Higher Order Nonlinear Neutral Delay Research Articles

Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients

Abstract In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 2013, 143–163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 2013, 11–28]. Examples are included to illustrate the validation of the results.

On the oscillation of bounded solutions of higher order nonlinear neutral delay differential equations with oscillating coefficients

On the oscillation of bounded solutions of higher order nonlinear neutral delay differential equations with oscillating coefficients

Oscillatory behaviour of higher-order nonlinear neutral delay dynamic equations on time scales

In this paper, new sufficient conditions are established for the oscillation of solutions of the higher order dynamic equations [r(t)(z?n-1(t))?]? + q(t) f(x(?(t)))=0, for t ?[t0,?)T, where z(t):= x(t)+ p(t)x(?(t)), n ? 2 is an even integer and ? ? 1 is a quotient of odd positive integers. Under less restrictive assumptions for the neutral coefficient, we employ new comparison theorems and Generalized Riccati technique.

Solvability and Mann iterative approximations for a higher order nonlinear neutral delay differential equation

The purpose of this paper is to study solvability of the higher order nonlinear neutral delay differential equation dndtn[x(t)+c(t)x(t−τ)]+(−1)n+1f(t,x(σ1(t)),x(σ2(t)),…,x(σk(t)))=g(t),t≥t0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& \\frac{d^{n}}{dt^{n}}\\bigl[x(t)+c(t)x(t-\\tau)\\bigr]+(-1)^{n+1}f\\bigl(t,x \\bigl(\\sigma _{1}(t)\\bigr),x\\bigl(\\sigma_{2}(t)\\bigr), \\ldots,x\\bigl(\\sigma_{k}(t)\\bigr)\\bigr) \\\\ & \\quad =g(t),\\quad t\\geq t_{0}, \\end{aligned}$$ \\end{document} where n and k are positive integers, tau>0 , t_{0}in{mathbb{R}}, fin C ([t_{0},+infty)times {mathbb{R}}^{k},{mathbb{R}} ) , c,g,sigma_{i}in C([t_{0},+infty),{mathbb{R}}) and lim_{trightarrow +infty}sigma_{i}(t)=+ infty for i in{1,2,ldots,k}. Under suitable conditions, several existence results of uncountably many nonoscillatory solutions and convergence of Mann iterative approximations for the above equation are shown. Three nontrivial examples are given to demonstrate the advantage of our results over the existing ones in the literature.

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.

Oscillation results of higher order nonlinear neutral delay differential equations

In this paper we investigate the oscillation and asymptotic behavior of solu- tions to the n-th order neutral nonlinear differential equations of the form (x(t) + g(t,x(t(t)))) (n) + f(t,x(s(t))) = 0.

Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

This paper is concerned with the higher order nonlinear neutral delay differential equation[a(t)(x(t)+b(t)x(t-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

Existence of Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Delay Difference Equations

This paper studies the existence of nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation \begin{align*} \Delta \big(a_{kn} \cdots & \Delta (a_{2n} \triangle (a_{1n} \Delta (x_n + b_n x_{n-d}) ) ) \big) \\ &{} + f(n,x_{n-r_{1n}}, x_{n-r_{2n}}, \ldots , x_{n-r_{sn}} ) = 0, \ \ n \ge n_0, \end{align*} where $n_0 \ge 0$, $n \ge 0$, $d > 0$, $k > 0$, $s > 0$ are integers, $\{ a_{in} \} _{n \ge n_0}$ ($i = 1, 2, \ldots , k)$) and $\{ b_n \} _{n \ge n_0}$ are real sequences, $f \colon \{ n : n \ge n_0 \} \times {\mathbb R}^n \to {\mathbb R}$ is a mapping and $\bigcup _{j=1}^s \{ r_{jn} \} _{n \ge n_0} \subseteq {\mathbb Z}$. By applying Krasnoselskii's Fixed Point Theorem, some sufficient conditions for the existence of nonoscillatory solutions of this equation are established and indicated through five theorems according to the range of value of the sequence $b_n$.

Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation , , where , , , and are integers, and are real sequences, , and is a mapping, is studied. Some sufficient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence . Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation

The paper is concerned with the higher order nonlinear neutral delay differential equation r ( t ) ( x ( t ) + p ( t ) x ( t - τ ) ) ( m ) ( n - m ) + ( - 1 ) n - m + 1 f t , x σ 1 ( t ) , … , x ( σ l ( t ) ) = g ( t ) , t ⩾ t 0 . Using the Banach fixed-point theorem, we establish five existence results of uncountably many bounded nonoscillatory solutions for the above equation, construct five Mann iterative sequences with mixed errors for approximating these nonoscillatory solutions and discuss five error estimates between the approximate solutions and these nonoscillatory solutions. To dwell the importance and applications of our results, five nontrivial examples are constructed.

