Lower and upper solutions method applied to systems of Stieltjes differential equations

In our work, we use the lower and upper solutions method to study systems of Stieltjes differential equations. We make use of the Stieltjes derivative, which involves differentiation with respect to a left-continuous nondecreasing function. This allows us to analyze various types of equations, including discrete equations, dynamic equations on time scales, and differential equations with impulses, and also to model phenomena presenting discontinuities and stationary periods. This general framework simplifies the right-hand side term, and incorporates properties of the derivator into the solution: the jumps and stationary periods. In this work, we extend prior existence results [2] to systems with several derivators, addressing both initial value and periodic boundary value problems. Systems with and without functional dependence are studied without requiring monotonicity conditions as seen in prior works [1, 5].

References:

1. R. López Pouso, I. Márquez Albés, G.A. Monteiro, Extremal solutions of systems of measure differential equations and applications in the study of Stieltjes differential problems. Electron. J. Qual. Theory Differ. Equ. 2018, Paper no. 38, 24 pp.

2. L. Maia, N. El Khattabi, M. Frigon, Existence and multiplicity results for first-order Stieltjes differential equations. Adv. Nonlinear Stud. 22 (2022), no. 1, 684–710.

3. L. Maia, N. El Khattabi, M. Frigon, Systems of Stieltjes differential equations and application to a predatorprey model of an exploited fishery. Discrete Contin. Dyn. Syst. 43 (2023), no 12 4244–4271.

4. I. Márquez Albés, F.A.F. Tojo, Existence and uniqueness of solution for Stieltjes differential equations with several derivators. Mediterr. J. Math. 18 (2021), no. 5, Paper no. 181, 31 pp.

5. G. A. Monteiro, A. Slavik, Extremal solutions of measure differential equations. J. Math. Anal. Appl. 444 (1), (2016), 568–597.