# Common Fixed-Point Theorems in Modular Function Spaces Endowed with Reflexive Digraph

Authors: Jaauad Jeddi; Mustapha Kabil; Samih Lazaiz
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

The purpose of this work is to extend the Knaster–Tarski fixed-point theorem to the wider field of reflexive digraph. We give also a DeMarr-type common fixed-point theorem in this context. We then explore some interesting applications of the obtained results in modular function spaces.


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## Body

## 1. Introduction

The study of fixed point results in partially ordered sets finds its root in the work of Knaster [1] and Tarski [2]. In 1955, Tarski published his work in the context of complete lattices; the result states that each monotone mapping from a complete lattice to itself has a fixed point. In [3], Abian and Brown extended the result of Knaster–Tarski to chain-complete poset with a least or largest element and showed that every order-preserving map has a fixed point.On the contrary, several fixed-point theorems in metric spaces endowed with a partial order have been stated and studied. In 2004, Ran and Reuring (see [4]) combined successfully the Banach Contraction Principle and Knaster–Tarski fixed point. They managed to prove that every monotone mapping in a complete metric space has a fixed point provided that it satisfies contraction condition only for comparable elements. Jachymski [5] managed to prove an equivalent result of Ran and Reuring’s in a metric space endowed with a graph.In the same vein, the authors in [6, 7] extended the result of Ran and Reuring to the case of monotone nonexpansive mappings. Their starting point was to approach the fixed point by iterative techniques and successive approximations. Recently, Espínola and Wiśnicki [8] generalized the above results in Hausdorff topological spaces endowed with partial order. The key ingredient in such a generalization is the compactness of the order intervals mixed with Knaster–Tarski fixed point. For more details, see [9, 10].In this work, we generalize several known results in the context of topological spaces endowed with a digraph instead of partial order. For this purpose, we introduce the concept ofG-regular monotone mapping and we give some applications in modular function spaces of the obtained results.


