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In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [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]

In this study, we present a mild solution to the Hilfer fractional differential equations with infinite delay. Firstly, we establish the results on an infinite interval; to achieve this, we use the generalized Ascoli–Arzelà theorem and Mönch's fixed point theorem via a measure of noncompactness. Secondly, we consider the existence of a mild solution when the semigroup is compact, and the Schauder fixed-point theorem is used. The outcome is demonstrated using an infinitesimal operator, fractional calculus, semigroup theory, and abstract space. Finally, we present an example to support the results. [ABSTRACT FROM AUTHOR]

This paper delves into the enhancement of asymptotic and oscillatory behaviors in solutions to even-order neutral differential equations with multiple delays. The main objective is to establish improved inequalities to advance the understanding of oscillation theory for these equations. The paper's approach is centered on improving the understanding of the intricate relationship between solutions and their corresponding functions. This is achieved by harnessing the modified monotonic properties of positive solutions, which provide valuable insights into oscillation behavior. Furthermore, leveraging the symmetry between positive and negative solutions, we derived criteria that ensure oscillation for all solutions, with a specific emphasis on excluding only positive solutions. To illustrate the significance of our findings, we provide an illustrative example. [ABSTRACT FROM AUTHOR]

This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation. [ABSTRACT FROM AUTHOR]

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept. [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]

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]

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]

In this paper, we explain the approximate controllability of Ψ -Hilfer fractional neutral differential equations with infinite delay. The outcome is demonstrated using the infinitesimal operator, fractional calculus, semigroup theory, and the Krasnoselskii's fixed point theorem. To begin, we emphasise the presence of the mild solution and show that the Ψ -Hilfer fractional system is approximately controllable. Additionally, we present theoretical and practical examples. [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 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]

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, we aim to study the neutral-type delayed Caputo fractional differential equations of the form C D α x t − g t , x t = f t , x t , t ∈ t 0 , ∞ , t 0 ≥ 0 with order 0 < α < 1 , which can be used to describe the growth processes in real-life sciences at which the present growth depends on not only the past state but also the past growth rate. Our ultimate goal in this study is to concentrate on the convergence of the solutions to a predetermined constant by establishing a linkage between the delayed fractional differential equation and an integral equation. In our analysis, the sufficient conditions for the asymptotic results are obtained due to fixed point theory. The utilization of the contraction mapping principle is a convenient approach in obtaining technical conditions that guarantee the asymptotic constancy of the solutions. [ABSTRACT FROM AUTHOR]

The present article aims to establish more effective criteria for testing the oscillation of a class of functional differential equations with delay arguments. In the non-canonical case, we deduce some improved monotonic and asymptotic properties of the class of decreasing positive solutions of the studied equation. Depending on both the new properties and the linear representation of the studied equation, we obtain new oscillation criteria. Moreover, we test the effectiveness of the new criteria by applying them to some special cases of the studied equation. [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]

The purpose of this paper is to determine the existence of tripled fixed point results for the tripled symmetry system of fractional hybrid delay differential equations. We obtain results which support the existence of at least one solution to our system by applying hybrid fixed point theory. Similar types of stability analysis are presented, including Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias. The necessary stipulations for obtaining the solution to our proposed problem are established. Finally, we provide a non-trivial illustrative example to support and enhance our analysis. [ABSTRACT FROM AUTHOR]

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

It is known that the method of Lyapunov functionals is a powerful method of stability investigation for functional differential equations. Here, it is shown how the previously proposed method of stability investigation for nonlinear stochastic differential equations with delay and a high order of nonlinearity can be extended to nonlinear mathematical models of a much more general form. An important feature is the combination of the method of Lyapunov functionals with the method of Linear Matrix Inequalities (LMIs). Some examples of applications of the proposed method of stability research to known mathematical models are given. [ABSTRACT FROM AUTHOR]

In this work, we study the asymptotic behavior of even-order delay functional differential equation. As an extension of the recent development in the study of oscillation, we obtain improved and simplified criteria that test the oscillation of solutions of the studied equation. We adopt an approach that improves the relationships between the solution with and without delay. The symmetry between the positive and negative solutions also plays a key role in simplifying the presentation of the main results. Finally, we attach an example to illustrate the results and compare them together with the previous results in the literature. [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]

In this article, we study some existence and controllability results for two classes of second order functional differential equations with delay and random effects. To begin, we employ a random fixed point theorem with a stochastic domain to demonstrate the existence of mild random solutions. Next, we prove that our problems are controllable. Finally, an example is given to validate the theory part. [ABSTRACT FROM AUTHOR]

In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the significance of our results, we apply them to delay differential equation of Euler-type. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

