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This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout. [ABSTRACT FROM AUTHOR]

The paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders. The method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. The numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional-differential equations with proportional delays including equations of neutral type. The method is applicable to both initial and boundary value problems. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with a class of partial functional differential equation with a small dissipating parameter on a rectangular domain. Classical numerical methods for solving this type of problems reveal disappointing behavior or are tremendously expensive in computer memory and processor time. This arises because the precision of an approximate solution depends inversely on perturbation parameter values and, thus, it deteriorates as a parameter decreases. Therefore, it is of particular interest to develop numerical methods whose error estimates would be independent of the perturbation parameter contaminating the solution. In order to overcome the said difficulty we derive robust parameter uniform error estimates for a class of partial functional differential equations. The analysis presented in this paper uses a suitable decomposition of the error into smooth and a singular component combined with the appropriate barrier functions and comparison principle. [Copyright &y& Elsevier]

Abstract: We give sufficient conditions for the system of nonlinear functional differential equations of the formwhere A is a matrix, , , , and functions , , , , , , are continuous, for the unique existence of continuously differentiable solutions which are bounded together with their first derivatives on and satisfy the condition . [Copyright &y& Elsevier]

Abstract: Some conditions that guarantee the existence of a family of solutions converging to zero of a system of nonlinear functional differential equations with iterated deviating arguments are given. [Copyright &y& Elsevier]

Abstract: Sufficient conditions for the existence of a unique bounded solution together with its first derivative on of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function, and satisfying the condition , are given. [Copyright &y& Elsevier]

Abstract: Some sufficient conditions which guarantee the existence of a bounded solution together with its first derivative on the whole real line of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function, are given. [Copyright &y& Elsevier]

Abstract: In this paper, we establish some new oscillation criteria for the following forced functional differential equationNo restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. Two cases have been taken into consideration: (i) and ; (ii) q(t) changes its sign, 0< λ <1 and . The main results generalize some existing results and can be applied to the oscillation problem that are not covered in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate stability of sets for a class of impulsive functional differential equations by using piecewise continuous Lyapunov functions with Razumickhin techniques. Some sufficient conditions for stability of sets are established, and some known stability theorems also are generalized. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of three positive T-periodic solutions of the first order functional differential equations of the formwhere and are positive T-periodic functions, and λ is a positive parameter. Leggett–Williams multiple fixed point theorem has been used to prove our results. Our results extend and improve some results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the existence of analytic solution of a functional differential equation , where are four complex numbers. We first discuss the existence of analytic solutions for some special cases of the above equation. Then, by reducing the equation with the Schröder transformation to the another functional equation with proportional delay, an existence theorem is established for analytic solutions of the original equation. For the constant given in the Schröder transformation, we discuss the case and on the unit circle , i.e., . We study is at resonance, i.e., at a root of the unity and is near resonance under the Brjuno condition. [Copyright &y& Elsevier]

Abstract: In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearitieswhere , , , , . Without assuming that , and are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation. [Copyright &y& Elsevier]

Abstract: We offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the formandwhen . The case when is also included. [Copyright &y& Elsevier]

Abstract: By improving the bracketing techniques for a functional differential inequality in a previous paper [H.W. Wu, Y.T Xu, The distribution of zeros of solutions of neutral differential equations, Appl. Math. Comput. 156 (3) (2004) 665–677], we are able to derive sharper upper bounds on the distance between zeros of solutions of a class of neutral functional differential equations with delays, and hence improve many known bounds in the literature. [Copyright &y& Elsevier]

Abstract: In this article, we investigate the existence of positive periodic solutions for a class of functional differential equations. By using fixed point index theory, we establish sufficient criteria that are easily verifiable and that generalize and improve related studies in the literature. Also, we give some examples in bio-mathematics to show that our main results are valid. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to the investigation of the oscillation of a kind of very extensively higher order nonlinear functional differential equations with impulses, some interesting results are obtained. The results extend and improve the earlier publications. Two examples are given to illustrate the theory. [Copyright &y& Elsevier]

We apply the generalized form of Leggett–Williams fixed point theorem to prove that the following nonlinear functional differential equationhas at least two positive T-periodic solutions, where a(t) is a T-periodic function satisfying exp(∫0Ta(u) du)>1, f(t,xt) is a nonnegative function defined on R×BC, where BC denotes the Banach space of bounded continuous functions. [Copyright &y& Elsevier]

This paper is concerned with a functional differential equation x′(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b2y′(z)=(y(μ2z)−ay(μz))(μy′(μz)−ay′(z)), analytic solutions of the form 1/b(y(μy−1(z))−az) for the original differential equation are obtained. [Copyright &y& Elsevier]

This paper investigate a class of higher order functional differential equations with damped, and derive two new oscillatory criteria of solution. [Copyright &y& Elsevier]

In this article, the stability problem of impulsive reaction–diffusion functional differential equations (IRFDEs) is discussed. Applying Lyapunov functional method and the comparison principle of IRFDEs, some useful criteria are acquired to assure uniform stability and uniformly asymptotical stability of the equilibrium point of IRFDEs, respectively. Finally, a numerical example is provided to show the validity of our findings. [ABSTRACT FROM AUTHOR]