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This paper focuses on the study of existence of solutions and periodic solutions for a general class of neutral measure functional differential equations (NMFDEs for short) with infinite delay. The relationship between solutions of NMFDEs with infinite delay and solutions of generalized ordinary differential equations is first discussed, based on this the existence, uniqueness and continuous dependence of solutions for NMFDEs with infinite delay are investigated. Then the existence of periodic solutions for the considered equation is also studied and the corresponding sufficient conditions are obtained by fixed-point principle. Finally two examples are given to illustrate the main obtained results. • Study the existence of solutions and periodic solutions for general NMFDEs with infinite delay. • Due to different approaches, assumptions in the main results are more applicable and concise. • Avoid the priori boundedness assumptions on periodic solutions. [ABSTRACT FROM AUTHOR]

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results. [ABSTRACT FROM AUTHOR]

This paper focuses on the oscillatory properties of the third-order semi-canonical nonlinear delay differential equation. By using the new canonical transform method, we transformed the studied equation into a canonical-type equation, which simplified the examination of the studied equation. The obtained oscillation results are new and complement the existing results mentioned in the literature. Examples are provided to illustrate the importance and novelty of the main results. [ABSTRACT FROM AUTHOR]

It is shown how to approximate the solution of functional differential equations in terms of Bell's polynomials. Some numerical checks are shown, by using the computer algebra system Mathematica © . [ABSTRACT FROM AUTHOR]

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth. [ABSTRACT FROM AUTHOR]

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method's effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results. [ABSTRACT FROM AUTHOR]

This article examines the stability properties of linear stochastic difference equations with delays. For this purpose, a novel approach is used that combines the theory of inverse-positive matrices and the asymptotic methods developed by N.V. Azbelev and his students for deterministic functional differential equations. Several efficient conditions for p-stability and exponential p-stability (2 ≤ p < ∞) of systems of linear Itô-type difference equations with delays and random coefficients are found. All results are conveniently formulated in terms of the coefficients of the equations. The suggested examples illustrate the feasibility of the approach. [ABSTRACT FROM AUTHOR]

This paper presents a numerical approach to solve fuzzy partial differential equation restrict to fuzzy boundary and initial conditions. The numerical solution to this problem is obtained by the finite difference method considering a particular type of fuzzy arithmetic called J 0 -interactive arithmetic. A study of the computational cost of the method is presented, as well as a comparison with the numerical solution obtained by the standard fuzzy arithmetic, which is associated with the Zadeh's extension principle. The paper focuses on the heat equation in order to illustrate the methodology. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

In this work, we study a nonlocal erosion equation, which simulates the process of nanorelief formation. For a periodic boundary-value problem for the functional differential equation with partial derivatives, we examine local bifurcations in the cases where homogeneous equilibrium states change their stability. We prove that in the problem considered, subcritical bifurcations occur. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a disc, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group onto the disc are found in explicit form for any right-hand side from the Lebesgue space. [ABSTRACT FROM AUTHOR]

The stability of systems of linear autonomous functional differential equations of neutral type is studied. The study is based on the well-known representation of the solution in the form of an integral operator, the kernel of which is the Cauchy function of the equation under study. The definitions of Lyapunov, asymptotic, and exponential stability are formulated in terms of the corresponding properties of the Cauchy function, which allows us to clarify a number of traditional concepts without loss of generality. Along with the concept of asymptotic stability, a new concept of strong asymptotic stability is introduced. The main results are related to the stability with respect to the initial function from the spaces of summable functions. In particular, it is established that strong asymptotic stability with initial data from the space L1 is equivalent to the exponential estimate of the Cauchy function and, moreover, exponential stability with respect to initial data from the spaces Lp for any p ≥ 1. [ABSTRACT FROM AUTHOR]

A boundary value problem for a nonlinear second-order functional-differential equation on the interval with integral boundary conditions is considered. Sufficient conditions for the existence of at least one positive solution of the problem under consideration are found by using special topological tools. A nontrivial example in which the sufficient conditions for the unique solvability of the problem are satisfied is given. [ABSTRACT FROM AUTHOR]

