First-order periodic coupled systems with orderless lower and upper solutions

Abstract

<abstract><p>We present some existence and localization results for periodic solutions of first-order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with not necessarily well-ordered upper and lower solutions. A real-case scenario shows the applicability of our results to some population dynamics models, describing the interaction between a criminal and a non-criminal population with a law enforcement component.</p></abstract>

First-order periodic coupled systems with generalized impulse conditions

“…By Corollary 2, T is relatively compact. Then, by Theorem 5, T has a fixed point (z * (t), w * (t)) ∈ (P C[0, T ]) 2 , which is solution of ( 17), (2), (18).…”

“…Step 3: The pair (z * (t), w * (t)), solution of ( 17), ( 2), (18), is a solution of the initial problem, (1), ( 2), (3).…”

“…Following a translation technique suggested in [18], we obtain the existence and the localization of an impulsive periodic solution. In such method, for C 1 lower and upper functions, there is a change of sign in the nonlinearities (see Definition 4).…”

“…Furthermore, results on equi-regulated functions are essential to deal with the discontinuities at the instants of impulse. A similar problem is studied in [19,20,21], and with similar techniques in [18,22].…”