International Journal of Mathematics and Mathematical Sciences

Volume 2011, Article ID 390720, 9 pages

http://dx.doi.org/10.1155/2011/390720

## Fixed-Point Theory on a Frechet Topological Vector Space

^{1}Departement de Mathématiques, Faculté des Sciences de Gafsa, Université de Gafsa, Cite Universitaire Zarrouk, Gafsa 2112, Tunisia^{2}Departement de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, Route de Soukra Km 3.5, B.P.1171, Sfax 3000, Tunisia

Received 6 December 2010; Revised 14 February 2011; Accepted 15 February 2011

Academic Editor: Genaro Lopez

Copyright © 2011 Afif Ben Amar et al. 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.

#### Abstract

We establish some versions of fixed-point theorem in a Frechet topological vector space . The main result is that every map (where is a continuous map and is a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem for *U*-contractions and weakly compact mappings, while the second one, by assuming that the family where and a compact is nonlinear equicontractive, we give a fixed-point theorem for the operator of the form .

#### 1. Introduction

Fixed-point theorems are very important in mathematical analysis. They are an interesting way to show that something exists without setting it out, which sometimes is very hard, or even impossible to do. Several algebraic and topological settings in the theory and applications of nonlinear operator equations lead naturally to the investigation of fixed-points of a sum of two nonlinear operators, or more generally, fixed-points of mappings on the cartesian product into , where is some appropriate space.

Fixed-point theorems in topology and nonlinear functional analysis are usually based on certain properties (such as complete continuity, monotonicity, contractiveness, etc.) that the operator, considered as a single entity must satisfy. We recall for instance the Banach fixed-point theorem, which asserts that a strict contraction on a complete metric space into itself has unique fixed-point, and the Schauder principle, which asserts that a continuous mapping on a closed convex set in Hausdorff locally convex topological vector space into such that is contained in a compact set, has a fixed-point. In many problems of analysis, one encounters operators which may be split in the form , where is a contraction in some sense, and is completely continuous, and itself has neither of these properties (see [1–3]). Thus neither the Schauder fixed-point theorem nor the Banach fixed-point theorem applies directly in this case, and it becomes desirable to develop fixed-point theorems for such situations. An early theorem of this type was given by Krasnosel’skiĭ [4]: “Let be a Banach space, be a bounded closed convex subset of , and be operators on into such that for every pair . If is a strict contraction and is continuous and compact, then the equation has a solution in .” This result has been extended to locally convex spaces in 1971 by Cain and Nashed [5]. There is also another theorem of this type which was given by Amar et al. [6] in 2005 and which extended the Schauder and Krasnoselskii fixed-point theorems in Dunford-Pettis spaces to weakly compact operators. Also in 2010, Amar and Mnif [7] established some new variants of Leray-Schauder type fixed-point theorems for weakly sequentially continuous operators.

In this paper, we give also a generalization of Krasnoselskii fixed-point theorems not in Dunford-Pettis Banach spaces but in Dunford-Pettis Frechet spaces. More precisely, let be a Frechet topological vector space having the property of Dunford-Pettis, a closed bounded convex subset of , and (where is a continuous map and is a linear weakly compact operator). If leaves invariant then has a fixed-point in (see Proposition 3.1). In addition, if is a -contraction map of into , for each with , there is a such that and is relatively weakly compact, then has a fixed-point in (see Proposition 3.3).

Based on our results and other theorems which was given by Sehgal and Singh in 1976 ([8]), we give also an extension of the Krasnoselskii fixed-point theorem: Let be a Frechet topological vector space having the property of Dunford-Pettis (DP), , a compact operator (An operator is said to be compact if it is continuous and maps bounded sets into precompact.) and a map defined on the set and having range in . By assuming that the family is nonlinear equicontractive we prove the existence of a point such that

Our paper is organized as follows. In Section 2, we give some important definitions and preliminaries which will be used in this paper. Among this preliminaries we cite definition of Dunford-Pettis space, the theorems of Schauder-Tychonoff, Krein-Smulian. The Section 3 is devoted to the generalization of the Krasnoselskii fixed-point theorem in Dunford-Pettis Frechet spaces where our proofs of our two results (Proposition 3.3 and Theorem 3.5) in this section are based on the theorem of Sehgal and Singh and the main result (Proposition 3.1).

