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Dirichlet Problem with Space Singularities

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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

We are interested in the solvability of the singular Dirichlet boundary value problem with impulses at fixed times

$$\begin{aligned} -u''(t)&=f(t,u(t),u'(t))\quad \text {for a.e. }t \in [0,T]\subset \mathbb {R},\\ u(t_i+)&=J_i(u(t_i-)),\quad u'(t_i+)=M_i(u'(t_i-)),\quad i=1,\ldots ,p,\\ u(0)&= u(T) = 0, \end{aligned}$$

under the assumptions that \(p \in \mathbb {N}\), \(0 < t_1 < \cdots < t_p < T\), the impulse functions \(J_i, M_i\in \mathbb {C}(\mathbb {R})\), \(i = 1,\ldots ,p\), are increasing and \(f \in \mathrm{{Car}}_\mathrm{loc}\left( [0,T]\times \left( (0,\infty )\times \mathbb {R}\right) \right) \) can have a space singularity at \(x=0\). The main goal is to find additional conditions for f, \(J_i\), \(M_i\), \(i=1,\ldots ,p\), which guarantee the existence of at least one positive solution of the problem.

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Authors and Affiliations

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

    Irena Rachůnková & Jan Tomeček

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  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 Space Singularities. In: State-Dependent Impulses. Atlantis Briefs in Differential Equations, vol 6. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-127-7_4

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