Almost periodic solutions of recurrent neural networks with continuously distributed delays

In this paper, a class of recurrent neural networks with continuously distributed delays is discussed. Without resorting to the theory of exponential dichotomy, several new sufficient conditions are obtained ensuring the existence of an almost periodic solution for this model based on a special functional and analysis technique. Moreover, by constructing suitable Lyapunov functions, the attractivity and exponential stability of the almost periodic solution are also considered for the system. The results obtained are helpful to design globally stable almost periodic oscillatory neural networks. A numerical example is given to show the feasibility of the results obtained.     

New product advertising in dynamic oligopolies

The success of the introduction of a new product in a market is very sensitive to the marketing decision variables adopted by the firm. In the present paper we are concerned with the question of new product advertising in a heterogeneous oligopoly market consisting of N firms. A dynamic game is formulated to model strategic as well as sales interactions in such a market. Optimal advertising strategies are identified as open-loop Nash solutions.The comments of two anonymous referees are appreciated. The first author wishes to acknowledge support from NSERC (Grant No. OGP0037342).     

A shunting inhibitory cellular neural network with continuously distributed delays of neutral type

In this paper, a shunting inhibitory cellular neural network with continuously distributed delays of neutral type is considered. We establish some new results about the existence and exponential stability of the almost periodic solution for the shunting inhibitory cellular neural network.     

Optimal dynamic advertising policies in the presence of continuously distributed time lags

The Nerlove-Arrow model of optimal dynamic advertising policies is generalized by incorporating a continuously distributed lag between advertising expenditures and increases in the stock of goodwill. This leads to a control problem where the equation of motion is given by an integro-differential equation. The transitory and steady-state properties of the optimal policies are examined, both for a general lag function and for a gamma distributed lag. The dependence of the steady-state solution on the parameters of the gamma distribution is also investigated. An example is given using specific demand and cost functions.     

Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms

Bidirectional associative memory (BAM) model is considered with the introduction of continuously distributed delays in the leakage (or forgetting) terms. By using continuation theorem in coincidence degree theory and the Lyapunov functional, some very verifiable and practical algebraic mean delay dependent criteria on the existence and global attractive periodic solutions are derived.     

Stability analysis of Cohen–Grossberg neural network with both time-varying and continuously distributed delays

In this paper, the Cohen–Grossberg neural network model with both time-varying and continuously distributed delays is considered. Without assuming both global Lipschitz conditions on these activation functions and the differentiability on these time-varying delays, applying the idea of vector Lyapunov function, M-matrix theory and inequality technique, several new sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of equilibrium point for Cohen–Grossberg neural network with both time-varying and continuously distributed delays. These results generalize and improve the earlier publications. Two numerical examples are given to show the effectiveness of the obtained results. It is believed that these results are significant and useful for the design and applications of the Cohen–Grossberg neural networks.     

A new criterion to global exponential stability for impulsive neural networks with continuously distributed delays

This paper considers the global exponential stability and exponential convergence rate of impulsive neural networks with continuously distributed delays in which the state variables on the impulses are related to the unbounded distributed delays. By establishing a new impulsive delay differential inequality, a new criterion concerning global exponential stability for these networks is derived, and the estimated exponential convergence rate is also obtained. The result extends and improves on earlier publications. In addition, two numerical examples are given to illustrate the applicability of the result. Copyright © 2010 John Wiley & Sons, Ltd.     

Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type

In this paper, by using the Lyapunov-Krasovskii functional method, we investigate the global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type. Some new stability criteria are presented in terms of linear matrix inequality (LMI). Two numerical examples are also given to show the effectiveness of the obtained results using LMI control toolbox in MATLAB.     

Existence and stability of almost periodic solutions of Hopfield neural networks with continuously distributed delays

In this paper, the global stability and almost periodicity are investigated for Hopfield neural networks with continuously distributed neutral delays. Some sufficient conditions are obtained for the existence and globally exponential stability of almost periodic solution by employing fixed point theorem and differential inequality techniques. The results of this paper are new and they complement the previously known ones. Finally, an example is given to demonstrate the effectiveness of our results.     

