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The oscillatory behavior of solutions of an even-order neutral differential equation with distributed deviating arguments is considered using Riccati, generalized Riccati transformations, integral averaging technique of Philos type and the theory of comparison. New sufficient conditions are established in both canonical and noncanonical cases. Two examples are given to support our results. [ABSTRACT FROM AUTHOR]

In this work, we establish the necessary and sufficient conditions for oscillation of a class of functional differential equations of the form ((x(t)+p(t)x(to))' + q(t)(x(tr)) + v(t)(x( - )) = 0 of a neutral coefficient p(t), by using the Knaster-Tarski fixed point theorem and Banach's fixed point theorem. [ABSTRACT FROM AUTHOR]

In this work, new oscillation criteria are established for a second-order differential equation with several sublinear neutral terms and in the canonical case. To determine the oscillation conditions, we followed the Riccati approach and also compared the studied equation with a first-order delay equation. Obtaining the oscillation conditions required deducing some new relationships linking the solution to the corresponding function as well as its derivatives. The paper addresses some interesting analytical points in the study of the oscillation of equations with several sublinear neutral terms. These new findings complement some well-known findings in the literature. Furthermore, an example is provided to show the importance of the results. [ABSTRACT FROM AUTHOR]

This article is dedicated to the oscillatory properties of a second order Emden-Fowler delay differential equation with a sublinear neutral term. The presented criteria extend and improve several well-known results reported recently in the literature. Four illustrating examples are included to verify the validity of the results derived. [ABSTRACT FROM AUTHOR]

The authors study the oscillatory behavior of solutions to a class of fifth-order differential equations with unbounded neutral coefficients. The results are obtained by a comparison with first-order delay differential equations whose oscillatory characters are known. Two examples illustrating the results are provided, one of which is applied to Euler type equations. [ABSTRACT FROM AUTHOR]

This paper presents a study on the oscillatory behavior of solutions to even-order neutral differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first-order delay differential equations, Riccati technique and integral averages technique. The results of this work contribute to a deeper understanding of even-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability for retarded neutral differential equations with unbounded delay, non-instantaneous impulses, and non-local conditions. First, we set the problem in a natural Banach phase space satisfying Hale-Kato axiomatic Theory about the phase space for retarded ordinary equations with unbounded delay. Second, we assume some conditions on the nonlinear terms to achieve the approximate controllability using the technique developed by Bashirov and Ghahramanlou [6], which avoids the use of fixed point theorems. [ABSTRACT FROM AUTHOR]

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study's methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using a comparison technique to connect the oscillation of the studied equation with some second-order equations, the second aspect takes advantage of the significant progress made in the study of the oscillation of second-order equations. Numerous applications of functional differential equations of the neutral type served as the inspiration for the study of a subclass of these equations. [ABSTRACT FROM AUTHOR]

In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to describe a variety of complex, multi-step, real-world situations that arise from nature. Using fixed point (FP) techniques, like Banach's FP theorem, Schauder's FP theorem, and Arzelá Ascoli's FP theorem, the Hyer–Ulam (HU) stability and the existence theorem conclusions are investigated for the considered problem. Eventually, a supportive example is given to demonstrate the applicability and efficacy of the applied concept. [ABSTRACT FROM AUTHOR]

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields. [ABSTRACT FROM AUTHOR]

We study the local and global existence and uniqueness of strict solution and the local well-posedness for a class of abstract "explicit" neutral integro-differential equations with state-dependent delay. In the last section some examples motivated by explicit neutral ordinary integro-differential equations arising in the theory of population dynamics are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of a third-order neutral-type differential equation with time-varying delays. Some sufficient conditions on the existence of a periodic solution are established for the considered system. Different from the previously reported research results, by utilizing the properties of neutral operators and a special variable substitution, we transform a high-order neutral equation into a first-order three-dimensional nonneutral system. The existence of a periodic solution such as the high-order neutral equation has not been given much attention in past papers due to the difficulty of estimation of prior bounds of solutions. This paper is devoted to the use of properties of neutral-type operators with a variable parameter and Mawhin's continuation theorem for overcoming the above difficulties. The neutral term in the third-order neutral differential equation in this paper contains a variable parameter which is different from third-order neutral-type equations that have been studied. The third-order neutral-type equation studied in this paper is more general, and similar equations studied in the past are special cases of the equations studied in this paper. Finally, an example is given to elucidate the effectiveness and values of the present results. Our results are new and complement the related results of third-order functional differential equations. [ABSTRACT FROM AUTHOR]

