Higher Order Neutral Differential Equations Research Articles

Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator

Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator

Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations

AbstractIn this article, sufficient conditions are obtained so that every solution of the neutral difference equationΔm(yn−pnL(yn−s))+qnG(yn−k)=0,$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$or every unbounded solution ofΔm(yn−pnL(yn−s))+qnG(yn−k)−unH(yα(n))=0,n≥n0,$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$oscillates, wherem=2 is any integer, Δ is the forward difference operator given by Δyn=yn+1−yn; Δmyn= Δ(Δm−1yn) and other parameters have their usual meaning. The non linear functionL∈C(ℝ, ℝ) inside the operator Δmincludes the caseL(x) =x. Different types of super linear and sub linear conditions are imposed onGto prevent the solution approaching zero or ±∞. Further, all the three possible cases,pn≥ 0,pn≤ 0 andpnchanging sign, are considered. The results of this paper generalize and extend some known results.

On Oscillatory and Asymptotic Behavior of Higher Order Neutral Differential Equations with Impulsive Conditions

On Oscillatory and Asymptotic Behavior of Higher Order Neutral Differential Equations with Impulsive Conditions

Oscillatory Behaviour of Higher Order Neutral Differential Equations with Several Delays and with a Super Linear Term

Oscillatory Behaviour of Higher Order Neutral Differential Equations with Several Delays and with a Super Linear Term

Oscillation Criteria of Higher-order Neutral Differential Equations with Several Deviating Arguments

This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using the technique of Riccati transformation and comparison principles with the second-order differential equations, we obtain a new Philos-type criterion. Our results extend and improve some known results in the literature. An example is given to illustrate our main results.

Characterization of Higher-Order Neutral Difference Equations with Several Delays

Characterization of Higher-Order Neutral Difference Equations with Several Delays

Existence of periodic solutions for a higher-order neutral difference equation

Based on a continuation theorem of Mawhin, the existence of a periodic solution for a higher-order nonlinear neutral difference equation is studied. Our conclusion is new and interesting.

Existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed coefficients and delays

In this paper we consider the existence of nonoscillatory solutions for a system of higher-order neutral differential equations with distributed coefficients and delays. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

Existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed delays

In this paper we consider the existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed coefficients and delays. We use the B a n a c h contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients

In this paper we consider the existence of nonoscillatory solutions of higher-order neutral differential equations with distributed coefficients and delays. We use the $Banach$ contraction principle to obtain new sufficient condition for the existence of nonoscillatory solutions.

Oscillation of certain higher-order neutral partial functional differential equations.

In this paper, we study the oscillation of certain higher-order neutral partial functional differential equations with the Robin boundary conditions. Some oscillation criteria are established. Two examples are given to illustrate the main results in the end of this paper.

Existence of nonoscillatory solutions of higher order neutral differential equations

This article is concerned with nonoscillatory solutions of higher order nonlinear neutral differential equations with deviating and distributed deviating arguments. By using Knaster-Tarski fixed point theorem, new sufficient conditions are established. Illustrative example is given to show applicability of results.

Oscillation results for higher order nonlinear neutral differential equations

In this paper, the authors investigated the oscillatory and asymptotic behaviour of solution of a class of nonlinear higher order neutral differential equations with positive and negative coefficients. We have established a new conditions which guarantees that every solutions either oscillatory or converges to zero. Moreover, using Schauder's fixed point theorem sufficient conditions are obtained for the bounded positive solutions. Examples are considered to illustrate the main results.

Existence of Nonoscillatory Solutions of Higher-Order Neutral Differential Equations with Distributed Coefficients and Delays

We consider the existence of nonoscillatory solutions of higher-order neutral differential equations with distributed coefficients. We use theBanachcontraction principle to obtain new sufficient condition for the existence of nonoscillatory solutions.

Convergence of solutions of higher order neutral difference equations with quasi-differences

Abstract A higher order nonlinear neutral difference equation with quasi-differences is studied. Sufficient conditions are established under which for every real constant there exists a solution of considered equation convergent to this constant.

Comparison theorems for higher-order neutral delay differential equations

In the paper, we combine new generalization of classical Philos and Staikos lemma together with suitable comparison technique to offer new oscillation criteria for \(n\)-th order neutral delay differential equation $$\begin{aligned} \Big (r(t)\left[ z'(t)\right] ^{\gamma }\Big )^{(n-1)} +q(t)x^{\gamma }(\sigma (t))=0. \end{aligned}$$ The presented technique essentially simplifies the examination of the higher order differential equations.

Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations

Abstract This paper deals with the solvability of the higher-order nonlinear neutral delay differential equation d n d t n [ x ( t ) + p ( t ) x ( t − τ ) ] + ( − 1 ) n + 1 ∑ i = 1 m q i ( t ) x ( α i ( t ) ) + ( − 1 ) n + 1 f ( t , x ( β 1 ( t ) ) , … , x ( β l ( t ) ) ) = r ( t ) , t ≥ t 0 , where τ > 0 , n , m , l ∈ N , p , r , q i , α i , β j ∈ C ( [ t 0 , + ∞ ) , R ) , and f ∈ C ( [ t 0 , + ∞ ) × R l , R ) satisfying lim t → + ∞ α i ( t ) = lim t → + ∞ β j ( t ) = + ∞ , i ∈ { 1 , 2 , … , m } , j ∈ { 1 , 2 , … , l } . With respect to various ranges of the function p, we investigate the existence of uncountably many bounded nonoscillatory solutions for the equation. The main tools used in this paper are the Krasnoselskii and Schauder fixed point theorems together with some new techniques. Six nontrivial examples are given to illustrate the superiority of the results presented in this paper. MSC:39A10, 39A20, 39A22.

Periodic solutions for a kind of higher-order neutral functional differential equation with variable parameter

In this paper, we consider a kind of higher-order neutral equation with distributed delay and variable parameter: . By using the classical coincidence degree theory of Mawhin, sufficient conditions for the existence of periodic solutions are established. Recent results in the literature are generalized and significantly improved. Furthermore, two examples are given to illustrate that the results are almost sharp. MSC:34K10, 30D05, 34B45.

Asymptotic behavior of higher-order neutral difference equations with general arguments

We study the asymptotic behavior of solutions of the higher-order neutral difference equation $$ {\varDelta^m}\left[ {x(n)+cx\left( {\tau (n)} \right)} \right]+p(n)x\left( {\sigma (n)} \right)=0,\quad \mathbb{N}\mathrel\backepsilon m\geq 2,\quad n\geq 0, $$ where τ (n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ $ \mathbb{R} $ ; (p(n)) n ≥ 0 is a sequence of real numbers, ∆ denotes the forward difference operator ∆x(n) = x(n+1) - x(n); and ∆ j denotes the jth forward difference operator ∆ j (x(n) = ∆ (∆ j-1(x(n))) for j = 2, 3,…,m. Examples illustrating the results are also given.

Oscillation of higher-order half-linear neutral differential equations

Abstract In this paper, we establish some new oscillation criteria for the higher-order half-linear neutral differential equation [ a ( t ) ( [ x ( t ) + p ( t ) x ( σ ( t ) ) ] ( n − 1 ) ) γ ] ′ + q ( t ) x γ ( τ ( t ) ) = 0 . $$\left[ {a(t)\,\,\left( {[x(t) + p(t)x(\sigma (t))]^{(n - 1)} } \right)^\gamma } \right]^\prime + q(t)x^\gamma (\tau (t)) = 0.$$ An example is given to illustrate the main results.