Retarded Arguments Research Articles

On the stability of one equation with a discrete retarded argument and a constant concentrated delay

A functional differential equation with a discrete retarded argument and a constant concentrated delay is considered. The problem of the asymptotic stability of this equation is reduced to the problem of the location of the spectrum of the shift operator. Coefficient sufficient conditions for the asymptotic stability of this equation are obtained. The domain in the parameter space such that these conditions are necessary is obtained.

Влияние деформаций инструмента и заготовки в направлении скорости резания на динамику обработки

Numerous studies were devoted to studying the dynamic cutting system considered in combination of the tool and the workpiece sub-systems interacting through a dynamic connection formed by the machining process. Dynamic coupling is simulated by the cutting forces represented in the system state coordinates, which determines the system properties. Several models were proposed to describe the dynamic connection reflecting various experimentally observed effects and contributing to the self-excitation. Regenerative effect of the tool trace left on the workpiece at the previous revolution was considered. The regenerative effect was studied on the basis of assumption that the retarding argument in the forces description was remaining unchanged. The paper studies the regenerative effect influence on dynamic properties of the cutting process taking into account (unlike the known works) the retarded argument dependence on the tool deformation displacement. This could fundamentally change properties of the system under consideration in the unity of stability and the attracted formed deformation displacement sets (limit cycles, invariant tori and chaotic attractors). Results of the mathematical simulation are presented taking into account the regenerative self-excitation, where the retarded argument is the state coordinates function. Bifurcation diagrams of attracting sets of the deformation displacements are considered, and conditions for formation of their superlow-frequency components of the complex spatial-temporal structure are discussed. Research results are aimed at determining the machining conditions based on requirements for ensuring the specified quality of parts manufacturing using the longitudinal turning example.

New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments

In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving \(\lim \sup\), which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions.

Oscillation tests for difference equations with non-monotone retarded arguments

This paper presents new sufficient conditions, involving limsup, for the oscillation of all solutions of linear difference equations with general retarded arguments.

Oscillation Criteria for Difference Equations with Several Retarded Arguments

We consider a difference equation$$ \varDelta u(k)+\sum \limits_{i-1}^m{p}_i(k)\;u\left({\tau}_i(k)\right)=0, $$ where m ϵ N, the functions, pi : N → R+ and τi : N → N, limk → + ∞τi(k) = + ∞ , τi(k) ≤ k − 1, i = 1, …, m, are defined on the set of natural numbers, and Δu(k) = u(k + 1) − u(k) is the difference operator. New oscillation criteria are established for all solutions of the equation.

Oscillation of first-order differential equations with several non-monotone retarded arguments

Abstract Consider the first-order linear differential equation with several non-monotone retarded arguments x ′ ⁢ ( t ) + ∑ i = 1 m p i ⁢ ( t ) ⁢ x ⁢ ( τ i ⁢ ( t ) ) = 0 {x^{\prime}(t)+\sum_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0} , t ≥ t 0 {t\geq t_{0}} , where the functions p i , τ i ∈ C ⁢ ( [ t 0 , ∞ ) , ℝ + ) {p_{i},\tau_{i}\in C([t_{0},\infty),\mathbb{R}^{+})} , for every i = 1 , 2 , … , m {i=1,2,\ldots,m} , τ i ⁢ ( t ) ≤ t {\tau_{i}(t)\leq t} for t ≥ t 0 {t\geq t_{0}} and lim t → ∞ ⁡ τ i ⁢ ( t ) = ∞ {\lim_{t\to\infty}\tau_{i}(t)=\infty} . New oscillation criteria which essentially improve the known results in the literature are established. An example illustrating the results is given.

Spectral properties of boundary-value-transmission problems with a constant retarded argument

In this work, spectra and asymptotics of eigenfunctions of a generalized class of boundary value problems with constant retarded argument are obtained. Contrary to previous works in the literature, the problem has nonclassical transmission conditions.

Computation of Trace and Nodal Points of Eigenfunctions for a Sturm-Liouville Problem with Retarded Argument

In this study, a formula for regularized sums of eigenvalues and nodal points of eigenfunctions for a discontinuous Sturm-Liouville problem with a constant retarded argument. Contrary to standart problems the spectral parameter appears not only in the differential equation, but also in one of the boundary conditions. Thus, we see whether the nodal points of eigenfunctions and the trace change or not.

Coupled Pettis-Hadamard Fractional Differential Systems with Retarded and Advanced Arguments

In this study, we present some results concerning the existence of weak solutions for some coupled systems of Hadamard fractional differential equations involving the both retarded and advanced arguments. Our results are obtained by using fixed point theory and the technique of measure of weak noncompactness.

A note on the continuability and boundedness of solutions to a class of vector differential equations of third order with finite delay

In this article, we use Lyapunov's second (or direct) method, by building an appropriate Lyapunov functional. Sufficient conditions which warrant the continuability and boundedness of all solutions to a kind of nonlinear vector differential equations of third order with constant retarded argument are established. Also, we give an example to describe the theoretical analysis in this study and to check the impressiveness of the method employed.