#### 2. Preliminaries

In this section, we give the following well-known definitions and results which will be used in this paper.

*Definition 2.1. *Suppose that and are locally convex spaces. A continuous linear operator from into is said to be weakly compact if is relatively weakly compact subset of whenever is a bounded subset of .

Theorem 2.2 (Eberlein-mulian, see [9]). *Let be a metrizable locally convex topological vector space, a weakly relatively compact sequence in . Then from may be extracted a weakly convergent subsequence.*

*Definition 2.3. *A subset in a vector space is called balanced if for all , if .

*Definition 2.4 (see [9, 10]). *A locally convex topological vector space is said to have the Dunford-Pettis (DP) property if any continuous linear map of into a complete locally convex topological vector space , which transforms bounded sets into weakly relatively compact sets, transforms each balanced and weakly compact subset of into a relatively compact subset of .

*Remark 2.5 (see [9]). *If is complete, we replace in the precedent definition each balanced and weakly compact subset of by each weakly compact subset of .

Theorem 2.6 (see [11]). *Let be a locally convex topological vector space and a convex subset of . Then is closed if and only if it is weakly closed.*

Theorem 2.7 (Krein-mulian). *Let be a metrizable and complete locally convex topological vector space and weakly compact. Then the closed convex hull of is weakly compact.*

Theorem 2.8 (Schauder-Tychonoff [12]). *Let be a closed and convex subset of a locally convex topological vector space and a continuous mapping such that the range is contained in a compact set. Then has a fixed-point.*

In the remainder of this section, denotes a Frechet topological vector space having the Dunford-Pettis (DP) property and is a neighborhood basis of the origin consisting of absolutely convex open subsets of . Let for each , the Minkowski's functional of .

Let be a nonempty subset of . A mapping is a -contraction if for each there is a such that if and if If is a -contraction for each , then is a -contraction.

Note that if is a -contraction, then is continuous. (For a related definition of -contraction, see Taylor [13].)

Lemma 2.9 (see [8]). *Let be a -contraction, then is -contractive, that is for each , if and 0, otherwise.*

Theorem 2.10 (Theorem of Sehgal and Singh [8]). *Let be a sequentially complete subset of a complete separated locally convex topological vector space and be a -contraction. If satisfies the condition:
**
Then has a unique fixed-point in .*

*Definition 2.11. *Let be a map such that be a nonempty subset of . The family is called -equicontractive if for each there is a such that if in the domain of and if

If is a -equicontractive for each , then the family is a -equicontraction. Note that if the family is a -equicontraction, then the operator is a -contraction for all .

*Definition 2.12. *let for some , the nonnegative reals and a family of mapping defined as such that is continuous and if . A mapping is a nonlinear contraction (see [14]) if for each , there is a such that for all . If this inequality holds with such that , then is called -contraction (see [5]).

Since a nonlinear contraction is a -contraction, the following result immediately follows by Theorem 2.10 and provides an extension of a result in [5]:

Theorem 2.13 (see [8]). *Let be a sequentially complete subset of a complete separated locally convex topological vector space and be a nonlinear contraction. If satisfies (2.2) then has a unique fixed-point in .*

*Definition 2.14. *The family is called nonlinear equicontractive if for each , there is a such that if in the domain of , then

*Remark 2.15. *Since any nonlinear contraction is a -contraction then any nonlinear equicontraction is a -equicontraction.

#### 3. Krasnoselskii's Type Theorems

In this section, we will give some new fixed-point results for the sum of two operators where is a Frechet topological vector space having the Dunford-Pettis property. Firstly, we give the following proposition which is a generalization of Theorem 2.1 in [6].

