# A Study of a Nonlinear Ordinary Differential Equation in Modular Function Spaces Endowed with a Graph

Authors: Jaauad Jeddi; Mustapha Kabil; Samih Lazaiz
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we prove by means of a fixed-point theorem an existence result of the Cauchy problem associated to an ordinary differential equation in modular function spaces endowed with a reflexive convex digraph.


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## Body

## 1. Introduction

It is well known that fixed-point theory is a powerful tool that was frequently exploited to prove existence of solutions of differential equations not only in Banach spaces but also in a wider range of spaces, particularly in Orlicz and Musielak-Orlicz spaces [1, 2] and more generally in modular function spaces.The Orlicz spaces were introduced in the early 1930s when the lack of flexibility of classical Lebesgue function spacesLp, in fact the lack of stability under some differential operators, leads Orlicz and Birnbaum to consider the space
(1)Lφ=f:ℝ⟶ℝ:∫ℝφλfxdx⟶0asλ⟶0,where φ:ℝ+ℝ+ is a convex increasing function such that limx→∞φx=∞ (the convexity of φ was subsequently very often omitted).Later, in the end of the 1950s, Orlicz and Musielak considered the space(2)Lϕ=f∈X:∫Ωϕx,λfxdμ⟶0asλ⟶0,where Ω,∑,μ is a measure space, X is the set of all real-valued (or complex-valued) Σ-measurable, μ-almost everywhere finite functions on Ω, and ϕ:Ω×ℝ+⟶ℝ+ is a Caratheodory function which means that it is Σ-measurable for first variable, nondecreasing continuous mapping on the second variable and such that ϕx,0=0,ϕx,u>0 if u>0.The theory of modular function spaces was initiated by Kozlowski [3–5], and those spaces were a sort of spaces situated in between the Musielak-Orlicz and modular ones that were both more concrete of ordinary modular spaces, as treating about functions sets, and offering much more flexibility than the Musielak-Orlicz spaces.Furthermore, in [6, 7] jointly with Khamsi, Kozlowski has initiated fixed-point results in modular function spaces. Recently, a new direction has been developed, combining fixed-point results and graph theory; see, for instance, [8–10].In the same vein, Kozlowski in [11] managed to prove the existence of solutions of the following differential equation of type:
(3)O.D.Eu0=f,u′t+I−Tut=0,∀t∈0,A,where ut has values in modular function spaces and T satisfies nonexpansiveness assumption.In this work, we intend to solve the equation (O.D.E) in modular function spaces endowed with a digraph, where nonexpansiveness ofT is restricted to connected points, which is a far more general result than the one obtained by Kozlowski. We will first establish a fixed-point result that will be employed to prove the existence of solutions of (O.D.E) with less restraint conditions over T.


