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

: 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.


Introduction
We study the following first-order coupled nonlinear system, z (t) = f (t, z(t), w(t)) , w (t) = g (t, z(t), w(t)) , The motivation for the study of systems (1.1) and (1.2) arise from the application of nonlinear models to several phenomena in different domains of life, namely population dynamics [1], neuroscience [2], biological systems [3], ecology [4], electronic circuits [5], celestial mechanics [6], among others. Periodic solutions are usually found in differential equations. For example: In [7], the authors present sufficient conditions for the solvability of a class of singular Sturm-Liouville equations with periodic boundary value conditions. The problem describes the periodic oscillations of the axis of a satellite in the plane of the elliptic orbit around its center of mass.
In [8], the authors study the stability of two coupled Andronov oscillators sharing a mutual discrete periodic interaction. The study was motivated by the phenomenon of synchronization observed in Huygens' clocks [9].
In [10], the authors prove the existence of positive periodic solutions in a generalized Nicholson's blowflies model, using Krasnoselskii cone fixed point theorem.
In [11], it is considered the dynamics of hematopoietic stem cell populations under some external periodic regulatory factors at the cellular cycle level. The existence of periodic solutions in the respective nonautonomous differential system is typically associated to the presence of some hematological disease.
However, nonlinear systems with non-oscillatory nonlinearities can hamper the search for periodic solutions.
To overcome that difficulty, in [12] the authors use well-ordered lower and upper solutions to guarantee the existence of solutions of a first-order fully coupled system. In [13], the authors introduce the idea of well-ordered coupled lower and upper solutions, to obtain an existence and localization result for periodic solutions of first-order coupled non-linear systems of n ordinary differential equations.
Motivated by these works, in this paper we present some novelties: A new translation technique to deal with lower and upper solutions, not necessarily well-ordered; A method to provide sufficient conditions to obtain existence and localization results, combining the adequate coupled lower and upper solutions definition with the monotonicity of the nonlinearities.
This work is organized as follows. In Section 2 we present the definitions and the solvability conditions for problems (1.1) and (1.2). In Section 3, we present the main results regarding the existence and localization of solutions for the problem, using the lower and upper solutions technique, together with Schauder's fixed point theorem, and a numerical example. In Section 4 we adapt the solvability conditions of Section 2 to the lack of monotonicity criteria in the nonlinearities. We find coupled lower and upper solutions for the original problem. Finally, in Section 5, we present an application to a real case scenario of population dynamics.

Definitions
In this work we consider the space of continuous functions in So, (C[0, T ]) 2 is a Banach space.
The nonlinearities assumed will be L 1 −Carathéodory functions, that is: (ii) for almost every t ∈ [0, T ], (y 1 , y 2 ) → h(t, y 1 , y 2 ) is continuous in R 2 ; (iii) for each L > 0, there exists a positive function A key argument in this work is the lower and upper solutions method, but not necessarily well-ordered as it is standard in the literature. This issue is overcome by a new type of lower and upper solutions' definition with some adequate translations: and The functions (β 1 , β 2 ) are upper solutions of the periodic problems (1.1) and (1. Theorem 2.1. Let Y be a nonempty, closed, bounded and convex subset of a Banach space X, and suppose that P : Y → Y is a compact operator. Then P has at least one fixed point in Y.

Existence and localization theorem
This method allows to obtain an existence and localization result for periodic non-trivial solutions in presence of well ordered lower and upper solutions: Theorem 3.1. Let (α 1 , α 2 ) and (β 1 , β 2 ) be lower and upper solutions of (1.1) and (1.2), respectively. Assume that f is a L 1 -Carathéodory function in the set g is a L 1 -Carathéodory function in for fixed t ∈ [0, T ] and y 1 ∈ R, and g (t, y 1 , y 2 ) nondecreasing in y 1 , for fixed t ∈ [0, T ] and y 2 ∈ R. Then problems (1.1) and (1.2) has, at least, a solution (z, and , and consider the auxiliary problem composed by together with the periodic boundary conditions (1.2).
(3.4) and, by (1.2), Therefore, the integral form of (3.3) and (1.2) is (3.7) Following Definition 2.1, the norm of the operator T , defined in (3.6), is given by Step 2. T has a fixed point.
The conditions for Theorem 2.1 require the existence of a nonempty, bounded, closed and convex subset D ⊂ (C [0, T ]) 2 such that T D ⊂ D.
As f, g are L 1 -Carathéodory functions, by Definition 2.1, there are positive Consider the closed ball of radius K, with K given by Similarly, (3.14) Since T 1 and T 2 are uniformly bounded, so is T . Consider t 1 , t 2 ∈ [0, T ] and t 1 < t 2 , without loss of generality. Then, proving that T 1 is equicontinuous.
In the same way, it can be proved that T 2 (z(t 1 ), Therefore, T is equicontinuous. By (3.12), it is clear that T D ⊂ D, then, by Schauder's Fixed Point Theorem, T has a fixed point (z * (t), w * (t)) ∈ (C[0, T ]) 2 , which is a solution of (3.3) and (1.2).

