Differential Equations Of Neutral Type Research Articles

Periodic solution for neutral-type differential equation with piecewise impulses on time scales

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

On Asymptotic Properties of Solutions for Differential Equations of Neutral Type

On Asymptotic Properties of Solutions for Differential Equations of Neutral Type

Second-order general Emden-Fowler differential equations of neutral type: Improved Kamenev-type oscillation criteria

<p>The study of the oscillatory behavior of a general class of neutral Emden-Fowler differential equations is the focus of this work. The main motivations for studying the oscillatory behavior of neutral equations are their many applications as well as the richness of these equations with exciting analytical issues. We obtained novel oscillation conditions in Kamenev-type criteria for the considered equation in the canonical case. We improve the monotonic and asymptotic characteristics of the non-oscillatory solutions to the considered equation and then utilize these characteristics to refine the oscillation conditions. We present, through examples and discussions, what demonstrates the novelty and efficiency of the results compared to previous relevant findings in the literature. In addition, we numerically represent the solutions of some special cases to support the theoretical results.</p>

Minimax solutions of Hamilton-Jacobi equations in dynamic optimization problems for hereditary systems

A survey of the results concerning the development of the theory of Hamilton-Jacobi equations for hereditary dynamical systems is presented. One feature of these systems is that the rate of change of their state depends not only on the current position, like in the classical case, but also on the full path travelled, that is, the history of the motion. Most of the paper is devoted to dynamical systems whose motion is described by functional differential equations of retarded type. In addition, more general systems described by functional differential equations of neutral type and closely related systems described by differential equations with fractional derivatives are considered. So-called path-dependent Hamilton-Jacobi equations are treated, which play for the above classes of systems a role similar to that of the classical Hamilton-Jacobi equations in dynamic optimization problems for ordinary differential systems. In the context of applications to control problems, the main attention is paid to the minimax approach to the concept of a generalized solution of the Hamilton-Jacobi equations under consideration and also to its relationship with the viscosity approach. Methods for designing optimal feedback control strategies with memory of motion history which are based on the constructions discussed are presented. Bibliography: 183 titles.

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.