University of Tabriz Computational Methods for Differential Equations 2345-3982 12 3 2024 05 01 Upper and lower solutions for fractional integro-differential equation of higher-order and with nonlinear boundary conditions 438 453 17165 10.22034/cmde.2023.56512.2363 EN Abdelati El Allaoui MISCOM, National School of Applied Sciences, Cadi Ayyad University, Marrakech, Morocco Youssef Allaoui LMACS, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, Morocco Said Melliani LMACS, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, Morocco Hamid El Khalfi Lab. LMRI, FP of Khouribga, Sultan Moulay Slimane University, Morroco Journal Article 2023 05 09 This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\`a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations. https://cmde.tabrizu.ac.ir/article_17165_fac8a371c1e287d299703d5691fceff5.pdf