Stability of periodic solutions of state-dependent delay-differential equations

We consider a class of autonomous delay-differential equations

     

Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations

We study the singularly perturbed state-dependent delay-differential equation

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Periodic solutions for general nonlinear state-dependent delay logistic equations

In this paper, by using the continuation theorem of coincidence degree theory, we investigate the existence of periodic solutions for more general state-dependent delay logistic equations. Several sufficient conditions are given, and the obtained conditions possess important significance in both theories and applications.     

New results 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$T$-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(t,x^′(t))+g(t,x(t))=e(t).$x^{″}+f(t,x^{\prime }\left(t\right))+g(t,x\left(t\right))=e\left(t\right)\text{.}$

Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space

Using spectral theory we obtain sufficient conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=A(t)x([t])+f(t),t∈R, where A(t) is an almost automorphy operator, f(t) is an X-valued almost automorphic function and X is a finite dimensional Banach space.     

A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist.The regularization is based on the replacement of the vector-field with its time average over an interval of length ε>0. The regularized solution converges as ε→⁰⁺ to the classical Filippov solution (Filippov, 1964, 1988 [13] and [14]). Several properties of the solutions corresponding to small ε>0 are presented.     

Existence of global solutions of delayed differential equations via retract approach

The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The topological approach known as the topological retract principle is used. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results.     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded. This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National Natural Science Foundation of China (Grant No. 10671020 and 10301006).     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations

x^″+f(x^′)+g(x)=e(t).     

Convergence of a class of discrete-time semiflows with application to neutral delay differential equations

In this paper, we introduce a class of pseudo-monotone maps on ordered topological spaces. By exploiting monotonicity methods and the invariance of the omega limit set, we establish a convergence principle for discrete-time semiflows generated by the maps introduced. The convergence principle is then applied to a class of periodic neutral delay differential equations, which leads to some novel and sharper results.     

Multiple periodic solutions of scalar second order differential equations

We prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity, thus generalizing previous results by Lazer and McKenna (1987) [1] and Del Pino, Manasevich and Murua (1992) [2]. The main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity. The proof is based on the use of the Poincaré-Birkhoff Fixed Point Theorem.     

On the oscillation of solutions of first-order delay differential inequalities and equations

Oscillation criteria generalizing a series of earlier results are established for first-order linear delay differential inequalities and equations.     

Theory of fractional functional differential equations

In this paper, the basic theory for the initial value problems for fractional functional differential equations is considered, extending the corresponding theory of ordinary functional differential equations.     

The existence of multiple positive solutions for singular functional differential equations with sign-changing nonlinearity

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form

     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular solutions to the equation −div(|Du|^p−2Du)=|u|^q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.