Order Nonlinear Delay Differential Equations Research Articles

Uniform Ultimate Boundedness Results for some System of Third Order Nonlinear Delay Differential Equations

Lyapunov functional, third-order delay differential system, Boundedness.

Lower bounds for the smallest singular values of generalized asymptotic diagonal dominant matrices

This article presents three classes of real square matrices. They are models of coefficient matrices of linearized Galerkin’s equations. These Galerkin’s equations are derived from first order nonlinear delay differential equation with smooth nonlinearity. This paper shows results of computer experiments stating that the minimum singular values of these matrices are unchanged even if orders of matrices are increased. A time variant Hutchinson equation is also considered. A computer experimental result is shown about this equation. This result shows that “the minimum singular values invariant-ness” holds also for the coefficient matrices of linearized Galerkin’s equations of this equation. A remark is presented that this property can be seen experimentally for a wide class of nonlinear delay differential equation with smooth nonlinearity. A theorem is presented based on the Schur complement. Through it, tight lower bounds are derived for the minimum singular values of such three matrices. It is proved that these lower bounds are unchanged even if orders of matrices are increased.

Nonlinear third-order differential equations with distributed delay: Some new oscillatory solutions

We consider a certain class of third order nonlinear delay differential equations in this work. The results that we obtained are an improvement and extension of some results mentioned in previous literature, as the criteria we obtained are less restrictive compared to the previous results reported in literature. An example is provided to illustrate new results.

Block generalized Störmer-Cowell methods applied to second order nonlinear delay differential equations

In this paper we deal with numerical solutions for a class of nonlinear second order delay differential equations. In order to approximate this problems, the block methods consisting of generalized Störmer-Cowell methods are proposed. The error analysis indicates, under appropriate conditions, the numerical schemes can have convergence of order p when they are consistent of order p. Finally, some numerical experiments are performed to illustrate accuracy of proposed methods and corresponding theoretical results.

On the Oscillation of second order nonlinear neutral delay differential equations

AbstractA class of second-order neutral delay differential equations is considered. New oscillation criteria are established to complement and improve some known results in the literature. Two examples are given to support our results.

Fixed solutions of second order nonlinear difference equations

The paper is concerned with asymptotic and stability behaviors of fixed solutions of the second order nonlinear delay difference equation of the form ?2(xn +pnxn-k -qnxn-l)+ f(xn) = 0, n = 0,1,2, · ··,. Examples are provided to illustrate the results.

An iterative scheme for the oscillation criteria of a nonlinear delay differential equation with several deviating arguments

This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [1, 2, 8]. Various examples are solved in MATLAB software to illustrate the relevance of the main results.

Nonexistence of Kneser solution for third order nonlinear neutral delay differential equations

The achieved sufficient conditions for nonexistence of so-called Kneser solutions are based on the new comparison principles, which help us decrease the problem of the wavering between the third and first-order equations. Examples are given to prove the significance of new theorems.

Delay dynamics of a levitating motor with two-limit control strategy

In a recent work (Shayak, 2019), I have proposed a new comparator-based control algorithm for a magnetically levitated motor. The rotor dynamics are governed by a sixth order nonlinear differential equation, whose stability analysis is treated as given. Here we consider this device from a dynamical systems viewpoint. We first present a simplified model which is a second order nonlinear delay differential equation. We then see that in this equation, a fixed point which is unstable in the absence of control gets converted to a small-amplitude stable limit cycle in its presence. Extrapolating from the simplified model, we find that insufficient damping can create an instability but it can be countered by increasing the inverter voltage, decreasing the inverter switching period, and relaxing the displacement tolerance value. Extensive simulation results show that all predictions made on the basis of the simplified model are applicable to the original system.

Oscillatory behavior of $n$-th order nonlinear delay differential equations with a nonpositive neutral term

We study the oscillation problem for solutions of a class of $n$-th order nonlinear delay differential equations with nonpositive neutral terms. The obtained results improve and correlate many of the known oscillation criteria in the literature for neutral and non-neutral equations.

Oscillatory behavior of second order nonlinear delay differential equations with positive and negative neutral terms

Oscillatory behavior of second order nonlinear delay differential equations with positive and negative neutral terms

Legendre wavelet solution of high order nonlinear ordinary delay differential equations

The purpose of this paper is to illustrate the use of the Legendre wavelet method in the solution of high-order nonlinear ordinary differential equations with variable and proportional delays. The main advantage of using Legendre polynomials lies in the orthonormality property, which enables a decrease in the computational cost and runtime. The method is applied to five differential equations up to sixth order, and the results are compared with the exact solutions and other numerical solutions when available. The accuracy of the method is presented in terms of absolute errors. The numerical results demonstrate that the method is accurate, effectual and simple to apply.

Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type

In this paper, we prove the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of a class of higher order nonlinear delay differential equations with multiple bounded variable delays on a compact interval. Result of existence and uniqueness of solution is obtained by fixed point approach. Meanwhile, for the first time, integral inequality of Grönwall-Bellman-Bihari’s type with delay is applied to prove the stability theorem which made our results more generalized and interesting.

Asymptotic dichotomy in a class of higher order nonlinear delay differential equations

Employing a generalized Riccati transformation and integral averaging technique, we show that all solutions of the higher order nonlinear delay differential equation \t\t\ty(n+2)(t)+p(t)y(n)(t)+q(t)f(y(g(t)))=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}$$ y^{(n+2)}(t)+p(t)y^{(n)}(t)+q(t)f\\bigl(y\\bigl(g(t)\\bigr)\\bigr)=0 $$\\end{document} will converge to zero or oscillate, under some conditions listed in the theorems of the present paper. Several examples are also given to illustrate the applications of these results.

English

Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form $$ \Delta (a_n(\Delta (b_n(\Delta y_n)^{\alpha })))+q_nf(y_{\sigma (n)})=0 $$ to have property ${(\rm A)}$ or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results.

Stability, Boundedness and periodic solutions to certain second order delay differential equations

Stability, boundedness and existence of a unique periodic solution to certain second order nonlinear delay differential equations is discussed. By employing Lyapunov's direct (or second) method, a complete Lyapunov functional is constructed and used to establish sufficient conditions, on the nonlinear terms, that guarantee uniform asymptotic stability, uniform ultimate boundedness and existence of a unique periodic solution. Obtained results complement many outstanding recent results in the literature. Finally, examples are given to show the effectiveness of our method and correctness of our results.

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.

Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

INTEGRAL OSCILLATION CRITERIA FOR THIRD-ORDER DIFFERENTIAL EQUATIONS WITH DELAY ARGUMENT

In the present paper, some new criteria for property A and the oscillation of third order nonlinear delay differential equations of the type � a(t) h b(t)y ' (t) � ' i�' + p(t)f (y(�(t))) = 0.

The distribution of zeros of oscillatory solutions for second order nonlinear neutral delay differential equations

In this paper, the distance between adjacent zeros of oscillatory solutions for second order nonlinear neutral delay differential equations ( a ( t ) z ′ ( t ) ) ′ + q ( t ) f ( x ( t − σ ) ) = 0 , t ≥ t 0 , where z ( t ) = x ( t ) + p ( t ) x ( t − τ ) is investigated. By means of inequality techniques, specific function sequences and nonincreasing solutions for corresponding first order differential inequality, some new estimates for the distribution of zeros of oscillatory solutions have been presented, which have extended and improved some known results.