Fixed Point Theory on a Frechet Topological Vector Space

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Volume 2011, Article ID 390720,9pages doi:10.1155/2011/390720

Research Article

Fixed-Point Theory on a Frechet Topological

Vector Space

Afif Ben Amar,

1

_{Mohamed Amine Cherif,}

2

_{and Maher Mnif}

2

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

Correspondence should be addressed to Maher Mnif,[email protected]

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

Academic Editor: Genaro Lopez

Copyrightq2011 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.

We establish some versions of fixed-point theorem in a Frechet topological vector spaceE. The

main result is that every mapA BCwhereB is a continuous map andC is a continuous

linear weakly compact operatorfrom 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{T·, y : y∈CMwhereM⊂EandC : M → Ea

compact operator}is nonlinearϕequicontractive, we give a fixed-point theorem for the operator

of the formEx:Tx, Cx.

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-fixed-points of mappings on

the cartesian productE×EintoE, whereEis some appropriate space.

Fixed-point theorems in topology and nonlinear functional analysis are usually based

on certain propertiessuch as complete continuity, monotonicity, contractiveness, etc.that

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itself has unique fixed-point, and the Schauder principle, which asserts that a continuous

mappingAon a closed convex setMin Hausdorﬀlocally convex topological vector spaceE

intoMsuch thatAMis contained in a compact set, has a fixed-point. In many problems

of analysis, one encounters operators which may be split in the formT AB, whereA

is a contraction in some sense, andBis completely continuous, and T itself has neither of

these propertiessee1–3. Thus neither the Schauder fixed-point theorem nor the Banach

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’ski˘ı

4: “Let E be a Banach space, M be a bounded closed convex subset of E, and A, B be

operators onM into E such that AxBy ∈ M for every pair x, y ∈ M. IfA is a strict

contraction andBis continuous and compact, then the equationAxBxxhas a solution

inM.” 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.6in 2005

and which extended the Schauder and Krasnoselskii fixed-point theorems in Dunford-Pettis

spaces to weakly compact operators. Also in 2010, Amar and Mnif7established 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

Ebe a Frechet topological vector space having the property of Dunford-Pettis,Ma closed

bounded convex subset ofE, andA BCwhereBis a continuous map andCis a linear

weakly compact operator. If A leaves M invariant then A has a fixed-point in M see

Proposition 3.1. In addition, if B is aϕ-contraction map ofM into E, for each x, y ∈ M

withBxAy /∈M, there is az∈x, BxAy∩Msuch thatBzAy∈MandI−B−1AM

is relatively weakly compact, thenABhas a fixed-point inMseeProposition 3.3.

Based on our results and other theorems which was given by Sehgal and Singh in

19768, we give also an extension of the Krasnoselskii fixed-point theorem: Let Ebe a

Frechet topological vector space having the property of Dunford-Pettis DP, M ⊆ E, C :

M → CM ⊆ Ea compact operatorAn operator C : M → E is said to be compact if

it is continuous and maps bounded sets into precompact.andT a map defined on the set

M×CMand having range in E. By assuming that the family {T·, y : y ∈ CM} is

nonlinearϕequicontractive we prove the existence of a pointx∈Msuch that

xTx, Cx. 1.1

Our paper is organized as follows. InSection 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 3is 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 3.5in this section are based on the theorem of Sehgal and Singh and the main

resultProposition 3.1.

2. Preliminaries

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Definition 2.1. Suppose thatEandFare locally convex spaces. A continuous linear operator

AfromEintoF is said to be weakly compact ifABis relatively weakly compact subset of

FwheneverBis a bounded subset ofE.

Theorem 2.2Eberlein-Smulian, see 9. LetEbe a metrizable locally convex topological vector

space,x_n_na weakly relatively compact sequence inE. Then fromx_n_nmay be extracted a weakly

convergent subsequence.

Definition 2.3. A subsetCin a vector spaceEis called balanced if for allx ∈ C,λx ∈ Cif

|λ| ≤1.

Definition 2.4see9,10. A locally convex topological vector spaceE is said to have the

Dunford-Pettis DP property if any continuous linear map of E into a complete locally

convex topological vector space F, which transforms bounded sets into weakly relatively

compact sets, transforms each balanced and weakly compact subset of E into a relatively

compact subset ofF.

Remark 2.5see9. IfEis complete, we replace in the precedent definition each balanced

and weakly compact subset ofEby each weakly compact subset ofE.

Theorem 2.6see11. LetEbe a locally convex topological vector space andMa convex subset of

E. ThenMis closed if and only if it is weakly closed.

Theorem 2.7Krein-Smulian. LetEbe a metrizable and complete locally convex topological vector

space andM⊂Eweakly compact. Then the closed convex hull ofMis weakly compact.

Theorem 2.8Schauder-Tychonoﬀ 12. LetMbe a closed and convex subset of a locally convex

topological vector spaceEandA : M → Ma continuous mapping such that the rangeAMis

contained in a compact set. ThenAhas a fixed-point.

