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Behavior of the solutions of functional equations. (English) Zbl 1512.34121

Daras, Nicholas J. (ed.) et al., Computational mathematics and variational analysis. Cham: Springer. Springer Optim. Appl. 159, 465-504 (2020).
Summary: In the last decades the oscillation theory of delay differential equations has been extensively developed. The oscillation theory of discrete analogues of delay differential equations has also attracted growing attention in the recent years. Consider the first-order delay differential equation, \[ x'(t) + p(t) \, x(\tau(t)) = 0, \quad t \ge t_0, \tag{1} \] where \(p, \tau \in C([t_0, \infty], \mathbb{R}^+)\), \(\tau (t)\) is nondecreasing, \( \tau (t) < t\) for \(t \geq t_0\) and \(\lim\limits_{t \to \infty} \tau (t) =\infty \), and the (discrete analogue) difference equation, \[ \varDelta x(n) + p(t) \, x(\tau(n)) = 0, \quad n = 0, 1, 2, \ldots, \tag{2} \] where \(\Delta x(n) = x(n + 1) - x(n)\), \(p(n)\) is a sequence of nonnegative real numbers and \(\tau (n)\) is a nondecreasing sequence of integers such that \(\tau (n) \leq n - 1\) for all \(n \geq 0\) and \(\lim\limits_{n \to \infty} \tau (n) = \infty \).
In this review chapter, a survey of the most interesting oscillation conditions is presented, along with numerical examples of delay and difference equations. We focus our attention on these examples, to illustrate the level of improvement in the oscillation criteria and the significance of the obtained results. The numerical calculations were made with the use of MATLAB software. These examples are relevant to many physical and biological applications.
For the entire collection see [Zbl 1446.65002].

MSC:

34K11 Oscillation theory of functional-differential equations
39A21 Oscillation theory for difference equations
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