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

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.


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 spaces L p , in fact the lack of stability under some differential operators, leads Orlicz and Birnbaum to consider the space where φ : ℝ + ℝ + is a convex increasing function such that lim x→∞ φðxÞ = ∞ (the convexity of φ was subsequently very often omitted). Later, in the end of the 1950s, Orlicz and Musielak considered the space where ðΩ,∑,μÞ is a measure space, X is the set of all realvalued (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 ϕðx, 0Þ = 0, ϕðx, uÞ > 0 if u > 0. The theory of modular function spaces was initiated by Kozlowski [3][4][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.
In the same vein, Kozlowski in [11] managed to prove the existence of solutions of the following differential equation of type: where uðtÞ 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 of T 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.

Preliminaries
We begin by recalling some elementary notions about graphs; see [12] for further properties. Definition 1. A directed graph or digraph G is determined by a nonempty set VðGÞ of its vertices and the set EðGÞ ⊂ VðG Þ × VðGÞ of its directed edges. A digraph is reflexive if each vertex has a loop. Given a digraph G = ðV, EÞ.
(i) If whenever ðx, yÞ ∈ EðGÞ ⇒ ðy, xÞ ∉ EðGÞ, then the digraph G is called an oriented graph (ii) A digraph G is transitive whenever ½ðx, yÞ ∈ EðGÞ and ðy, zÞ ∈ EðGÞ ⇒ ðx, zÞ ∈ EðGÞ, for any x, y, z ∈ VðGÞ (iii) A dipath of G is a sequence a 0 , a 1 , ⋯, a n , ⋯ with ð a i , a i + 1Þ ∈ EðGÞ for each i ∈ ℕ (iv) A finite dipath of length n from x to y is a sequence of n + 1 vertices ða 0 , a 1 , ⋯, a n Þ with ða i , a i+1 Þ ∈ EðGÞ and x = a 0 , y = a n (v) A closed directed path of length n > 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 if G ∼ is connected (vii) ½x G is the set of all vertices which are contained in some path beginning at x (i.e., y ∈ ½x G ⇔ there exist ða 0 , a 1 , ⋯, a n Þ with ða i , a i+1 Þ ∈ EðGÞ and x = a 0 , y = a n ) 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].
Let Ω 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 K n ↑Ω where K n ∈ P, for all n.
E is the linear space of P-simple functions, and M ∞ is the set of measurable functions. We denote by 1 A the characteristic function of A, where A ⊂ Ω.
Definition 2 [11]. An even convex function ρ : , g n ∈ E, and jg n j↓0 implies ρjg n j↓0 Let ρ be a regular convex function pseudomodular, the following notions are borrowed from [11].
We will identify pair of measurable sets whose symmetric difference is ρ-null, as well as pair of measurable function differing only on a ρ-null set We denote by R the set of all nonzero regular convex function modulars on Ω Definition 3 [11]. Let ρ ∈ R.
(a) We say that ð f n Þ n ∈ L ρ ρ converges to f and write f n Definition 4. [11]. Let ρ ∈ R.

Journal of Function Spaces
The modular function space is the vector space L ρ ðΩ, ∑Þ or briefly L ρ is defined The map k⋅k ρ : L ρ ⟶ ½0, +∞Þ defined by is called norm of Luxembourg on L ρ .
The following properties play a prominent role in the study of modular function spaces.
The following definitions and results could be found in [6].  (v) If ρ has the Δ 2 -property and ρðαf n Þ ⟶ 0 for α > 0, then k f n k ρ ⟶ 0 Theorem 8. Let ρ ∈ R, then E ρ is a k:k ρ -closed subspace of L ρ . Moreover, E ρ is the k:k ρ closure of E the set of all ðP Þ simple functions.
Definition 9. We say that a set C ⊂ L ρ possesses the Vitali property if C ⊂ E ρ , and for any g ⊂ L ρ and ðg n Þ n ⊂ C with g n ⟶ gðρÞ, there exists a subsequence ðg n k Þ k of ðg n Þ n such that for every α > 0 the subadditive measures ρ ðαg n k , :Þ are order equicontinuous. That is, if ðE p Þ p ⊂ Σ such that ðE p Þ p ↓ ∅ then ∀α > 0 lim ρ→∞ sup kεN ρð αg nk , E p Þ0.
The following statement characterizes sets with the Vitali property as subsets of E ρ where the ρ convergence is equivalent to the k•k ρ convergence.
Theorem 10. Let ρ ∈ R. A set C ∈ L ρ has the Vitali property if and only if the following conditions are satisfied: (i) C ⊂ E ρ (ii) If g ∈ L ρ and ðg n Þ n ⊂ C and ðg n Þ n ⟶ gðρÞ, then kg n − gk ρ ⟶ 0 Definition 11. A convex function modular ρ ∈ R is said separable if ∀ƒ ∈ E, ðkf 1 ð:Þ k ρ Þ 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 ðA k Þ k of elements of A with We recall this important result, which states that if ρ is separable, then ðL ρ , k:k ρ Þ is a separable Banach space; it is then a Polish space.