The work in this article provides literature with some results concerning the uniform asymptotic stability, the boundedness and the square integrability of solutions for some non autonomous equations of third order supplied with delay and neutral parameters. The main tool used in this work is the second method of Lyapunov. The article is finished by giving a concrete example that ensure the application of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate the existence and uniqueness of solutions for neutral functional differential equations with sequential fractional orders, specifically involving the G -Caputo operator. To obtain the desired results, we employ the Banach fixed point theorem (BFPT), a nonlinear variation of the Leray-Schauder fixed point theorem (SFPT), and the Krasnoselski fixed point theorem (KFPT). Additionally, we provide illustrative examples that demonstrate the key findings. Furthermore, we address a scenario where an initial value integral condition is considered. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the International Journal of Robust & Nonlinear Control on the topic of time-delay systems. It provides a historical overview of functional differential equations and highlights the contributions of Nikolay Nikolayevich Krasovskii and Aristide Halanay to the field. The special issue includes 21 research papers covering stability, robustness, observer theory, predictors, and delay-based approaches to partial differential equation models. Each paper is briefly described, covering topics such as stabilization of uncertain systems, sampled-data control for nonlinear systems, observer theory for differential delay systems, sensor fault estimation, and control approaches for vehicular platoons. [Extracted from the article]

The article starts from the challenge of the critical cases in difference operator stability for neutral functional differential equations (NFDE). Such cases occur in the NFDE associated to 1D$$ 1D $$ hyperbolic partial differential equations (PDE) dynamics in mechanical and hydraulic engineering. For some of such applications it resulted that the aforementioned critical (nonasymptotic) stability is connected to the character and level of the energy losses. It is shown that suitable choice of the losses to be taken into account can remove the critical stability properties and give the difference operator the asymptotic stability thus ensuring asymptotic stability for the system's dynamics and also other asymptotic properties. [ABSTRACT FROM AUTHOR]

The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including H∞$$ {H}_{\infty } $$ stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example. [ABSTRACT FROM AUTHOR]

The partial neutral functional fractional differential equation described by the fractional operator is considered in the present investigation. The used fractional operator is the Caputo derivative. In the present paper, the fractional resolvent operators have been defined and used to prove the existence of the unique solution of the fractional neutral differential equations. The fixed point theorem has been used in existence investigations. For an illustration of our results in this paper, an example has been provided as well. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of Lp space. This exploration involved the utilization of Jensen’s, Gröenwall-Bellman’s, Hölder’s, Burkholder-Davis-Gundy’s inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this work, we proposed some generalized Darbo type fixed point theorem by employing the concept of generalized operators and measure of non-compactness within the framework of partial order Banach spaces. Some fixed point theorems and their consequences are presented as part of our exploration. These theorems encompass the presentation of various fixed point theorems and the subsequent implications they entail. Further, we demonstrate the practical application of our findings by establishing the existence of solutions for delay differential equations. To substantiate our conclusions, we provide numerical estimations based on a real-world example. This research contributes to the advancement of fixed point theory in the context of partial order Banach spaces and showcases its practical relevance through a concrete application and empirical support. [ABSTRACT FROM AUTHOR]

