Abstract
The chapter generalizes and extends the previous results contained in Chaps. 6–9. We study the solvability of a boundary value problem with state-dependent impulses of the form
where the impulse instants \(t \in (a,b)\) are a priori unknown and depend on z. Here, \(n,p\in \mathbb {N}\), \(\mathbf {c} \in \mathbb {R}^n\), the vector function \(\mathbf {f}\) satisfies the Carathéodory conditions on \([a,b]\times \mathbb {R}^n\). The impulse functions \(\mathbf {J}_i\), \(i=1,\ldots ,p\), are continuous on \([a,b]\times \mathbb {R}^n\), and the barrier functions \(\gamma _i\), \(i = 1,\ldots ,p\), are continuous on \(\mathbb {R}^n\). The operator \(\ell \) is an arbitrary linear and bounded operator on the space of left-continuous regulated on [a, b] vector valued functions. Arbitrary linear local and nonlocal boundary conditions are covered by \(\ell \). Under the assumptions of boundedness of data functions \(\mathbf {f}\) and \(\mathbf {J}_i\) and transversality conditions, the existence result for a solution of the investigated boundary value problem having exactly one cross through each barrier \(t = \gamma _i(\mathbf {x})\) is guaranteed.
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Rachůnková, I., Tomeček, J. (2015). First Order System with Linear Boundary Conditions. In: State-Dependent Impulses. Atlantis Briefs in Differential Equations, vol 6. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-127-7_10
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DOI: https://doi.org/10.2991/978-94-6239-127-7_10
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