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This work focuses on functional differential equations subject to wideband noise perturbations. Modeling using a white noise is often an idealization of the actual physical process, whereas a wideband noise can be easily realized in applications and well approximates a white noise. Using functional derivatives together with the combined perturbed test function methods and martingale techniques, this paper demonstrates that when a small parameter tends to zero, the underlying process converges to a limit that is the solution of a stochastic functional differential equation. To illustrate, an integro-differential system with wideband noise perturbation is examined as an example. Not only are the results interesting from a mathematical point of view, but also they are of utility to a wide range of applications.

We study the Dirichlet problem for a functional differential equation containing shifted and contracted argument under the Laplacian sign. We establish conditions for the unique solvability and demonstrate also that the problem may have an infinite dimensional solution manifold.

In this paper, under Acquistapace–Terreni conditions, we study the existence of periodic solutions for some non-autonomous semilinear partial functional differential equation with delay. We assume that the linear part is non-densely defined. The delayed part is assumed to be ω -periodic with respect to the first argument. Using a fixed point theorem for multivalued mapping, some sufficient conditions are given to prove the existence of periodic solutions. An example is shown to illustrate our results.

This paper is concerned with the oscillation of a class of partial functional population model, and some sufficient conditions for all positive solutions of the model to oscillate about the positive equilibrium are obtained. The approach is based on the upper- and lower-solution method of the partial functional differential equations and the oscillation theory of the functional differential equation. Moreover, some numerical simulations are also given to illustrate our results.

In this paper we stochastically perturb the functional Kolmogorov-type system x ˙ ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) f ( x t ) into the stochastic functional differential equation d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ f ( x t ) d t + g ( x t ) d w ( t ) ] . This paper studies existence and uniqueness of the global positive solution, and its asymptotic bound properties and moment average in time. These properties are natural requirements from the biological point of view. As the special cases, we discuss the various stochastic Lotka–Volterra systems.

In this paper, sufficient conditions are given for asymptotic stability and uniformly asymptotic stability of impulsive functional differential equation of the form { x ′ ( t ) = f ( t , x t ) , t ⩾ t 0 , Δ x = I k ( t , x ( t − ) ) , t = t k , k ∈ Z + , and the conditions in theorems extend or improve the corresponding ones in [J. Yan, J. Shen, Impulsive stabilization of functional differential equation by Liapunov–Razumikhin functions, Nonlinear Anal. 37 (1999) 245–255; J. Shen, Z. Luo, Impulsive stabilization of functional differential equations via Liapunov functionals, J. Math. Anal. Appl. 240 (1999) 1–15; L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Liapunov functionals, Trans. Amer. Math. Soc. 354 (2002) 3555–3571].

By using the coincidence degree theory of Mawhin, we study a kind of high-order neutral functional differential equation with distributed delay as follows: ( x ( t ) − c x ( t − σ ) ) ( n ) + f ( x ( t ) ) x ′ ( t ) + g ( ∫ − r 0 x ( t + s ) d α ( s ) ) = p ( t ) . Some new results on the existence of periodic solutions are obtained.

With the Lyapunov second method, we study the abstract functional differential equation, d u d t = f ( t , u t ) . We obtain inequalities of solutions and exponential stability with conditions like: (i) W 1 ( | u ( t ) | X ) ⩽ V ( t , u t ) ⩽ W 2 ( D ( t , u t ) ) + ∫ t − h t L ( s ) W 1 ( | u ( s ) | X ) d s , (ii) V ( 1 ) ′ ( t , u t ) ⩽ − η ( t ) W 2 ( D ( t , u t ) ) + P ( t ) . Applications in ordinary and partial functional differential equations are given.

Asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established. In the scalar case, it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation. The proofs are based on a recent Perron type theorem for functional differential equations.

A Perron type theorem about the existence of the strict Lyapunov exponents of the solutions of retarded functional differential equations is established.

This paper considers the existence of extreme solutions of the periodic boundary value problems for second order functional differential equations. A new concept of lower and upper solutions is introduced. And we present a new comparison principle. Meanwhile, we extend previous results.