## 2. Main Results

Since the main result of this work relates topological properties to graphs, the following definition is needed. The interested reader can consult [11], for more details.Definition 1.
A directed graph or digraphG is determined by a nonempty set VG of its vertices and the set EG⊂VG×VG of its directed edges. A digraph is reflexive if each vertex has a loop. Given a digraph G=V,E,(i)
If wheneverx,y∈EG⇒y,x∉EG, then the digraph G is called an oriented graph.(ii)
A digraph G is transitive wheneverx,y∈EG and y,z∈EG⇒x,z∈EG, for any x,y,z∈VG.(iii)
A dipath ofG is a sequence a0, a1,…an,…with ai,ai+1∈EG for each i∈ℕ.(iv)
A finite dipath of lengthn from x to y is a sequence of n+1 vertices a0,a1,…,an with ai,ai+1∈EG and x=a0 and y=an.(v)
A closed directed path of lengthn>1 from x to y, i.e., x=y, is called a directed cycle.(v)
A digraph is connected if there is a finite (di) path joining any two of its vertices, and it is weakly connected ifG˜ is connected.(vi)
xG is the set of all vertices which are contained in some path beginning at x, i.e., y∈xG⇔ there exist a0,a1,…,an with ai,ai+1∈EG and x=a0.
We extend the notions of upper and lower bound and supremum and infimum known in the case of ordered sets to graph structures.Definition 2.
LetG=VG,EG be a reflexive digraph and a,b∈VG.(i)
We define theG-intervals as follows:(1)a,⟶=x∈VG,x∈aG,⟵,a=x∈VG,a∈xG,and(2)a,b=a,⟶∩⟵,b.(ii)
For a subsetA of VG, we say that a∈VG is aG-upper bound of A if a∈x,⟶,∀x∈AG-lower bound of A if a∈⟵,x,∀x∈A(iii)
AG-upper bound of A that belongs to A is called G-maximal element of A, and a G-lower bound of A that belongs to A is called G-minimal element of A(iv)
We say thata∈VG is a G-supremum of A ifa is a G-upper boundFor everyG-upper bound b of A, a∈⟵,b(v)
We say thata∈VG is a G-infimum of A ifais aG-lower bound andFor everyG-lower boundbofA,a∈b,⟶
The following example illustrates this last definition.Example 1.
(1)
LetVG=ℝ and EG=x,y:x≤y. G is a reflexive digraph. Set A1=−∞,0.a∈A1 is a G-upper bound if for each x∈A1(3)a∈x,⟶⇔x≤a,
thus,A1 has no G-upper bound.
a
∈
A
1 is a G-lower bound if for each x∈A1
(4)
a
∈
⟵
,
x
⇔
a
≤
x
,
thus, 0 is the onlyG-lower bound of A1. Moreover, since 0∈A1 it is the only G-minimal element and it is the G-infimum.
SinceA1 has no G-upper bound the sets of G-maximal and G-supremum are empty.
(2)
Unlike the case of partially order, theG-supremum of set may be not unique. Indeed, we consider A2=0,1, thus we have
The set ofG-upper bound of A2 is −∞,−1∪1,∞
1 is theonlyG-maximal element of A2
a is G-supremum of A2 if and only if a∈−1,1
Recall that a collection of setsℰ has the finite intersection property (f.i.p.), if, for every family ℱ of members of ℰ, the intersection of ℱ is nonempty provided that the intersection of all finite subfamilies of ℱ are nonempty.Definition 3.
LetG=VG,EG be a reflexive digraph; a subset L of VG is said G-directed, if every finite subset of L has a G-upper bound in L.
We then get the following generalization of the result obtained in [8] for graphs.Lemma 1.
LetG=VG,EG be a reflexive digraph; if G has the finite intersection property for G-intervals, then every G-directed subset L of VG has a G-supremum.Proof 1.
We consider the setM=∩x∈Lx,⟶, as L is G-directed; every finite intersection ∩i=1nxi,⟶, where x1,x2,…xn∈L, is nonempty since it contains every G-upper bound that is in L of the finite subset x1,x2,…xn and as G has the finite intersection property, and M is nonempty.
Let us consider now the setM′=∩x∈L,y∈Mx,y, then, again, every finite intersection ∩i=1nxi,yi, where x1,x2,…xn∈L and y1,y2,…yn∈M is nonempty (also it contains any G-upper bound that is in L of the finite subset x1,x2,…xn) and G has the finite intersection property M′ which is nonempty. And, it is clear that every element of M′ is a G-supremum of L.
