Nonlinear Neutral Functional Differential Equations Research Articles

Stability and boundedness criteria for certain second-order nonlinear neutral stochastic functional differential equations

This paper presents stochastic stability and stochastic boundedness to certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is weakened to a neutral stochastic system of first-order equations and used together with a second-order quadratic function to obtain perfect Lyapunov-Krasovskii functional. This functional is adapted and applied to obtain criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, fresh results on stochastic boundedness and uniform stochastic boundedness of solutions are obtained. The results of this paper are original, new, essentially improving, complementing, and simplifying several related ones in the literature. Two special cases of the theoretical results are supplied to demonstrate the applicability of the hypothetical results.

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay

Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay

An efficient IMEX method for nonlinear functional differential equations with state-dependent delay

Implicit-explicit midpoint (IMEX MP) formula is proposed to solve numerically nonlinear neutral functional differential equations (NFDEs) with state-dependent delay. Since the deviating argument will be explicitly treated, the proposed method is extremely effective. Second order convergence of this method is shown theoretically and numerically for NFDEs with smooth solutions and Volterra functional differential equations (VFDEs). For general NFDEs with nonsmooth solutions, regularization problem by smoothing the initial data is introduced and analyzed. The regularization error is derived in maximum norm. The global errors of IMEX MP method are derived and reveal that it is only first order of convergence for this regularization error or for the non-smoothness of the original problems. Numerical examples are provided to support the theoretical analysis, compared with IMEX Euler method.

Delay‐dependent stability of highly nonlinear neutral stochastic functional differential equations

AbstractThis article focuses on the delay‐dependent stability of highly nonlinear hybrid neutral stochastic functional differential equations (NSFDEs). The delay dependent stability criteria for a class of highly nonlinear hybrid NSFDEs are derived via the Lyapunov functional. The stabilities discussed in this article include stability, asymptotically stability and exponential stability. A numerical example is given to illustrate the criteria established.

Complete controllability of nonlinear fractional neutral functional differential equations

This paper is concerned with the complete controllability of a nonlinear fractional neutral functional differential equation. Some sufficient conditions are established for the complete controllability of the nonlinear fractional system. The conditions are established based on the fractional power of operators and the fixed-point theorem under the assumption that the associated linear system is completely controllable. Finally, an example is presented to illustrate our main result.

The stability with general decay rate of neutral stochastic functional hybrid differential equations with Lévy noise

This paper is concerned with the existence and uniqueness, the almost sure stability with general decay rate (including almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability) for the global solution of nonlinear neutral stochastic functional hybrid differential equations with Lévy noise. The key technique used is the method of Lyapunov function, nonnegative semi-martingale convergence theorem and the theory of M-matrix. We use auxiliary functions to dominate the corresponding Lyapunov function and the diffusion operator. Our conditions on the diffusion operator are weaker than those in the related existing works. Finally, one example is given to show the effectiveness of the obtained theory.

Asymptotic Stability of Neutral Set-Valued Functional Differential Equation by Fixed Point Method

This paper studies a class of nonlinear neutral set-valued functional differential equations. The globally asymptotic stability theorem with necessary and sufficient conditions is obtained via the fixed point method. Meanwhile, we give an example to illustrate the obtained result.

Three positive periodic solutions of second order nonlinear neutral functional differential equations with delayed derivative

This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity (x(t)-cx(t-delta)){''}+a(t)g(x(t))x(t)=lambda b(t)f(t,x(t),x(t-tau_{1}(t)),x'(t-tau_{2}(t))). By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.

Analytical and numerical dissipativity of neutral functional differential equations

In this paper, the dissipativity of a class of nonlinear neutral functional differential equations is studied. A new generalized discrete Halanay-type inequality is derived and the dissipativity of implicit Euler method for the above-mentioned equations is given by applying this inequality.

Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing variable delay

The purpose of this paper is to obtain the error bounds of fully geometric mesh one-leg methods for solving the nonlinear neutral functional differential equation with a vanishing delay. For this purpose, we consider Gq-algebraically stable one-leg methods which include the midpoint rule as a special case. The error of the first-step integration implemented by the midpoint rule on [0,T0] is first estimated. The optimal convergence orders of the fully geometric mesh one-leg methods with respect to T0 and the mesh diameter \(h_{\max }\) are then analyzed and provided for such equation. Numerical studies reported for several test cases confirm our theoretical results and illustrate the effectiveness of the proposed method.

Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second Order Hybrid Neutral Functional Differential Equations

Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second Order Hybrid Neutral Functional Differential Equations

A new generalization of Halanay-type inequality and its applications

In this paper, in order to study the dissipativity of nonlinear neutral functional differential equations, a generalization of the Halanay inequality is given. We apply this generalized Halanay inequality to an analysis of the dissipativity of two classes of nonlinear neutral delay integro-differential equations and the sufficient conditions are presented to ensure these systems are dissipative.

Periodicity and stability in neutral nonlinear differential equations by Krasnoselskii’s fixed point theorem

The nonlinear neutral functional differential equation with variable delay is investigated. By using Krasnoselskii’s fixed point theorem we obtain the existence and the asymptotic stability of periodic solutions. Sufficient conditions are established for the existence and the stability of the above equation. Our results extend some results obtained in the work [19].

Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of G-Brownian motion

In the past decades, quantitative study of different disciplines such as system sciences, physics, ecological sciences, engineering, economics and biological sciences, have been driven by new modeling known as stochastic dynamical systems. This paper aims at studying these important dynamical systems in the framework of G-Brownian motion and G-expectation. It is demonstrated that, under the contractive condition, the weakened linear growth condition and the non-Lipschitz condition, a neutral stochastic functional differential equation in the G-frame has at most one solution. Hölder’s inequality, Gronwall’s inequality, the Burkholder-Davis-Gundy (in short BDG) inequalities, Bihari’s inequality and the Picard approximation scheme are used to establish the uniqueness-and-existence theorem. In addition, the stability in mean square is developed for the above mentioned stochastic dynamical systems in the G-frame.

On stability of nonlinear neutral functional differential equations

We address the challenging problem of the exponential stability of nonlinear time-varying functional differential equations of neutral type. By a novel approach, we present explicit sufficient conditions for the exponential stability of nonlinear time-varying neutral functional differential equations. A discussion of the obtained results and illustrative examples are given.

Neutral functional differential equations and systems of conservation laws

A natural way of introducing neutral functional differential equations is integration along the characteristics of the Riemann invariants of the hyperbolic partial differential equations. Up to now the problem has been discussed by A.D. Myshkis, K.L. Cooke and their followers in the linear or quasilinear case. The conservation laws are nonlinear hyperbolic partial differential equations whose study goes back to the fifties of the previous century, being due to the pioneering papers of O.A. Oleinik and P.D. Lax. These equations describe important phenomena in Physics and Engineering, being also subject to control issues. The present paper is an attempt to extend the method of integrating along the characteristics to the systems of conservation laws. The main purpose of this attempt is to emphasize the occurring nonlinear neutral functional differential equations and to construct the augmented validation (existence, uniqueness, continuous data dependence, Stability Postulate) at least at the level of the classical solutions.

Stability Analysis of Nonlinear Neutral Functional Differential Equations

We present a novel approach to analyze the exponential stability of nonlinear neutral functional differential equations. Our approach is simple and based on a system transformation, the comparison principle, and the spectral properties of Metzler matrices. Consequently, we derive some new explicit criteria for the exponential stability of general nonlinear neutral functional differential equations. It is important to note that our approach leads to both delay-dependent and delay-independent stability criteria. Several illustrative examples with simulations and a brief comparison between the obtained results with some existing results in the literature are presented.

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

Abstract In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation d dt x ( t ) = − a ( t ) h ( x ( t ) ) + d dt Q ( t , x ( t − τ ( t ) ) ) + G ( t , x ( t ) , x ( t − τ ( t ) ) ) . $${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.