Theorem 12.
Let ρ ∈ R. The space ðL ρ , k:k ρ Þ is separable if and only if ρ is separable.
Remark 13. Let Z be a separable linear subspace of E ρ , k:k ρ and let C ⊂ Z have the Vitali property. Assume that the function u : ½a, b ⟶ C ða, b ∈ ℝÞ, is ρ-continuous. Then, u is Bochner integrable function with respect to the Lebesgue measure m on ½a, b. That is, if for τ ≔ t 0 < t 1 < ⋯ < t m a subdivision of ½a, b, we define jτj = sup and we write
(i) A function u : ½a, b ⟶ C is said to be continuous if uðt n Þ ⟶ uðtÞðρÞ provided t n ⟶ t. We denote by Cð½a:b, CÞ the set of all these continuous functions 3

Journal of Function Spaces (ii) A mapping T : C ⟶ C is said to be ρ-continuous if
Tð f n Þ ⟶ Tð f ÞðρÞ provided f n ⟶ f ðρÞ Proposition 15. Let ρ ∈ R be separable and C ⊂ E ρ 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 ∈ Cð½0, A, CÞ the mapping ϕðuÞ defined by takes values in C and is continuous, i.e., ϕðuÞ ∈ Cð½0, A, CÞ.
Proof. Let u ∈ Cð½0, A, CÞ. We will first prove that ϕðuÞðtÞ ∈ C for every t ∈ ð0, AÞ. Let t ∈ ð0, AÞ, for τ ≔ t 0 < t 1 < ⋯ < t m , a subdivision of ½0, t, we define By definition of Bochner integral We have f ∈ C, Tðuðt i ÞÞ ∈ C for every i ∈ f1, ⋯:,ng and Using convexity of C, we get Since C is ρ-closed, it is also closed with respect to k:k ρ . Thus, Observing that we conclude Furthermore, as C has the Vitali property, T is continuous with respect to the norm of Luxembourg, and then ϕðu Þ is continuous as t ↦ Ð t 0 e s−t TðuðsÞÞds is continuous; it is even differentiable.
The following notion of convex digraph was already introduced in [15].
Definition 16. Let C ∈ L ρ be a convex subset and G a digraph ðE ðgÞ = CÞ, we say that G is convex if ∀λ ∈ ½0, 1 and ∀f , g, u, v, ∈C such that f ∈ ½u G and g ∈ ½v G ; then Definition 17. Let ρ ∈ R and G a digraph EðGÞ ⊂ L ρ . We say that G has the ðP 0 Þ property, if for all ðf n Þ n , ðg n Þ n , f , g ∈ L ρ , g ∈ ½ f G provided g n ∈ ½ f n G for everyn ∈ ℕ and f n ⟶ f ð ρÞ, g n ⟶ gðρÞ.
The following lemma will play a preponderant role in the proof of the next theorem; its proof can be found in [11].
Lemma 19. Let ρ ∈ R be separable. Let x, y : ½0, A ⟶ L p two Bochner integrable k:k ρ bounded functions, where A > 0. Then, for every t ∈ ½0, A, we have Now, we are able to state the main result.
Theorem 20. Let ρ ∈ R be separable, C ⊂ E ρ a nonempty convex, ρ-bounded, ρ-closed set that has the Vitali property, and let G ðEðGÞ = CÞ be a reflexive, convex digraph, with property ðP 0 Þ. Let T : C ⟵ C be a ρ-continuous and G-monotone ρ -nonexpansive mapping and suppose there exists f ∈ C such that Tðf Þ ∈ ½ f G ; then, the mapping has a fixed point, where ϕðuÞðtÞ = e −t f + Ð t 0 e s−t TðuðsÞÞds, for every t ∈ ½0, A.

Journal of Function Spaces
Proof. Note that the mapping φ is well defined by Proposition 15. We define the sequence ðu n Þ n by It is easy to see that ðu n Þ n ⊂ Cð½0, A, CÞ. We will prove by induction over n ∈ ℕ that For n = 0, it comes for every t in ½0, A, Lemma 19 applied for y = 0 and xðtÞ = Tðu p−1 ðtÞ Þ − f for every t ∈ ½0, A gives From i.e., s ↦ e ðs−tÞ TðuðsÞÞ is Bochner integrable and Ð t 0 e s−t TðuðsÞ Þds = uðtÞ − e −t f . Finally, we get for every t ∈ ½0, Α 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 that ρ has the Δ 2 -property.
Theorem 21. Let ρ ∈ R be separable and has the Δ 2 -property, C ⊂ L ρ a nonempty convex, ρ-bounded, ρ-closed set, and let G ðEðGÞ = CÞ be a reflexive, convex digraph, with property ð P 0 Þ. Let T : C ⟶ C be a ρ-continuous and G-monotone ρ -nonexpansive mapping, and suppose that there exist f ∈ C such that Tð f Þ ∈ ½ f G ; then, the mapping has a fixed point where ϕðuÞðtÞ = e −t f + Ð t 0 e s−t TðuðsÞÞds,, for every t ∈ ½0, Α.
Proof. Since ρ has the Δ 2 -property, the ρ-convergence is equivalent to the convergence with respect to k:k ρ all over in L ρ ; 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. Let ρ ∈ R be separable, C ⊂ E ρ a nonempty convex, ρ-bounded, ρ-closed set that has the Vitali property, and let GðEðGÞ = CÞ be a reflexive, convex digraph, with property ðP 0 Þ. Let T : C ⟶ C be a ρ-continuous and G-monotone ρ -nonexpansive mapping, and suppose that there exists f ∈ C such that Tð f Þ ∈ ½ f G ; then, the differential equation where u : ½0, Α ⟶ C, Α > 0, has a solution.
Proof. The application ϕ defined above has a fixed point u ∈ Cð½0, Α, CÞ, that is, then u is differentiable and Þds + e −t e t T u t ð Þ ð Þ À Á that is, u is the solution of (O.D.E).
Note that the result of Theorem 22 remains true if ρ has the Δ 2 -property instead of C having the Vitali property.

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