Under suitable conditions on f(t,y(t+θ)), the boundary value problem of higher-order functional differential equation (FDE) of the form (BVP) (FDE) y (n) (t)+f(t,y(t+θ))=0, t∈[0,1], θ∈[−τ,a], (BC) y (i) (0)=0, 0⩽i⩽n−3, αy (n−2) (t)−βy (n−1) (t)=η(t), t∈[−τ,0], γy (n−2) (t)+δy (n−1) (t)=ξ(t), t∈[1,1+a], has at least one positive solution, where θ∈[−τ,a] is a fixed constant. Moreover, we also apply this main result to establish several existence theorems which guarantee (BVP) has multiple positive solutions.

In this paper, we consider a kind of neutral functional differential equation as follows: x(t)−cx(t−τ)′=gt,xt−μ(t)+e(t) in the critical case |c|=1. By employing the topological degree theory and some analysis techniques, we obtain some new results on the existence of periodic solutions.

In this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y ′( t )=− a ( t ) y ( t )+ λh ( t ) f ( y ( t − τ ( t ))). By the eigenvalue problems of completely continuous operators and theory of α -concave or − α -convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution y λ ( t ) of the above equation depends continuously on the parameter λ . Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model.

In this paper, we deal with the boundary value problem x″=f t,x,x t ,x′,x′ t , x 0 ,x′ 0 ∈ (φ+c 1 ,ψ+c 2 ): c 1 ,c 2 ∈R , α(x| J )=0, β x′(1)−δx′| J =0, where f ∈Car( J × R × C r × R × C r ), φ , ψ ∈ C r , J :=[0,1], δ ≠1, α∈ A J , β∈ A [0,1) , A J and A [0,1) are two special sets of functionals, and x | J is the restriction of x to J . We find sufficient conditions for the existence of solutions of the above problem. The proof is based on the Leray–Schauder degree theory.

We obtain sufficient conditions for the existence of periodic solutions of the following third order nonlinear functional differential equations x‴(t)+ax″ 2k−1 (t)+bx′ 2k−1 (t)+x 2k−1 (t)+g t,x(t−τ 1 ),x′(t−τ 2 ) =p(t)=p(t+2π). Our approach is based on the continuation theorem of the coincidence degree, and the a priori estimate of periodic solutions.

In this paper, we employ Avery–Henderson fixed point theorem to study the existence of positive periodic solutions to the following nonlinear nonautonomous functional differential system with feedback control: dx dt =−r(t)x(t)+F(t,x t ,u(t−δ(t))), du dt =−h(t)u(t)+g(t)x(t−σ(t)). We show that the system above has at least two positive periodic solutions under certain growth condition imposed on F.

We study the existence of positive solutions for the following boundary value problem on infinite interval for second-order functional differential equations: x″(t)−px′(t)−qx(t)+f(t,xt)=0,t∈[0,∞),αx(t)−βx′(t)=ξ(t),t∈[−τ,0],limt→∞x(t)=0, and x″(t)−px′(t)−qx(t)+f(t,xt,x′t)=0,t∈[0,∞),αx(t)−βx′(t)=ξ(t),t∈[−τ,0],limt→∞x(t)=0, where p,α,β⩾0, α2+β2>0, and q>0. The fixed point theorem on cone is used.

In this paper we investigate the oscillation of forced functional differential equations x ( n ) t + a t f x q t = e t , when the forcing term e(t) is not required to be the nth derivative of an oscillatory function. Some new oscillatory criteria are given. We also apply our technique to the forced neutral differential equation of the form x t + cx t − τ ( n ) + a t x t + b t x t − τ = e t + c t f 1 x t + d t f 2 x t − τ , where xf1(x) > 0 and xf2(x) > 0 for x ≠ 0; n ≥ 1; τ, δ are nonnegative constants; a(t) > 0; b(t) > 0; c(t) > 0; d(t) > 0; which includes the special case f1(x) = |x|λ sgn x, f2(x) = |x|θ sgn x, λ ≠ 1 and θ ≠ 1.

This paper is concerned with an iterative functional differential equation x ′( x [ r ] ( z )) = c 0 z + c 1 x ( z ) + c 2 x ( x ( z )) + ⋯ + c m x [ m ] ( z ), where r and m are nonnegative integers, x [0] ( z ) = z , x [1] ( z ) = x ( z ), x [3] ( z ) = x ( x ( x ( z ))), etc. are the iterates of the function x ( z ), and ∑ j = 0 m c j ≠ 0. By constructing a convergent power series solution y ( z ) of a companion equation of the form α y ′(α r + 1 z ) = y ′(α r z ) ∑ j = 0 m c j y (α j z ), analytic solutions of the form y (α y − 1 ( z )) for the original differential equation are obtained.

