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Razumikhin type theorems for impulsive differential equations with piecewise constant argument of generalized type

Razumikhin type theorems for impulsive differential equations with piecewise constant argument of generalized type

In the 1980s, differential equations with piecewise constant argument (DEPCA) that contain deviation of arguments were initially proposed for investigation by Cooke, Wiener, Busenberg, and Shah [10–12]. Later, many interesting results have been obtained and ap- plied efficiently to approximation of solutions and various models in biology, electronics, and mechanics [13–17]. Such equations represent a hybrid of continuous and discrete dy- namical systems and combine the properties of both differential and difference equations. Akhmet [18–20] generalized the concept of DEPCA by considering arbitrary piecewise constant functions as arguments; the proposed approach overcomes the limitations in the previously used method of study, namely reduction to discrete equations. Afterward, the results of the theory have been further developed [21, 22] and applied for qualitative anal-
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Invariant curves for a delay differential equation with a piecewise constant argument

Invariant curves for a delay differential equation with a piecewise constant argument

The study of differential equations with piecewise constant argument (EPCA) initiated in [, ]. These equations represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations, hence, they are of importance in control theory and in certain biomedical models []. In this paper the second order delay differential equation with a piecewise constant argument

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Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument

Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument

Differential equations with piecewise constant arguments (DEPCA) exist in a widely expanded areas such as biomedicine, chemistry, mechanical engineering, physics, and so on. To the best of our knowledge, the first mathematical model that includes a piecewise constant argument was proposed by Busenberg and Cooke [1]. They investigated the fol-

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8. Convergence in an impulsive advanced  differential equations with piecewise constant argument

8. Convergence in an impulsive advanced differential equations with piecewise constant argument

was initiated in ([13],[30]) where h (t) = [t] , [t − n] , [t + n] , etc. These types of equations have been intensively investigated for twenty five years. Systems de- scribed by DEP CA exist in a large area such as biomedicine, chemistry, physics and mechanical engineering. Busenberg and Cooke [11] first established a mathemati- cal model with a piecewise constant argument for analyzing vertically transmitted diseases. Examples in practice include machinery driven by servo units, charged particles moving in a piecewise constantly varying electric field, and elastic systems impelled by a Geneva wheel.
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Asymptotically Antiperiodic Solutions for a Nonlinear Differential Equation with Piecewise Constant Argument in a Banach Space

Asymptotically Antiperiodic Solutions for a Nonlinear Differential Equation with Piecewise Constant Argument in a Banach Space

In this paper, we give sufficient conditions for the existence and uniqueness of asymp- totically ω -antiperiodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space when ω is an integer. This is done using the Banach fixed point theorem. An example involving the heat operator is discussed as an illustration of the theory.

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On a class of third order neutral delay differential equations with piecewise constant argument

On a class of third order neutral delay differential equations with piecewise constant argument

In this paper we study existence, uniqueness and asymptotic stability the solutions of a class of third order neutral delay differential equations.. piecewise constant argument..[r]

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Almost Periodic Weak Solutions of Second Order Neutral Delay Differential Equations with Piecewise Constant Argument

Almost Periodic Weak Solutions of Second Order Neutral Delay Differential Equations with Piecewise Constant Argument

Differential equations with piecewise constant argument, which were firstly considered by Cooke and Wiener 1, and Shah and Wiener 2, usually describe hybrid dynamical systems a combination of continuous and discrete and so combine properties of both differential and difference equations. Over the years, great attention has been paid to the study of the existence of almost-periodic-type solutions of this type of equations. There are many remarkable works on this field see 3–10 and references therein. Particularly, the second-order neutral delay- differential equations with piecewise constant argument of the form
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Existence and Uniqueness of Periodic Solutions for a Second Order Nonlinear Differential Equation with Piecewise Constant Argument

Existence and Uniqueness of Periodic Solutions for a Second Order Nonlinear Differential Equation with Piecewise Constant Argument