## 2. Preliminaries

We begin by recalling some elementary notions about graphs; see [12] for further properties.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 digraphG is transitive whenever x,y∈EGandy,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,y=an(v)
A closed directed path of lengthn>1 from x to y, i.e., x=y, is called a directed cycle(vi)
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(vii)
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,y=an)We also need to introduce some properties of modular function spaces and tools that will be often used later. For more details, one can consult [3–5, 13, 14].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∈∑; and Kn↑Ω where Kn∈P, for all n.E is the linear space of P-simple functions, and M∞ is the set of measurable functions. We denote by 1A the characteristic function of A, where A⊂Ω.Definition 2 [11].
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∈∑ and A∩B=ϕ and f∈∑(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, the following notions are borrowed from [11].
(i)
A setA∈∑ is said to be ρ-null if ρg.1A=0, ∀g∈E(ii)
A propertyP 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,∑,P,ρ=f∈M∞:fω<∞ρ−a.e. briefly noted M(v)
ρ is said to be 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 ΩDefinition 3 [11].
Letρ∈R.
(a)
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⟶0 as n,m⟶∞(b)
A setB⊂Lρ is called ρ-closed, if for any sequence fnn∈B,fn⟶fρ implies f∈B(c)
A setB⊂Lρ is called ρ-bounded, if its diameter δρB=supρf−g:f,g∈B is finite(d)
A setB⊂Lρ is called ρ-compact, if for any sequence fnn∈B there exists a subsequence fknn and f∈B such that fknρ-converges to f(e)
A setB⊂Lρ is called ρ-a.e.-closed, if for any sequence fnn∈B,fn⟶f, ρ-a.e. implies f∈B(f)
A setB⊂Lρ is called ρ-a.e.compact, if for any sequence fnn∈B there exists a subsequence fnkk and f∈B such that fnk⟶f, ρ-a.eDefinition 4. [11].
Letρ∈R.
The modular function space is the vector spaceLρΩ,∑ or briefly Lρis defined
(4)Lρ=f∈M:limλ→0ρλ.f=0.
The map⋅ρ:Lρ⟶0,+∞ defined by
(5)fρ=infα>0:ρfα≤1is called norm of Luxembourg on Lρ.The following properties play a prominent role in the study of modular function spaces.Definition 5 [11].
Letρ∈R.
We say thatρ has the Δ2-property, if ρ2fn⟶0 whenever ρfn⟶0, fnn∈Lρ. We say that ρ has the Δ2-type condition, if there exists k∈0,+∞ such that ρ2f≤kρf, for any f∈Lρ.The following definitions and results could be found in [6].Theorem 6.
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⟶0Definition 7.
Letρ∈R, we define
(6)Lρo=f∈Lρ:ρf,.isordercontinuous,Eρ=f∈Lρ:λf∈Lρ0,∀λ>0.Theorem 8.
Letρ∈R, then Eρ is a .ρ-closed subspace of Lρ. Moreover, Eρ is the .ρ closure of E the set of all P simple functions.Definition 9.
We say that a setC⊂Lρ possesses the Vitali property if C⊂Eρ, and for any g⊂Lρ and gnn⊂C with gn⟶gρ, there exists a subsequence gnkk of gnn such that for every α>0 the subadditive measures ραgnk,. are order equicontinuous. That is, if Epp⊂Σ such that Epp↓∅ then ∀α>0limρ→∞supkεNραgnk,Ep0.The following statement characterizes sets with the Vitali property as subsets ofEρ where the ρ convergence is equivalent to the ∙ρ convergence.Theorem 10.
Letρ∈R. A set C∈Lρ has the Vitali property if and only if the following conditions are satisfied:
(i)
C⊂Eρ(ii)
Ifg∈Lρ and gnn⊂C and gnn⟶gρ, then gn−gρ⟶0Definition 11.
A convex function modularρ∈R is said separable if ∀ƒ∈E, f1.ρ is a separable set function for each ƒ∈E, which means that there exists a countable A⊂P such that to every A∈P there corresponds a sequence Akk of elements of A with
(7)∀α>0,ραf,AΔAk⟶0k→∞.We recall this important result, which states that ifρ is separable, then Lρ,.ρ is a separable Banach space; it is then a Polish space.Theorem 12.
Letρ∈R. The space Lρ,.ρ is separable if and only if ρ is separable.Remark 13.
LetZ be a separable linear subspace of Eρ,.ρ and let C⊂Z have the Vitali property. Assume that the function u:a,b⟶Ca,b∈ℝ, is ρ-continuous. Then, u is Bochner integrable function with respect to the Lebesgue measure m on a,b. That is, if for τ≔t0<t1<⋯<tm a subdivision of a,b, we define τ=sup0≤i≤m−1ti+1−ti (called the step of τ), then
(8)limτ→0∑i=0m−1ti+1−tiutiexists,and we write
(9)∫abusds=limτ→0∑i=0m−1ti+1−tiuti.


