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In this paper we consider a class of nonlinear neutral functional differential equations with state-dependent delays. We study well-posedness and differentiability of the solution with respect to the parameters in a pointwise sense and also in the supremum norm. [ABSTRACT FROM AUTHOR]

In this paper we study ventilation patterns in a set of parameter dependent nonlinear delay equations with two transport delays modeling the human respiratory control system with peripheral and central control loops. We present a convergent numerical scheme suitable to perform simulations when all disturbances and system parameters are known, then we consider the numerical identifiability of various system parameters based on ventilation data. We are especially interested in the identification of the transport delays in the control loops because these parameters are not measurable directly, but they have a strong influence on system stability/instability. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of delay differential equations of the form x(t) = α(t)h(x(t - τ), x(t - σ)) - β(t) f (x(t)), where h is a mixed monotone function. Sufficient conditions are presented for the permanence of the positive solutions. Our results give also lower and upper estimates of the limit inferior and the limit superior of the solutions via a special solution of an associated nonlinear system of algebraic equations. The results are generated to a more general class of delay differential equations with mixed monotonicity. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of functional differential equations with time-dependent delay. We show continuous differentiability of the solution with respect to the time delay function for each fixed time value assuming natural conditions on the delay function. As an application of the differentiability result, we give a numerical study to estimate the time delay function using the quasilinearization method. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay. As an application, we discuss asymptotic properties of perturbed nonlinear differential equations with state-dependent delays. [ABSTRACT FROM AUTHOR]

The nonlinear delay differential equation x ˙ ( t ) = r ( t ) [ g ( t , x t ) − h ( x ( t ) ) ] , t ≥ 0 is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the equation. In several particular cases, explicit formulas are given for the upper and lower limit of the solutions. In some special cases, we give conditions which imply that all solutions have the same asymptotic behavior, in particular, when they converge to a periodic or constant steady-state. [ABSTRACT FROM AUTHOR]

• Delay differential equations with impulses. • Self-support condition. • Numerical approximation using equations with piecewise constant arguments. • Existence of periodic solutions. In this paper we consider a scalar linear delay equation with constant delay associated with an impulsive self-support condition. We define a numerical approximation scheme using a sequence of approximate delay equations with piecewise constant arguments, and we show its theoretical convergence. We present numerical examples to illustrate the applicability of the method, and we also observe existence of periodic solutions of the impulsive delay equation using numerical studies. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the $$C$$ C -norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the growth/decay rate of solutions of a class of nonlinear Volterra difference equations. Our results can be applied for the case when the characteristic equation of an associated linear difference equation has complex dominant eigenvalue with higher than one multiplicity. Illustrative examples are given for describing the asymptotic behaviour of solutions in a class of linear difference equations and in several discrete nonlinear population models. [ABSTRACT FROM PUBLISHER]

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is strictly monotone increasing. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the growth/decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation-of-constants formula. These results are applicable to some abstract equations which appear in the theory of age-dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays. Examples are given to illustrate the sharpness of the results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we formulate a stability theorem by means of linearization around a trivial solution in the case of autonomous neutral functional differential equations with state-dependent delays. We prove that if the trivial solution of the linearized equation is exponentially stable, then the trivial solution of the nonlinear equation is exponentially stable as well. As an application of the main result, explicit stability conditions are given. [Copyright &y& Elsevier]

Abstract: In this paper we consider a class of nonlinear neutral differential equations with state-dependent delays. We study well-posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite-dimensional parameters in a pointwise sense and also in the C- and -norms. [Copyright &y& Elsevier]

Abstract: Unwanted relative vibrations between the tool and the workpiece represent significant challenges in high-speed machining. In order to avoid this problem, one needs to specify ranges for system parameters (spindle speed, depth of cut) for which the process is stable, i.e., to obtain a so-called stability chart. Such stability charts usually can only be given by numerical means which is illustrated in the paper for a single degree of freedom model of milling. In this paper, we establish the convergence of the semi-discretization approximation method for a class of delay equations modeling the milling process. Moreover, we show that semi-discretization preserves asymptotic stability of the original equation, thus it can be used to obtain good approximations for the stability charts. [Copyright &y& Elsevier]

In this paper, we study stability of periodic solutions of a class of nonlinear functional differential equations (FDEs) with state-dependent delays using the method of linearization. We show that a periodic solution of the nonlinear FDE is exponentially stable, if the zero solution of an associated linear periodic linear homogeneous FDE is exponentially stable. [Copyright &y& Elsevier]

In this paper we study the asymptotic behavior and numerical approximation of the single neuron model equation x˙(t)=−dx(t)+af(x(t))+bf(x(t−τ))+I, t⩾0 (1), where d>0 and f(x)=0.5(|x+1|−|x−1|). We obtain new sufficient conditions for global asymptotic stability of constant equilibriums of (1), give several numerical examples to illustrate our results, and formulate conjectures on the asymptotic behavior of the solutions based on our numerical experiments. [Copyright &y& Elsevier]

In this paper we study numerical approximation of linear neutral differential equations on infinite interval using equations with piecewise constant arguments. As an application of our approximation results, we obtain stability theorems for some classes of linear delay and neutral difference equations. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the bounded input bounded output (BIBO) stability in a class of control system of nonlinear differential equations with time-delay. The proofs are based on our studies on the boundedness of the solutions of a general class of nonlinear Volterra integral equations. [Copyright &y& Elsevier]

Generalized Laguerre polynomials, L n (α) , verify the well-known Rodrigues' formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues' formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them. [ABSTRACT FROM AUTHOR]

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results. [ABSTRACT FROM AUTHOR]

The motivation for this paper is to create new Philos-type oscillation criteria that are established for third-order mixed neutral differential equations with distributed deviating arguments. The key idea of our approach is to use the triple of the Riccati transformation techniques and the integral averaging technique. The established criteria improve, simplify and complement results that have been published recently in the literature. An example is also given to demonstrate the applicability of the obtained conditions. [ABSTRACT FROM AUTHOR]

This article aims to mark out new conditions for oscillation of the even-order Emden–Fowler neutral delay differential equations with neutral term β 1 ı Φ α [ ζ r − 1 ı ] ′ + β 3 ı Φ α [ ς ξ ı ] = 0. The obtained results extend, and simplify known conditions in the literature. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]