However, there are reasons for studying higher-order equations with piecewise constant arguments. Indeed, as mentioned in 10, a potential application of these equations is in the stabilization of hybrid control systems with feedback delay, where a hybrid system is one with a continuous plant and with a discrete sampled controller. As an example, suppose that a moving particle with time variable mass rt is subjected to a restoring controller −φxt which acts at sampled time t. Then Newton’s second law asserts that

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Advanced differential equations with piecewise constant argument deviations

Advanced differential equations with piecewise constant argument deviations

established between differential equations with piecewise constant deviations and difference equations of an integer-valued argument... may be included in our scheme too..[r]

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BOUNDEDLY SOLVABILITY OF FIRST ORDER DELAY DIFFERENTIAL OPERATORS WITH PIECEWISE CONSTANT ARGUMENTS

BOUNDEDLY SOLVABILITY OF FIRST ORDER DELAY DIFFERENTIAL OPERATORS WITH PIECEWISE CONSTANT ARGUMENTS

Abstract. Using the methods of operator theory, we investigate all boundedly solvable extensions of a minimal operator generated by first order delay differential-operator ex- pression with piecewise constant argument in the Hilbert space of vector-functions at finite interval. Also spectrum of these extensions is studied.

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Stability of the logistic population model with generalized piecewise constant delays

Stability of the logistic population model with generalized piecewise constant delays

We see that a differential equation with piecewise constant argument, whose distance between the two consecutive switching moments is equal, can be reduced into an au- tonomous difference equation. However, if we have a differential equation with general- ized piecewise constant argument whose switching moments are ordered arbitrarily, then it generates a nonautonomous difference equation. This fact stimulates us to study the effects of generalized piecewise constant arguments on the stability of the fixed points of the logistic equation. Our results show that the existence of a generalized piecewise con- stant argument influences the behavior of the solutions. As far as we know, it is the first time in the literature that one reduces a differential equation with piecewise constant ar- gument of generalized type into a nonautonomous difference equation. This idea can be used for the investigation of differential equations with piecewise constant argument of generalized type.
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Discrete Pseudo Almost Periodic Solutions for Some Difference Equations

Discrete Pseudo Almost Periodic Solutions for Some Difference Equations

This work is organized as follows. In Section , we consider geometrical properties of the shift operator in general case and, we deal with the properties of shift operator the spaces of almost periodic and on ergodic sequences. In Section 3, we a consider the existence and uniqueness solutions of some difference equations using polynomial functions. In the last section, we deal with the application of the previous results to some second order differential equation with a piecewise constant argument.

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Partial differential equations with piecewise constant delay

Partial differential equations with piecewise constant delay

In Section 2, the initial value problems IVP are discussed for differential equations with piecewise constant argument EPCA in partial derivatives.. A class of loaded partial differentia[r]

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On the Spectrum of Almost Periodic Solution of Second Order Neutral Delay Differential Equations with Piecewise Constant of Argument

On the Spectrum of Almost Periodic Solution of Second Order Neutral Delay Differential Equations with Piecewise Constant of Argument

Differential equations with piecewise constant argument, which were firstly considered by Cooke and Wiener 1 and Shah and Wiener 2, combine properties of both differential and difference equations and usually describe hybrid dynamical systems and have applications in certain biomedical models in the work of Busenberg and Cooke 3. Over the years, more attention has been paid to the existence, uniqueness, and spectrum containment of almost periodic solutions of this type of equations see, e.g., 4–12 and reference there in.

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Vol 3, No 11 (2012)

Vol 3, No 11 (2012)

Differential equations with piecewise constant argument (For detailed study see [1]) are worthwhile studying since describe hybrid dynamical systems (a combination of continues and discrete) and therefore, combine properties of both differential and difference equations. These equations are considerable applied interest since differential equations with piecewise constant argument include, as particular cases, impulsive and loaded equations of control theory and are similar to those found in biomedical models .The initial value of a differential equation, is uncertain and a fuzzy approach is required often.
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Ergodic type solutions of differential equations with piecewise constant arguments

Ergodic type solutions of differential equations with piecewise constant arguments

[23] G. Papaschinopoulos, On asymptotic behavior of the solutions of a class of perturbed differential equations with piecewise constant argument and variable coefficients, J. Math. Anal. Appl. 185 (1994), no. 2, 490–500. MR 95f:34064. Zbl 810.34079. [24] G. Seifert, Almost periodic solutions for delay-differential equations with infinite delays, J.

