Functional Differential Equations Of Neutral Type Research Articles

Asymptotic behaviour of the non-oscillating solutions of first order nonlinear impulsive functional-differential equations of neutral type

This paper deals with nonlinear first-order impulsive functional-differential equations of neutral type with variable coefficients. The asymptotic behaviour of the non-oscillating solutions of such equations is investigated as a function of the values of the variable coefficient in the neutral term to serve as a basis for further studies of the oscillation of the solutions of such equations.

On Some Nonlinear Functional-Differential Equations of Neutral Type with Linear Deviations of the Argument

On Some Nonlinear Functional-Differential Equations of Neutral Type with Linear Deviations of the Argument

Nonlinear Functional-Differential Equation of Neutral Type with Linear Deviation of the Argument

Nonlinear Functional-Differential Equation of Neutral Type with Linear Deviation of the Argument

Solvability of the Pursuit Problem for One Differential Game in a Banach Space

In a Banach space, we study the solvability of the pursuit problem in the sense of L.S. Pontryagin for a differential game whose dynamics is described by a functional-differential equation of neutral type

Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients

<p style='text-indent:20px;'>In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.</p>

Second-order Nonlinear Differential Equations: Oscillation Tests and Applications

The 2nd order Differential Equations (DEs) are applied to analyze various phenomena in physics and it is extended to engineering. Specifically 2nd order linear DEs are having a big role in this field. We already observed this working rule in moving systems like a vertical spring attached with a mass. Another example is an electric circuit with a resistor, an inductor, and a capacitor in series connection. These models are applicable to approximate more complex situations like bond structure between atoms or molecules. They are generally patterned as vibrating springs which is explained by the same DEs. Qualitative theory analysis of DEs is an active area of investigation in theory and application point of view. For instance, oscillatory behavior of 2nd order DEs have many applications in the research related to distributed networks where in high-speed computations lossless transmission lines evolve. These lossless transmission lines are employed to connect switching circuits. Recently the study the functional DEs of neutral type is an active area of research. Many researchers are working in the field of oscillation theory and studying only sufficient conditions. Merely few are considering the necessary and sufficient conditions. We have attempted to set up the necessary and sufficient conditions for oscillation of solutions of 2nd order nonlinear neutral DEs. Mathematics Subject Classification (2010): 34K.

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.

On Basic Equation of Differential Games for Neutral-Type Systems

For a conflict-controlled dynamical system described by neutral-type functional differential equations in Hale’s form, a differential game is considered in the classes of control strategies with a guide for a minimax-maximin of the quality index, which evaluates the system’s motion history implemented by the terminal time moment. The differential game is associated with the Cauchy problem for a functional Hamilton-Jacobi type equation in coinvariant derivatives. It has been proven that the game value functional coincides with the minimax solution of this problem. A method of constructing the optimal strategies of players is given. The approximation by ordinary Hamilton-Jacobi equations in partial derivatives is proposed for this functional Hamilton-Jacobi equation in coinvariant derivatives.

Non-stationary Differential-Difference Games of Neutral Type

We consider the pursuit problem for 2-person conflict-controlled process with single pursuer and single evader. The problem is given by the system of the linear functional-differential equations of neutral type. The players pursue their own goals and choose controls in the form of functions of a certain kind. The goal of the pursuer is to catch the evader in the shortest possible time. The goal of the evader is to avoid the meeting of the players’ trajectories on a whole semiinfinite interval of time or if it is impossible to maximally postpone the moment of meeting. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain moment of time for any counteractions of the evader.

On Hamilton-Jacobi equations for neutral-type differential games

We consider a two-person zero-sum differential game in which a motion of the dynamical system is described by neutral-type functional-differential equations in Hale’s form and the quality index estimates a motion history realized up to the terminal instant of time and includes integral estimations of control realizations of the players. The formalization of the game in the class of pure positional strategies is given, the corresponding notions of the value functional and optimal control strategies of the players are defined. For the value functional, we derive a Hamilton-Jacobi type equation with coinvariant derivatives. It is proved that, if a solution of this equation satisfies certain smoothness conditions, then it coincides with the value functional. On the other hand, it is proved that, at the points of coinvariant differentiability, the value functional satisfies the derived Hamilton-Jacobi equation. Therefore, this equation can be called the Hamilton-Jacobi-Bellman-Isaacs equation for neutral-type systems.