Recall that a mapT:X⟶X is said to be G-monotone if for all x,y∈X, whenever y∈xG, then Ty∈TxG. Next, we introduce the notion of G-regular monotone.Definition 4.
LetX be a set endowed with a graph G=VG,EG a map T:X⟶X is said to be G-regular monotone; if T is G-monotone and for every x,y∈X, if x∈yG and y∈xG, then Tx=Ty.
The following theorem is the cornerstone of what follows.Theorem 1.
LetX be a topological space endowed with a reflexive digraph G=VG,EG such that G-intervals are compact, and let T:X⟶X be G-regular monotone map; if there exists x0∈X such that Tx0∈x0G, then the set of fixed points of T is not empty and has a G-maximal element.Proof.
LetL=Tnx0:n∈ℕ. Then, L is G-directed and for all x∈L, Tx∈L and Tx∈xG. Set now(5)ℱ=JG−directedL⊂J,and∀x∈J,Tx∈JandTx∈xG,then ℱ is nonempty inductive set with respect to the ⊂ order (if Jii∈I is a chain in ℱ then ∪i∈IJi is an upper bound of Jii∈I in ℱ).
By Zorn’s lemma, there exists a maximalG-directed set L0 such that L⊂L0 and for all x∈L0, Tx∈L0, and Tx∈xG. Since the G-intervals are compacts, G has the finite intersection property for G-intervals; thus, L0 has a G-supremum s.
Now, for allx∈L0, we have s∈xG and Ts∈TxG and Tx∈xG for all x∈L0. Hence, Ts is a G-upper bound of L0, that is, Ts∈sG. We then claim that s∈L0. Indeed if s∉L0, then(6)L˜=L0∪Tns,n∈ℕ,is a G-directed subset of X that contains strictly L0 (and thus J as well) and such that, for all x∈L˜, Tx∈L˜ and Tx∈xG, which is in contradiction with maximality of L0.
Ass∈L0 and Ts∈L0 too, we have s∈TsG and as Ts∈sG; we get then TTs=Ts, that is, Ts is a fixed point of T.
Notice that ifz is a fixed point of T, z∈L0, and thus s∈zG and Ts∈Tz as T is G-monotone.Example 2.
LetX the unit disc in ℂ, i.e., X=z∈ℂ,z=1, endowed with the graph G=VG,EG, where VG=X and x,y∈EGiffx≤y.
ConsiderT:X⟶X defined Tx=x for all x∈X with the usual topology.
It is easy then to check thatG-intervals are compact, and that T is G-regular monotone mapping and that 1 is a fixed point for T (every real positive number in X is indeed a fixed point for T).
In the same way, we obtain a common fixed point for commuting family ofG-monotone mappings.Theorem 2.
LetX be a topological space endowed with a reflexive digraph G=VG,EG such that G-intervals are compact, and let Tλ:X⟶X,λ∈Λ be a family of commuting G-regular monotone mappings; if there exists x0∈X such that Tλx0∈x0G for all λ∈Λ then the set of common fixed points of the family Tλλ∈Λ is nonempty and has a G-maximal element.Proof.
LetL=Tλ1⋅Tλ2⋅…⋅Tλnx0:n∈ℕandλ1…λn∈Λ; then, L is G-directed and we have Tλx∈LandTλx∈xG for all x∈L and λ∈Λ. Let(7)ℱ=JG−directedL⊂J,and∀x∈J,∀λ∈Λ,Tλx∈JandTλx∈xG.
Then,ℱ is a nonempty inductive set with respect to the inclusion order. Indeed, if Jii∈I is a chain in ℱ, then ∪i∈IJi is an upper bound of Jii∈I in ℱ. By Zorn’s lemma, there exists a maximal G-directed set L0 such that L⊂L0, and for all x∈L0 and λ∈Λ, we get Tλx∈L0 and Tλx∈xG. As G has the finite intersection property for G-intervals, L0 has a G-supremum s.
For allx∈L0, we have s∈xG and Tλs∈TλxG and Tλx∈xG for all x∈L0 and for all λ∈Λ. Hence, Tλs is a G-upper bound of L0, for any λ∈Λ. That is, Tλs∈sG,∀λ∈Λ. We then claim that s∈L0, indeed if s∉L0, then(8)L˜=L0∪Tλ1∘Tλ2∘…∘Tλns:n∈ℕandλ1…λn∈Λ,is a G-directed subset of X that contains strictly L0 (and thus J as well) and such that ∀x∈L˜, ∀λ∈Λ,Tλx∈L˜,andTλx∈xG, which is in contradiction with maximality of L0.
Now again, ass∈L0, then, for each λ∈Λ, Tλs∈L0 and s∈TλsG; hence, TμTλs=Tμs,∀λ,μ∈Λ. By commutativity, Tμs is a common fixed point for the family of mappings Tλλ∈Λ.
Finally, ifz is a common fixed point of the family of mappings Tλλ∈Λ then z∈L thus s∈zG and Tμs∈TμzG, for all μ∈Λ. Then, Tμs is G-maximal element of the set of common fixed points of Tλλ∈Λ.


