Existence principle for BVPs with state-dependent impulses

Irena Rachůnková, Jan Tomeček

Abstract


The paper provides an existence principle for the Sturm-Liouville boundary value problem with state-dependent impulses
\begin{gather}
z''(t) = f(t,z(t),z'(t)) \quad \text{for a.e. } t \in [0,T] \subset \re, \nonumber\\
z(0) - az'(0) = c_1, \quad z(T) + bz'(T) = c_2, \nonumber\\
z(\x{\tau}{i}+) - z(\x{\tau}{i}) = J_i(\x{\tau}{i},z(\x{\tau}{i})),
\quad z'(\x{\tau}{i}+) - z'(\x{\tau}{i}-) = \m_i(\x{\tau}{i},z(\x{\tau}{i})), \nonumber
\end{gather}
where the points $\x{\tau}{1}, \ldots, \x{\tau}{p}$ depend on $z$ through the equations
\begin{equation*}
\x{\tau}{i} = \gamma(z(\x{\tau}{i})), \quad i = 1,\ldots,p, \ p \in \en.
\end{equation*}
Provided $a$, $b \in [0,\infty)$, $c_j \in \re$, $j = 1,2$, and the data functions $f$, $J_i$, $\m_i$, $i=1,\ldots,p$, are bounded, transversality conditions for barriers $\gamma_i$, $i = 1,\ldots,p$, which yield the solvability of the problem, are delivered. An application to the problem with unbounded data functions is demonstrated.

Keywords


Impulsive differential equation; state-dependent impulses; Sturm-Liouville problem; second order ODE; transversality conditions

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