On delay-differential equations with meromorphic solutions of hyper-order less than one

A Method and New Results for Stability and Instability of Autonomous Functional Differential Equations

Aimed at resolving the problem of the random time delay of the network-based control system (NCS), this paper gives a state feed-back controller in a concise form which is expressed as the problem of the feasibility of a certain linear matrix inequality system based on Lyapunov stability theory. For any time delay satisfying 0lesdlesdmacr, the system with the control u=Kx(t-d) is globally asymptotically stable. And the emulational results show a good effect of this controller

Meromorphic solutions of delay differential equations related to logistic type and generalizations

Special functions are functions encountered in the process of solving various problems in physics. In a natural way one of the directions concerning analysis of properties of special function is to examine them as solutions of linear differential equations. This line of thought can be extended not only to nonlinear ordinary differential equations, but also to difference equations and delay-differential equations. Well known examples of nonlinear ordinary differential equations with solutions expressed in terms of special functions include Riccati equations (Airy functions) and Painlevé equations (Airy functions for P 2 P_2 , Bessel functions for P 3 P_3 or Whittaker functions for P 4 P_4 ). Painlevé transcendents have been thoroughly examined by means of global Nevanlinna theory, resulting in estimates of growth, descriptions of distribution of zeros and poles, analysis of existence of exceptional and ramified values. It has also provided insight into behaviour of special functions involved. Recent development of difference Nevanlinna theory opened similar possibilities of analysis of behaviour of solutions of difference and delay-differential equations in general, and various involved special functions in particular.

Using a Razumikhin ‐ type theorem, we deduce sufficient conditions that guarantee the uniform asymptotic stability and boundedness of solutions of a scalar real fourth‐order delay differential equation. The Lyapunov function constructed for an ordinary fourth‐order differential equation is seen to work for the delay system.

In this paper, we investigate the differential-difference equation $f^{2}(z)+p(z)f(z+c)+h(z)f'(z)+g(z)$ $=p_{1}{\\rm~e}^{\\alpha_{1}z^{n}}+p_{2}{\\rm~e}^{\\alpha_{2}z^{n}},~$where $n\\in\\mathbb{N}^{+},~$ $c\\in\\mathbb{C}\\setminus\\{0\\}$, $\\alpha_{1},~\\alpha_{2}$ are two distinct nonzero constants, with coefficients being small functions of ${\\rm~e}^{z^{n}}$. We study properties of meromorphic solutions of the above equation,which have extended and improved some known results obtained recently.

This paper is devoted to exploring the properties of meromorphic solutions on complex differential–difference equations using Nevanlinna theory. We state some relationships between the exponent of convergence of zeros with the order of meromorphic solutions on linear or non-linear differential–difference equations.

The celebrated Malmquist theorem states that a differential equation, which admits a transcendental meromorphic solution, reduces into a Riccati differential equation. Motivated by the integrability of difference equations, this paper investigates the delay differential equations of form $ w(z+1)-w(z-1)+a(z)\frac{w'(z)}{w(z)} = R(z, w(z))(*), $ where $ R(z, w(z)) $ is an irreducible rational function in $ w(z) $ with rational coefficients and $ a(z) $ is a rational function. We characterize all reduced forms when the equation $ (*) $ admits a transcendental entire solution with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc. Amer. Math. Soc. 145, no.6 (2017)], we obtain the reduced forms without the assumptions that the denominator of rational function $ R(z, w(z)) $ has roots that are nonzero rational functions in $ z $. The value distribution and forms of transcendental entire solutions for the reduced delay differential equations are studied. The existence of finite iterated order entire solutions of the Kac-van Moerbeke delay differential equation is also detected.

We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function B ( r , s ) B(r,s) of the natural number parameters r r and s s so that for any system of algebraic ordinary differential-difference equations in the variables x = x 1 , … , x q \mathbfit {x} = x_1, \ldots , x_q and y = y 1 , … , y r \mathbfit {y} = y_1, \ldots , y_r , each of which has order and degree in y \mathbfit {y} bounded by s s over a differential-difference field, there is a nontrivial consequence of this system involving just the x \mathbfit {x} variables if and only if such a consequence may be constructed algebraically by applying no more than B ( r , s ) B(r,s) iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over C \mathbb {C} is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.

Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay

The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.

We introduce a family of solutions to a linear delay differential equation with continuous and piecewise constant arguments depending on four parameters, and we give an asymptotic expansion of any solution of this equation with respect to this family. The fundamental solutions are indexed by the zeros (counting multiplicities) of a meromorphic function in the complex plane. This function generalizes the characteristic equation which was known to characterize the oscillatory behaviour of the delay differential equation for certain values of the parameters. The proof uses the Laplace transform, Fourier series, and the adjoint equation.

<abstract> In this article, we mainly use Nevanlinna theory to investigate some differential-difference equations. Our results about the existence and the forms of solutions for these differential-difference equations extend the previous theorems given by Wang, Xu and Tu ^{[<xref ref-type="bibr" rid="b19">19</xref>]}. </abstract>

By using Nevanlinna theory and linear algebra, we show that the number one is a lower bound of the hyper-order of any meromorphic solution of a nonlinear differential–difference equation under certain conditions.

Membership-difference-dependent switching control for time-delay T–S fuzzy systems