We are concerned with the study of a first-order nonlinear periodic boundary value problem

involving the Stieltjes derivative with respect to a left-continuous nondecreasing function. Based on Schaeffer’s fixed point theorem and making use of a notion of partial Stieltjes derivative (along with its natural properties), we prove the existence of regulated solutions and provide a useful characterization in terms of Stieltjes integrals. The generality of our result is coming from the impressive number of particular cases of the described problem. Thus, first-order periodic differential equations, impulsive differential problems (including also the possibility to have Zeno points, i.e. accumulations of impulse moments), dynamic equations on time scales or generalized differential equations can all be studied through the theory of Stieltjes differential equations.

我们关注一阶非线性周期边值问题的研究

关于左连续非递减函数涉及Stieltjes导数。基于Schaeffer的不动点定理，并利用部分Stieltjes导数（以及其自然性质）的概念，我们证明了调节解的存在并提供了有关Stieltjes积分的有用描述。我们得出的结论的普遍性来自于所描述问题的特殊情况数量之多。因此，一阶周期微分方程，脉冲微分问题（还包括拥有Zeno点的可能性，即脉冲力矩的累积），时标上的动力学方程或广义微分方程都可以通过Stieltjes微分方程理论进行研究。