Abstract
In this paper, we consider a first-order coupled impulsive system of equations with functional boundary conditions, subject to the generalized impulsive effects. It is pointed out that this problem generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and integrodifferential ones, or global arguments, as maxima or minima, among others. Our method is based on lower and upper solution technique together with the fixed point theory. The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and the asymptotic behavior of the unknown functions.
Highlights
The study of impulsive boundary value problems is richer than the related differential equation theory without impulses and has strategic importance in multiple current scientific fields, from sociology and medical sciences to generalized industry production or in any other real-world phenomena where sudden variations occur
Functional problems composed by differential equations and conditions with global dependence on the unknown variable generalize the usual boundary value problems and can include equations and/or conditions with deviating arguments, delays or advances, nonlinear, or nonlocal, increasing in this way the range of applications
(2) The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and asymptotic behavior of the unknown functions
Summary
The study of impulsive boundary value problems is richer than the related differential equation theory without impulses and has strategic importance in multiple current scientific fields, from sociology and medical sciences to generalized industry production or in any other real-world phenomena where sudden variations occur. (1) Condition (2) generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and/or integrodifferential ones, or global arguments, as maxima or minima, among others. In this way, new types of problems and applications could be considered, enabling greater and wider information on the problems studied (2) The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and asymptotic behavior of the unknown functions. The main result is obtained studying a perturbed and truncated system, with modified boundary and impulsive conditions and applying Schauder’s fixed point theorem to a completely continuous vectorial operator. The last section contains an application to a vital dynamic SIRS-type model, representing the dynamic epidemiological evolution of susceptible (S), infected (I), Recovered (R), and newly infected individuals in a population on a normalized period, subject to impulsive effects and global restrictions
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