In the remainder of this section,Edenotes a Frechet topological vector space having

the Dunford-PettisDPproperty andϑ is a neighborhood basis of the origin consisting of

absolutely convex open subsets ofE. Let for eachU∈ϑ,p_Uthe Minkowski’s functional ofϑ.

LetMbe a nonempty subset ofE. A mappingA:M → Eis aU-contractionU∈ϑ

if for each >0 there is aδ >0 such that ifx, y∈Mand if

x−y∈δU, thenAx−Ay∈U. 2.1

IfA:M → Eis aU-contraction for eachU∈ϑ, thenAis aϑ-contraction.

Note that if Ais a ϑ-contraction, thenA is continuous. For a related definition of

ϑ-contraction, see Taylor13.

Lemma 2.9see8. LetA:M → Ebe aϑ-contraction, thenAisϑ-contractive, that is for each

U∈ϑ,pUAx−Ay< pUx−yifpUx−y/0 and 0, otherwise.

Theorem 2.10Theorem of Sehgal and Singh8. LetMbe a sequentially complete subset of a

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satisfies the condition:

for eachx∈M withAx∈/M,

there is a z∈x, Ax∩M such thatAz∈M. 2.2

ThenAhas a unique fixed-point inM.

Definition 2.11. LetT : M×E → Ebe a map such thatMbe a nonempty subset ofE. The

family{T·, y:y ∈E}is calledU-equicontractiveU∈ϑif for each >0 there is aδ >0

such that ifx1, y,x2, yin the domain ofT and if

x1−x2∈δU, then Tx1, y−Tx2, y∈U. 2.3

If{T·, y : y} is aU-equicontractive for each U ∈ ϑ, then the family{T·, y : y}

is aϑ-equicontraction. Note that if the family{T·, y : y}is a ϑ-equicontraction, then the

operatorx → Tx, yis aϑ-contraction for ally.

Definition 2.12. letϕ {p pUfor someU ∈ ϑ},R the nonnegative reals andψ a family

of mapping defined asψ {Φ :R → Rsuch thatΦis continuous andΦt< tift >0}.

A mappingA : M → Eis a nonlinearϕcontractionsee14if for eachp ∈ ϕ, there is a

Φp∈ψsuch thatpAx−Ay≤Φppx−yfor allx, y∈M. If this inequality holds with

Φpt αptsuch that 0< αp<1, thenAis calledϕ-contractionsee5.

Since a nonlinear ϕcontraction is a ϑ-contraction, the following result immediately

follows byTheorem 2.10and provides an extension of a result in5:

Theorem 2.13see8. LetMbe a sequentially complete subset of a complete separated locally

convex topological vector spaceFandA:M → Fbe a nonlinearϕcontraction. If Asatisfies2.2

thenAhas a unique fixed-point inM.

Definition 2.14. The family{T·, y :y ∈ E}is called nonlinearϕequicontractive if for each

p∈ϕ, there is aΦp∈ψsuch that ifx1, y,x2, yin the domain ofT, then

pTx1, y−Tx2, y≤Φppx1−x2. 2.4

Remark 2.15. Since any nonlinear ϕ contraction is a ϑ-contraction then any nonlinear ϕ

equicontraction is aϑ-equicontraction.

3. Krasnoselskii’s Type Theorems

In this section, we will give some new fixed-point results for the sum of two operators where

Eis a Frechet topological vector space having the Dunford-Pettis property. Firstly, we give

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Proposition 3.1. LetEbe a Frechet topological vector space having the Dunford-Pettis property,M

a closed, bounded and convex subset ofEandB, Ctwo operators such that:

iB:E→Ea continuous map;

iiC:E→Ea linear weakly compact operator onE;

iiiBCMis relatively weakly compact;

ivAM⊂M.

ThenABChas a fixed-point inM.

Proof. We denote byNcoAM, the closed convex hull ofAM. Firstly, we show thatN

is a weakly compact subset ofE. Indeed, we haveAM⊂BCM. This implies thatAM

is relatively weakly compact and thereforeAMis weakly compact. We have

AM⊂AM ⇒NcoAM⊂coAM 3.1

and since AM is weakly compact, then by Krein-Smulian’s theorem coAM is also

weakly compact. SinceNis a closed convex subset ofE, therefore it is weakly closed and this

implies thatNis a weakly closed subset of a weakly compact. Consequently,N is weakly

compact.

Now, we show thatCNis relatively compact. We haveNis a weakly compact set

inE andCis a weakly compact operator on Eand sinceEis a Frechet topological vector

space having the Dunford-Pettis property, then byDefinition 2.4, we haveCNis a relatively

compact set inE. SinceBis a continuous map, thenBCNis a relatively compact set inE.

Moreover, we have

AM⊂M so coAM⊂coM. 3.2

Therefore

NcoAM⊂M 3.3

and this implies that

AN⊂AM⊂coAM N, 3.4

where N is a closed convex and AN BCN is a relatively compact set. SinceC is a

weakly compact oprator onE, then byDefinition 2.1Cis continuous and soA:N → Nis

continuous. Finally, the use of Schauder-Tychonoﬀ’s fixed-point theorem shows thatAhas at

least one fixed-point inN⊂M.

