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Dirichlet Problem with One Impulse Condition

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State-Dependent Impulses

Part of the book series: Atlantis Briefs in Differential Equations ((ABDE,volume 6))

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Abstract

The chapter deals with the second order Dirichlet boundary value problem with one state-dependent impulse condition

$$\begin{aligned}&z''(t) = f(t,z(t)) \quad \text {for a.e. }t \in [0,T]\subset \mathbb {R}, \\&z'(t+) - z'(t-) = M(z(t)), \quad t = \gamma (z(t)), \\&\qquad \quad z(0) = 0, \quad z(T) = 0. \end{aligned}$$

Proofs of the main results make use of a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space \({\mathbb C}^1([0,T])\times {\mathbb C}^1([0,T])\). Sufficient conditions for the existence of solutions to the problem are given. The presented approach is extended to more impulses and to other boundary conditions in the next chapters.

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Authors and Affiliations

  1. Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, Czech Republic

    Irena Rachůnková & Jan Tomeček

Authors
  1. Irena Rachůnková
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  2. Jan Tomeček
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Correspondence to Irena Rachůnková .

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Rachůnková, I., Tomeček, J. (2015). Dirichlet Problem with One Impulse Condition. In: State-Dependent Impulses. Atlantis Briefs in Differential Equations, vol 6. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-127-7_6

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