On common fixed points in modular vector spaces

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R E S E A R C H

Open Access

On common ﬁxed points in modular

vector spaces

Afrah AN Abdou

1

_{and Mohamed A Khamsi}

2,3*

*_{Correspondence:}

[email protected] 2_{Department of Mathematical}

Sciences, University of Texas at El Paso, El Paso, Texas, USA 3_{Department of Mathematics and}

Statistics, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this work, we discuss the concept of Banach operator pairs in modular vector spaces. We prove the existence of common ﬁxed points for these type of operators which satisfy a modular continuity in modular compact sets. On the basis of our result, we are able to give an analog of DeMarr’s common ﬁxed point theorem for a family of symmetric Banach operator pairs in modular vector spaces.

MSC: Primary 47H09; secondary 46B20; 47H10; 47E10

Keywords: Banach operator pair; electrorheological ﬂuids; ﬁxed point; KKM-mapping; modular function space; modular metric space; retraction

1 Introduction

In recent years, there was an surge of interest in the study of electrorheological fluids, sometimes referred to as ‘smart fluids’ (for instance lithium polymetachrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces,Lp

andW,p_{, where}_p_{is a ﬁxed constant is not adequate, rather the exponent}_p_{should be}

able to vary [–]. One of the most interesting problem discussed in these spaces is the Dirichlet energy problem [, ]. One way to discuss this problem is to convert the energy functional, deﬁned by a modular, to a problem which involves a convoluted and compli-cated norm (the Luxemburg norm). The modular approach is natural and has not been used.

The purpose of this paper is to discuss the existence of common fixed points of map-pings defined on subsets of modular vector spaces, as introduced by Nakano [] which are natural generalizations of many classical function spaces. The common fixed point problem of a pair of commuting mappings was investigated as early as the first fixed point results were proved [, ]. This problem becomes more challenging in view of the his-torically significant and negatively settled question that a pair of commuting continuous self-mappings defined on [, ] may not have a common fixed point []. Over the years, many mathematicians have tried to find weaker forms of commutativity that imply the ex-istence of a common fixed point for a pair of self-mappings. Weakly compatible mappings [] and Banach operator pairs [–] were introduced and provided generalizing results in metric fixed point theory for single-valued mappings. In this work, we discuss some of these results for multi-valued in modular vector spaces.

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A good reference for metric ﬁxed point theory is the book []. For modular spaces, the interested reader may consult the books [–].

2 Preliminaries

Modular vector spaces have been studied in [, ].

Deﬁnition .[] LetXbe a real vector space. A functionalρ:X→[, +∞] is called a modular if

() ρ(x) = if and only ifx= ; () ρ(x) =ρ(–x);

() ρ(αx+ ( –α)y)≤ρ(x) +ρ(y), for anyx,y∈Xandα∈[, ]. If we replace () by

() ρ(αx+ ( –α)y)≤αρ(x) + ( –α)ρ(y), for anyx,y∈Xandα∈[, ], thenρis called a convex modular.

The concept of modular ﬁnds its roots in the work of Nakano [] who expanded the earlier ideas of Orlicz and Birnbaum who tried to generalize the classical function spaces of the Lebesgue typeLp_{. Orlicz and Birnbaum’s ideas consisted in considering spaces of}

functions with some growth properties diﬀerent from the power type growth. In other words, they considered the space:

Lϕ₌

f :R→R;∃λ>  such that

Rϕ

λf(x)dx<∞

,

where ϕ: [,∞]→[,∞] was assumed to be a convex function increasing to inﬁnity,

i.e., the function which to some extent behaves similar to power functionsϕ(t) =tp_{. The}

functionalρ:Lϕ_→_{[, +}_∞_{] deﬁned by}

ρ(f) =

Rϕ

_f₍_x₎dx,

is a modular onLϕ_.

The classical book [] contains many examples of modular function spaces. The inter-ested reader may also consult the most recent book [].

Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. The set

Xρ=

x∈X; lim

λ→+ρ(λx) = 

is a vector subspace ofXknown as the associated modular vector space. On it we deﬁne the Luxemburg norm

xρ=inf

λ;ρ

x λ

≤

,

for anyx∈Xρ.

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Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. () The sequence{xn}n∈NinXρis said to beρ-convergent tox∈Xρif and only if

ρ(xn–x)→, asn→ ∞.xwill be called theρ-limit of{xn}.

() The sequence{xn}n∈N inXρis said to beρ-Cauchy ifρ(xm–xn)→, asm,n→ ∞.

() A subsetMofXρis said to beρ-closed if theρ-limit of aρ-convergent sequence of

Malways belong toM.

() A subsetMofXρis said to beρ-complete if anyρ-Cauchy sequence inMis a

ρ-convergent sequence and itsρ-limit is inM. () A subsetMofXρis said to beρ-bounded if we have

δρ(M) =sup

ρ(x–y);x,y∈M<∞.

() A subsetMofXρis said to be sequentiallyρ-compact if for any{xn}inMthere

exists a subsequence{xnk}andx∈Msuch thatρ(xnk–x)→.

() A nonempty subsetKofXρis said to beρ-compact if for any family{Aα;α∈}of

ρ-closed subsets ofXρwithK∩Aα∩ · · · ∩Aαn=∅, for anyα, . . . ,αn∈, we have

K∩

α∈

Aα

=∅.

() ρis said to satisfy the Fatou property if and only if for any sequences{xn}and{yn}in Xρwhichρ-converge, respectively, toxandy, we have

ρ(x–y)≤lim inf

n→∞ ρ(xn–yn).

In general iflimn→∞ρ(λ(xn–x)) = , for someλ> , then we may not havelimn→∞ρ(λ

(xn–x)) = , for allλ> . Therefore, as it is done in modular function spaces, we will say

thatρsatisﬁes-condition if this is the case,i.e.limn→∞ρ(λ(xn–x)) = , for someλ> 

implieslim_n_→∞ρ(λ(xn–x)) = , for allλ> . In particular, we have

lim

n→∞xn–xρ=  if and only if nlim→∞ρ

λ(xn–x)

= , for allλ> ,

for any{xn} ∈Xρandx∈Xρ. In other words, we see thatρ-convergence and · ρ

con-vergence are equivalent if and only if the modularρsatisﬁes the-condition.

In the next section, we will prove Brouwer’s fixed point theorem in modular vector spaces and obtain a common fixed point result as was done in []. The following defi-nition is therefore needed.

Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. LetC⊂Xρ

be nonempty andρ-closed. LetT:C→Xρbe a map. We will say thatT isρ-continuous

if{T(xn)}ρ-converges toT(x) if{xn}ρ-converges tox. Moreover, we will sayTis strongly ρ-continuous ifT isρ-continuous and

lim inf

n→∞ ρ

y–T(xn)

=ρy–T(x),

for any{xn} ⊂Csuch that{xn}ρ-converges toxand for anyy∈C. A pointx∈Cis called

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We now introduce the concept of Banach operator pairs [, ] in modular vector spaces.

Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. LetC⊂Xρ

be nonempty. LetS,T:C→Cbe two self-mappings. The ordered pair (T,S) is said to be a Banach operator pair whenever the setFix(T) isS-invariant,i.e.,

SFix(T)⊆Fix(T).

Note that ifSandTcommute,i.e.,S◦T=T◦S, then the ordered pairs (S,T) and (T,S) are Banach operator pairs. Therefore the concept of Banach operator pairs is seen as a weakening of the commutativity condition.

3 Common ﬁxed points for Banach operators pairs

We prove the Brouwer ﬁxed point theorem [] in modular vector spaces via the theorem of Knaster-Kuratowski-Mazurkiewicz (KKM) [].

LetXbe a vector space andρa convex modular deﬁned onX. LetC⊂Xρbe nonempty.

The set of all subsets ofCis denoted C_{. The notation}_conv₍_A_{) describes the convex hull}

ofA, whileconvρ(A) describes the smallestρ-closed convex subset ofXρwhich containsA.

