Second-order Neutral Functional Differential Equations Research Articles

Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

Oscillation Results for Second-Order Neutral Functional Differential Equations

In this article, we obtain several comparison theorems for the second-order neutral differential equation , t ⩾ t 0, where γ, λ, β are ratios of positive odd integers. We compare above equation with the first-order differential inequalities in the sense that the absence of the eventually positive solutions of these first-order inequalities implies the oscillation of the studied equation.

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

Abstract In this paper, under some ordered conditions, we investigate the existence of positive ω-periodic solutions for a class of second-order neutral functional differential equations with delayed derivative in nonlinearity of the form (x(t) − cx(t − δ))″ + a(t)g(x(t))x(t) = λb(t)f(t, x(t), x(t − τ 1(t)), x′(t − τ 2(t))). By utilizing the perturbation method of a positive operator and the fixed point index theory in cones, some sufficient conditions are established for the existence as well as the non-existence of positive ω-periodic solutions.

Oscillation theorems of solution of second-order neutral differential equations

<abstract><p>In this paper, we aim to explore the oscillation of solutions for a class of second-order neutral functional differential equations. We propose new criteria to ensure that all obtained solutions are oscillatory. The obtained results can be used to develop and provide theoretical support for and further develop the oscillation study for a class of second-order neutral differential equations. Finally, an illustrated example is given to demonstrate the effectiveness of our new criteria.</p></abstract>

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.

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.

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.

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.

Multiple Periodic Solutions for a Class of Second-Order Neutral Impulsive Functional Differential Equations

In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.

Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments

We study oscillatory properties of a class of second-order nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results. MSC:34C10, 34K11.

Existence of Subharmonic Periodic Solutions to a Class ofSecond‐Order Non‐Autonomous Neutral Functional Differential Equations

By introducing subdifferentiability of lower semicontinuous convex function φ(x(t), x(t − τ)) and its conjugate function, as well as critical point theory and operator equation theory, we obtain the existence of multiple subharmonic periodic solutions to the following second‐order nonlinear nonautonomous neutral nonlinear functional differential equation x″(t) + x″(t − 2τ) + f(t, x(t), x(t − τ), x(t − 2τ)) = 0, x(0) = 0.

Positive periodic solution of second-order neutral functional differential equations

In this paper, we consider two types of second-order neutral functional differential equations. By choosing available operators and applying Krasnoselskii’s fixed point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations.

Infinite periodic and subharmonic solutions to a class of second-order nonlinear non-autonomous neutral functional differential equations

In this paper, by setting up a new Hilbert space H&#422γ τ and the variational method, we finded energy functions for a class of second-order nonlinear non-autonomous neutral functional differential equations. Then by critical point and Z2-group index theory, we get the sufficient condition for the existence of infinite periodic solutions for the system. Simultaneity, we put up an open question.

Existence results for abstract impulsive second-order neutral functional differential equations

We establish the existence of mild solutions for a class of impulsive second-order partial neutral functional differential equations with infinite delay in a Banach space.

Some results on second-order neutral functional differential equations with infinite distributed delay

In this paper, we consider two types of second-order neutral functional differential equations with infinite distributed delay. By choosing available operators and applying Krasnoselskii’s fixed-point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations.

Two periodic solutions of second-order neutral functional differential equations

In this paper, we consider a type of second-order neutral functional differential equations. We obtain some existence results of multiplicity and nonexistence of positive periodic solutions. Our approach is based on a fixed point theorem in cones.

Triple positive solutions for a class of boundary value problems for second-order neutral functional differential equations

We obtain sufficient conditions for the existence of at least three positive solutions for the second-order neutral functional differential equation x ″ ( t ) + q ( t ) f ( t , x ( t ) , x ( t − τ ) , x ′ ( t ) , x ′ ( t − τ ) ) = 0 , 0 < t < 1 , τ > 0 , subject to one of the following two pairs of boundary conditions: { x ( t ) = ξ ( t ) , − τ ≤ t ≤ 0 , x ( 1 ) = 0 { x ( t ) = ξ ( t ) , − τ ≤ t ≤ 0 , x ′ ( 1 ) = 0 and generalize and correct some conditions of Theorem 3.2 in [X.B. Shu, Y.T. Xu, Triple positive solutions for a class of boundary value problems of second-order functional differential equations, Nonlinear Anal. 61 (8) (2005) 1401–1411]. This is an application of a new fixed-point theorem introduced by Avery and Peeterson [R.I. Avery, A.C. Peeterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42 (2001) 313–322].

Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space

The well-known Krasnoselskii twin fixed point theorem is used to investigate the existence of mild solutions for first- and second-order impulsive semilinear functional and neutral functional differential equations in Hilbert spaces.

Oscillation criteria for second-order neutral equations with distributed deviating arguments

In this paper, we establish several sufficient conditions for the oscillation of solutionsof second-order neutral functional differential equations with distributed deviating arguments, which generalize and improve some known results.

Oscillatory criteria of a class of second-order neutral functional differential equations

In this paper, we investigate a class of second-order neutral functional differential equations, by using a generalized Riccati transformation, introducing parameter function and integral averaging technique, we obtain some new criteria that ensure the oscillation of solutions. The results generalize and improve some known results.