## 3. Main Results

Definition 14.
LetC⊂Lρ, a,b,∈ℝ and C⊂Lρ.
(i)
A functionu:a,b⟶C is said to be continuous if utn⟶utρ provided tn⟶t. We denote by Ca.b,C the set of all these continuous functions(ii)
A mappingT:C⟶C is said to be ρ-continuous if Tfn⟶Tfρ provided fn⟶fρProposition 15.
Letρ∈R be separable and C⊂Eρ be a nonempty convex and ρ-closed set that has the Vitali property. Let T:C⟶C be a ρ-continuous mapping and f∈C. Then, for every u∈C0,A,C the mapping ϕu defined by
(10)ϕut=e−tf+∫0tes−tTusdsforeveryt∈0,Atakes values in C and is continuous, i.e., ϕu∈C0,A,C.Proof.
Letu∈C0,A,C. We will first prove that ϕut∈C for every t∈0,A. Let t∈0,A, for τ≔t0<t1<⋯<tm, a subdivision of 0,t, we define
(11)SτTut=∑i=0m−1ti+1−tietiT−tut.
By definition of Bochner integral(12)∫0tes−tTusds=limτ→0SτTut.
We havef∈C,Tuti∈C for every i∈1,⋯.,n and
(13)e−t+∑i=0m−1ti+1−tieti−t≤e−t+∫0tes−tds=1.
Using convexity ofC, we get
(14)e−tf+∑i=0m−1ti+1−tieti−tTuti+1−e−t+∑i=0m−1ti+1−tieti−tf∈C.
SinceC is ρ-closed, it is also closed with respect to .ρ. Thus,
(15)limτ→0e−tf+∑i=0m−1ti+1−tieti−tTuti+1−e−t+∑i=0m−1ti+1−tieti−tf∈C.
Observing that(16)limτ→01−e−t+∑i=0m−1ti+1−tieti−t=1−e−t−∫0tes−tds=0,we conclude
(17)ϕut=e−tf+∫0tes−tTusds∈C.
Furthermore, asC has the Vitali property, T is continuous with respect to the norm of Luxembourg, and then ϕu is continuous as t↦∫0tes−tTusds is continuous; it is even differentiable.The following notion of convex digraph was already introduced in [15].Definition 16.
LetC∈Lρ be a convex subset and G a digraph Eg=C, we say that G is convex if ∀λ∈0,1 and ∀f,g,u,v,∈C such that f∈uG and g∈vG; then
(18)λf+1−λg∈λu+1−λvG.Definition 17.
Letρ∈R and G a digraph EG⊂Lρ. We say that G has the P0 property, if for all fnn,gnn,f,g∈Lρ,g∈fG provided gn∈fnG for everyn∈ℕ and fn⟶fρ,gn⟶gρ.Definition 18.
LetC⊂Lρ,G a digraph EG=C and T:C⟶C. We say that T is G-monotone ρ-nonexpansive if for all f,g∈C such that g∈fG we have
(19)Tg∈TfG,ρTf−Tg≤ρf−g.The following lemma will play a preponderant role in the proof of the next theorem; its proof can be found in [11].Lemma 19.
Letρ∈R be separable. Let x,y:0,A⟶Lp two Bochner integrable .ρ¯ bounded functions, where A>0. Then, for every t∈0,A, we have
(20)ρe−tyt+∫0tes−txsds≤e−tρyt+1−e−tsups∈0,tρxs.Now, we are able to state the main result.Theorem 20.
Letρ∈R be separable, C⊂Eρ a nonempty convex, ρ-bounded, ρ-closed set that has the Vitali property, and let GEG=C be a reflexive, convex digraph, with property P0. Let T:C⟵C be a ρ-continuous and G-monotone ρ-nonexpansive mapping and suppose there exists f∈C such that Tf∈fG; then, the mapping
(21)ϕ:C0,A,C⟶C0,A,Cu↦ϕuhas a fixed point, where ϕut=e−tf+∫0tes−tTusds, for every t∈0,A.Proof.
Note that the mappingφ is well defined by Proposition 15. We define the sequence unn by
(22)u0t=f,∀t∈0,A,un+1=ϕun,∀n∈ℕ.
It is easy to see thatunn⊂C0,A,C. We will prove by induction over n∈ℕ that
(23)∀t∈0,A,∀n,p∈ℕ,ρun+pt−unt≤1−eAn+1δρC.
Forn=0, it comes for every t in 0,A,
(24)upt−u0t=∫0tes−tTup−1sds−1−e−tf=∫0tes−tTup−1s−fds.
Lemma19 applied for y=0 and xt=Tup−1t−f for every t∈0,A gives
(25)∀t∈0,A,ρupt−u0≤1−e−tsups∈0,tρTup−1s−f≤1−e−AδρC.
We suppose now that for allt∈0,A and for all p∈ℕ(26)ρun+pt−unt≤1−e−An+1δρC.
Let us first prove thatun+1t∈untG,∀n∈ℕ and ∀t∈0,A. By induction on n, for n=0, we have ∀t∈0,A,
(27)u1t=e−tf+1−e−tTf∈u0tG=fG,as G is convex and Tf∈fG. If we suppose that for every t∈0,A,un+1t∈untG, then for τ≔t0<t1⋯<tm a subdivision of 0,t. Set for k≥1(28)ukτ=e−tf+∑i=0m−1ti+1−tieti−tTuk−1ti,then
