Monotone method for first order functional differential equations with retardation and anticipation

This paper discusses first order functional differential equations with retardation and anticipation. By using the method of upper and lower solutions coupled with monotone iterative technique, new existence results are obtained which improve known results in the literature.     

Monotone iterative technique for functional differential equations with retardation and anticipation

The method of the monotone iterative technique together with coupled lower and upper solutions is employed to prove existence and uniqueness results for functional differential equations involving retardation and anticipation.     

Monotone iterative technique for delay differential equations

In proving existence of extremal solutions for delay differential equations, one usually assumes nondecreasing property on the function involved without which the proof breaks down. Our main purpose, in this paper, is to remove the monotone assumption by proving a new comparison result which is required in the analysis and which avoids the standard line of argument. Furthermore, our comparison theorem shows that the choice of initial functions cannot be arbitrary for the results to hold. Using this result we then develop monotone technique to obtain extremal solutions.     

The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order

In this paper, we show that the monotone technique produces two monotone sequences that converge uniformly to extremal solutions of second order functional differential equations and ?$?$-Laplacian equations with Neumann boundary value conditions. Moreover, we obtain existence results assuming upper and lower solutions in the reverse order.     

New results for the periodic boundary value problem for impulsive integro-differential equations

This paper discusses the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations. We prove some new comparison principles, and then establish new existence results for extremal solutions by using these principles and the monotone iterative technique. These results improve and extend all known relevant conclusions from the literature. Some examples are also given to illustrate the advantage of our results.     

Existence of global solutions of delayed differential equations via retract approach

The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The topological approach known as the topological retract principle is used. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results.     

Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces

This paper deals with the existence of mild L-quasi-solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered Banach space E. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. An example is also given.     

Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking

In this paper, by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a generalized delayed predator–prey system with stocking is established. When our result is applied to a delayed predator–prey system with nonmonotonic functional response and stocking, we establish the sufficient condition for the existence of multiple positive periodic solutions for the system.     

Continuous transformations of differential equations with delays

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformationz(t)=g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of thenth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions are considered.     

Stability of a critical nonlinear neutral delay differential equation

This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of constant and periodic solutions. A complete analysis of the stability diagram is given in the linear homogeneous case. Under periodic forcing, existence of periodic solutions is discussed, involving a Diophantine condition on the period of the source.     

The numerical solution of forward-backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x^′(t)=ax(t)+bx(t−1)+cx(t+1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.     

A model of immune system with time-dependent immune reactivity

In this paper we deal with the Marchuk model of an immune system. Among the main parameters of the model are the coefficients which describe the state of infected organism and the rate of production of antibodies. In the classical model these coefficients are constants. We consider the case when these coefficients are time-dependent. In particular, we are interested in the case of periodic coefficients which can describe periodic changes of the immune reactivity due to periodic changes of the environment. We examine the asymptotic behaviour of solutions. Under some assumptions we prove that the solutions tend to periodic functions. We also present the results of numerical simulations to illustrate the behaviour of solutions.     

New convergence behavior of high-order hopfield neural networks with time-varying coefficients

In this paper the convergence behavior of the delayed high-order Hopfield neural networks (HHNNs) with time-varying coefficients are considered. Some sufficient conditions are established to ensure that all solutions of the networks converge to zero point, which are new and complement of previously known results.     

Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces

In this work, we study the existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces. We suppose that the undelayed part admits a resolvent operator in the sense of Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society 273 (1982) 333–349]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of strict solutions.     

On the initial value problem for functional differential systems

For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied.     

Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

Two periodic solutions of neutral difference systems depending on two parameters

In this paper, we establish existence, multiplicity and nonexistence of periodic solutions for a class of first-order neutral difference systems. Our approach is based on a fixed point theorem in cones as well as some analysis techniques.     

Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz activation functions

In this paper the shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, sufficient conditions for the existence and local exponential stability of the almost periodic solutions are established by using a fixed point theorem, Lyapunov functional method and differential inequality techniques. The results of this paper are new and they complement previously known results.     

Periodic solutions for generalized high-order neutral differential equation in the critical case

By applying Mawhin’s continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for a generalized high-order neutral differential equation in the critical case. Moreover, an example is given to illustrate the results.