Ordered lower and upper-solutions and monotonicity
In this section we present a technique to deal with the lack of monotonicity requirements in Theorem 3.1 on the nonlinearities, by defining coupled lower and upper solutions α i and β i , conveniently, as in the next definition: The functions (β 1 , β 2 ) are coupled upper solutions of the periodic problems (1.1) and (1.2) if and Assume that f is a L 1 -Carathéodory function in the set g is a L 1 -Carathéodory function in with f (t, y 1 , y 2 ) nonincreasing in y 2 , (4.1) for t ∈ [0, T ], y 1 ∈ R, and g (t, y 1 , y 2 ) nondecreasing in y 1 , for t ∈ [0, T ], y 2 ∈ R. Then problems (1.1) and (1.2) has, at least, a solution (z, w) ∈ C 1 [0, T ] 2 such that Proof. For i = 1, 2, define the continuous functions δ i : [0, T ] × R → R, given by and and consider the auxiliary problem composed by together with the periodic boundary conditions (1.2). Problems (4.4) and (1.2) can be addressed using the arguments for the proof of Theorem 3.1. The corresponding operator has a fixed point (z,w).
Using the same technique, one can prove that

Application to a criminal vs non-criminal population dynamics
In [15], the authors present an analysis of the dynamics of the interaction between criminal and non-criminal populations based on the pray-predator Lotka-Volterra models. Motivated by this work, we consider a variant of the constructed model with a logistic growth of the non-criminal population and a law enforcement term, that is, where C and N are, respectively, the criminal and non-criminal population as functions of time, µ is the growth rate of N in the absence of C, a and b are the variation rates of N and C, respectively, due to their interaction, γ is the natural mortality rate of C, K is the carrying capacity for N in the absence of C, and l c is the measure of enforced law on C.
We note that the interaction term could have different effects in each of the populations, hence, the, eventually, different multiplying factors a and b.
As a numerical example, we consider the parameter set: We choose a normalized period T = 1, representing the evolution dynamics in one year, and the periodic boundary conditions N(0) = N(1), C(0) = C(1).
As all the assumptions of Theorem 4.1 are verified, then there is at least a solution (N, C) of (5.1), moreover,

Discussion
The existence of periodic solutions for problems where the nonlinearities have no periodicity at all is scarce in the literature, not only on differential equations but also on coupled systems of differential equations. This work aims to contribute towards fulfilling this gap, by presenting sufficient conditions to guarantee the existence of periodic solutions in the presence of lower and upper solutions and some adequate monotonicity properties on the nonlinearities.
The localization ensured by the lower and upper solutions method has been undervalued, as well as the estimation and some qualitative properties that it provides for the oscillation and variation of the solutions. The introduction of a practical example, and an application to a real social phenomenon, was intended to stress the potentialities of this technique.

Conclusions
In the literature, in general, the lower and upper solutions method requires the well-ordered case, that is, the lower function below the upper one, and a specific local monotonicity on the nonlinearity. This work suggests techniques to overcome both restrictions, so that the search for functions that verify lower and upper solutions properties becomes easier.
For the first issue, we apply a translation, such that, regardless of the order relation between the lower and upper solutions, the shifted functions are always well-ordered.
Section 4 presents a technique that combines the definition of coupled lower and upper-solutions (Definition 4.1) with the monotonicity properties verified by the nonlinearities. In this way, this method can be applied to other problems where the different types of monotonicity hold.