Recall that a family{Aα⊂X;α∈}is said to have the finite intersection property if the intersection of each finite subfamily is not empty. Throughout this section, we will assume thatρis finite,i.e.,ρ(x) < +∞, for anyx∈X, andρsatisfies the Fatou property.

Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. LetC⊂ Xρ be nonempty. A multi-valued mappingG:C→Xρ is called a

Knaster-Kuratowski-Mazurkiewicz mapping (in short KKM-mapping) if

conv{x, . . . ,xn}

⊂

i∈[,n]

G(xi),

for anyf, . . . ,fn∈C.

A subset A ⊂Xρ is called ﬁnitely ρ-closed if for every x,x, . . . ,xn ∈Xρ, the set

convρ({x, . . . ,xn})∩Aisρ-closed.

We are now ready to prove the following result.

Theorem . Let X be a vector space andρa convex modular deﬁned on X.Let C⊂Xρbe

nonempty,and G:C→Xρ _{be a KKM-mapping such that for any x}∈_C_,_G₍_x₎_{is nonempty} and ﬁnitelyρ-closed.Then the family{G(x);x∈C}has the ﬁnite intersection property.

Proof Assume not, i.e. there exist x, . . . ,xn ∈ C such that

≤i≤nG(xi) =∅. Set L=

convρ({xi}) inXρ. Our assumptions imply thatL∩G(xi) isρ-closed for everyi= , , . . . ,n.

Sinceρ-closedness implies closedness for the Luxemburg norm · ρ,L∩G(xi) is closed

for · ρ, for anyi∈ {, . . . ,n}. Thus for everyx∈L, there existsisuch thatxdoes not belong toL∩G(xi) sinceL∩(

≤i≤nG(xi)) =∅. Hence dx,L∩G(xi)

=infx–yρ;y∈L∩G(xi)

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becauseL∩G(xi) is closed. We use the function α(x) =

≤i≤n

dx,L∩G(xi)

> ,

wherex∈K=conv{x, . . . ,xn}, to deﬁne the mapT:K→Kby

T(x) = 

α(x)

≤i≤n

dx,L∩G(xi)

xi.

It is obvious thatTis a continuous map. Using the compactness of the convex subsetKof (Xρ, · ρ), we can use Brouwer’s theorem to ensure the existence of a ﬁxed pointx∈K ofT,i.e. T(x) =x. Set

I=i;dx,L∩G(xi)

= .

Clearly, we have

x= 

α(x)

i∈I

dx,L∩G(xi)

xi.

Hencex∈/

i∈IG(xi) andx∈conv({xi;i∈I}) contradict the assumption

conv{xi;i∈I}

⊂

i∈I

G(xi).

As an immediate consequence of Theorem ., we obtain the following result.

Theorem . Let X be a vector space andρ a convex modular deﬁned on X. Let C⊂ Xρ be nonempty,and G:C→Xρ be a KKM-mapping such that for any x∈C,G(x)is

nonempty andρ-closed.Assume there exists f∈C such that G(f)isρ-compact.Then we

have_x_∈_CG(x)=∅.

The following lemma is needed in proving the analog of Ky Fan ﬁxed point result [] in modular vector spaces.

Lemma . Let X be a vector space andρa convex modular deﬁned on X.Let K⊂Xρbe a

nonempty convex,ρ-compact subset.Let T:K→Xρbe stronglyρ-continuous.Then there

exists x∈K such that

ρx–T(x)

=inf

x∈Kρ

x–T(x)

.

Proof LetG:K→K_⊂_Xρ_{be deﬁned by} G(y) =x∈K;ρx–T(x)≤ρy–T(x).

SinceTis stronglyρ-continuous andρis assumed to satisfy the Fatou property, we have

ρx–T(x)≤lim inf

n→∞ ρ

xn–T(xn)

≤lim inf

n→∞ ρ

y–T(xn)

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for any sequence{xn} inG(y) whichρ-converges tox. HenceG(y) isρ-closed for any y∈K. Next, assume thatGis not a KKM-mapping. Then there exists{y, . . . ,yn} ⊂Kand x∈conv({yi}) such thatx∈/

≤i≤nG(yi). This clearly implies

ρyi–T(x)

<ρx–T(x), fori= , . . . ,n.

