Noncanonical fourth-order nonlinear neutral differential equations of Emden-Fowler type: Oscillation via canonical transform

In this paper, we establish the solution of the fourth-order nonlinear homogeneous neutral functional difference equation. Moreover, we study the new oscillation criteria have been established which generalize some of the existing results of the fourth-order nonlinear homogeneous neutral functional difference equation in the literature. Likewise, a few models are given to represent the significance of the primary outcomes.

In this work first we transform the semi-noncanonical fourth order neutral delay differential equations into canonical type. This simplifies the investigations of finding the relationships between the solution and its companion function which plays an important role in the oscillation theory of neutral differential equations. Moreover, we improve these relationships based on the monotonic properties of positive solutions. We present new conditions for the oscillation of all solutions of the corresponding equation which improve the oscillation results already reported in the literature. Examples are provided to illustrate the importance of our main results. For moreinformation see https://ejde.math.txstate.edu/Volumes/2023/70/abstr.html

This work is concerned with the study of oscillation of a class of fourth-order neutral differential equations with unbounded delay. Sufficient conditions are obtained for oscillation and they are classified as C−1, C0 and C1 type oscillatory criteria.

This paper focuses on the study of the oscillatory behavior of fourth-order nonlinear neutral delay difference equations. The authors use mathematical techniques, such as the Riccati substitution and comparison technique, to explore the regularity and existence properties of the solutions to these equations. The authors present a new form of the equation: Δ(a(m)(Δ3z(m))p1−1)+p(m)wp2−1(σ(m))=0, where z(m)=w(m)+q(m)w(m−τ) with the following conditions: ∑s=m0∞1a(1p1−1(s))=∞. The equation represents a system where the state of the system at any given time depends on its current time and past values. The authors demonstrate new insights into the oscillatory behavior of these equations and the conditions required for the solutions to be well-behaved. They also provide a numerical example to support their findings.

Existence and nonexistence of eventually positive solutions for nonlinear neutral differential equations

In this paper we continue our developments in Hernandez and O'Regan (J Funct Anal 261:3457---3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernandez and O'Regan (J Funct Anal 261:3457---3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial "nonlinear" neutral differential equations. Some applications involving partial neutral differential equations are presented.

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

Stochastic differential equations can always simulate the scientific problem in practical truthfully. They have been widely used in Physics, Chemistry, Cybernetics, Finance, Neural Networks, Bionomics, etc. So far there are not many results on the numerical stability of nonlinear neutral stochastic delay differential equations. The purpose of our work is to show that the Euler method applied to the nonlinear neutral stochastic delay differential equations is mean square stable under the condition which guarantees the stability of the analytical solution. The main aim of this paper is to establish new results on the numerical stability. It is proved that the Euler method is mean-square stable under suitable condition, i.e., assume the some conditions are satisfied, then, the Euler method applied to the nonlinear neutral stochastic delay differential equations with initial data is mean-square stable. Moreover, the theoretical result is also verified by a numerical example.

The main purpose of this research was to use the comparison approach with a first-order equation to derive criteria for non-oscillatory solutions of fourth-order nonlinear neutral differential equations with p Laplacian operators. We obtained new results for the behavior of solutions to these equations, and we showed their symmetric and non-oscillatory characteristics. These results complement some previously published articles. To find out the effectiveness of these results and validate the proposed work, two examples were discussed at the end of the paper.

This paper considers a nonlinear fourth-order ordinary differential equation. The study of this class of equations is conducted using an analytical approximation method based on dividing the solution domain into two parts: the region of analyticity and the vicinity of a movable singular point. This work focuses on investigating the equation in the region of analyticity and solving two problems. The first problem is a classical problem in the theory of differential equations: proving the theorem of existence and uniqueness of a solution in the region of analyticity. The structure of the solution in this region takes the form of a power series. To transition from formal series to series converging in a neighborhood of the initial conditions, a modification of the majorant method is used, which is applied in the Cauchy – Kovalevskaya theorem. This method allows determining the domain of validity of the theorem. Within this domain, error estimates for the analytical approximate solution are obtained, enabling the solution to be found with any predefined accuracy. When leaving the domain of the theorem’s validity, analytical continuation is required. To do this, it is necessary to solve the second task of the study: to study the effect of perturbation of the initial data on the structure of the analytical approximate solution.

A comparison theorem on oscillation behavior is firstly established for a class of even-order nonlinear neutral delay difference equations. By using the obtained comparison theorem, two oscillation criteria are derived for the class of even-order nonlinear neutral delay difference equations. Two examples are given to show the effectiveness of the obtained results.

Consider a linear neutral integrodelay differential equation of the form $$\dot{x}(t)+\sum_{j=1}^{m}b_{j}\dot{x}(t-\sigma_{1})+\beta \int_{0}^{\infty}K_{2}(s)\dot{x}(t-s)ds+a_{0}x(t)+\sum_{j=1}^{n}a_{j}x(t-\tau_{j})+\alpha \int_{0}^{\infty}K_{1}(s)x(t-s)ds=0$$ (5.1.1) in which ẋ(t) denotes the right derivative of x at t. (Throughout this chapter we use an upper dot to denote right derivative and this is convenient in writing neutral differential equations systematically). Asymptotic stability of the trivial solution of (5.1.1) and several of its variants have been considered by many authors. There exists a well developed fundamental theory for neutral delay differential equations (e.g. existence, uniqueness, continuous dependence of solutions on various data; see, for instance, the survey article by Akhmerov et al. [1984]); however, there exist no “easily verifiable” sufficient conditions for the asymptotic stability of the trivial solution of (5.1.1). By the phrase “easily verifiable” we mean a verification which is as easy as in the case of Routh-Hurwitz criteria, the diagonal dominance condition or the positivity of principal minors of a matrix etc. Certain results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [1986] that a linear neutral differential equation can have unbounded solutions even though the associated characteristic equation has only purely imaginary roots (see also Snow [1965], Gromova [1967], Zverkin [1968], Brumley [1970], and Datko [1983]); such a behavior is not possible in the case of ordinary or (non-neutral) delay differential equations. It is known (Theorem 6.1 of Henry [1974]) that if the characteristic equation associated with a linear neutral equation has roots only with negative real parts and if the roots are uniformly bounded away from the imaginary axis, then the asymptotic stability of the trivial solution of the corresponding linear autonomous equation can be asserted. However, verification of the uniform boundedness away from the imaginary axis of all the roots of the characteristic equation is usually difficult. An alternative method for stability investigations is to resort to the technique of Lyapunov-type functional and functions; this will be amply illustrated in this chapter.

The purpose of this paper is to present several new results concerning relations between linear differential equations of the fourth order and nonlinear differential equations of the fourth order. These equations are involved in the description of models of building structures, where there are beams with small deflections or curved axes. We consider linear differential equations of the second, the third and the fourth order and nonlinear fourth order differential equations related via the Schwarzian derivative. As a result, we obtain new relations between the solutions of these linear and nonlinear equations. For example, assuming that we know a solution of linear differential equation of the fourth order and a solution of the third order linear differential equation, then the Schwarzian derivative of their ratio solves certain nonlinear differential equation of the fourth order. We also present some conditions on the coefficients when this statement holds. Two more similar statements are presented. To illustrate theorems and our constructive approach we give two examples. The given method may be generalized to differential equations of higher orders.

Asymptotic trichotomy for positive solutions of a class of fourth-order nonlinear neutral difference equations with quasidifferences

By using of the critical point method, the existence of periodic solutions for fourth-order nonlinear functional difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear functional difference equations. Results obtained generalize and complement the existing one.