Necessary and sufficient conditions for oscillation of higher order nonlinear neutral delay differential equations

We first show that the n order superlinear neutral delay differential equation with unstable type $$ [x(t) - px(t - \tau )]^{(n)} = q(t)|x(t - \sigma )|^{\beta - 1} x(t - \sigma ),t \geqslant t_0 $$ always has an unbounded and nonoscillatory solution, where n > 1 is an even integer, σ > 0, τ > 0, β > 1, 0 < p < 1 and q ∈ C([t0, ∞), [0, ∞)). Then a necessary and sufficient condition is obtained for all bounded solution of the above equation to be oscillatory.

New oscillatory criteria for higher-order nonlinear neutral delay differential equation

In this paper, we consider the higher-order nonlinear neutral delay differential equation (1.1) ( x ( t ) − p ( t ) x ( t − τ ) ) ( n ) + f ( t , x ( σ ( t ) ) ) = 0 where n is an odd number and n ⩾ 3 , τ > 0 , p , σ ∈ C ( [ t 0 , ∞ ) , R + ) , σ ( t ) ⩽ t , lim t → ∞ σ ( t ) = ∞ , f ∈ C ( [ t 0 , ∞ ) × R , R ) , f ( t , u ) is nondecreasing in u .

Existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations

This paper studies a few existence results of nonoscillatory solutions for the following higher order nonlinear neutral delay differential equation: dndtnx(t)+cx(t-τ)+(-1)n-i+1didtih(t,x(h1(t)),x(h2(t)),…,x(hk(t)))+(-1)n+1f(t,x(f1(t)),x(f2(t)),…,x(fk(t)))=g(t),t⩾t0, where n is a positive integer, 0⩽i⩽n-1, c∈R⧹{-1}, τ>0, h∈Ci([t0,+∞)×Rk,R), f∈C([t0,+∞)×Rk,R), g∈C([t0,+∞),R), hl∈Ci([t0,+∞),R) and fl∈C([t0,+∞),R) with limt→+∞hl(t)=limt→+∞fl(t)=+∞,l=1,…,k constructs several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. In addition sufficient conditions for the existence of infinitely many nonoscillatory solutions for the above equation are given. Three nontrivial examples are given to illustrate the advantages of our results.

Oscillation results for higher order nonlinear neutral delay difference equations

In this work, we shall consider higher order nonlinear neutral delay difference equation of the type Δ m ( y n + p n y n − l ) + q n y n − k α = 0 , where { p n } , { q n } are sequences of nonnegative real numbers, k and l are positive integers and α ∈ ( 0 , 1 ) is a ratio of odd positive integers. We obtain sufficient conditions for the oscillations of all solutions of this equation.

Oscillatory Behaviour of a Higher Order Nonlinear Neutral Delay Type Functional Differential Equation with Oscillating Coefficients

In this paper we are concerned with the oscillation of solutions of a certain more general higher order nonlinear neutral type functional differential equation with oscillating coefficients. We obtain two sufficient criteria for oscillatory behaviour of its solutions.

Classification and existence of nonoscillatory solutions of higher order nonlinear neutral difference equations

In this paper, we consider the higher order nonlinear neutral delay difference equation of the form $$\Delta ^r (x_n + px_{n - \tau } ) + f(n,x_{n - \sigma _1 (n)} ,x_{n - \sigma _2 (n)} ,...,x_{n - \sigma _m (n)} ) = 0.$$ We give an integrated classification of nonoscillatory solutions of the above equation according to their asymptotic behaviours. Necessary and sufficient conditions for the existence of nonoscillatory solutions with designated asymptotic properties are also established.

Non-Oscillatory Solutions of Higher Order Nonlinear Neutral Delay Difference Equation

Consider the forced higher order nonlinear neutral delay difference equation L_m( x(n) + cx(n - τ) ) + F ( n, x(σ(n) ) ) = g(n)\quad (n ≥ n_0). We obtain a global result (with respect to c ), which consists of some sufficient conditions for the existence of a non-oscillatory solution of the above equation. Our results improve and extend a number of existing results.

Oscillatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations

AbstractThe authors consider the higher-order nonlinear neutral delay difference equationand obtain results on the asymptotic behavior of solutions when (pn) is allowed to oscillate about the bifurcation value –1. We also consider the case where the sequence {pn} has arbitrarily large zeros. Examples illustrating the results are included, and suggestions for further research are indicated.