We consider an initial-boundary value problem for a system of two differential equations, one of which is ordinary and the other a partial differential equation, where the connection between the equations is carried out through an integral functional. In this case, the boundary conditions contain higher-order time derivatives of the unknown functions. The initial-boundary value problem models the rotation of a mechanical system consisting of two rigid bodies connected by an elastic rod. The rotation is considered around the center of mass of one of the rigid bodies and carried out by a moment of external forces (control torque) applied to the axis of rotation of the rigid body. For the initial-boundary value problem, we introduce the concept of a generalized solution and prove theorems on the existence and uniqueness of a generalized solution and on the well-posedness of the problem formulation. We solve problems of optimal control of the rotation of a mechanical system from the initial state to the final state at a given point in time, minimizing the value of the control torque and minimizing the energy functional of the control torque. In this formulation, we also solve the performance problems under restrictions on the value of the control torque and on the value of the energy integral of the control torque. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for an iterative system of nonlinear two‐point tempered fractional boundary value problem. Utilizing Krasnoselskii's fixed point theorem in a cone, we establish criteria for the existence of positive solutions. The proofs involve transforming the problem into an equivalent Fredholm integral equation of the second kind. We further explore solution uniqueness using Rus's theorem and examine Hyers–Ulam stability, particularly for the case when only a single fractional differential equation is considered, (m = 1). Our study represents a significant departure from previous works by including Riemann–Liouville tempered fractional derivative operators and an iterative equation. This research sheds light on the diverse applications of iterative functional differential equations, extending beyond noniterative counterparts. Throughout the paper, presumptive conditions are applied, and the results are validated through illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we aim to establish an averaging principle for neutral stochastic functional differential equations driven by G-Lévy processes with non-Lipschitz coefficients. Utilizing the theory of sub-linear expectations, we show that the solutions of the considered stochastic systems can be approximated by the solutions of averaged neutral stochastic functional differential equations driven by G-Lévy processes in the mean square convergence and convergence in the sense of capacity. Finally, we give an example to support our main results. [ABSTRACT FROM AUTHOR]

In this paper we investigate the mean square asymptotic stability of a perturbed scalar linear stochastic functional differential equation. Specifically, we are able to give necessary and sufficient conditions on the forcing terms for convergence of the mean square, exponential convergence of the mean square, and integrability of the mean square of solutions. It is also essential that the underlying unperturbed SFDE is mean–square asymptotically stable for these results to hold. [ABSTRACT FROM AUTHOR]

We study a problem of damping a control system described by functional-differential equations of natural order and neutral type with nonsmooth complex coefficients on an arbitrary tree with global delay. The latter means that the delay propagates through internal vertices of the tree. Minimization of the energy functional of the system leads to a variational problem. We establish its equivalence to a certain self-adjoint boundary value problem on the tree for equations of order with nonlocal quasi-derivatives and multidirectional shifts of the argument as well as Kirchhoff-type conditions emerging at the internal vertices. The unique solvability of both problems is proved. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients. [ABSTRACT FROM AUTHOR]

We study inverse spectral problems for Dirac-type functional-differential operators with two constant delays greater than two fifths the length of the interval, under Dirichlet boundary conditions. The inverse problem of recovering operators from four spectra has been studied. We consider cases when delays are greater or less than half the length of the interval. The main result of the paper refers to the proof that in both cases operators can be recovered uniquely from four spectra. [ABSTRACT FROM AUTHOR]

In this work, we mainly focus on the local existence and regularity of integral solutions for a class of nondensely defined partial neutral functional integrodifferential equations with unbounded delay. We use the theory of integrated resolvent operators introduced by Oka [H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 1995, 2, 193–232]. Finally, we provide an example to demonstrate the basic findings of our work. [ABSTRACT FROM AUTHOR]

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p = 2 to results in the L p space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. [ABSTRACT FROM AUTHOR]

In this paper, we study the differentiability of solutions of neutral stochastic differential equations driven by G-Brownian motion with respect to parameter. Under suitable assumptions, we show that solutions are differentiable with respect to the parameter which appears in the initial data. In addition, the stochastic differential equation of the derivative is given and the existence-uniqueness of solution is proved. Moreover, an example to illustrate the theoretically obtained results is presented. [ABSTRACT FROM AUTHOR]

In this work, we prove a nonlocal Halanay inequality with an exact decay rate. This enables us to analyze behavior of solutions to some classes of nonlocal ODEs and PDEs involving unbounded delays. The obtained results extend and improve the previous ones proved for fractional differential equations and other nonlocal subdiffusion equations. [ABSTRACT FROM AUTHOR]