In this work, new sufficient conditions for the oscillation of all solutions of the second-order neutral delay differential equations with the non-canonical operator are established. Using a generalized Riccati substitution, we obtained criteria that complement and extend some previous results in the literature. [ABSTRACT FROM AUTHOR]

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term r ı y ı + m ı y g ı n − 1 γ ′ + ∑ i = 1 j q i ı y γ μ i ı = 0 . We use Riccati transformation and the comparison with first-order differential inequalities to obtain theses criteria. Moreover, the presented oscillation conditions essentially simplify and extend known criteria in the literature. To show the importance of our results, we provide some examples. Symmetry plays an essential role in determining the correct methods for solutions to differential equations. [ABSTRACT FROM AUTHOR]

In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples. [ABSTRACT FROM AUTHOR]

In this study, a new oscillation criterion for the fourth-order neutral delay differential equation r u x u + p u x δ u ‴ α ′ + q u x β ϕ u = 0 , u ≥ u 0 is established. By introducing a Riccati substitution, we obtain a new criterion for oscillation without requiring the existence of the unknown function. Furthermore, the new criterion improves and complements the previous results in the literature. The results obtained are illustrated by an example. [ABSTRACT FROM AUTHOR]

The functional abstract second order impulsive differential equation with state dependent delay is studied in this paper. First, we consider a second order system and use a control to determine the controllability result. Then, using Sadovskii's fixed point theorem, we get sufficient conditions for the controllability of the proposed system in a Banach space. The major goal of this study is to demonstrate the controllability of an abstract second-order impulsive differential system with a state dependent delay mechanism. The wellposed condition is then defined. Next, we studied whether the defined problem is wellposed. Finally, we apply our results to examine the controllability of the second order state dependent delay impulsive equation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established. [ABSTRACT FROM AUTHOR]

Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system's past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufficient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results. [ABSTRACT FROM AUTHOR]

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included. [ABSTRACT FROM AUTHOR]

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results. [ABSTRACT FROM AUTHOR]

This article aims to mark out new conditions for oscillation of the even-order Emden–Fowler neutral delay differential equations with neutral term β 1 ı Φ α [ ζ r − 1 ı ] ′ + β 3 ı Φ α [ ς ξ ı ] = 0. The obtained results extend, and simplify known conditions in the literature. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

In this article, we obtain oscillation conditions for second-order differential equation with neutral term. Our results extend, improve, and simplify some known results for neutral delay differential equations. Several effective and illustrative implementations are provided. [ABSTRACT FROM AUTHOR]

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u = u (x , t) , also contain the same functions with dilated or contracted arguments of the form w = u (p x , t) , w = u (x , q t) , and w = u (p x , q t) , where p and q are the free scaling parameters (for equations with proportional delay we have 0 < p < 1 , 0 < q < 1 ). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations). [ABSTRACT FROM AUTHOR]

In this paper, we establish new sufficient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term, which are under the non-canonical condition. The results obtained complement and simplify some known results in the relevant literature. Example illustrating the results is included. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included. [ABSTRACT FROM AUTHOR]

The purpose of this article is to explore the asymptotic properties for a class of fourth-order neutral differential equations. Based on a comparison with the differential inequality of the first-order, we have provided new oscillation conditions for the solutions of fourth-order neutral differential equations. The obtained results can be used to develop and provide theoretical support for and to further develop the study of oscillation for a class of fourth-order neutral differential equations. Finally, we provide an illustrated example to demonstrate the effectiveness of our new criteria. [ABSTRACT FROM AUTHOR]

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results. [ABSTRACT FROM AUTHOR]

In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic and oscillatory properties of a certain class of third-order neutral delay differential equations with middle term. We obtain new characterizations of oscillation of the third-order neutral equation in terms of oscillation of a related, well-studied, second-order linear equation without damping. An Example is provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the oscillatory properties of 4th-order neutral differential equations. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve well-known oscillation results in the literate. Symmetry plays an important role in determining the right way to study these equation. An example to illustrate the results is given. [ABSTRACT FROM AUTHOR]

Some new sufficient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form r t N x ‴ t α ′ + ∫ a b q t , ϑ x β δ t , ϑ d ϑ = 0 , where t ≥ t 0 and N x t : = x t + p t x φ t . An example is provided to show the importance of these results. [ABSTRACT FROM AUTHOR]

The aim of this study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and complement some well-known results. We obtain Hille and Nehari type oscillation criteria to ensure the oscillation of the solutions of the equation. One example is provided to illustrate these results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a certain class of third-order nonlinear delay differential equations r w ″ α ′ v + q v x β ς v = 0 , for v ≥ v 0 , where w v = x v + p v x ϑ v . We obtain new criteria for oscillation of all solutions of this nonlinear equation. Our results complement and improve some previous results in the literature. An example is considered to illustrate our main results. [ABSTRACT FROM AUTHOR]