The problem of global stability in scalar delay differential equations of the form x/spl dot/(t)=f(x(t-/spl tau/))-g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size /spl tau/ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as /spl tau//spl rarr//spl infin/) difference equation g(x/sub n+1/)=f(x/sub n/): the latter always implying the global stability in the differential equation for all values of the delay /spl tau//spl ges/0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay /spl tau/.
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机译：研究了形式为x / spl dot /（t）= f（x（t- / spl tau /））-g（x（t））的标量延迟微分方程的全局稳定性问题。函数f和g是连续的，因此方程式具有唯一的平衡。建立了两种类型的唯一平衡的全局渐近稳定性的充分条件：（i）与延迟无关，以及（ii）涉及延迟的大小/ splau的条件。时滞无关的稳定性条件在极限差分方程g（x / sub n + 1 /）= f（x / sub n /）中利用全局稳定性（如/ spl tau // spl rarr // spl infin /）：对于延迟/ spl tau // spl ges / 0的所有值，后者总是隐含在微分方程中的全局稳定性。依赖于延迟的条件包括在特殊构造的一维映射（差分方程）中的全局吸引力，该一维映射包括非线性f和g，以及延迟/ spl tau /。
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