Differential Equations Of Neutral Type Research Articles

Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

We present sufficient conditions for the existence of a solution x to an equation Δm(xn-unxn-k)=anf(xn-τ)+bn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\ au })+b_n, \\end{aligned}$$\\end{document}which is “close” to a given solution y to the linear homogeneous equation of neutral type Δm(yn-λyn-k)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta ^m(y_n-\\lambda y_{n-k})=0$$\\end{document}, where λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} is the limit of the sequence u. Closeness of solutions to above equations is understood as xn-yn=o(ωn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_n-y_n=\ extrm{o}(\\omega _n)$$\\end{document}, where ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution x to Δm(xn-unxn-k)=anf(xn-τ)+bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\ au })+b_n$$\\end{document} and a given nonincreasing sequence with positive values ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} there exists a polynomial sequence φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} of degree less than m such that xn=φ(n)+o(ωn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_n=\\varphi (n)+\ extrm{o}(\\omega _n)$$\\end{document}. Presented conditions strongly depend on λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}.

Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

Stepanov Eberlein almost periodic functions and applications

In this paper, we prove that the class of Eberlein weakly almost periodic functions in the Stepanov's sense is strictly larger than both Eberlein weak almost periodicity (E‐w.a.p.) and Stepanov's almost periodicity. Some examples are provided. We present a composition result for this class of functions, and using Banach fixed‐point theorem, we prove the existence and uniqueness of E‐w.a.p. solution for a differential equation of neutral type with nondense domain when the forcing terms are only E‐w.a.p. in the Stepanov's sense. We mention that our results can extend and improve previous several works in this direction. An application is given to illustrate the result.

Exponential Stability of Autonomous Differential Equations of Neutral Type. II

Exponential Stability of Autonomous Differential Equations of Neutral Type. II

On asymptotic properties of solutions for differential equations of neutral type

The stability of systems of linear autonomous functional differential equations of neutral type is studied. The study is based on the well-known representation of the solution in the form of an integral operator, the kernel of which is the Cauchy function of the equation under study. The definitions of Lyapunov, asymptotic, and exponential stability are formulated in terms of the corresponding properties of the Cauchy function, which allows us to clarify a number of traditional concepts without loss of generality. Along with the concept of asymptotic stability, a new concept of strong asymptotic stability is introduced. The main results are related to the stability with respect to the initial function from the spaces of summable functions. In particular, it is established that strong asymptotic stability with initial data from the space \(L_1\) is equivalent to the exponential estimate of the Cauchy function and, moreover, exponential stability with respect to initial data from the spaces \(L_p\) for any \(p\ge1.\)

Exponential Stability of Autonomous Differential Equations of Neutral Type. I

Exponential Stability of Autonomous Differential Equations of Neutral Type. I

Oscillation Test for Second-Order Differential Equations with Several Delays

In this paper, the oscillatory properties of certain second-order differential equations of neutral type are investigated. We obtain new oscillation criteria, which guarantee that every solution of these equations oscillates. Further, we get conditions of an iterative nature. These results complement and extend some beforehand results obtained in the literature. In order to illustrate the results we present an example.

ON THE EXACT ESTIMATION FOR THE EXPONENT OF SOLUTIONS OF LINEAR AUTONOMOUS DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

We consider linear autonomous differential equations of neutral type. We obtain the exact estimation for the exponent of solutions of these equations

Even-Order Neutral Delay Differential Equations with Noncanonical Operator: New Oscillation Criteria

The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and reflection of developments in the study of second-order equations. We also derive criteria for improving conditions that exclude the decreasing positive solutions of the considered equation.

