On the distribution of zeros of solutions of first order delay differential equations

This paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation

x^′(t)+p(t)x(t−τ)=0     

Oscillation of solutions of second-order nonlinear differential equations of Euler type

We consider the nonlinear Euler differential equation t²x^″+g(x)=0. Here g(x) satisfies xg(x)>0 for x≠0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t²x^″+a(t)g(x)=0.     

Oscillation theorems for nth-order differential equations with deviating arguments

This paper is concerned with the study of the oscillatory behavior of solutions of the nth-order nonlinear differential equation (a(t)x^{(n ? v)}(t))^(v) + q(t)f(x[g(t)]) = 0, where n is even and 1 ? v ? n ? 1. A systematic study is attempted which extends and correlates a number of existing results.     

A discontinuous functional differential equation

In this paper we give a sufficient condition for the existence of maximal and minimal solutions to a discontinuous functional differential equation x′(t)=f(t,x(t),x(·)), x(0)=0. We apply the result to establish an existence theorem for the Darboux problem for a partial differential equation.     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

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)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

Oscillation of solutions of second-order nonlinear self-adjoint differential equations

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equation (a(t)x′)′+b(t)g(x)=0. Here g(x) satisfied the signum condition xg(x)>0 if x≠0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Liénard type. We also explain an analogy between our results and an oscillation criterion of Kneser-Hille type for linear differential equations.     

Limit-point type solutions of nonlinear differential equations

We are concerned with the nonexistence of L²-solutions of a nonlinear differential equation x″=a(t)x+f(t,x). By applying technique similar to that exploited by Hallam [SIAM J. Appl. Math. 19 (1970) 430-439] for the study of asymptotic behavior of solutions of this equation, we establish nonexistence of solutions from the class L²(t₀,∞) under milder conditions on the function a(t) which, as the examples show, can be even square integrable. Therefore, the equation under consideration can be classified as of limit-point type at infinity in the sense of the definition introduced by Graef and Spikes [Nonlinear Anal. 7 (1983) 851-871]. We compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations.     

On the oscillation of Hill's equations under periodic forcing

We use the Floquet theory of the Hill's equation to prove the conjecture that all solutions of the second order forced linear differential equation y^″+c(sint)y=cost, are oscillatory on [0,∞) for all c≠0.     

On the existence and multiplicity of positive periodic solutions for first-order vector differential equation

In this paper, we consider the existence and multiplicity of positive periodic solutions for first-order vector differential equation x^′(t)+f(t,x(t))=0, a.e. t∈[0,ω] under the periodic boundary value condition x(0)=x(ω). Here ω is a positive constant, and is a Carathéodory function. Some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones.     

On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity

It is well known that a scalar differential equation , where f(t,x) is continuous, T-periodic in t and weakly convex or concave in x has no, one or two T-periodic solutions or a connected band of T-periodic solutions. The last possibility can be excluded if f(t,x) is strictly convex or concave for some t in the period interval. In this paper we investigate how the actual number of T-periodic solutions for a given equation of this type in principle can be determined, if f(t,x) is also assumed to have a continuous derivative . It turns out that there are three cases. In each of these cases we indicate the monotonicity properties and the domain of values for the function P(ξ)=S(ξ)−ξ, where S(ξ) is the Poincaré successor function. From these informations the actual number of periodic solutions can be determined, since a zero of P(ξ) represents a periodic solution.     

Periodic solutions of a nonlinear second-order differential equation with deviating argument

With the help of the coincidence degree continuation theorem, the existence of periodic solutions of a nonlinear second-order differential equation with deviating argument

x^″(t)+f₁(x(t))x^′(t)+f₂(x(t))(x^′(t)⁾²+g(x(t−τ(t)))=0,     

Asymptotic behavior of a differential equation with distributed delays

In this paper, sufficient conditions are established for the asymptotical behavior of solutions of the delay differential equation

x^′(t)=F(t,xt)+G(t,xt)     

On the complete group classification of the reaction-diffusion equation with a delay

The reaction-diffusion delay differential equation

ut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ))     

Note on forced oscillation of nth-order sublinear differential equations

In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation x(n)+q(t)λ|x|sgnx=e(t), t∈[t₀,∞). No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation x^″+tαsintλ|x|sgnx=mtβcost, t?0, 0<λ<1 are oscillatory for all m≠0 if β>(α+2)/(1−λ). This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673].     

On the asymptotic behavior of the solutions of a class of second order nonlinear differential equations

In this paper, we investigate properties of the solutions of a class of second-order nonlinear differential equation such as [p(t)f(x(t))x′(t)]′ + q(t)g(x′(t))e(x(t)) = r(t)c(x(t)). We prove the theorems of monotonicity, nonoscillation and continuation of the solutions of the equation, the sufficient and necessary conditions that the solutions of the equation are bounded, and the asymptotic behavior of the solutions of the equation when t → ∞ on condition that the solutions are bounded. Also we provide the asymptotic relationship between the solutions of this equation and those of the following second-order linear differential equation: [p(t)u′(t)]′ = r(t)u(t)     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Global branches of periodic solutions for forced delay differential equations on compact manifolds

We prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x^′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ?0. The approach is based on the fixed point index theory for maps on ANRs.     

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(x^′(t))+g(t,x(t))=e(t).     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.