Research Article A Study of a Nonlinear Ordinary Differential Equation in Modular Function Spaces Endowed with a Graph Jaauad Jeddi , 1 Mustapha Kabil, 1 and Samih Lazaiz 2 1 Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies Mohammedia, University Hassan II Casablanca, Morocco 2 LASMA Laboratory, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Fes, Morocco Correspondence should be addressed to Jaauad Jeddi; jaauadjeddi@gmail.com Received 4 November 2020; Revised 18 December 2020; Accepted 8 January 2021; Published 30 January 2021 Academic Editor: Huseyin Isik Copyright © 2021 Jaauad Jeddi 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. 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. 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 spaces L p , in fact the lack of stability under some differential operators, leads Orlicz and Birnbaum to consider the space Lφ = f : ℝ ⟶ ℝ : ð ℝ φλ fx ðÞ j j ð Þdx ⟶ 0 as λ ⟶ 0   , ð1Þ 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 con- sidered the space Lϕ = f ∈ X : ð Ω ϕ x, λ fx ðÞ j j ð Þdμ ⟶ 0 as λ ⟶ 0   , ð2Þ where ðΩ,∑,μÞ is a measure space, X is the set of all real- valued (or complex-valued) Σ-measurable, μ-almost every- where finite functions on Ω, and ϕ : Ω × ℝ + ⟶ ℝ + is a Car- atheodory function which means that it is Σ-measurable for first variable, nondecreasing continuous mapping on the sec- ond variable and such that ϕðx,0Þ = 0, ϕðx, uÞ >0 if u >0. The theory of modular function spaces was initiated by Kozlowski [3–5], and those spaces were a sort of spaces situ- ated 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 flexi- bility 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: O:D:E ð Þ u 0 ðÞ = f , u′ t ðÞ + I − T ð Þut ðÞ = 0, ∀t ∈ 0, A ½ , ( ð3Þ where uðt Þ has values in modular function spaces and T sat- isfies nonexpansiveness assumption. Hindawi Journal of Function Spaces Volume 2021, Article ID 6654057, 7 pages https://doi.org/10.1155/2021/6654057