Bachir2001@gmail Division of Industrial Engineering, OSTIM Technical University, 06374 Ankara, Turkey Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan 6517838695, Iran; [email protected] or mesamei@gmail DL-AP4 Epigenetics Correspondence: [email protected] Existing address: Division of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.Citation: Boutiara, A.; Benbachir, M.; Alzabut, J.; Samei, M.E. Monotone Iterative and Upper ower Remedy Methods for Solving Nonlinear -Caputo Fractional Boundary Worth Challenge. Fractal Fract. 2021, five, 194. ten.3390/ fractalfract5040194 Academic Editor: JosFrancisco G ez Aguilar Received: eight September 2021 Accepted: 27 October 2021 Published: three NovemberAbstract: The objective of this paper will be to study the existence of extremal options for nonlinear boundary value troubles of fractional differential equations involving the -Caputo Riodoxol Biological Activity derivative C D; (t) = V (t, (t)) below integral boundary situations ( a) = I; . Our key benefits a are obtained by applying the monotone iterative approach combined with the system of upper and reduce solutions. Additional, we look at three situations for (t) as t, Caputo, 2t , t, and Katugampola (for = 0.five) derivatives and examine the validity of your acquired outcomes with the assistance of two distinctive certain examples. Keywords and phrases: extremal solutions; monotone iterative method; -Caputo fractional derivative; upper and reduced solutions MSC: 26A33; 34A08; 34B1. Introduction The notion of fractional calculus refers towards the last 3 centuries and it might be described as the generalization of classical calculus to orders of integration and differentiation that happen to be not necessarily integers. Quite a few researchers have applied fractional calculus in different scientific regions [1]. Within the literature, various definitions from the fractional-order derivative happen to be suggested. The oldest as well as the most popular ones advocate for the usage of the RiemannLiouville and Caputo settings. One of the most recent definitions of a fractional derivative was delivered by Kilbas et al., exactly where the fractional differentiation of a function with respect to a further function within the sense of Riemann iouville was introduced [5]. They further defined proper weighted spaces and studied some of their properties by using the corresponding fractional integral. In [6], Almaida defined the following new fractional derivative and integrals of a function with respect to some other function:Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Copyright: 2021 by the authors. Licensee MDPI, Basel, Switzerland. This short article is definitely an open access write-up distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (licenses/by/ 4.0/).D, (x) := a1 d dx d 1 dxn- In , (x) an ax= ( (x) -)n–1 d, (n -)(1)Fractal Fract. 2021, five, 194. ten.3390/fractalfractmdpi/journal/fractalfractFractal Fract. 2021, five,2 ofwhere n = [ ] 1 andI, (x) := ax a ( (x) -)-1 d, (two)respectively. He referred to as the fractional derivative the -Caputo fractional operator. In the above definitions, we get the Riemann iouville and Hadamard fractional operators whenever we consider (x) = x or (x) = ln x, respectively. Lots of researchers employed this -Caputo fractional derivative (see [73] and the references therein). Abdo et al., in [14], investigated the BVP to get a fractional differential equation (FDE) involving operator and.