The major goal of this study is to demonstrate the existence and controllability of a second-order abstract differential system with state-dependent delay. We also discuss the well-posedness of the problem at hand. Finally, some examples are provided for the illustration of the main results. [ABSTRACT FROM AUTHOR]

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in order to determine the oscillation terms of the studied equation. In order to contribute to this subject, we create new and significant relationships and inequalities. We use these relationships to create conditions in which positive and N-Kneser solutions of the considered equation are excluded. To obtain these terms, we employ the comparison method and the Riccati technique. Furthermore, we use the relationships obtained to create new criteria, so expanding the existing literature on the field. Finally, we provide an example from the general case to demonstrate the results' significance. The findings given in this work provide light on the behavior of odd-order neutral differential equation solutions with multiple delays. [ABSTRACT FROM AUTHOR]

In this paper, new criteria for a class oscillation of second-order delay differential equations with distributed deviating arguments were established. Our method mainly depends on making sharper estimates for the non-oscillatory solutions of the studied equation. By using the Ricati technique and comparison theorems that compare the studied equations with first-order delay differential equations, we obtained new and less restrictive conditions that ensure the oscillation of all solutions of the studied equation. Further, we give an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations. [ABSTRACT FROM AUTHOR]

In this work, by considering a third-order differential equation with delay-neutral arguments, we investigate the oscillatory behavior of solutions. It is known that the relationships between the solution and its derivatives of different orders, as well as between the solution and its corresponding function, can help to obtain more efficient oscillation criteria for differential equations of neutral type. So, we deduce some new relationships of an iterative nature. Then, we test the effect of these relationships on the criteria that exclude positive solutions to the studied equation. By comparing our results with previous results in the literature, we show the importance and novelty of the new results. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the monotonic properties of the solutions of a class of neutral delay differential equations. The importance of this study lies in the fact that the monotonic properties largely control the study of the oscillation and asymptotic behaviour of the solutions to delay differential equations. Then, by using the new properties, we create improved criteria for testing the oscillation of solutions to the studied equation. We also find new criteria that can be applied more than once. Moreover, we discuss the importance and novelty of the results through the application to a special case of the studied equation. [ABSTRACT FROM AUTHOR]

Using operator‐valued Lp$$ {L}^p $$‐Fourier multiplier theorem, weighted Sobolev spaces, and the Carleman transform, we characterize the well‐posedness of second‐order degenerate differential equations with finite delay (Mu)′′(t)=Au(t)+Fut+Gut′+f(t)$$ {(Mu)}^{\prime \prime }(t)= Au(t)+F{u}_t+G{u}_t^{\prime }+f(t) $$ on Lp(ℝ;X)$$ {L}^p\left(\mathbb{R};X\right) $$, where A:D(A)⊆X→X$$ A:D(A)\subseteq X\to X $$ and M:D(M)⊆X→X$$ M:D(M)\subseteq X\to X $$ are closed linear operators defined on a Banach space X$$ X $$, the operators F$$ F $$ and G$$ G $$ are in B(Lp([−τ,0];X);X)$$ B\left({L}^p\left(\left[-\tau, 0\right];X\right);X\right) $$ for some fixed τ>0$$ \tau >0 $$, and ut(s)=u(t+s),ut′(s)=u′(t+s)$$ {u}_t(s)=u\left(t+s\right),{u}_t^{\prime }(s)={u}^{\prime}\left(t+s\right) $$ when t∈ℝ$$ t\in \mathbb{R} $$ and s∈[−τ,0]$$ s\in \left[-\tau, 0\right] $$. These results are used to study the well‐posedness of the associated second‐order neutral degenerate differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the oscillatory properties of certain second-order differential equations of neutral type are investigated. We obtain new oscillation criteria, which guarantee that every solution of these equations oscillates. Further, we get conditions of an iterative nature. These results complement and extend some beforehand results obtained in the literature. In order to illustrate the results we present an example. [ABSTRACT FROM AUTHOR]

In this paper, the oscillatory and nonoscillatory qualities for every solution of fourth-order neutral delay equation are discussed. Some conditions are established to ensure that all solutions are either oscillatory or approach to zero asદ →∞. Two examples are provided to demonstrate the obtained findings. [ABSTRACT FROM AUTHOR]

This study purposes to present some new comparison theorems that guarantee the oscillation of all solutions of even order functional differential equations with a neutral term including both advanced and delayed arguments. The obtained results are based on comparisons with associated first‐order delay differential inequalities and first‐order delay differential equations, and they are applicable to the both cases where the neutral coefficients are unbounded and/or bounded. In the last section, several examples showing the applicability of the results are provided. [ABSTRACT FROM AUTHOR]