Oscillation criteria for nth order differential equations with deviating arguments of the form x ( n − 1) ( t ) α − 1 x ( n − 1) ( t ) ′ + F(t, x[g(t)]) = 0, n even are established, where g ∈ C([t0, ∞), R ), F ∈ C([t0, ∞) × R , R ), and α > 0 is a constant.

Our aim is to obtain conditions under which every solutionyof the vector functional differential equation[formula]can be estimated in terms of a solution ω of a scalar functional equation[formula]wherea(x) is a suitable real positive function andbis a suitable real negative constant. Moreover, ifB(x) is diagonal matrix, we derive more precise estimates ofyin terms of a solution ν of a vector functional equation[formula] where the elements ofA + (x) are formed by the absolute values of the corresponding elements ofA(x). These asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems.

It is shown that every solution of the nonhomogeneous functional differential equation d dt (x(t)−px(t−τ))+Q(t)G(x(t−σ))=f(t), wheref, Q∈C([T, ∞), (0, ∞)), σ, τ∈(0, ∞), 0≤p 0 forx≠0,Gis nondecreasing, Lipschitzian, and satisfy a sublinear condition ∫ 0 ±k dx G(x) and ∫ ∞ f(s) ds is either oscillatory or tends to zero asymptoticallyif and only if ∫ T ∞ Q(s) ds=∞.

In this paper we consider a sufficient condition for W ( t , x ( t )) to approach zero as t →∞, where x ( t ) is a solution of a non-autonomous functional differential equation with finite delays and W ( t , x ) is a so-called Lyapunov function. We shall show that in the applications this provides useful information for asymptotic behavior of the solution x ( t ). For example, we generalize examples given by J. R. Haddock and J. Terjeki ( J. Differential Equations 48 , 1983, 95–122) to the case of non-autonomous systems.

We consider the periodic scalar neutral functional differential equation ( d / dt )[ x ( t )− c ( t ) x ( t −τ)]=− h ( t , x ( t ))+ h ( t −σ, x ( t −σ)), where c is continuously differentiable, h is increasing in its second argument, and both c and h are 1-periodic in the t -variable. The two time-lags τ and σ are not required to be the same. It is shown that, under certain conditions, (i) the set of 1-periodic solutions is an ordered arc and each solution is convergent to a periodic solution; (ii) the asymptotic and oscillatory behaviors of each solution are completely classified in terms of the value of the first integral at the initial condition.

In this paper we deal with the linear neutral functional differential equation d[f(xt)]/dt + g(xt) = 0. We conclude that all small components of solutions vanish after a finite time, as do all small solutions, and that the equation is oscillatory if and only if its characteristic equation has no real roots.

In this paper, we establish the theorems of existence of an almost periodic solution of an almost periodic neutral functional differential equation and its disturbed systems by Liapunov functional.

Our aim in this paper is to obtain new necessary and sufficient conditions under which every solution of the functional differential equation with several deviating arguments of the delay y (t)+py(t)+ ∑ i=1 n p i y(t−τ i )=0 will be oscillating. Here pi 0, τi > 0, and p are all constants, i = 1, 2, …, n. Our results are more explicit and more precise than those of Refs. [5, 6, 8, 9] related to Eq. (2).

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 1, he has the following results: (i) If B is diagonalizable with eigenvalues bi such that Re bi>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(tα) as t → ∞. (ii) If B is diagonalizable with eigenvalues bi such that Re b1 ⩽ Re b2 ⩽ ··· ⩽ Re bn

Some new oscillation criteria for the forced functional differential equation x (n) (t) + a(t) ƒ(x[q(t)]) = e(t) are established.

In this paper we present an extension of the Alekseev variation of constant formula for a nonlinear neutral functional differential equation and make an application to the relative asymptotic equivalence between the solutions of the systems: (1) (d dt) Dy 1 = f(t, y t ) and (2) (d dt) Dx t = f(t, x t ) + g(t, x t ) , where the equation (1) is a nonlinear system of neutral functional differential equation with f and g mapping bounded closed sets into bounded sets.

