Abstract
Existence criteria are derived for the eventually periodic solutions of a class of differential equations with piecewise constant argument whose solutions at consecutive integers satisfy nonlinear recurrence relations. The proof characterizes the initial values of periodic solutions in terms of the coefficients of the resulting difference equations. Sufficient conditions for the unboundedness, boundedness, and symmetry of general solutions also follow from the corresponding properties of the difference equations.
Highlights
Since the seminal works of Shah and Wiener [1] and Cooke and Wiener [2], differential equations with piecewise constant arguments of the form x (t) = f (t, x (t), x ([t])), (1)where f is continuous and [⋅] is the greatest integer function, have been treated widely in the literature and applied to certain biomedical models
We consider a class of equations of the above form but where f is discontinuous: the chaotic and eventually periodic behavior and symmetry of solutions of initial-value problems of the form x (t) = Ax (t) + Bx ([t]) + CF (x ([t])), x (0) = x0 (2)
We show that x(t) > 0 for all t in the interval (n − 1, n): note that x(t) = x(n − 1+)eA(t−[t])
Summary
(a) Suppose by way of contradiction that x = x(t) is oscillatory but 0 is not a stationary state of x(n) It follows that x(n − 1) ≠ 0 for all positive integers n. We assume that a∗ > 1 and b∗ > 0; and Proposition 1 applies to the resulting solution x = x(t) of (2) It will follow from the third section that if λ is outside the interval (b∗/a∗2, b∗/(a∗ − 1) − b∗/a∗2), x(t) is either eventually constant or unbounded. B, x0, and λ > 0, periodic solutions of difference equation (18) were used in [13] to determine the real eigenvalues of certain arbitrarily large, sparse matrices
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