In this work first we transform the semi-noncanonical fourth order neutral delay differential equations into canonical type. This simplifies the investigations of finding the relationships between the solution and its companion function which plays an important role in the oscillation theory of neutral differential equations. Moreover, we improve these relationships based on the monotonic properties of positive solutions. We present new conditions for the oscillation of all solutions of the corresponding equation which improve the oscillation results already reported in the literature. Examples are provided to illustrate the importance of our main results. [ABSTRACT FROM AUTHOR]

This paper presents a novel class of third order nonlinear nonautonomous neutral differential equation with delay. The third order neutral differential equation is cut down to a system of first order, a suitable complete Lyapunov-Krasovskii's functional is constructed and used, to obtain standard conditions on the nonlinear functions to ensure stability and uniform asymptotic stability of the trivial solutions, the existence of a unique periodic solution, uniform boundedness and uniform ultimate boundedness of solutions when the forcing term is nonzero. The obtained results are new and include many prominent results on neutral and nonneutral delay differential equations in literature. Finally, the practicability and reliability of the theoretical results are demonstrated. [ABSTRACT FROM AUTHOR]

Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence of periodic or nonnegative periodic solutions for a totally nonlinear neutral differential equation with infinite delay. In the process, we convert the given neutral differential equation into an equivalent integral equation. Then, we employ Krasnoselski˘ı-Burton’s fixed point theorem to prove the existence of periodic or nonnegative periodic solutions. Two examples are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

When a real‐life problem is mathematically modeled by differential equations or another type of equation, there are always intrinsic phenomena that are not taken into account and can affect the behavior of such a model. For example, external forces can abruptly change the model; impulses and delay can cause a breakdown of it. Considering these intrinsic phenomena in the mathematical model makes the difference between a simple differential equation and a differential equation with impulses, delay, and nonlocal conditions. So, in this work, we consider a semilinear nonautonomous neutral differential equation under the influence of impulses, delay, and nonlocal conditions. In this paper we study the controllability of these semilinear neutral differential equations with some of these intrinsic phenomena taking into consideration. Our aim is to prove that the controllability of the associated ordinary linear differential equation is preserved under certain conditions imposed on these new disturbances. In order to achieve our objective, we apply Rothe's fixed point Theorem to prove the exact controllability of the system. Finally, our method can be extended to the evolution equation in Hilbert spaces with applications to control systems governed by PDE's equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay. [ABSTRACT FROM AUTHOR]

In this paper, we establish oscillation theorems for all solutions to fourth-order neutral differential equations using the Riccati transformation approach and some inequalities. Some new criteria are established that can be used in cases where known theorems fail to apply. The approach followed depends on finding conditions that guarantee the exclusion of positive solutions, and as a result of the symmetry between the positive and negative solutions of the studied equation, we therefore exclude negative solutions. An illustrative example is given. [ABSTRACT FROM AUTHOR]

In this study, new asymptotic properties of positive solutions of the even-order neutral delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Using these properties, we obtain new criteria to exclude a class from the positive solutions of the studied equation, using the comparison principles. [ABSTRACT FROM AUTHOR]

A linear system of delayed differential equations with multiple delays x ̇ (t) = − ∑ i = 1 s c i (t) A i (t) x (t − τ i (t)) , t ∈ [ t 0 , ∞) , is considered where x is an n -dimensional column vector, t 0 ∈ R , s is a fixed integer, delays τ i are positive and bounded, entries of n by n matrices A i as well as functions c i are nonnegative, and the sums of columns of the matrix A i (t) are identical and equal to a function α i (t). It is proved that, on [ t 0 , ∞) , the system has a solution with positive coordinates if and only if the scalar equation y ̇ (t) = − ∑ i = 1 s c i (t) α i (t) y (t − τ i (t)) , t ∈ [ t 0 , ∞) , has a positive solution. Some asymptotic properties of solutions related to both equations are also discussed. Illustrative examples are considered and some open problems formulated. • System of delayed differential equations. • Scalar delayed differential equations. • Multiple delays. • Novel positivity criterion. • Topological principle. [ABSTRACT FROM AUTHOR]

