Dense Short Solution Segments from Monotonic Delayed Arguments

We construct a delay functional d on an open subset of the space $$C^1_r=C^1([-r,0],\mathbb {R})$$ C r 1 = C 1 ( [ - r , 0 ] , R ) and find $$h\in (0,r)$$ h ∈ ( 0 , r ) so that the equation $$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$ x ′ ( t ) = - x ( t - d ( x t ) ) defines a continuous semiflow of continuously differentiable solution operators on the solution manifold $$\begin{aligned} X=\{\phi \in C^1_r:\phi '(0)=-\phi (-d(\phi ))\}, \end{aligned}$$ X = { ϕ ∈ C r 1 : ϕ ′ ( 0 ) = - ϕ ( - d ( ϕ ) ) } , and along each solution the delayed argument $$t-d(x_t)$$ t - d ( x t ) is strictly increasing, and there exists a solution whose short segments$$\begin{aligned} x_{t,short}=x(t+\cdot )\in C^2_h,\quad t\ge 0, \end{aligned}$$ x t , s h o r t = x ( t + · ) ∈ C h 2 , t ≥ 0 , are dense in an infinite-dimensional subset of the space $$C^2_h$$ C h 2 . The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments.

Optimisation problem for some class of hybrid differential-difference systems with delay

In the paper, the linear differential-difference dynamic systems with delayed arguments are considered. Such systems have a lot of application areas, in particular, processes with repetitive and learning structure. We apply the method of the separation hyperplane theorem for convex sets to establish optimality conditions for the control function to drive the trajectory to zero equilibrium state in the fastest possible way. For the special case of the integral control constraints, the proposed method is detailed to establish an analytical form of the optimal control function. The illustrative example is given to demonstrate the obtained results with the step-by-step calculation of the basic elements of the optimal control.

Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem

Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators.

New Oscillation Criteria for Advanced Differential Equations of Fourth Order

The main objective of this paper is to establish new oscillation results of solutions to a class of fourth-order advanced differential equations with delayed arguments. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Four examples are provided to illustrate the main results.

Asymptotic Behavior of Solutions of the Third Order Nonlinear Mixed Type Neutral Differential Equations

The objective of our paper is to study asymptotic properties of the class of third order neutral differential equations with advanced and delayed arguments. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided.

On Stationary Solutions Of Delay Differential Equations: A Model Of Local Translation In Synapses

The results of analytical analysis of stationary solutions of a differential equation with two delayed arguments τ1 and τ2 are presented. Such equations are used in modeling of molecular-genetic systems where the delay of arguments appear naturally. Conditions of existence of non-negative solutions are described, and dependence of stability of these solutions on the values of delayed arguments is studied. This stability theory allows to give complete characterization of these solutions for all values of the parameters of the model, and ensures instability of a positive equilibrium point for any values of the delays τ2 ≥ τ1 ≥ 0 in the case when it is unstable for τ2 = τ1 = 0 (absolute instability). If this positive equilibrium point is stable only for τ2 = τ1 = 0, then this domain τ2 ≥ τ1 ≥ 0 is the domain of absolute instability as well. For positive equilibrium points which are stable at τ2 = τ1 = 0, we find domains of absolute stability were the equilibrium points remain stable for all values of the parameters τ1 and τ2. The domains of relative stability, where these points become unstable for some values of these parameters are also described. We show that when the efficiency of translation, and non-linearity and complexity of its regulation mechanisms grow, the domains of the absolute and relative stability of the positive equilibrium point shrink, while the domains of its instability expand. So, enhanced activity of the local translation system can be a factor of its instability and that of the risk of neuro-psychical diseases related to distortions of plasticity of the synapse and memory, where importance of stability of the proteome in the synapse is postulated.

Some New Oscillation Criteria for Second Order Neutral Differential Equations with Delayed Arguments

In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given.