Periodic solutions of measure functional differential equations
Let \(f,g \colon \mathbb{R} \times G_T^- (\mathbb{R}, \mathbb{R}^n) \to \mathbb{R}^n\) be \(T\)-periodic functions in the first variable, \(T >0\) and \(G_T^- (\mathbb{R}, \mathbb{R}^n)\) denote the space of all \(T\)-periodic regulated left-continuous functions. For \(x \in G_T^- (\mathbb{R}, \mathbb{R}^n)\) and \(t \in \mathbb{R}\) the symbol \(x_t\) denotes the function from \(G_T^- (\mathbb{R}, \mathbb{R}^n)\) given as \(x_t(\theta) = x(t+ \theta)\), \(\theta \in \mathbb{R}\). The authors study the existence of regulated \(T\)-periodic solutions to the following measure functional equation \[ x(t) = x(0) + \int_0^t f(s,x_s)ds + \int_0^t g(s,x_s)du(s), \] where \(u \colon \mathbb{R} \to \mathbb{R}\) is a left-continuous regulated function and the integrals in the right-hand side are taken in the Perron and Perron-Stieltjes sense, respectively. As the main tool, the topological transversality theory which is a version of the topological degree theory (see, e.g., [\textit{A. Granas} and \textit{J. Dugundji}, Fixed point theory. New York, NY: Springer (2003; Zbl 1025.47002)]) is used. The obtained results are applied to a periodic problem for a class of impulsive functional differential equations.
measure functional differential equations
scientific article; zbMATH DE number 7450436
scientific article