Periodic boundary value problems involving Stieltjes derivatives
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Journal of Fixed Point Theory and Applications
Periodic boundary value problems involving Stieltjes derivatives Bianca Satco
and George Smyrlis
Abstract. We are concerned with the study of a first-order nonlinear periodic boundary value problem ug (t) + b(t)u(t) = f (t, u(t)), t ∈ [0, T ] u(0) = u(T )
(1)
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. Mathematics Subject Classification. 34B15, 34A06, 47H10, 26A24, 26A42. Keywords. Periodic boundary value problem, Stieltjes derivative, Schaeffer’s fixed point theorem, regulated function, Kurzweil–Stieltjes integral.
1. Introduction The theory of differential equations driven by measures has been continuously growing over the last decade (e.g. [2,4,7–9,21,27]) since it offers a tool to study in a unified way several classical problems: first-order differential equations (in the case when the driving measure is absolutely continuous with respect to the Lebesgue measure), impulsive differential problems (when we take into consideration a measure which can be written as a sum of Lebesgue measure with a discrete measure) with no limitations on the impulse moments, dynamic equations on time scales (see [4,8,9]) and generalized differential equations (e.g. [16,21,28,29]). 0123456789().: V,-vol
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B. Satco and G. Smyrlis
On the other hand, an equivalent formulation in terms of a notion of (Stieltjes) derivative with respect to a nondecreasing function is available (c.f. [22], see also [19]). This derivative, considered in [22] (even if the idea is not really new in literature, c.f. [31]) has found recent interesting applications in biology, population dynamics or chemistry (see [12,13] or [23]). At the same time, it is well known that differential problems with periodic boundary conditions have wide applicability in various areas of science. Relying on these considerations, we focus on first order nonlinear periodic boundary value problems of the form (1): ug (t) + b(t)u(t) = f (t, u(t)), t ∈ [0, T ] u(0) = u(T ) involving the Stieltjes derivative with respect to a function g : [0, T ] → R left-continuous and nondecreasing. The maps b : [0, T ] → R and f : [0, T ] × R → R are supposed to be continuous at the continuity points of g. In two steps (first, for the linear and then, applying Schaeffer’s fixed point theorem, for the general case), we
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