Oscillation criteria for fractional differential equations with a distributed delay

This paper deals with obtaining some sufficient conditions for oscillation of high-order neutral fractional integro-differential equations. The obtained results are mentioned for the first time in the literature for the oscillation of Caputo–Fabrizio fractional integro-differential equations. Finally, some illustrative examples are given to verify our main results.


Introduction
The theory of fractional differential equations has become very important in recent years due to the increasing effects of their applications in various fields of science. For the fractional derivative, we refer the reader to the related books (Kilbas et al. 1996;Podlubny 1988;Samko et al. 1993). Other than the different areas of pure mathematics, fractional differential equations can be considered in modeling diverse areas of engineering and science, such as self-similar dynamical processes, viscoelasticity, fluid flows, electrochemistry, electromagnetic theory, control theory, and many other disciplines. To find out the details, see (Singh et al. 2019;Luo et al. 2018;Failla and Zingales 2020;Sierociuk et al. 2013;Baleanu et al. 2020;Khan et al. 2019;Lazopoulos et al. 2016;Bushnaq et al. 2018;Owolabi 2019). The Riemann-Liouville type is the most widely used definition for the fractional derivative and has many applications. But this type of fractional differential equation had some disadvantages. For example, the derivative of the constant function is not zero, and we need the initial values in prac-Sermin Öztürk have contributed equally to this work. tical examples. The disadvantages given above do not apply to the Caputo fractional derivative, and this is why it is considered one of the most influential definitions of fractional derivative. Therefore, it is applied in the fields of science and engineering. Following these studies, Caputo and Fabrizio introduced a new definition with all the characteristics of the old definitions (Caputo and Fabrizio 2015;Al-Refai and Pal 2019;Caputo and Fabrizio 2016). It supposes two different representations for the spatial and temporal variables. Caputo gives the classic definition, especially suitable for mechanical phenomena related to plasticity, fatigue, damage. They asserted that using Caputo-Fabrizio is more convenient than Caputo if such effects are not available. The essential advantage of the Caputo-Fabrizio method is the boundary conditions that admits the same form as for the integer-order differential equations. It is well-known that the oscillation theory is one of the most important topics for differential equations and dynamic equations. This theory first emerged thanks to the Sturm-Liouville theorems. Together with the spectral theory, these theorems have become quite worthy of the scientific world's attention. These days, there is a lack of studies on these topics in the literature (Zhang et al. 2012;Grace et al. 2010;Li et al. 2011;Öztürk 2015, 2018;Öcalan et al. 2020;Cesarano and Bazighifan 2019;Tunc and Bazighifan 2019). In other words, in this regard, comprehensive studies have not been carried out yet. Furthermore, as it is known, the role of fractional differential equations in the theory of oscillation has started to increase considerably in the last few years (Uzun et al. 2019;Yalçın Uzun 2020;Grace et al. 2012;Abdalla and Abdeljawad 2019;Bayram et al. 2016). In particular, many articles have appeared on fractional integrodifferential equations with Riemann-Liouville, Caputo and Caputo-Fabrizio types. Nonetheless, there are many open problems with developing the oscillation theory of fractional integro-differential equations for such types (Aslıyüce et al. 2017;Feng and Sun 2022;Suragan 2020, 2021). Cesarano and Bazighifan (2019) studied the oscillation of fourth-order functional differential equations with distributed delay Tunc and Bazighifan (2019) have considered the fourth-order neutral differential equation with a continuously distributed delay Restrepo and Suragan (2020), studied a high-order neutral differential fractional equation with a continuously distributed delay a + is Caputo fractional derivative with respect to another function. Restrepo and Suragan (2021) focused on a high-order neutral differential fractional equation with a continuously distributed delay of the form where C D α θ,β,ω;a + is regularized Prabhakar derivative. And also they considered similar higher-order neutral differential equation with a continuously distributed delay for Atangana-Baleanu-Caputo fractional derivative operator.
This paper aims to obtain oscillation criteria for solutions of high-order neutral Caputo-Fabrizio fractional differential equations with a distributed delay. For our proofs, we use techniques from the literature Suragan 2020, 2021; Tunc and Bazighifan 2019; Cesarano and Bazighifan 2019). The paper organized as follows: In Sect. 2 we focus on some basic definitions and lemmas, and auxiliary results about fractional calculus. Section 3 is dedicated to obtain of oscillation criteria for the solutions of Caputo-Fabrizio fractional integro-differential equations and some concrete examples are given to conclude the paper.

Preliminaries
This section will briefly focus on some basic definitions and auxiliary results about fractional calculus. H 1 (a, b), b > a and α ∈ [0, 1] then the Caputo-Fabrizio fractional derivative (CFD) of order α is defined as

Definition 1 (Caputo and Fabrizio 2015) Let x be a function
Lemma 1 (Nchama 2020 Losada and Nieto (2015) modified CFD as (4) and defined the fractional integral associated to the CFD as Definition 3 Assume 0 < α < 1. The fractional integral of order α of a function is defined by We consider high-order neutral fractional integro-diffe rential equation where a ≥ 0, h : R + → R + , 0 < α < 1 and and the conditions below are met.
(i) λ is a quotient of odd positive integers.
A solution y(x) of (6), is a nontrivial real function in C x, +∞) , R ,x ≥ a and provides Eq.
Theorem 2 Let us assume that the conditions (7) and (i)-(v) satisfy, where λ is a quotient of odd positive integers, α ∈ (0, 1) and 1/3 < ν ≤ 1. If for some large values of x ν such that x ν > a and where 3λ > λ/ν and δ(x, c)/x ≤ 1 for any x ∈ [x ν , +∞). If we suppose that y(x) is eventually positive and a nonoscillatory solution of Eq. (6) over [a, +∞), then we arrive at two possible cases using the same methodology to prove Theorem 1. In this case, we reach the desired result thanks to the contradictory arguments to be obtained.

Discussion
The new results obtained in this paper are related to the oscillation of Caputo-Fabrizio fractional integro-differential equations. As it is known, it is essential in the literature to analyze the convergence and stability analysis of differential equations. Therefore, convergence and stability analysis of some fractional equations will be among our research topics in our future studies. Also, numerical approximations for such equations can yield exciting results in some cases.