Nonlinear Functional Differential Equations Research Articles

Method of Lines for Quasilinear Functional Differential Equations

We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the numerical method of lines for evolution functional differential equations. The initial boundary-value problems for quasilinear equations are transformed (by means of discretization in spatial variables) into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given.

Variation formulas of solution for a functional differential equation with delay function perturbation

Variation formulas of solution for a nonlinear functional differential equation with variable delay and continuous initial condition are proved. The effects of delay function perturbation and continuous initial condition are detected in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

Oscillation Criteria for Higher order Nonlinear Functional Differential Equations with Advanced Argument

We consider a differential equation $$ {u^{(n) }}(t)+p(t)\left| {u\left( {\sigma (t)} \right)\left| {^{{\mu (t)}}} \right.\mathrm{sign}\,u\left( {\sigma (t)} \right)=0,} \right. $$ where p ∈ L loc (R +; R +), μ ∈ C(R +; (0, +∞)), σ ∈ C(R +; R +), and σ(t) ≥ t for t ∈ R +. We say that the equation is almost linear if the condition lim t→+∞ μ(t) = 1 is satisfied. At the same time, if lim sup t→+∞ μ(t) ≠ 1 or lim inf t→+∞ μ(t) ≠ 1, then Eq. is called an essentially nonlinear differential equation. The oscillatory properties of almost linear differential equations have been extensively studied. In the paper, new sufficient (necessary and sufficient) conditions are established for a general class of essentially nonlinear functional differential equations to have Property A.

LaSalle-Type Theorems for General Nonlinear Stochastic Functional Differential Equations by Multiple Lyapunov Functions

We investigate LaSalle-type theorems for general nonlinear stochastic functional differential equations. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions. For the typical special case with estimations involving|xt|pfor the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem. At the end of the paper, an example is given to illustrate the usage of the method proposed and show the advantage of the results obtained.

Stability Analysis of Runge-Kutta Methods for Nonlinear Functional Differential and Functional Equations

This paper is concerned with the numerical stability of Runge-Kutta methods for a class of nonlinear functional differential and functional equations. The sufficient conditions for the stability and asymptotic stability of(k,l)-algebraically stable Runge-Kutta methods are derived. A numerical test is given to confirm the theoretical results.

Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions

We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional dierential equations. Dierence methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite dierence systems. Methods of second type are truncation of the infinite dierence system, so that the resulting dierence problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-dierenti al problems can be obtained from the general model by specifying the given operators.

Asymptotic Properties of Solutions to Third-Order Nonlinear Neutral Differential Equations

The aim of this work is to discuss asymptotic properties of a class of third-order nonlinear neutral functional differential equations. The results obtained extend and improve some related known results. Two examples are given to illustrate the main results.

Stability Analysis of Impulsive Stochastic Functional Differential Equations with Delayed Impulses via Comparison Principle and Impulsive Delay Differential Inequality

The problem of stability for nonlinear impulsive stochastic functional differential equations with delayed impulses is addressed in this paper. Based on the comparison principle and an impulsive delay differential inequality, some exponential stability and asymptotical stability criteria are derived, which show that the system will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous stochastic flows. The obtained results complement ones from some recent works. Two examples are discussed to illustrate the effectiveness and advantages of our results.

On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order

The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.

New Oscillation Criteria for Third-Order Nonlinear Functional Differential Equations

This paper discusses oscillatory and asymptotic behavior of solutions of a class of third-order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, three new sufficient conditions which insure that the solution is oscillatory or converges to zero are established. The results obtained essentially generalize and improve the earlier ones.

Existence of Gevrey solutions for some polynomially nonlinear functional differential equations

Existence of Gevrey solutions for some polynomially nonlinear functional differential equations

On a class of impulsive functional-differential equations with nonatomic difference operator

We establish conditions for the existence and uniqueness of the solutions of nonlinear functional-differential equations with impulsive action in a Banach space. The equation under consideration is not solved for the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in the right half-plane. Applications to partial functional-differential equations not of Kovalevskaya type are considered.

Periodic solutions for fourth‐order nonlinear functional difference equations

By using the critical point method, some new criteria are obtained for the existence and multiplicity of periodic solutions for fourth‐order nonlinear functional difference equations. The proof is based on the linking theorem in combination with variational technique. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.

Homoclinic orbits of nonlinear functional difference equations with Jacobi operators

By using the critical point theory, the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for difference equations containing both advance and retardation is obtained. The proof is based on the mountain pass lemma in combination with periodic approximations. Our results extend the results of 2007.

Method of lines for nonlinear first order partial functional differential equations

Classical solutions of initial problems for nonlinear functional differential equations of Hamilton--Jacobi type are approximated by solutions of associated differential difference systems. A method of quasilinearization is adopted. Sufficient conditions for the convergence of the method of lines and error estimates for approximate solutions are given. Nonlinear estimates of the Perron type with respect to functional variables for given operators are assumed. The proof of the stability of differential difference problems is based on a comparison technique. The results obtained here can be applied to differential integral problems and differential equations with deviated variables.

On the instability of a kind of vector functional differential equations of the eighth order with multiple deviating arguments

In this paper, we investigate the instability of solutions to a certain class of nonlinear vector functional differential equations of the eighth order with n-deviating arguments. We employ the Lyapunov-Krasovskii functional approach and base on the Krasovskii criteria to prove two new theorems on the topic. Our results improve certain results in the literature from scalar functional differential equations to their vectorial forms.

Stability and convergence of Runge-Kutta methods for nonlinear neutral functional differential equations

This paper concerns the stability and convergence of Runge-Kutta methods for nonlinear neutral functional differential equations (NFDEs). Based on the B-theory of Runge-Kutta methods for Volterra functional differential equations (VFDEs), we introduce the concepts of EB-stability and EB-convergence of Runge-Kutta methods for NFDEs and obtain EB-stability and EB-convergence results on variable stepsize Runge-Kutta methods for this class of equations. These results are new even for neutral delay differential equations and neutral delay integro-differential equations.

Novel criteria for exponential stability of functional differential equations

We first give explicit criteria for exponential stability of general linear nonautonomous functional differential equations. Then the obtained results are extended to nonlinear functional differential equations. Two examples are given to illustrate the results. To the best of our knowledge, the results of this note are new.

Approximation of nonlinear differential functional equations

We have investigated the approximation of an element with delay in $ {{\mathbb{R}}^n} $ for the case of a continuous input function. We have constructed and substantiated a scheme of the approximation of solutions of an initial-value problem for a nonlinear differential functional equation by the solutions of the Cauchy problem for a system of ordinary differential equations. Calculations for the model example based on the proposed scheme of approximation are in good agreement with well-known results in the literature.

On q-difference asymptotic solutions of a system of nonlinear functional differential equations

The asymptotic behavior and the existence of a continuously differentiable solution bounded together with its first derivative of a system of nonlinear functional differential equations is studied here.