Massera Problem for Some Nonautonomous Functional Differential Equations of Neutral Type with Finite Delay

Massera Problem for Some Nonautonomous Functional Differential Equations of Neutral Type with Finite Delay

Oscillation criteria for sublinear and superlinear first-order difference equations of neutral type with several delays

<abstract><p>The purpose of this paper is to investigate the oscillatory behaviour of a class of first-order sublinear and superlinear neutral difference equations. Some conditions are established by applying Banach's Contraction mapping principle, Knaster-Tarski fixed point theorem and using several inequalities. We provide some examples to illustrate the outreach of the main results.</p></abstract>

Stability results for the functional differential equations associated to water hammer in hydraulics

There is considered a system of two sets of partial differential equations describing the water hammer in a hydroelectric power plant containing the dynamics of the tunnel, turbine penstock, surge tank and hydraulic turbine. Under standard simplifying assumptions (negligible Darcy–Weisbach losses and dynamic head variations), a system of functional differential equations of neutral type, with two delays, can be associated to the aforementioned partial differential equations and existence, uniqueness and continuous data dependence can be established. Stability is then discussed using a Lyapunov functional deduced from the energy identity. The Lyapunov functional is "weak" i.e. its derivative function is only non-positive definite. Therefore only Lyapunov stability is obtained while for asymptotic stability application of the Barbashin–Krasovskii–LaSalle invariance principle is required. A necessary condition for its validity is the asymptotic stability of the difference operator associated to the neutral system. However, its properties in the given case make the asymptotic stability non-robust (fragile) in function of some arithmetic properties of the delay ratio.

Controllability of neutral impulsive fractional differential equations with Atangana-Baleanu-Caputo derivatives

In this article, we aim to establish the controllability results for fractional differential equations of neutral type with Atangana-Baleanu-Caputo derivatives. We develop these results with the help of the theory of semigroup operators and fixed point theorem coupled with a measure of noncompactness. An example is also presented to verify the applicability of the obtained results.

Asymptotic Properties of Solutions to Differential Equations of Neutral Type

Asymptotic Properties of Solutions to Differential Equations of Neutral Type

Bernstein collocation method for neutral type functional differential equation

Functional differential equations of neutral type are a class of differential equations in which the derivative of the unknown functions depends on the history of the function and its derivative as well. Due to this nature the explicit solutions of these equations are not easy to compute and sometime even not possible. Therefore, one must use some numerical technique to find an approximate solution to these equations. In this paper, we used a spectral collocation method which is based on Bernstein polynomials to find the approximate solution. The disadvantage of using Bernstein polynomials is that they are not orthogonal and therefore one cannot use the properties of orthogonal polynomials for the efficient evaluation of differential equations. In order to avoid this issue and to fully use the properties of orthogonal polynomials, a change of basis transformation from Bernstein to Legendre polynomials is used. An error analysis in infinity norm is provided, followed by several numerical examples to justify the efficiency and accuracy of the proposed scheme.

A pseudo-spectrum based characterization of the robust strong H-infinity norm of time-delay systems with real-valued and structured uncertainties

Abstract This paper provides a mathematical characterization of the robust (strong) H-infinity norm of an uncertain linear time-invariant system with discrete delays in terms of the robust distance to instability of an associated characteristic matrix. The considered class of uncertainties consists of real-valued, structured, Frobenius norm-bounded matrix uncertainties that act on the coefficient matrices. The robust H-infinity norm, defined as the worst-case value of the H-infinity norm over all admissible uncertainty values, is an important measure of robust performance to quantify the worst-case disturbance rejection of an uncertain dynamical system. For the considered system class, this robust H-infinity norm is however a fragile measure, as for a particular instance of the uncertainties, the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the uncertainties on the system matrices and infinitesimal delay perturbations. This quantity is a continuous function of both the elements of the system matrices and the delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the robust distance to instability of an associated characteristic matrix. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems.

Dynamics of a Coupled Chua’s Circuit with Lossless Transmission Line

This paper proposes a coupled-circuit system composed of two Chua’s circuits with lossless transmission lines. By applying the Kirchhoff’s voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point.

Neutral fuzzy fractional functional differential equations

In this paper, a class of fuzzy fractional functional differential equations of neutral type is investigated. The existence, uniqueness and iterative formula in the case of linear equations are obtained. Meanwhile, the continuous dependence of solutions on initial data, parameters and functions involved in right hand side of equation is also discussed. Several examples are given to illustrate the applications of our results.

ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

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