## 3. Application to Modular Function Spaces

For the sake of completeness, we begin by recalling some definitions and properties of modular function spaces that we used later. For more details, see [12].LetΩ be a nonempty set and P a nontrivial δ-ring of subsets of Ω, and let Σ be the smallest σ-algebra of subsets of Ω such that Σ contains P such that E∩A∈P for every E∈P and A∈Σ; Kn↑Ω, where Kn∈P, for all n.ℰ is the linear space of P-simple functions; ℳ∞ is the set of measurable functions. We denote by 1A the characteristic function of A, where A⊂Ω.Definition 5.
An even convex functionρ:ℳ∞⟶0,+∞ is called regular convex function pseudomodular if(i)
ρ0=0.(ii)
ρ is monotone, i.e., if for f,g∈ℳ∞, fω≤gω for all ω∈Ω, then ρf≤ρg.(iii)
ρ is orthogonally subadditive, i.e., ρf.1A∪B=ρf.1A+ρf.1B, whenever A,B∈Σ and A∩B=∅ and f∈Σ.(iv)
ρ has the Fatou property, i.e., if fnωn↑fω, for all ω∈Ω, fn,f∈ℳ∞ then ρfn↑ρf.(v)
ρ is order continuous in ℰ, i.e.,gn∈ℰ and gn↓0 implies ρgn↓0.
Letρ be a regular convex function pseudomodular; we then introduce these notions:(i)
A setA∈Σ is said ρ-null, if ρg.1A=0,∀g∈ℰ.(ii)
ApropertyP is said to hold ρ almost everywhere if the exceptional set is ρ-null.(iii)
We will identify pair of measurable sets whose symmetric difference isρ-null, as well as pair of measurable function differing only on a ρ-null set.(iv)
ℳΩ,Σ,P,ρ=f∈ℳ∞:fω<∞ρ−a.e. briefly noted ℳ.(v)
ρ is said a regular convex function modular if ρf=0 implies f=0ρ-a.e.(vi)
We denote byℜ the set of all nonzero regular convex function modulars on Ω.Definition 6.
Let
ρ
∈
ℜ:(i)
We say thatfnn∈Lρρ-converges to f, and write fn⟶fρ, if ρfn−f⟶0, and a sequence fnn∈Lρ is called ρ-Cauchy if ρfn−fm⟶0asn,m⟶∞.(ii)
A setB⊂Lρ is called ρ-closed, if for any sequence fnn∈B,fn⟶fρ implies f∈B.(iii)
A setB⊂Lρ is called ρ-bounded, if his diameter δρB=supρf−g/f,g∈B is finite.(iv)
A setB⊂Lρ is called ρ-compact, if for any sequence fnn∈Lρ there exists a subsequence fknn and f∈B such that fknρ-converges to f.(v)
A setB⊂Lρ is called ρ-a.e.-compact, if for any sequence fnn∈Lρ there exists a subsequence fknn and f∈B such that fknρ-a.e.-converges to f.(vi)
A setB⊂Lρ is called ρ-a.e.-closed, if for any sequence fnn∈B,fn⟶f, ρ-a.e. implies f∈B.Definition 7.
Letρ∈ℜ.
The modular function space is the vector spaceLρΩ,Σ or briefly Lρ defined by(9)Lρ=f∈ℳ;limλ⟶0ρλ.f=0.
Themap.ρ:Lρ⟶0,+∞ is defined by(10)fρ=infα>0;ρfα≤1,which is called norm of Luxembourg on Lρ.
The following properties play a prominent role in the study of modular function spaces.Definition 8.
Letρ∈ℜ:(i)
We say thatρ has the Δ2-property, if ρ2fn⟶0 whenever ρfn⟶0, fnn∈Lρ.(ii)
We say thatρ has the Δ2-type condition, if there exists k∈0,+∞ such that ρ2f≤kρf, for any f∈Lρ.
We need the following definition of the growth function.Definition 9.
(see [12]). Let ρ be a function modular; the function ωρ:0,+∞⟶0,+∞ is defined by(11)ωρt=supρtfρf:f∈Lρand0<ρf<∞,which is called the growth of ρ.
The growth function has the following properties.Proposition 1.
