 2006,7(2):175–190.MathSciNetMATHGoogle Scholar
  18. Dzedzej Z: Fixed point index theory for a class of nonacyclic multivalued maps. Dissertationes Mathematicae 1985, 253: 1–58.MathSciNetMATHGoogle Scholar

Copyright

© R. P. Agarwal and D. O’Regan 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.