Uncertain differential equations with time-varying delay are a special type of uncertain functional differential equation. Stability in measure and stability in mean for this type of differential equation have been investigated. For the sake of completeness, this article explores the other two types of stability. Firstly, the concepts of the almost sure stability and the pth moment stability of uncertain differential equations with time-varying delay are proposed. Next, sufficient conditions, in the form of theorems, for uncertain differential equations with time-varying delay to be stable almost surely and in pth moment are deduced. Finally, the relationships between several types of stability for uncertain differential equations with time-varying delay are discussed. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior of even-order neutral delay differential equation (a ⋅ (u + ρ ⋅ u ∘ τ) (n − 1) ) ′ (ℓ) + h (ℓ) u (g (ℓ)) = 0 , ℓ ≥ ℓ 0 , {(a\cdot {(u+\rho \cdot u\circ \tau)}^{(n-1)})}^{^{\prime} }(\ell)+h(\ell)u(g(\ell))=0,\hspace{1.0em}\ell \ge {\ell }_{0}, where n ≥ 4 n\ge 4 , and in noncanonical case, that is, ∫ ∞ a − 1 (s) d s < ∞. \mathop{\int }\limits^{\infty }{a}^{-1}\left(s){\rm{d}}s\lt \infty. To the best of our knowledge, most of the previous studies were concerned only with the study of n n -order neutral equations in canonical case. By using comparison principle and Riccati transformation technique, we obtain new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Examples are presented to illustrate our new results. [ABSTRACT FROM AUTHOR]

In this work, we consider a fuzzy delay differential system with non-instantaneous impulses and investigate the existence, uniqueness and total controllability results. Non-linear functional analysis, Banach fixed point theorem and fuzzy theory are the main techniques to establish these results. In support, an example is given to validate the obtained analytical outcomes. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtaining a new oscillation criterion for a class of neutral delay differential equations of the type u(x)ym−1(x)κ′+q(x)zκγ(x)=0, where m ≥ 4 is an even integer and y(x):=z(x)+g(x)zξ(x). It should be noted here that the results obtained improve and complement those given by Xing et al. (2016). An example illustrating the applicability of the new criterion is presented. [ABSTRACT FROM AUTHOR]

In this work, we aim at studying the asymptotic and oscillatory behavior of even-order neutral delay noncanonical differential equations. To the best of our knowledge, most of the related previous works are concerned only with neutral equations in the canonical case. Our new oscillation criteria essentially improve, simplify, and complement related results in the literature, especially those from a paper by Li and Rogovchenko (Abstr. Appl. Anal. 2014:395368, 2014). Some examples are presented that illustrate the importance of the new criteria. [ABSTRACT FROM AUTHOR]

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems. [ABSTRACT FROM AUTHOR]

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of a class of fourth-order Emden–Fowler delay differential equations with neutral term. Using the Riccati transformation and comparison method, we establish several new oscillation conditions. These new conditions complement a number of results in the literature. We give examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Using Krasnoselskii's fixed‐point theorem, we obtain some results for the existence of periodic solutions to the third‐order neutral differential equations with delay. The results obtained essentially improve, include, and complement the results in the literature. [ABSTRACT FROM AUTHOR]

Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented. [ABSTRACT FROM AUTHOR]

Uncertain fractional-order delay differential equation is a class of fractional-order functional differential equations driven by Liu process. This paper devotes to studying linear uncertain fractional-order delay differential equation. The explicit representation and iterative formula of the solution to linear uncertain fractional-order delay differential equations are obtained. Meanwhile, the inverse uncertainty distribution of the solution to linear uncertain fractional-order delay differential equation by the α -path is presented. [ABSTRACT FROM AUTHOR]

In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the ⊙∗-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the ⊙∗-shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system "mimics" the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the ⊙∗-shift operator and of the ⊙∗-generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research. [ABSTRACT FROM AUTHOR]

In this article, we study a class of non-linear neutral delay differential equations of third order. We first prove criteria for non-existence of non-Kneser solutions, and criteria for non-existence of Kneser solutions. We then use these results to provide criteria for the under study differential equations to ensure that all its solutions are oscillatory. An example is given that illustrates our theory. [ABSTRACT FROM AUTHOR]

Studying the behavior of solutions to the Cauchy problem for a family of nonlinear functional differential equations with delay which arise in the living system models, we establish the conditions that provide some exponential decay estimates for components of solutions. We find the parameters of simultaneous exponential estimates as solutions to nonlinear inequalities built from the majorants of the mappings on the right-hand sides of the differential equations. We present the results of constructing the exponential estimates for the variables in an epidemic dynamics model. [ABSTRACT FROM AUTHOR]

In this paper, sucient conditions are established ensuring asymptotic stability, boundedness and square integrability of solutions and their derivatives to a nonlinear third order neutral delay differential equations. An example is given to show applicability of the established assumptions. The obtained results extend and generalize some existing ones in the related literature. [ABSTRACT FROM AUTHOR]