Order Nonlinear Neutral Differential Equations Research Articles

Existence results for positive periodic solutions to first order neutral differential equations with distributed deviating arguments

We take into account the first order nonlinear neutral differential equation with distributed deviating arguments.
 Using Krasnoselskii's fixed point theorem, we give some new criteria for the existence of positive periodic solutions to this equation. The theorems we have established are illustrated by an example.

Existence of Nonoscillatory Solutions of Higher Order Nonlinear Neutral Differential Equations

In this paper, an n-th order neutral nonlinear differential equation is studied. By using the Banach contraction principle, some sufficient conditions are established for the existence of nonoscillatory solutions of nonlinear n-th order neutral differential equation. An example is included to illustrate the results obtained.

Nonoscillatory Properties of Fourth Order Nonlinear Neutral Differential equation

In this paper, the oscillatory and nonoscillatory qualities for every solution of fourth-order neutral delay equation are discussed. Some conditions are established to ensure that all solutions are either oscillatory or approach to zero as . Two examples are provided to demonstrate the obtained findings.

Oscillation of Second Order Nonlinear Neutral Differential Equations

The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential equations with noncanonical operators are established. The results extend and improve on those reported in the literature. Moreover, some examples are provided to show the significance of the results.

On the behaviour of solutions to a kind of third order nonlinear neutral differential equation with delay

This paper presents a novel class of third order nonlinear nonautonomous neutral differential equation with delay. The third order neutral differential equation is cut down to a system of first order, a suitable complete Lyapunov-Krasovskii's functional is constructed and used, to obtain standard conditions on the nonlinear functions to ensure stability and uniform asymptotic stability of the trivial solutions, the existence of a unique periodic solution, uniform boundedness and uniform ultimate boundedness of solutions when the forcing term is nonzero. The obtained results are new and include many prominent results on neutral and nonneutral delay differential equations in literature. Finally, the practicability and reliability of the theoretical results are demonstrated.

On the distribution of zeros of all solutions of a first order nonlinear neutral differential equation

We investigate the distribution of zeros of all solutions of a non-autonomous nonlinear neutral differential equation that generalizes a lossless transmission network model. The neutral term is taken to be positive. We give several new estimations of the gap between adjacent zeros. The obtained results are supported by illustrative examples.

Application of certain Third-order Non-linear Neutral Difference Equations in Robotics Engineering

We have obtained some results on oscillatory behavior of third order nonlinear neutral difference equations of the form where β and γ are odd integers with γ ≥ 1. Example is provided to illustrate the results.

Oscillation of solutions of third order nonlinear neutral differential equations

The main objective of this article is to improve and complement some of the oscillation criteria published recently in the literature for third order differential equation of the form (r(t)(z″(t))α)′+q(t)f(x(σ(t)))=0,t≥t0>0,\\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}$$ \\bigl( r(t) \\bigl( z^{\\prime \\prime }(t) \\bigr) ^{\\alpha } \\bigr) ^{\\prime }+q(t)f \\bigl(x \\bigl(\\sigma (t) \\bigr) \\bigr)=0,\\quad t\\geq t_{0}>0, $$\\end{document} where z(t)=x(t)+p(t)x(tau (t)) and α is a ratio of odd positive integers in the two cases int _{t_{0}}^{infty }r^{frac{-1}{alpha } }(s),mathrm {d}s<infty and int _{t_{0}}^{infty }r^{frac{-1}{alpha } }(s),mathrm {d}s=infty . Some illustrative examples are presented.

Three positive periodic solutions of second order nonlinear neutral functional differential equations with delayed derivative

This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity (x(t)-cx(t-delta)){''}+a(t)g(x(t))x(t)=lambda b(t)f(t,x(t),x(t-tau_{1}(t)),x'(t-tau_{2}(t))). By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.

On oscillatory second order impulsive neutral difference equations

The present paper deals with the problem of oscillation for a class of second order nonlinear neutral impulsive difference equations with fixed moments of impulse effect. The technique employed here is due to the classical impulsive inequalities. Some examples are given to illustrate our results.

On oscilatory fourth order nonlinear neutral differential equations – IV

AbstractIn this paper, oscillation of all solutions of fourth order functional differential equations of neutral type of the form$$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\sigma))=0 \end{array}$$are studied under the assumption$$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}t \lt \infty \end{array}$$for various ranges ofp(t)

On the oscillation of higher order nonlinear neutral difference equations

In this paper, we shall investigate some oscillation criteria for the solutions of mth-order nonlinear neutral difference equation where mgeq 1. The results presented here complement some of the known results reported in the literature. Examples are included to illustrate the importance of the main results.

Existence of Nonoscillatory Solutions of First Order Non-Linear Neutral Differential Equations

In this paper, some sufficient conditions have been obtained for the existence of positive solutions of the first order nonlinear neutral differential equations bounded by a ratio of bounded functions. For this purpose we used Krasnoselskii’s fixed point theorem and Lebesgue’s dominated convergence theorem to establish new sufficient conditions for the existence of nonoscillatory positive solutions. An example is included to illustrate the main result.

A higher order nonlinear neutral differential equation

A higher order nonlinear neutral differential equation

Oscillation criteria for even order neutral difference equations

In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form \[\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,\] where \(m\geq 2\) is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results.

Existence of nonoscillatory solutions for second-order nonlinear differential equations of neutral type

In this work, we consider the existence of nonoscillatory solutions of second order nonlinear neutral differential equations. Our results include as special cases some well-known results for linear and nonlinear equations. We use the Lebesgue’s dominated convergence theorem and Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

On the oscillatory behavior of solutions of third order nonlinear neutral differential equations

On the oscillatory behavior of solutions of third order nonlinear neutral differential equations

On oscillatory fourth order nonlinear neutral differential equations – III

Abstract In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form ( r ( t ) ( y ( t ) + p ( t ) y ( t − τ ) ) ″ ) ″ + q ( t ) G ( y ( t − σ ) ) = 0 $$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption ∫ 0 ∞ t r ( t ) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$

Oscillation of Even Order Nonlinear Neutral Differential Equations of E

Oscillation of Even Order Nonlinear Neutral Differential Equations of E

Positive solutions and iterative approximations of a third order nonlinear neutral delay difference equation

This paper deals with the third order nonlinear neutral delay difference equation with a forced term \t\t\tΔ2(a(n)Δ(x(n)+c(n)x(n−τ)))+f(n,x(n−b1(n)),x(n−b2(n)),…,x(n−bk(n)))=d(n),n≥n0.\\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} $$\\begin{aligned}& \\Delta^{2} \\bigl(a(n)\\Delta \\bigl(x(n)+c(n)x(n-\\tau) \\bigr) \\bigr)+f \\bigl(n,x \\bigl(n-b_{1}(n) \\bigr),x \\bigl(n-b_{2}(n) \\bigr), \\ldots,x \\bigl(n-b_{k}(n) \\bigr) \\bigr) \\\\& \\quad =d(n),\\quad n\\geq n_{0}. \\end{aligned}$$ \\end{document} Using the Banach fixed point theorem, we prove the existence of uncountably many bounded positive solutions for the equation, suggest some Mann iterative schemes and obtain the error estimates between these bounded positive solutions and the sequences generated by the iterative schemes. Five nontrivial examples are also included.