Abstract
Equivalence between distributional differential equations and periodic problems with state-dependent impulses
Highlights
Let m ∈ N and τi, Ji, i = 1, . . . , m, be functionals defined on the set of 2π-periodic functions of bounded variation
Tomecek where x and y denote the classical derivatives of the functions x and y, respectively, ∆y(t) = y(t+) − y(t−). These connections make possible to transfer results reached for the classical impulsive periodic problem (1.2)–(1.4) to the distributional differential equation (1.1) and vice versa
We show it here and extend the lower and upper functions method to distributional equations
Summary
Tomecek where x and y denote the classical derivatives of the functions x and y, respectively, ∆y(t) = y(t+) − y(t−) These connections make possible to transfer results reached for the classical impulsive periodic problem (1.2)–(1.4) to the distributional differential equation (1.1) and vice versa. In [6] and [7] the authors reached interesting results for distributional equations which contain first derivatives and delay Their approach essentially depends on the global Lipschitz conditions for data functions in order to get a contractive operator corresponding to the problem. In [40], a second order differential equation with state-dependent impulses is studied using lower and upper solutions method. Let us emphasize that our differential equations are nonautonomous with state-dependent impulses and we need no global or local Lipschitz conditions, see Theorems 6.1 and 6.2. The novelty of our results is documented by Example 6.3, where no previously published theorem can be applied
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