Abstract
Consider the delay differential equation \begin{equation*} \dot{x}(t)=a \Bigg(\sum_{i=1}^n b_i\big[x(t-s_i)- x(t-r_i)\big]\Bigg)-g(x(t)), \end{equation*} where $a>0$, $b_i>0$ and $0\leq s_i 0} g(\xi)/\xi>2a$. This equation can be interpreted as a price model, where $x(t)$ represents the price of an asset (e.g. price of share or commodity, currency exchange rate etc.) at time $t$. The first term on the right-hand side represents the positive response for the recent tendencies of the price and $-g(x(t))$ is responsible for the instantaneous negative feedback to the deviation from the equilibrium price. We study the local and global stability of the unique, non-hyperbolic equilibrium point. The main result gives a sufficient condition for global asymptotic stability of the equilibrium. The region of attractivity is also estimated in case of local asymptotic stability.
Highlights
In this paper we consider the delay differential equation n x (t) = a bi x(t − si) − x(t − ri) i=1 − g(x(t)), (1.1)where n is a positive integer, a > 0, bi > 0 and 0 ≤ si < ri (i ∈ {1, . . . , n}) are parameters such that max1≤i≤n ri = 1, n i=1 bi = 1, and g: R → is an oddC1 function with g (0) = 0; we assume that the map (0, ∞) ξ → g(ξ)/ξ ∈ R2010 Mathematics Subject Classification
In this paper we prove that the unique, non-hyperbolic equilibrium of (1.1)
It is clear from the assumptions on g (given in equation (1.1)), that function h is even, continuous and it is positive and strictly increasing on (0, ∞)
Summary
Delay differential equation; multiple delay; global stability; price model; neutral equation; infinite delay; stable D operator
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