In the present paper, we shall consider the following impulsive delay system for modeling the price fluctuations in single-commodity markets:
View the MathML source{ṗ(t)=F(p(t),p(t−h))p(t),t≠τk,p(t)=φ0(t),t∈[t0−h,t0],Δp(t)=Ik(p(t)),t=τk,k∈Z.
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Impulsive delay differential equations have conspicuously occupied a great part of researchers’ interests for well over the last three decades. Indeed, it has been recently recognized that these equations do only generalize the corresponding theory of impulsive differential equations but also provide better mathematical descriptions for many real life applications. The publications [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and [15] are devoted to the theory and applications of impulsive differential equations with or without delay.
Specifically, the dynamics of the economy is one of most actively developing research areas that can be represented by using impulsive delay differential equations of a certain type [16], [17] and [18]. In particular, it has been noticed that these equations can provide adequate visualization for modeling the process of price fluctuations in single-commodity markets. Early authors often attributed these fluctuations to random factors such as weather change for agricultural commodities [19], [20] and [21]. Other authors, however, speculated that fluctuations might be caused by dynamical characteristics of unstable economic systems [22], [23], [24] and [25]. Apart from some diversities in the authors’ beliefs regarding this discussion, their work and that of others has played a fundamental role in the development of theory of nonlinear dynamics [26], [27], [28], [29], [30] and [31].
Searching the literature, one can realize that there has been intensive work regarding the study of periodic impulsive dynamical systems with or without delay; see for instance the Refs. [32], [33], [34], [35], [36] and [37] in which the existence of periodic solutions has been the main concern of the authors. Although it is known to be a natural generalization to the periodicity, the notion of almost periodicity has rarely been considered. The reader can easily figure out that a few results exist in this direction [38], [39], [40], [41] and [42].
The purpose of this paper is to study the almost periodic behavior of solutions for the impulsive delay model for price fluctuations in commodity markets. Piecewise continuous functions of the Lyapunov type as well as the Razumikhin technique have been utilized to prove the main results.
