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Second order BVPs with state dependent impulses via lower and upper functions. (English) Zbl 1302.34049

Assuming the existence of lower and upper solutions, the authors establish the existence of a solution to the second order Dirichlet boundary value problem with \(p\) impulses on the derivative depending on the state: \[ \begin{aligned} &z''(t) = f(t,z(t)), \\ &z(0)=z(T)=0, \\ &z'(\tau_i^+) - z'(\tau_i^-) = I_i(\tau_i,z(\tau_i)), \quad \tau_i = \gamma_i(z(\tau_i)),\quad i=1,\dots,p. \end{aligned} \] The proof relies on the Schauder fixed point theorem applied to a compact operator \(F\) defined on the space \((C^1([0,T]))^{p+1}\). For \((u_1,\dots,u_{p+1})\) a fixed point of \(F\), the solution \(z\) to this problem is such that \(z(t) = u_i(t)\) for \(t \in (\tau_{i-1},\tau_i)\), where \(\tau_i\) is the time where \(u_i\) hits the \(i\)-th barrier, i.e. \(\tau_i=\gamma_i(u(\tau_i))\).

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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