Letε>  be such thatρ(yi–T(x))≤ρ(x–T(x)) –ε, fori= , , . . . ,n. Using the convexity

ofρ, we get

ρy–T(x)≤ρx–T(x)–ε,

for any y∈conv({yi}). Sincex∈conv({yi}), we getρ(x,T(x))≤ρ(x,T(x)) –ε. This is a

contradiction which implies thatGis a KKM-mapping. SinceKisρ-compact, we deduce thatG(y) is alsoρ-compact for anyy∈K. Using Theorem ., there existsxin

y∈KG(y).

Henceρ(x–T(x))≤ρ(y–T(x)), for anyy∈K, which implies

ρx–T(x)

=inf

y∈Kρ

y–T(x)

.

Before we state the Ky Fan fixed point theorem [] in modular vector spaces, we recall the definition of modular balls. Letx∈Xρandr≥. The modular ballBρ(x,r) is defined

by

Bρ(x,r) =

y∈X;ρ(x–y)≤r.

Sinceρis convex and satisﬁes the Fatou property, the modular balls are convex andρ -closed.

Theorem . Let X be a vector space andρa convex modular deﬁned on X.Let K⊂Xρ

be a nonempty convex,ρ-compact subset.Let T:K→Xρbe stronglyρ-continuous such

that for any x∈K,with x=T(x),there existsα∈(, )such that K∩Bρ

x,αρx–T(x)∩Bρ

T(x), ( –α)ρx–T(x)=∅. (.)

Then T has a ﬁxed point.

Proof By Lemma ., there existsx∈Ksuch that

ρx–T(x)

=inf

x∈Kρ

x–T(x)

.

Assume thatxis not a ﬁxed point ofT,i.e.,x=T(x). Then there existsα∈(, ) such that

K=K∩Bρ

x,αρ

x–T(x)

∩Bρ

T(x), ( –α)ρ

x–T(x)

=∅.

Let y∈K. Then ρ(y–T(x))≤( –α)ρ(x–T(x)). This implies a contradiction to the property satisﬁed byx. Therefore we must haveT(x) =x,i.e.,xis a ﬁxed point

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Note that ifT(K)⊂K, thenT satisﬁes the condition (.). The following theorem is an analog to Brouwer’s ﬁxed point theorem in modular vector spaces.

Theorem . Let X be a vector space andρa convex modular deﬁned on X.Let K⊂Xρ

be a nonempty convex,ρ-compact subset.Let T:K→K be stronglyρ-continuous.Then Fix(T)is a nonemptyρ-compact subset.

Proof The existence of a ﬁxed point ofTis a direct consequence of Theorem .. SoFix(T) is nonempty. Next we show thatFix(T) isρ-compact. Let us prove thatFix(T) isρ-closed. Let{xn}be a sequence inFix(T) such that{xn}ρ-converges tox. SinceTisρ-continuous,

{T(xn)}ρ-converges toT(x). SinceT(xn) =xn, we see that{xn}ρ-converges toxandT(x).

Using the uniqueness of theρ-limit, we getT(x) =x,i.e.,x∈Fix(T). SinceKisρ-compact andFix(T) isρ-closed, we conclude thatFix(T) isρ-compact.

In order to prove our main result of this section, we need the following deﬁnition.

Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. LetC⊂Xρ

be a nonempty subset. A mapping T:C→Cis said to be anR-map if Fix(T) is aρ -continuous retraction ofC,i.e., there exists aρ-continuous mappingR:C→Fix(T) such thatR◦R=R. Such a mapping is known as a retraction.

Next we discuss our ﬁrst common ﬁxed point result of this work.