Lemma 3.2. LetEbe a Frechet topological vector space,Ma sequentially complete subset ofEand B:

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there is a z∈x, Bxy∩Msuch thatBzy∈M. Then, there exists a uniqueuy∈Mwith

Buy yuy, that isI−B−1_y_uy_∈_M.

Proof. Consequence ofTheorem 2.13see8.

The following proposition is a generalization of Theorem 2.2 in6.

Proposition 3.3. LetEbe a Frechet topological vector space having the Dunford-Pettis property,M

a closed, bounded and convex subset ofEandA, Btwo operators such that:

iA:E→Ea linear weakly compact operator onE;

iiB:M→Ebe aϕ-contraction;

iiiFor each x, y ∈ Mwith BxAy /∈ M, there is a z ∈ x, BxAy∩Msuch that

BzAy∈M;

iv I−B−1AMis relatively weakly compact.

Then there existsyinMsuch thatAyByy

Proof. Firstly, we haveBis aϕ-contraction thenBis a continuous function and for any x, y∈

Mwe have

pUI−Bx−I−By≥pUx−y−pUBx−By≥1−αppUx−y 3.5

withα_p∈0,1which gives the continuity ofI−B−1.

Now, byLemma 3.2equationzBzAyhas a unique solutionz∈Mfor ally∈M.

It follows, that

z I−B−1Ay∈M, 3.6

so

I−B−1AM⊂M. 3.7

For conclusion, we have I − B−1 is a continuous mapping, A a linear weakly compact

operator onEandI−B−1AMis relatively weakly compact onEwhereI −B−1AM⊂

M. So, byProposition 3.1, we prove thatI − B−1Ahas a fixed-point inMand this implies

that, there existsy∈Msuch thatAyByy.

We will now takeC:M → CM⊆Ea compact operator andTa map defined on the

setM×CMand having range inE. We are interested to the existence of a pointx∈M⊂E

such that

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Proposition 3.4. Let Ebe a Frechet topological vector space, M a bounded sequentially complete

subset ofEand

T :M×E−→E 3.8

a map such that the family{T·, y : y ∈ E}is nonlinearϕequicontractive, for all x ∈ M,y →

Tx, yis continuous and which satisfies the condition: for eachx, y∈M×EwithTx, y∈/M,

there is a

z∈x, Tx, y∩M such thatTz, y∈M. 3.9

Then there exists a continuous mapF_T:E → Msuch that

TFTy, yFTy. 3.10

Proof. We start from an arbitrary pointy∈E. Since the family{T·, y:y∈E}is a nonlinear

ϕequicontractive then the operator

x−→Tx, y:M−→Eis a nonlinearϕcontraction 3.11

which satisfy for eachx∈MwithTx, y/∈M, there is a

z∈x, Tx, y∩M such thatTz, y∈M. 3.12

Then byTheorem 2.13, there is a unique point x FTy ∈ M that satisfies the operator

equation:

TFTy, yFTy. 3.13

We will show that the mappingy→F_Ty:E → Mis continuous. To do this we lety_nbe

a sequence inE, with limyn y0 ∈E. We suppose thatFTyndoes not converge toFTy0.

Then there existp∈ϕ, an >0 and nsuch that

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Since{pFTy n, FTy0> ,n∈N}is a bounded real subsequence, it has a subsequence

{pFTy 1n, FTy0, n∈N} → r ≥0. However, we have

pFTy 1n

−FTy0pTFTy 1n

, y 1n

−TFTy0, y0

≤pTFTy 1n

, y 1n

−TFTy0, y n

pTFTy0, y 1n

−TFTy0, y0

≤ΦppFTy 1n

−FTy0

pTFTy0, y 1n

−TFTy0, y0

3.15

which implies thatr0. This contradicts3.14and consequentlyFTis continuous.

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

Theorem 3.5. LetMbe a closed, bounded and convex subset of a Frechet topological vector space

having the Dunford-Pettis propertyEandC :M → Ea linear weakly compact operator such that

the image ofCMby any continuous mapping is contained in a weakly compact subset ofE. Let

T :M×CM−→E 3.16

be a map such that the family{T·, y: y ∈CM}is nonlinearϕequicontractive, for allx ∈M,

y → Tx, yis continuous onCMand which satisfies that for each x, y ∈ M×CMwith

Tx, y∈/M, there is a

z∈x, Tx, y∩M such thatTz, y∈M. 3.17

ThenHadmits a solution inM.

Proof. We start from an arbitrary pointy ∈ CM. ByProposition 3.4 we prove that there

exists a unique pointxF_Ty∈Mthat satisfies the operator equation

TFTy, yFTy, 3.18

where the mappingy → F_Ty :CM → Mis continuous. Then the operatorF_TCmaps

the setMinto itself. We have by hypothesis thatFTCMis contained in a weakly compact

subset ofE. Therefore, byProposition 3.1, we prove the existence of a pointx∈Msuch that

FTCx x. This means that

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