Second-order Functional Differential Equations Research Articles

Existence results for second-order neutral stochastic equations driven by Rosenblatt process

In this paper we consider a class of second-order impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and a standard Brownian motion in a Hilbert space. We prove an existence and uniqueness result under non-Lipschitz condition which is weaker than Lipschitz one and we establish some conditions ensuring the controllability for the mild solution by means of the Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result. <br><br> Розглянуто клас імпульсних стохастичних функціонально-диференціальних рівнянь другого порядку, які керуються процесом Розенблата і стандартним броунівський рухом у ґільбертовому просторі одночасно за умови, яка є слабкішою за умови Ліпшица. Також встановлено умови керованості для помірного розв'язку за допомоги принципу Банаха про нерухому точку. Наведено приклад з практики, що ілюструє отримані результати.

EXISTENCE AND EXPONENTIAL STABILITY OF MILD SOLUTIONS FOR SECOND-ORDER NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATION WITH RANDOM IMPULSES

In this paper, we consider the existence and exponential stability in mean square of mild solutions to second-order neutral stochastic functional differential equations with random impulses in Hilbert space. Firstly, the existence of mild solutions to the equations is proved by using the noncompact measurement strategy and the Mönch fixed point theorem. Then, the mean square exponential stability for the mild solution of the considered equations is obtained by establishing an integral inequality. Finally, an example is given to illustrate our results.

Oscillation of Emden–Fowler-Type Neutral Delay Differential Equations

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.

Explicit criteria for the oscillation of second-order differential equations with several sub-linear neutral coefficients

In this work, we present sufficient conditions for oscillation of all solutions of a second-order functional differential equation. We consider two special cases when gamma >beta and gamma <beta . This new theorem complements and improves a number of results reported in the literature. Finally, we provide examples illustrating our results and state an open problem.

Oscillation of damped second-order linear mixed neutral differential equations

This article concerns the oscillatory behavior of solutions to damped second-order linear functional differential equations with a mixed neutral term. The authors present new oscillation criteria that improve and extend some existing ones in the literature. Examples to illustrate the main results are included.

Qualitative Properties of Solutions of Second-Order Neutral Differential Equations

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.

Necessary and Sufficient Condition for Oscillatory and Asymptotic Behaviour of Second-Order Functional Differential Equations

Necessary and Sufficient Condition for Oscillatory and Asymptotic Behaviour of Second-Order Functional Differential Equations

English

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $$ (r(t)(z'(t))^\gamma )' +\sum _{i=1}^m q_i(t)x^{\alpha _i}(\sigma _i(t))=0, \quad t\geq t_0, $$ where $z(t)=x(t)+p(t)x(\tau (t))$. Under the assumption $\int ^{\infty }(r(\eta ))^{-1/\gamma } {\rm d}\eta =\infty $, we consider two cases when $\gamma >\alpha _i$ and $\gamma <\alpha _i$. Our main tool is Lebesgue's dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

Existence of solutions for an ordinary second-order hybrid functional differential equation

In this paper, we study the existence of solutions for an initial value problem of an ordinary second-order hybrid functional differential equation (SHDE) using a fixed point theorem of Dhage. Example is given to illustrate the obtained result.

Linear difference operator with multiple variable parameters and applications to second-order differential equations

In this article, we first investigate the linear difference operator (Ax)(t):=x(t)-sum_{i=1}^{n}c_{i}(t)x(t- delta _{i}(t)) in a continuous periodic function space. The existence condition and some properties of the inverse of the operator A are explicitly pointed out. Afterwards, as applications of properties of the operator A, we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.

Oscillation criteria for second-order quasi-linear neutral functional differential equation

In this work, new sufficient conditions for oscillation of solution of second order neutral delay differential equation are established. One objective of our paper is to further simplify and complement some results which were published lately in the literature. In order to support our results, we introduce illustrating examples.

On the oscillation of second-order half-linear functional differential equations with mixed neutral term

In this article, the authors establish new sufficient conditions for the oscillation of solutions to a class of second-order half-linear functional differential equations with mixed neutral term. The results obtained improve and complement some known results in the relevant literature. Examples illustrating the results are included.

Existence of periodic solutions for two types of second-order nonlinear neutral integro-differential equations with infinite distributed mixed-delays

We consider two types of second-order neutral functional differential equations with infinite distributed delays and offer existence criteria for periodic solutions. During the process we invert the integro-differential equations into equivalent integral equations and derive suitable fixed point mappings. We show that these mappings fit into the framework of Schauder's fixed point theorem so that periodic solutions are readily obtained.

Asymptotic problems for functional differential equations via linearization method

We study the existence of positive decreasing solutions (the so-called Kneser solutions) for a class of second-order functional differential equations with a damping term. A linearization approach based on a general fixed point theorem is used to achieve this goal. The existence of zero-decaying Kneser solutions is also proved. Finally, the role of the deviating argument to the asymptotic behavior of solutions is illustrated together with some discrepancies between equations with or without delay.

Periodic Solutions of a Second-Order Functional Differential Equation with State-Dependent Argument

In this paper, we use Schauder and Banach fixed point theorem to study the existence, uniqueness and stability of periodic solutions of a class of iterative differential equation $$\begin{aligned} c_0x''(t)+c_1x'(t)+c_2x(t)=x(p(t)+bx(t))+h(t). \end{aligned}$$

On One Class of Multipoint Boundary-Value Problems for a Second-Order Linear Functional-Differential Equation

We state solvability conditions for a multipoint boundary-value problem for linear second-order functional-differential equation.

An exponential stability criterion for nonlinear second-order functional differential equations with time-variable delays

This paper is concerned with the exponential stability for nonlinear second-order functional differential equations (FDEs) with time-variable delays. An exponential stability relationship between the FDEs and the corresponding ordinary differential equations (ODEs) is derived. It is proved under some appropriate conditions that the second-order FDEs can preserve the exponential stability of the corresponding ODEs. This stability result is also illustrated with a numerical approach.

Global attracting set and exponential decay of second-order neutral stochastic functional differential equations driven by fBm

In this paper, we are concerned with a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter 1/2<hbar <1 on the Hilbert space. By combining some stochastic analysis theory and new integral inequality techniques, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the pth moment to the stochastic systems are obtained. Last, an example is presented to illustrate our theory in the work.

On the Unique Solvability of a Nonlinear Nonlocal Boundary-Value Problem for Systems of Second-Order Functional Differential Equations

We establish some optimal, in a certain sense, general conditions sufficient for the unique solvability of the boundary-value problem for a system of nonlinear second-order functional differential equations. The considered class of equations covers, in particular, equations of the neutral type. Specific example is presented to illustrate the general theory.

Multiple periodic solutions for a class of second-order neutral functional differential equations

In this paper we consider a class of second-order neutral functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of Z_{2} group index theory and variational methods. The main result is also illustrated with an example.