has no nontrivial T-periodic solution (see the books of Krasnosel’skii [6], Reissig, Sansone, and Conti [14], and Roseau [15]). This result has been extended to the case of the vector retarded functional differential equation by Fennel1 [2]. When the requirement upon the linear part is not satisfied, supplementary conditions upon f are needed to ensure the existence of T-periodic solutions, as follows immediately from the Fredholm alternative [4] for the special case off independent of x. Such auxiliary conditions appear in papers of Lazer [7] (for a second order scalar differential equation), Ezeilo [l], Sedsiwy [16], Villari [19], and the author [8] (f or a third order scalar differential equation), Sedsiwy [17] and Reissig [ll, 121 (for the n-th order scalar case) and the same authors [18, 131 f or the corresponding vector case. Also, Fennel1 [2] has extended Lazer’s result and method to the retarded case. Those results are proved by various methods (Brouwer, Schauder, or Leray-Schauder fixed point theorems) which make the arguments rather tedious. On the other hand, all the equations considered in those papers are

In this paper, we consider linear and nonlinear perturbations of a linear autonomous functional differential equation which has infinitely many equilibria. We give sufficient conditions under which the solutions of the perturbed equation tend to the equilibria of the unperturbed equation at infinity. As a consequence, we obtain sufficient conditions for systems of delay differential equations to have asymptotic equilibrium.

The aim of the present paper is to investigate the asymptotic behavior of the solutions of the asymptotically autonomous retarded functional differential equation ẋ(t) = Lxt + M(t)xt. We obtain a convergence result without assuming that ‖M(t)‖ is in Lp, 1 ≤ p < + ∞. Moreover, we give bounds on the rate of convergence as well.

By means of the abstract continuation theory for k-contractions, some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation: dNdt=N(t)a(t)−β(t)N(t)−∑j=1nbj(t)Nt−σj(t)−∑i=1mci(t)N′t−τi(t).

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form ( x ( t ) + B x ( t − δ ) ) ′ = g 1 ( t , x ( t ) ) + g 2 ( t , x ( t − τ ) ) + p ( t ) .

Existence and asymptotic behavior of solutions are given for the equation u ′( t ) = − A ( t ) u ( t ) + F ( t , u t ) ( t ⩾ 0) and u 0 = ϑ ϵ C ([− r ,0]; X )  C . The space X is a Banach space; the family {A(t) ¦ 0 ⩽ t ⩽ T} of unbounded linear operators defined on D ( A ) ⊂ X → X generates a linear evolution system and F : C → X is continuous with respect to a fractional power of A ( t 0 ) for some t 0 ϵ [0, T ].

There have been a lot of investigations about stability of the linear scalar functional differential equation x′(t)=a(t)x(t)+b(t)x(t−h), where a,b:R+→R continuous and h>0 a constant. However, almost all investigations require a(t)⩽0 for asymptotic stability and a(t)⩾0 for instability. In this paper, we investigate Wazewski inequalities of solutions of the equation. As a consequence, we offer some sufficient conditions for asymptotic stability if a(t)⩾0 and instability if a(t)⩽0. In the case that a(t) and b(t) are constant, we offer a region showing uniform asymptotic stability and instability of the zero solution of the equation. This region is different from J. Hale's (1977).

This paper concerns the regularity of a functional differential equation in the form: u ˙ ( t ) = Au ( t ) + B 1 u ( t − r ) + ∫ − r 0 a ( s ) B 2 u ( t + s ) d s , t > 0 , where A is the generator of an analytic semigroup on a Banach space X , and B 1 , B 2 are ( γ − A ) α -bounded linear operator for 0 α 1 . By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable.

By using the theory of coincidence degree, we study a kind of periodic solutions to p-Laplacian neutral functional differential equation with multiple deviating arguments, a new result on the existence of periodic solutions is obtained.

In this paper, we employ a well-known fixed-point index theorem to study the existence and non-existence of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. Several existence and non-existence results are established.

We resolve some conjectures concerning the zeros t n of the solution f of the initial value problem f ′( t ) = − f ( qt ), 0 q f (0) = 1. In particular, lim n → ∞ t n + 1 / t n = 1/ q .

The convergence of all solutions of the scalar equation x =f(t,x t ) is studied under the main assumption that every constant is its solution. The criterion and the sufficient conditions for convergence are proved. For the considered classes of equations it is, among others, proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution. As applications, criteria for some classes of linear equations as well as for nonlinear equations are obtained. Some comparisons with already known results are given too.

In this paper, we consider the impulsive stabilization problems for a class of impulsive functional differential equation of the form x′ t = f t, x t , t ≥ t 0 , Δx = I k t, x t − , t = t k , k ∈ Z + . Our method is based on the application of the Liapunov second method together with Liapunov functionals.