(29)un+2τ=e−tf+∑i=0m−1ti+1−tieti−tTun+1ti,as G is convex and Tun+1ti∈TuntiG for every i=1,⋯,m we have
(30)un+2τ+1−e−t+∑i=0m=1ti+1−tieti−tf∈un+1τ+1−e−t∑i=0m−1ti+1−tieti−tfG,since G has the P0 property it follows that
(31)limτ→0un+2τ+1−e−t+∑i=0m−1ti+1−tieti−tf∈limτ→0un+1τ+1−e−t+∑i=0m−1ti+1−tieti−tfG,which is exactly un+2t∈un+1tG. And then un+1t∈untG and un+pt∈untG for every n,p∈ℕ and t∈0,A.
Now, asρun+1+pt−un+1t=ρ∫0tes−tTun+ps−Tunsds, applying again Lemma 19 for yt=0,xt=Tun+ps−Tuns, we get
(32)ρun+1+pt−un+1t≤1−e−tsups∈0,tρTun+ps−Tuns,∀t∈0,A,but ρTun+ps−Tuns≤ρun+ps−uns,∀s∈0,t (as un+ps∈unsG and T is G-monotone ρ-nonexpansive); consequently,
(33)ρun+1+pt−un+1t≤1−etsups∈0,tρun+ps−uns,∀t∈0,A,by the inductive assumption, we get
(34)ρun+1+pt−un+1t≤1−e−t1−e−Αn+1δpC,∀t∈0,Α,i.e.,
(35)ρun+1+pt−un+1t≤1−e−Αn+2δρC,∀t∈0,Α.
Using inequality (23), it is clear that for every t∈0,Α,untn is a ρ-Cauchy sequence in C. Since C is ρ-closed in Lρ, then it is ρ-complete and then untnρ− converges to some ut∈C, and thus, it converges to ut with respect to .ρ. We also have ∀t∈0,Α, and ∀n∈ℕ,ut∈untG as G has P0 property. Indeed, for n∈ℕ,un+pt∈untG,∀p∈ℕ and when p⟶∞,ut∈untG.
Now, fort∈0,Α, let τ≔t0<t1<⋯<tm be a subdivision of 0,t. We have
(36)ρSτTunt−SτTut=ρ∑i=0m−1ti+1−tieti−tTunti−Tuti,and by convexity of ρ(37)ρSτTunt−SτTut≤∑i=0m−1ti+1−tieti−tρTunti−Tuti.
Sinceuti∈untiG for every i=0,1,⋯,m and T is G-monotone ρ-nonexpansive
(38)ρSτTunt−SτTut≤∑i=0m−1ti+1−tieti−tρunti−uti→n→∞0,that is, limn→∞ρSτTunt−SτTut=0.
Hence,(39)limn→∞SτTunt−SτTutρ=0.
Now, for everyt∈0,Α and n∈ℕ(40)SτTut−ut+e−fρ≤SτTut−SτTuntρ+SτTunt−∫0tes−tTunsdsρ+∫0tes−tTunsds−ut+e−tfρ.
From∫0tes−tTunsds−ut+e−tfρ=un+1t−utρ, we get
(41)limτ→0SτTut−ut+e−tfρ=0,i.e., s↦es−tTus is Bochner integrable and ∫0tes−tTusds=ut−e−tf. Finally, we get for every t∈0,Α(42)ϕut=e−tf+∫0tes−tTusds=ut,that is, u is a fixed point of ϕ.A similar result can be obtained without assuming the Vitali property, but we need to assume thatρ has the Δ2-property.Theorem 21.
Letρ∈R be separable and has the Δ2-property, C⊂Lρ a nonempty convex, ρ-bounded, ρ-closed set, and let GEG=C be a reflexive, convex digraph, with property P0. Let T:C⟶C be a ρ-continuous and G-monotone ρ-nonexpansive mapping, and suppose that there exist f∈C such that Tf∈fG; then, the mapping
(43)ϕ:C0,Α,C⟶C0,Α,Cu↦ϕuhas a fixed point where ϕut=e−tf+∫0tes−tTusds,, for every t∈0,Α.Proof.
Sinceρ has the Δ2-property, the ρ-convergence is equivalent to the convergence with respect to .ρ all over in Lρ; the proof of this corollary runs along similar lines to the proof of Theorem 20.The last result is devoted to prove the existence of solution of the equation (O.D.E).Theorem 22.
Letρ∈R be separable, C⊂Eρ a nonempty convex, ρ-bounded, ρ-closed set that has the Vitali property, and let GEG=C be a reflexive, convex digraph, with property P0. Let T:C⟶C be a ρ-continuous and G-monotone ρ-nonexpansive mapping, and suppose that there exists f∈C such that Tf∈fG; then, the differential equation
(44)O.D.Eu0=f,u′t+I−Tut=0,∀t∈0,Α,where u:0,Α⟶C,Α>0, has a solution.Proof.
The applicationϕ defined above has a fixed point u∈C0,Α,C, that is,
(45)ut=e−tf+∫0tes−tTusds,∀∈0,Α,then u is differentiable and
(46)u′t=−e−tf−∫0tes−tTusds+e−tetTut=Tut−ut,∀t∈0,Α,that is, u is the solution of (O.D.E).Note that the result of Theorem22 remains true if ρ has the Δ2-property instead of C having the Vitali property.



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