Abstract
The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.
Highlights
IntroductionIn this paper two different problems are analyzed. Part one is concerned with the study of a fully nonlinear coupled system of equations u1 (t) = f 1 (t, u1 (t), u2 (t), u3 (t))
The final section presents an application of (1)–(2) to an epidemic SIRS model to illustrate the applicability of the problem discussed and to show the potentialities of the functional boundary conditions, exploring global initial boundary conditions on the system
The second case, with periodic boundary conditions, which is not covered by the first result, is an existence and location result
Summary
In this paper two different problems are analyzed. Part one is concerned with the study of a fully nonlinear coupled system of equations u1 (t) = f 1 (t, u1 (t), u2 (t), u3 (t)). Given that the conditions on Li , do not allow the problem (1)–(2) to cover the periodic case, a different approach for the problem (1)–(3) is required In this case, in order to obtain the existence and location of periodic solutions, the upper and lower solutions method, along with some adequate local monotone assumptions on the nonlinearities, is used. To the best of our knowledge, it is the first time where coupled systems are considered with coupled functional boundary conditions. This feature allows to generalize the classical boundary data in the literature, such as two-point or multi-point, nonlinear, nonlocal, integro-differential conditions, among others. The final section presents an application of (1)–(2) to an epidemic SIRS model to illustrate the applicability of the problem discussed and to show the potentialities of the functional boundary conditions, exploring global initial boundary conditions on the system
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