Theorem . Let X be a vector space andρa convex modular deﬁned on X.Let K⊂Xρ

be a nonempty convex,ρ-compact subset.Let S,T:K→K be two self-mappings.Assume that S and T are stronglyρ-continuous such that(S,T)is a Banach operator pair.If we assume that T is an R-map,thenFix(T)∩Fix(S)is nonempty andρ-compact.

Proof Theorem . implies thatFix(T) is not empty andρ-compact. Since T is anR -map, there exists a retraction R:K →Fix(T) which is ρ-continuous. Since (S,T) is a Banach pair of operators, S(Fix(T))⊂Fix(T). It is clear thatS◦R:K→K is strongly

ρ-continuous. To see this, let{xn} ⊂Kbeρ-convergent tox. Then{R(xn)}ρ-converges to R(x) sinceRisρ-continuous. Using the strongρ-continuity ofS, we have

lim inf

n→∞ ρ

y–SR(xn)

=ρy–SR(x),

for anyy∈K,i.e.,S◦Ris stronglyρ-continuous. Theorem . implies thatFix(S◦R) is nonempty andρ-compact. Moreover, ifx∈Fix(S◦R), thenS◦R(x) =S(R(x)) =x∈Fix(T) becauseS◦R(K)⊂Fix(T). In particular, we haveR(x) =x. HenceS(x) =x,i.e. x∈Fix(T)∩

Fix(S). It is easy to see thatFix(T)∩Fix(S) =Fix(S◦R). HenceFix(T)∩Fix(S) is nonempty

andρ-compact.

Next we extend the conclusion of Theorem . to a family of Banach operator mappings. The following concept will be needed.

Deﬁnition . LetXbe a vector space andρa convex modular deﬁned onX. LetC⊂Xρ

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Banach operator pair if (S,T) and (T,S) are Banach operator pairs,i.e.,

TFix(S)⊆Fix(S) and SFix(T)⊆Fix(T).

Recall that the set of common ﬁxed points of a family of mappingsFis the setFix(F) =

T∈FFix(T). The main result of this work states:

Theorem . Let X be a vector space andρa convex modular deﬁned on X.Let K⊂Xρ

be nonempty convex andρ-compact.LetFbe a family of self-mappings deﬁned on K such that any map inFis a stronglyρ-continuous R-map.Assume that any two mappings inF form a symmetric Banach operator pair.ThenFix(F)is a nonemptyρ-compact subset.

Proof Theorem . implies thatFix(T)∩ · · · ∩Fix(Tn) is a nonemptyρ-compact subset

ofK, for anyT,T, . . . ,TninF. Therefore any ﬁnite family of the subsets{Fix(T);T∈F}

has a nonempty intersection. Since these sets are allρ-closed andK isρ-compact, we conclude that Fix(F) =_T_∈_FFix(T) is not empty and isρ-closed. ThereforeFix(F) is

ρ-compact.

As commuting mappings are symmetric Banach operator pairs, we obtain an analog to DeMarr’s common ﬁxed point theorem [] in modular spaces as follows.

Corollary . Let X be a vector space andρa convex modular deﬁned on X.Let K⊂Xρ

be a nonempty convex,ρ-compact subset.LetFbe a family of commuting self-mappings deﬁned on K such that any map inF is a stronglyρ-continuous R-map.ThenFix(F)is nonempty andρ-compact.

Remark . Since convexity plays a crucial role in the proof of Brouwer’s ﬁxed point the-orem, an analog of this profound theorem does not exist in metric spaces. In order to obtain an extension of this fundamental ﬁxed point result, one needs to use some kind of convexity therein. This is the case for example of hyperconvex metric spaces []. So to obtain an extension of all the results obtained in this work for modular metric spaces [–], we will have to introduce something like hyperbolicity in modular metric spaces. This concept will be new and has not been introduced or investigated so far.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the manuscript.

Author details

1_{Department of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah, 21589, Saudi Arabia.}2_{Department of}

Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, USA.3_{Department of Mathematics and Statistics,} King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia.

Acknowledgements

This work was supported by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, Jeddah, under grant No. 372-501-D1435. The authors, therefore, gratefully acknowledge the DSR technical and ﬁnancial support.

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