New necessary and sufficient stability tests for neutral‐type linear time delay systems are presented. They are based on functionals with prescribed derivative depending on the delay Lyapunov matrix and recent stability/instability theorems. A piecewise linear approximation of the initial function argument in the functional is employed. As a result, the functional is approximated via a quadratic form whose matrix provides stability criteria. This matrix involves the integrals based on the delay Lyapunov matrix which are handled via an efficient computational procedure. The main feature of these tests is that they are verified in a finite number of operations, which is achieved through rigorous bounding of the approximation error on a specific class of functions. Some examples which validate the obtained results are presented and discussed. [ABSTRACT FROM AUTHOR]

This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional Lyapunov stability theory, practical stability can portray qualitative behaviour and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov–Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi-sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behaviour of the addressed system. Two numerical examples are given to verify the validity of our developed results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of asymptotic stability of a kind of nonlinear perturbed neutral differential system with variable delay is discussed. The Lyapunov-Krasovskiˇı functional constructed, is used to obtain conditions for asymptotic stability of the nonlinear perturbed neutral differential system in terms of linear matrix inequality (LMI). The two new results (delay-independent and delay-dependent criteria) include and extend the existing results in the literature. Finally, an example of delay-dependent criteria is supplied and the simulation result is shown to justify the effectiveness and reliability of the used techniques. [ABSTRACT FROM AUTHOR]

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature. [ABSTRACT FROM AUTHOR]

This paper presents novel criteria for investigating the oscillatory behavior of even-order neutral differential equations. By employing a comparative approach, we established the oscillation properties of the studied equation through comparisons with well-understood first-order equations with known oscillatory behavior. The findings of this study introduce fresh perspectives and enrich the existing body of oscillation criteria found in the literature. To illustrate the practical application of our results, we provide an illustrative example. [ABSTRACT FROM AUTHOR]

In this study, we aim to contribute to the increasing interest in functional differential equations by obtaining new theorems for the oscillation of second-order neutral differential equations of mixed type in a non-canonical form. The results obtained here improve and extend those reported in the literature. The applicability of the results is illustrated by several examples. [ABSTRACT FROM AUTHOR]

This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results. [ABSTRACT FROM AUTHOR]

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results. [ABSTRACT FROM AUTHOR]

In this manuscript, we consider an inverse problem (IP) for an abstract impulsive differential equation with constant delay. We identify the parameter using an over-determined condition on a mild solution. The main techniques to find the results are a direct approach using Volterra integral (VI) equations for sufficiently regular data and an optimal control approach for less regular data. In optimal control approach, the characterization of the limit of the sequence of approximate solutions demonstrates that it is a solution to the original IP, under certain hypotheses. Finally, we provide an example to showcase the precision and reliability of our findings. [ABSTRACT FROM AUTHOR]

The oscillatory behavior of solutions of an even-order differential equation with a superlinear neutral term is considered using Riccati and generalized Riccati transformations, the integral averaging technique, and the theory of comparison. New sufficient conditions are established in the noncanonical case. An example is given to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a more general class of equations called functional Volterra integral equations involving measures, which encompass many types of equations such as functional Volterra equations, functional Volterra equations with impulses, functional Volterra delta integral equations on time scales, functional fractional differential equations with and without impulses, among others. Also we present some important results such as: local existence, uniqueness and prolongation of solutions, which play an important role for further investigations. We provide many examples and applications. [ABSTRACT FROM AUTHOR]

A periodic boundary-value problem for an equation with a deviating spatial argument is considered. Using the Poincaré–Dulac method of normal forms, the method of integral manifolds, and asymptotic formulas, we examine a number of bifurcation problems of codimension 1 and 2. For homogeneous equilibrium states, we analyze possibilities of implementing critical cases of various types. The problem on the stability of homogeneous equilibrium states is studied and asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the existence and uniqueness of continuous solutions of the following iterative functional differential equation ... By using of Boyd-Wong's fixed point theorem and under suitable conditions, we establish the existence and uniqueness of a continuous solution. [ABSTRACT FROM AUTHOR]