A note on stability of Mackey–Glass equations with two delays

Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro‐differential equations

“…where a(t), f(t, u), (t) are as in (29) and all the impulses are given by nonnegative continuous functions I k (u) satisfying (A2). For b(t) = a(t) and h(u) as in (31), from the arguments above and Corollary 1(a), we get the existence of at least one positive periodic solution if ∞ C * 2 < 1 < 0 C * 1 . As verified in many applications, next suppose also that f ∞ = 0.…”

“…Recently, there has been an increasing interest in DDEs with mixed monotonicity where the nonlinear terms involve one or more functions with different delays, e.g., of the form f ( t , x ( t − τ ( t )), x ( t − σ ( t ))), with f ( t , x , y ) monotone increasing in the variable x and monotone decreasing in y . Although small delays are in general harmless, in the sense that the delayed model has the same global properties of the equation without delays, the presence of two or more delays in the same nonlinear function may change this situation drastically, as illustrated in Berezansky and Braverman . However, DDEs with different delays in the same nonlinear term appear naturally in real‐world models, as shown in and references therein.…”

“…Proof. From (33), choose such that f 0 > and (e A( ) − 1)B 0 > e A( ) B 0 − 1, define b(t) = a(t) and h(u) as in (31).…”

“…Although small delays are in general harmless, in the sense that the delayed model has the same global properties of the equation without delays, the presence of two or more delays in the same nonlinear function may change this situation drastically, as illustrated in Berezansky and Braverman. 31 However, DDEs with different delays in the same nonlinear term appear naturally in real-world models, as shown in 32,33 and references therein. The global dynamics of such equations have been the subject of a few recent studies, where questions of stability, persistence, permanence, and existence of periodic solutions were addressed.…”

“…The global dynamics of such equations have been the subject of a few recent studies, where questions of stability, persistence, permanence, and existence of periodic solutions were addressed. 4,19,29,31,33,34 Nevertheless, as far as the authors know, only the case of discrete delays has been considered.…”