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L0 Regularization for the Estimation of Piecewise Constant Hazard Rates in Survival Analysis

L0 Regularization for the Estimation of Piecewise Constant Hazard Rates in Survival Analysis

While this model can be used in a nonparametric setting, it is often used in combination with covariates effects. This is the case for instance for the popular Poisson regression model (see [2] [3]) which assumes a proportional effect on the covariates and a piecewise constant hazard model for the baseline hazard. This model is widely used in practice typically when dealing with register data. On one hand it allows to perform survival analysis with large computational savings (and save considerable data storage requirements) and, on the other hand, it allows to easily estimate the baseline hazard rate as a piecewise constant function and to give a very easy interpretation of the baseline hazard rate. Among many practical examples, we refer the reader to [4] [5] [6]. In practice, as noticed by [7] for Poisson regression, “ the choice of time intervals should generally be guided by subject matter aspects , but the numbers of events and numbers at risk within intervals may also be considered when specifying the number and lengths of the intervals. A study of a rare event and/or a small exposure group may require longer intervals .” While this might be true, it is clear that in some situations there might be no a priori knowledge for the choice of the time intervals and then they are usually arbitrarily chosen. This is the case for example in [1] where the time intervals in Table 1 were arbitrarily chosen as ten years length.
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ARCH-COMP19 Category Report: Hybrid Systems with Piecewise Constant Dynamics

ARCH-COMP19 Category Report: Hybrid Systems with Piecewise Constant Dynamics

Lyse Lyse is a tool for the reachability analysis of convex hybrid automata, namely hybrid automata with piecewise constant dynamics, whose constraints are possibly non-linear but re- quired to be convex. In this class are HPWC whose flow is contrained in rectangles, polyhedra, but also ellipses and parabolae. Linear hybrid automata are a special case. Lyse performs for- ward reachability analysis by means of template-polyhedra, whose directions are incrementally extracted from spurious counterexamples. The extraction is performed by a novel technique that generates interpolants by means of convex programming [9].
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Universal Algorithms for Learning Theory Part I : Piecewise Constant Functions

Universal Algorithms for Learning Theory Part I : Piecewise Constant Functions

In this paper, we propose an approach which allows us to circumvent these difficulties, while staying in spirit close to the ideas of wavelet thresholding. In our approach, the hypothesis classes H are spaces of piecewise constant functions associated to adaptive partitions Λ. Our partitions have the same tree structure as those used in the CART algorithm (Breiman et al., 1984), yet the selection of the appropriate partition is operated quite differently since it is not based on an optimization problem which would have to be re-solved when a new sample is added: instead our algorithm selects the partition through a thresholding procedure applied to empirical quantities computed at each node of the tree which play a role similar to wavelet coefficients. While the connection between CART and thresholding in one or several orthonormal bases is well understood in the fixed design denoising context (Donoho, 1997), this connection is not clear to us in our present context. As it will be demonstrated, our estimation schemes enjoy the following properties:
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Chaos and bifurcation of the Logistic discontinuous dynamical systems with piecewise constant arguments

Chaos and bifurcation of the Logistic discontinuous dynamical systems with piecewise constant arguments

The discontinuous dynamical systems generated by the retarded functional equations have been defined in [1]-[4]. The dynamical systems with piecewise constant arguments have been studied in [5]-[8] and the references therein. In this work we define the discontinuous dynamical systems generated by functional equations with piecewise constant arguments. The dynamic properties of two discontinuous dynamical systems of the Logistic equation will be discussed. Comparison with the corresponding discrete dynamical systems of the Logistic equation

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