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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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**

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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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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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.

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]

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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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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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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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.

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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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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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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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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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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[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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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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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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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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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