Oscillation of differential equations with non-monotone retarded arguments

Consider the first-order retarded differential equation $$\begin{eqnarray}x^{\prime }(t)+p(t)x({\it\tau}(t))=0,\quad t\geqslant t_{0},\end{eqnarray}$$ where $p(t)\geqslant 0$ and ${\it\tau}(t)$ is a function of positive real numbers such that ${\it\tau}(t)\leqslant t$ for $t\geqslant t_{0}$, and $\lim _{t\rightarrow \infty }{\it\tau}(t)=\infty$. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $\liminf$, is established when the well-known oscillation condition $$\begin{eqnarray}\liminf _{t\rightarrow \infty }\int _{{\it\tau}(t)}^{t}p(s)\,ds>\frac{1}{e}\end{eqnarray}$$ is not satisfied. An example illustrating the result is also given.

Oscillations of difference equations with non-monotone retarded arguments

Consider the first-order retarded difference equationΔx(n)+p(n)xτ(n)=0,n∈N0where (p(n))n⩾0 is a sequence of nonnegative real numbers, and (τ(n))n⩾0 is a sequence of integers such that τ(n)⩽n-1, n⩾0, and limn→∞τ(n)=∞. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving liminf, is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail.

Asymptotic behavior in neutral difference equations with several retarded arguments

In this paper, we study the asymptotic behavior of the solutions of a neutral type difference equation of the form% \[ \Delta \bigg[ x(n)+\sum_{j=1}^{w}c_{j}x(\tau _{j}(n))\bigg] +p(n)x(\sigma (n))=0,\quad n\geq 0 \] where $\tau _{j}(n)$, $j=1,\ldots,w$, are general retarded arguments, $\sigma (n) $ is a general deviated argument (retarded or advanced), $c_{j}\in \mathbb{R}$, $j=1,\ldots,w$, $(p(n))_{n\geq 0}$ is a sequence of positive real numbers such that $p(n)\geq p$, $p\in\mathbb{R}_{+}$, and $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$. We also examine the convergence of the solutions when these are continuous and differentiable with respect to $c_{j}$, $j=1,\ldots,w$.

Oscillation Criteria for Differential Equations with Several Retarded Arguments

Consider the first-order linear differential equation with several retarded arguments $$ x^{\prime}(t)+\sum\limits_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0,\;\;\;t\geq t_{0}, $$ where the functions $p_{i},\tau_{i}\in C([t_{0,}\infty),\mathbb{R}^{+}),$ for every $i=1,2,\ldots,m,$ $\tau_{i}(t)\leq t$ \ for $t\geq t_{0}$ and $% \lim_{t\rightarrow \infty}\tau_{i}(t)=\infty $. The state of the art on the oscillation of all solutions to these equations are established especially in the case of non-monotone arguments. Examples illustrating the results are given.

Oscillation of differential equations with non-monotone retarded arguments

This paper, which was published on 3 November 2015, has been withdrawn.

A survey on the oscillation of differential equations with several deviating arguments

Consider the first-order linear differential equation with several retarded arguments x ′ (t)+ ∑ i = 1 m p i (t)x ( τ i ( t ) ) =0,t≥ t 0 , where the functions p i , τ i ∈C([ t 0 , ∞), R + ) for every i=1,2,…,m, τ i (t)≤t for t≥ t 0 and lim t → ∞ τ i (t)=∞. A survey on the oscillation of all solutions to this equation is presented in the case of several non-monotone arguments and especially when well-known oscillation conditions are not satisfied. Examples illustrating the results are given.MSC:34K11, 34K06.

Oscillatory Conditions on Both Directions for a Nonlinear Impulsive Differential Equation with Deviating Arguments

It has been observed that most investigations on the oscillations of impulsive differential equations are one-directional. No explanation has hitherto been contemplated for such restrictions. In this paper, we propose some sufficient conditions for both-directional oscillation of a nonlinear delay impulsive differential equation with several retarded arguments. An example of a one-dimensional delay impulsive equation is given to further demonstrate the efficiency of the approach.

New Retarded Integral Inequalities with Applications

Some new nonlinear integral inequalities of Gronwall type for retarded functions are established, which extend the results Lipovan (2003) and Pachpatte (2004). These inequalities can be used as basic tools in the study of certain classes of functional differential equations as well as integral equations. A existence and a uniqueness on the solution of the functional differential equation involving several retarded arguments with the initial condition are also indicated.

The modified two sided approximations method and Padé approximants for solving the differential equation with variant retarded argument

The aim of this paper is to present an efficient numerical procedure for solving boundary value problems for a differential equation with retarded argument: x ′′(t)+a(t)x(t−τ(t))=f(t), x(t)=ϕ(t) (λ 0⩽t⩽0), x(T)=x T, where 0⩽ t⩽ T and a(t),f(t),τ(t)⩾0 (0⩽t⩽T) and ϕ(t) (λ 0⩽t⩽0) are known continuous functions. A differential equation with retarded argument is computed by converting the obtained series solution into Padé (approximants) series. First we calculate power series of the given equation system then transform it into Padé (approximants) series form, which give an arbitrary order for solving differential equation numerically.

The ordinary successive approximations method and Padé approximants for solving a differential equation with variant retarded argument

The aim of this paper is to present an efficient numerical procedure for solving boundary value problems for a differential equation with retarded argument: x ″(t)+a(t)x(t−τ(t))=f(t) x(t)=ϕ(t) (λ 0⩽t⩽0), x(T)=x T, where 0⩽ t⩽ T and a(t),f(t),τ(t)⩾0 (0⩽t⩽T) and ϕ(t) (λ 0⩽t⩽0) are known continuous functions. A differential equation with retarded argument is computed by converting the obtained series solution into Padé (approximants) series. First we calculate power series of the given equation system then transform it into Padé (approximants) series form, which give an arbitrary order for solving differential equation numerically.