Convergence dynamics of stochastic reaction–diffusion recurrent neural networks with continuously distributed delays

Convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with continuously distributed delays and stochastic influence are considered. Some sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution are obtained, respectively. Lyapunov functional method, M-matrix properties, some inequality technique and nonnegative semimartingale convergence theorem are used in our approach. These criteria ensuring the different exponential stability show that diffusion and delays are harmless, but random fluctuations are important, in the stochastic continuously distributed delayed reaction–diffusion RNNs with the structure satisfying the criteria. Two examples are also given to demonstrate our results.     

Positive almost periodic solutions for shunting inhibitory cellular neural networks with time-varying and continuously distributed delays

In this paper shunting inhibitory cellular neural networks (SICNNs) with time-varying and continuously distributed delays are considered. Some new sufficient conditions for the existence and exponential stability of the positive almost periodic solutions are established.     

Fields with continuously distributed mass

We describe local field theories with continuously distributed mass. Such models can be realized as models in d > 4 space-time with Poincaré invariance only in four-dimensional space-time. We discuss some possible phenomenological consequences. Namely, we show that the Higgs boson phenomenology in the Standard Model extension with continuously distributed Higgs boson mass can differ in a drastic way from the standard Higgs boson phenomenology that makes the Higgs boson discovery at the Large Hadron Collider (LHC) an extremely difficult task. We also study the LHC discovery potential for Z′ models with continuously distributed mass for √2 = 7, 10 and 14 TeV center-of-mass energies. One of possible LHC signatures for such models is the existence of a broad resonance structure in the Drell-Yan reaction pp → Z′ + … → l ⁺ l ⁻ + ….     

Existence and global exponential stability of periodic solution for impulsive Cohen-Grossberg-type BAM neural networks with continuously distributed delays

This paper is concerned with the existence and global exponential stability of periodic solution for a class of impulsive Cohen-Grossberg-type BAM neural networks with continuously distributed delays. Some sufficient conditions ensuring the existence and global exponential stability of periodic solution are derived by constructing a suitable Lyapunov function and a new differential inequality. The proposed method can also be applied to study the impulsive Cohen-Grossberg-type BAM neural networks with finite distributed delays. The results in this paper extend and improve the earlier publications. Finally, two examples with numerical simulations are given to demonstrate the obtained results.     

New results on the existence and uniqueness of almost periodic solution for BAM neural networks with continuously distributed delays

In this paper, a class of BAM neural networks with continuously distributed delays are studied. Some new sufficient conditions are established for the existence and uniqueness of the almost periodic solution, which generalize and improve the previously known results. Moreover, the method used in the proof is original. An example is presented to illustrate the feasibility and effectiveness of the new results.     

Basic identities for additive continuously distributed sequences

For an additive sequence ξ(n), we establish basic factorization identities and express the distributions of limiting Junctionals (extremum values of ξ(n), the time and value of the first jump over a fixed level, etc.) in terms of the components of factorization.     

Systems with distributed parameters and time delays

Ukrainian Mathematical Journal -     

Stability and persistence in plankton models with distributed delays

In this paper a model with two independent distributed delays is proposed to describe a population of microorganism feeding on a limiting nutrient which is supplied at a constant rate and is recycled after the death of the species by decomposer action. We obtain sufficient conditions for local and global stability of the positive equilibrium of the model. A fairly general function for nutrient uptake is considered. Stability changes of the positive equilibrium as the nutrient supply increases are studied by the Hopf bifurcation theorem.     

Profit loss in differentiated oligopolies

We provide sharp lower and upper bounds on the ratio of decentralized to centralized profits when multiproduct firms offering differentiated products engage in price competition. The bounds depend on the demand sensitivity matrix but are independent of marginal costs.     

On a general nonlinear problem with distributed delays

The paper considers a general system of ordinary differential equations appearing in the neural network theory. The activation functions are assumed to be continuous and bounded by power type functions of the states and distributed delay terms. These activation functions are not necessarily Lipschitz continuous as it is commonly assumed in the literature. We obtain sufficient conditions for exponential decay of solutions.