(see [12]). Let ρ∈ℜ that has the Δ2-type condition, and ωρ its growth function; then,(i)
ωρt<∞,∀t∈0,+∞.(ii)
ωρ:0,+∞⟶0,+∞ is convex, and strictly increasing, it is then also continuous.(iii)
ωραβ≤ωραωρβ, for all α,β∈0,+∞.(iv)
ωρ−1αωρ−1β≤ωρ−1αβ, for all α,β∈0,+∞, where ωρ−1 is the inverse function of ωρ.(v)
fρ≤1/ωρ−11/ρf, for any f∈Lρ.
Proofs of following theorems could be found in [12].Theorem 3.
Letρ∈ℜ:(i)
Lρ,.ρ is a complete normed space, and Lρ is ρ-complete.(ii)
fnρ⟶0 iff ραfn⟶0 for every α>0.(iii)
Ifρfn−f⟶0 there exists fnkk subsequence of fnn such that fnk⟶f, ρ-a.e.(iv)
Iffn⟶fρ-a.e, then ρf≤liminfn⟶+∞ρfn (the Fatou property).(v)
Ifρ has the Δ2-property and ραfn⟶0 for α>0, then fnρ⟶0.
A modularρ is said σ-finite if there exists an increasing sequence of sets Kn∈P such that, for every n∈ℕ,0<ρKn<∞ and Ω=∪n∈ℕKn.
Letd:Lρ×Lρ⟶0,+∞ defined by(12)df,g=∑k=1+∞12k1ρ1Kkρf−g1+f−g1Kk.Theorem 4.
Letρ∈ℜ if ρ is σ-finite and has the Δ2- type condition; then, for any f,g∈Lρ,(1)
df,g=0 if f f=gρ-a.e.(2)
df,g=dg,f(3)
df,g≤ω2/2df,h+dh,gand if fnn is a sequence in Lρ that is ρ-a.e. convergent to f, then limn⟶+∞dfn,f=0.
Moreover, iflimn⟶+∞dfn,f=0, then there exists a subsequence fnkk that converges ρ-a.e. to f.
Here is the first application of our main result to modular function spaces.Theorem 5.
Letρ∈ℜ, that has the Δ2-property and G a digraph on Lρ such that G-intervals are ρ-compact. Let Tλ:Lρ⟶Lρ,λ∈Λ be a family of commuting G-regular monotone mappings; if there exists f0∈Lρ such that Tλf0∈f0G,∀λ∈Λ, then the set of common fixed points of the family Tλλ∈Λ is not empty and has a G-maximal element.Proof.
Asρ has the Δ2-property, then the ρ-convergence is equivalent to convergence in the Banach space Lρ,.ρ, which means that every ρ-compact subset of Lρ is a compact in Lρ,.ρ; we can then apply Theorem 2.
Requiring more conditions on the function modularρ one can suppose that G-intervals are only ρ-a.e. compact.Theorem 6.
Letρ∈ℜ, a σ-finite function modular, that has the Δ2-type condition and G a digraph on Lρ such that G-intervals are ρ-a.e.-compact. Let Tλ:Lρ⟶Lρ,λ∈Λ be a family of commuting G-regular monotone mappings; if there exists f0∈Lρ such that Tλf0∈f0G,∀λ∈Λ, then the set of common fixed points of the family Tλλ∈Λ is not empty and has a G-maximal element.Proof.
Indeed, Theorem4 states that Lρ,d is a b-metric space, and then sequential compactness is equivalent to compactness (the usual argument, which proves that fact for metric spaces, still holds in b-metric spaces). Now, if a subset K of Lρ is ρ.a.e.-compact, then from every sequence of elements of K one can extract a subsequence that converges ρ-a.e.; then, by Theorem 4, one can extract a subsequence that converges in the b-metric space Lρ,d. That is, K is sequentially compact in Lρ,d and thus compact, which implies that G-intervals are compact for the topology of the b-metric space Lρ,d, then using Theorem 2 we get the result.



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