Finite-dimensional approximations of neutral-type conflict-controlled systems

The paper deals with a dynamical system described by a neutral-type functional differential equation in Hale’s form which is controlled under conditions of unknown disturbances. This system is approximated by a controlled system of ordinary differential equations. An aiming procedure between the initial and approximating systems is elaborated. An application of such procedure is given for solving a control problem for linear neutral-type dynamical systems. An illustrative example is considered, simulation results are shown.

Differential games for neutral-type systems: An approximation model

For a conflict-controlled dynamical system whose motion is described by neutraltype functional differential equations in Hale’s form and for a quality index that evaluates the motion history realized up to the terminal instant of time, we consider a differential game in the class of control-with-guide strategies. We construct an approximating differential game in the class of pure positional strategies in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index is terminal. We show that the value of the approximating game gives the value of the original game in the limit, and that the optimal strategies in the original game can be constructed by using the optimal motions of the approximating game as guides.

Neutral Mixed Type Functional Differential Equations

We extend the existing Fredholm theory for mixed type functional differential equations developed by Mallet-Paret (J Dyn Differ Equ 11:1–47, 1999) to the case of implicitly defined mixed type functional differential equations. We present analogous results for the Fredholm alternative theorem, the cocycle property, and spectral flow property. In particular, we apply the theory to examples, one of which arises from modeling signal propagation in nerve fibers, and show the existence of traveling wave solutions via local continuation.

Finite-dimensional guides for conflict-controlled linear systems of neutral type

We develop approximations to systems of functional-differential equations of the neutral type by ordinary differential equations as applied to conflict control problems. We develop and justify a procedure of mutual feedback tracking between the motion of the original conflict-controlled plant described by linear functional-differential equations of neutral type and the motion of the modeling guide plant described by ordinary differential equations.

The $$L^p$$ L p -Version of the Generalized Bohl–Perron Principle for Neutral Type Functional Differential Equations

We consider a vector linear neutral type homogeneous functional differential equation. It is proved that the considered equation is exponentially stable, provided the corresponding non-homogeneous equation with the zero initial function and an arbitrary free term from \(L^p([0,\infty ), \mathbb {C}^n)\), has a solution belonging to \(L^p([0,\infty ), \mathbb {C}^n)\).

On the dichotomy of linear autonomous systems of functional-differential equations

We study the exponential dichotomy of solutions to the Cauchy problem for a subclass of linear autonomous systems of functional-differential equations of neutral type by reducing this problem to the same problem for a difference equation of the form xn = Γxn−1 with a compact operator Γ, in Banach space. The spectrum of Γ is described. The dichotomy of the solutions of the Cauchy problem is a consequence of the dichotomy of the spectrum. In a particular case, the result obtained provides a criterion for the dichotomy for a linear autonomous system of functional-differential equations of retarded type of general form.

On the asymptotic properties of the solutions of a linear functional-differential equation of neutral type with constant coefficients and linearly transformed argument

We establish new properties of solutions of a functional-differential equation x′(t) = ax(t) + bx(qt) + cx′(qt).

An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type

In this paper we consider an infinite dimensional bifurcationequation depending on a parameter $ \varepsilon>0 . $ By means ofthe theory of condensing operators, we prove the existence of abranch of solutions, parametrized by $ \varepsilon , $ bifurcatingfrom a curve of solutions of the bifurcation equation obtained for$\varepsilon =0 . $ We apply this result to a specific problem,namely to the existence of periodic solutions bifurcating from thelimit cycle of an autonomous functional differential equation ofneutral type when it is periodically perturbed by a nonlinearperturbation term of small amplitude.

The generalized Bohl–Perron principle for the neutral type vector functional differential equations

We consider a vector homogeneous neutral type functional differential vector equation of a certain class. It is proved that, if the corresponding nonhomogeneous equation with the zero initial conditions and an arbitrary free term bounded on the positive half-line, has a bounded solution, then the considered homogeneous equation is exponentially stable.

Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay

This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation.