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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]

Abstract: In this note we make some observations relating two papers recently published in Nonlinear Analysis Theory, Methods & Applications. [Copyright &y& Elsevier]

This paper is devoted to investigating a class of stochastic neutral-type neural networks with delays. By using the fixed point theorem and the properties of neutral-type operator, we obtain the existence conditions for periodic solutions of stochastic neutral-type neural networks. Furthermore, we obtain the conditions for the exponential stability of periodic solutions using Gronwall-Bellman inequality and stochastic analysis technique. Finally, a numerical example is given to show the effectiveness and merits of the present results. Our results can be used to obtain the existence and exponential stability of periodic solution to the corresponding deterministic systems. [ABSTRACT FROM AUTHOR]

Many techniques have been recently employed by researchers to address the challenges posed by fractional differential equations. In this paper, we investigate the concept of Ulam-Hyers stability for a class of neutral fractional stochastic differential equations by using the Banach fixed point theorem and the stochastic analysis techniques. An example is presented at the end of the paper to show the interest and the applicability of the results. [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, 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 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]

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 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 paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solution and extremal solutions for an initial-value problem of perturbed functional fractional differential equations with Caputo-Hadamard derivative. Our analysis relies on the fixed point theorem of Burton and Kirk and the concept of upper and lower solutions combined with a fixed point theorem in ordered Banach space established by Dhage and Henderson. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay principles combined with a technique based on vector-valued metrics and convergent to zero matrices. [ABSTRACT FROM AUTHOR]

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk's fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). [ABSTRACT FROM AUTHOR]

We propose necessary and sufficient conditions for the negativity of the two-point boundary value problem in the form of the Vallée-Poussin theorem about differential inequalities for the Hadamard fractional functional differential problem ... Here, the operator T:C→L∞ can be an operator with deviation (of delayed or advanced type), an integral operator or various linear combinations and superpositions. For example, the operator can be of the forms (Tx)(t)=q(t)x(t-τ(t)), (Tx)(t)=∫e1Q(t,s)x(θ(s))ds or (Tx)(t)=∫e1x(s)dsQ(t,s). We obtain explicit tests of negativity of Green's function in the form of algebraic inequalities. Our paper is the first one where a general form of the operator is considered with Hadamard fractional derivatives. [ABSTRACT FROM AUTHOR]

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al. 2015 Solutions to an advanced functional partial differential equation of the pantograph type (Proc. R. Soc. A471, 20140947 (doi:10.1098/rspa.2014.0947)). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups. [ABSTRACT FROM AUTHOR]

In this paper we provide sufficient conditions for the existence as well as the uniqueness of solution of general nonlinear functional differential equations, Our results will be obtained using suitable fixed point theorems, We give also some remarks about uniqueness theorem due to Dhage. [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. The summaries aim to provide library patrons with a quick understanding of the papers' contents and their relevance to their research interests. [Extracted from the article]

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 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]

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]

In the present paper, the evolutionary differential equations are investigated; the conditions for the solvability and uniqueness of the Cauchy problem for evolutionary differential equations are proved. [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 work focuses on stochastic functional differential equations (SFDEs) with wide-band noise perturbation, which is a class of actual physical process and has wide application in the modeling of communication and signal system. Using functional derivatives together with martingale methods and weak convergence techniques, this paper examines the asymptotic properties of the underlying systems as the small parameter tends to zero. Based on the techniques, this paper also establishes the average principle for a class of SFDEs with two-time scales. As a special case, this paper also considers a class of integro-differential equations with wide-band noise perturbation and examines its approximate properties. As an example, a scalar stochastic Lotka–Volterra integro-differential system with wide-band noise perturbation is investigated. [ABSTRACT FROM AUTHOR]

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This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of exponential stabilization of a one-dimensional thermoelastic diffusion problem of type II for isotropic and homogeneous materials with frictional damping and time delay. The system of equations under consideration is a coupling of three hyperbolic equations involving three variables considering frictional damping and time delay on each variable. Using semigroup theory, we prove the well-posedness by the Lumer–Phillips theorem and the exponential stability by exploring the dissipative properties of the linear operator associated with the model through the Gearhart–Huang–Pruss theorem. [ABSTRACT FROM AUTHOR]

This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov--Krasovskii function approach and the theory of stochastic analysis. The stability in pth-moment, the asymptotical stability in pth-moment, and the exponential stability in pth-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained. [ABSTRACT FROM AUTHOR]

In this paper, a study is carried out on the spatiotemporal dynamics of a Lengyel--Epstein model describing the chlorite-iodine-malonic-acid (CIMA) reaction with time delay and the Neumann boundary condition in a two-dimensional region. The existences for Turing, Hopf, Turing--Turing, Turing--Hopf, and Bogdanov--Takens bifurcations are derived by analyzing the dispersion relation between eigenvalues and wave numbers. In particular, to study the dynamics around a Turing--Hopf bifurcation singularity, the amplitude equations near a codimension-two bifurcation point are derived by employing the weakly nonlinear analysis method. Different spatiotemporal patterns for the system in parameter space are classified and various patterns identified, including spatially homogeneous periodic solutions, mixed mode, coexistence mode, bistable phenomenon, square, hexagon, black eye, two-phase oscillating staggered hexagon lattice, and other complex spatiotemporal patterns. The theoretical predictions are verified by numerical simulations showing an excellent agreement with many reported experiment results not only in chemistry but also in physics and biology. Results presented in this article reveal the mechanism of generating the spatiotemporal patterns of the CIMA reaction. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of d x a (s) + g (s , x a (s - ω (s))) = I x a (s) + f (s , x a (s - ϱ (s))) d s + ς (s) d ϖ H (s) , 0 ≤ s ≤ T , x a (s) = ζ (s) , - ρ ≤ s ≤ 0. The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence and uniqueness, the pth moment and almost sure stability with a general decay rate and the practical stability with general decay rate of a class of hybrid neutral stochastic functional differential equations driven by Lévy noise. Our crucial techniques include Lyapunov functions and the nonnegative semi-martingale convergence theorem. The conditions on the diffusion operator and neutral term are weaker than those in the related existing works. Examples are given to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper starts from several applications described by initial/boundary value problems for 1D (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties. [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]

This paper is devoted to investigating a class of neutral-type integral differential equations arising in an epidemic model. By using Mawhin's continuation theorem and the properties of neutral-type operators, we obtain the existence conditions for positive periodic solutions of the considered neutral-type integral differential equation. Compared with previous results, the existence conditions in this paper are less restricted, thus extending the results of the existing literature. Finally, two examples are given to show the effectiveness and merits of the main results of this paper. Our results can be used to obtain the existence of a positive periodic solution to the corresponding non-neutral-type integral differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the -mean for and apply the result to obtain confidence interval estimations for the approximate solutions. [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 paper, the functional differential equation (D C a + α x) (t) + ∑ i = 0 m (T i x (i)) (t) = f (t) , t ∈ [ a , b ] , with Caputo fractional derivative D C a + α is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green's functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green's functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality (D C a + α ν) (t) + ∑ i = 0 m (T i v (i)) (t) < 0 , t ∈ [ a , b ] . Choosing the function v, we obtain explicit sufficient tests of sign-constancy of Green's functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for "small" deviations. [ABSTRACT FROM AUTHOR]

In this paper, we establish the truncated Euler-Maruyama (EM) method for stochastic functional differential equation (SFDE) and consider the strong convergence theory for the numerical solutions of SFDEs under the local Lipschitz condition plus Khasminskii-type condition instead of the linear growth condition. The type of convergence specifically addressed in this paper is strong- convergence for , and p is a parameter in Khasminskii-type condition. We also discussed the rates of -convergence for the truncated EM method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of (ω , Q) -periodic mild solutions for the following problem x ′ (t) = A x (t) + f (t , x (t)) , t ∈ R , on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable C 0 -semigroup and the nonlinearity f satisfies suitable properties. The approaches are based on the well-known Banach contraction principle. In addition, a sufficient criterion is established for the existence and uniqueness of (ω , Q) -periodic mild solutions to the Hopfield-type neural network model. [ABSTRACT FROM AUTHOR]

The paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin-type condition and principle of comparison, this article first gives a lemma, discusses the simple system that does not consider impulse in the original system, then directly apply the conclusion of the lemma and use mathematical induction to get the main results of the theorem. In this paper, when allowing the original system to be unbounded and unstable, some sufficient conditions for pth moment globally exponentially ultimate boundedness and pth moment globally exponential stability are presented, and the linear coefficient of the upper bound of Lyapunov differential operator is time-varying function, and is not required to be negative definite. Finally, we use an example to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, new effective technique for investigation of the higher order differential equation (E) y(n)(t) = p(t)y(τ(t)). is established. We offer new criteria for excluding certain types of nonoscillatory solutions which improve all existing results from the literature. Examples are provided to illustrate the importance of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a size-stage-structured cooperation model which has two distinct life stages in facultative cooperator. The primary feature of this model is to consider size structure, stage structure and obligate and facultative symbiosis at the same time in a cooperation system. We use the method of characteristic to show that this new model can be reduced to a threshold delay equations (TDEs) model, which can be further transformed into a functional differential equations (FDEs) model by a simple change of variables. Such simplification allows us to apply the classical theory of FDEs and establish a set of sufficient conditions to investigate the qualitative analysis of solutions of the FDEs model, including the global existence and uniqueness, positivity and boundedness. What's more, we use the geometric criteria to get the conclusions about stability and Hopf bifurcation of positive equilibrium because the coefficients of the characteristic equation depend on the bifurcation parameter. Finally, numerical simulations are carried out as supporting evidences of our analytical results. Our results show that the presence of size structure and stage structure plays an important role in the dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem for a mean-field linear stochastic differential equation with a quadratic functional in the infinite time horizon. Under suitable conditions, including the stabilizability, the (strong) exponential, integral, and mean-square turnpike properties for the optimal pair are established. The keys are to correctly formulate the corresponding static optimization problem and find the equations determining the correction processes. These have revealed the main feature of the stochastic problems which are significantly different from the deterministic version of the theory. [ABSTRACT FROM AUTHOR]

In this paper, by taking Zvonkin's transformation, we investigate parameter estimation for a class of multidimensional stochastic differential equations with small perturbation parameters in diffusion coefficients, where the drift coefficients not only have an unknown parameter θ but also are Hölder continuous. These processes may enhance the applicability of our results to considerable practical models. Due to the irregular drift, the primary challenge is dealing with the mean square error between an accurate and numerical solution. Under these settings, we demonstrate the consistency and asymptotic normality of error concerning the least squares estimator in probability when stepsize $ \delta \rightarrow 0 $ δ → 0 and small parameter $ \varepsilon \rightarrow 0 $ ε → 0 simultaneously. Moreover, we extend the results to the case of stochastic functional differential equations. [ABSTRACT FROM AUTHOR]

The equation G (x , x) = y ˜ , where G : X × X → Y , and X , Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E + , M + of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a "model" equation, namely, g (t , t) = 0 , where a continuous map g : E + × E + → M + is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with "simpler" ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the "model" equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation. [ABSTRACT FROM AUTHOR]

Summary: How a network of multiple vehicles gets to a goal formation from an initial formation is as important as reaching the goal formation. Despite that stress matrix has been well applied to construct distributed affine formation control, most works have focused on the stabilization or stability issues and paid little attention to the performance regulation or the control parameterization issues. In this paper, we aim to design an error‐model based distributed control for the affine formation maneuver of second‐order multi‐agent systems (MASs), and to demonstrate the advantages of the control approach in performance tuning. The basic idea behind the control design is to force the actual error‐trajectory dynamics of each controlled follower to approximate that of an ideal second‐order error model. The designed controller possesses a notable feature that the historical acceleration command (HAC) is used to enhance system performance during the formation maneuvers. By a theory tool of neutral functional differential equations, we develop a delay‐independent condition on the design parameters, under which the system stability is robust to the actively introduced HAC. Furthermore, we derive an inequality describe the relationship between the ultimate bounds of tracking errors and the delay parameter. This paper presents a clear mechanism to tune the comprehensive performance of affine formation control. Simulation results show the effectiveness of the stability condition and the performance‐tuning mechanism, as well as the possible fragility of the system stability to the delay when the condition is not satisfied. [ABSTRACT FROM AUTHOR]

In this study, we pay attention to a functional differential equation (FDE) of first order including N-variable delays. We construct new sufficient conditions in relation to the Hyers-Ulam stability (HUS) and the generalized Hyers-Ulam-Rassias stability (GHURS) of the FDE of first order including N-variable delays. By using Banach contraction principle (BCP), Picard operator and Gronwall's lemma, we confirm two new theorems with regard to the HUS and the GHURS. The results of this study are new, they extend and improve some earlier results of the HUS and the GHURS. [ABSTRACT FROM AUTHOR]