Proposition 3.1. *Let be a Frechet topological vector space having the Dunford-Pettis property, a closed, bounded and convex subset of and two operators such that:*(i)* a continuous map;*(ii)* a linear weakly compact operator on ;*(iii)* is relatively weakly compact;*(iv)*. **Then has a fixed-point in .*

*Proof. *We denote by , the closed convex hull of . Firstly, we show that is a weakly compact subset of . Indeed, we have . This implies that is relatively weakly compact and therefore is weakly compact. We have
and since is weakly compact, then by Krein-mulian's theorem is also weakly compact. Since is a closed convex subset of , therefore it is weakly closed and this implies that is a weakly closed subset of a weakly compact. Consequently, is weakly compact.

Now, we show that is relatively compact. We have is a weakly compact set in and is a weakly compact operator on and since is a Frechet topological vector space having the Dunford-Pettis property, then by Definition 2.4, we have is a relatively compact set in . Since is a continuous map, then is a relatively compact set in .

Moreover, we have
Therefore
and this implies that
where is a closed convex and is a relatively compact set. Since is a weakly compact oprator on , then by Definition 2.1 is continuous and so is continuous. Finally, the use of Schauder-Tychonoff's fixed-point theorem shows that has at least one fixed-point in .

Lemma 3.2. *Let be a Frechet topological vector space, a sequentially complete subset of and a nonlinear contraction. Suppose that for we have: for each with , there is a such that. Then, there exists a unique with , that is .*

*Proof. *Consequence of Theorem 2.13 (see [8]).

The following proposition is a generalization of Theorem 2.2 in [6].

Proposition 3.3. *Let be a Frechet topological vector space having the Dunford-Pettis property, a closed, bounded and convex subset of and two operators such that:*(i)* a linear weakly compact operator on ;*(ii)*be a -contraction;*(iii)*For each with , there is a such that ;*(iv)* is relatively weakly compact.**Then there exists in such that *

*Proof. *Firstly, we have is a -contraction then is a continuous function and for any we have
with which gives the continuity of .

Now, by Lemma 3.2 equation has a unique solution for all . It follows, that
so
For conclusion, we have is a continuous mapping, a linear weakly compact operator on and is relatively weakly compact on where . So, by Proposition 3.1, we prove that has a fixed-point in and this implies that, there exists such that .

We will now take a compact operator and a map defined on the set and having range in . We are interested to the existence of a point such that

Proposition 3.4. *Let be a Frechet topological vector space, a bounded sequentially complete subset of and
**
a map such that the family is nonlinear equicontractive, for all , is continuous and which satisfies the condition: for each with , there is a
**Then there exists a continuous map such that*

*Proof. *We start from an arbitrary point . Since the family is a nonlinear equicontractive then the operator
which satisfy for each with , there is a
Then by Theorem 2.13, there is a unique point that satisfies the operator equation:
We will show that the mapping is continuous. To do this we let be a sequence in , with . We suppose that does not converge to . Then there exist , an and such that
Since , is a bounded real subsequence, it has a subsequence . However, we have
which implies that . This contradicts (3.14) and consequently is continuous.

In what follows, we give also another result of Krasnoselskii type.

Theorem 3.5. *Let be a closed, bounded and convex subset of a Frechet topological vector space having the Dunford-Pettis property and a linear weakly compact operator such that the image of by any continuous mapping is contained in a weakly compact subset of . Let
**
be a map such that the family is nonlinear equicontractive, for all , is continuous on and which satisfies that for each with , there is a
**
Then (H) admits a solution in .*

*Proof. *We start from an arbitrary point . By Proposition 3.4 we prove that there exists a unique point that satisfies the operator equation
where the mapping is continuous. Then the operator maps the set into itself. We have by hypothesis that is contained in a weakly compact subset of . Therefore, by Proposition 3.1, we prove the existence of a point such that . This means that

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