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Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. (English) Zbl 1103.34015

By using the theory of fixed-point index in cones, the authors prove the existence of multiple positive solutions for the Dirichlet boundary value problem with impulse effect \[ -x''=f(t,x), \quad t\neq t_{k}, \quad k=1,2,\ldots,m, \;t\in J:=[0,1], \]
\[ x'(t_{k}^{-})-x'(t_{k}^{+})=I_{k}(x(t_{k})), \]
\[ x(0)=x(1)=0, \] where \(f\in C(J\times {\mathbb R}^{+},{\mathbb R}^{+}),\) \(I_{k}\in C({\mathbb R}^{+},{\mathbb R}^{+}),\) \(0<t_1<t_2<\ldots<t_m<1\) and \(x'(t_{k}^{+}), x'(t_{k}^{-})\) denote the right and left limits of \(x'(t)\) at \(t=t_{k}.\)

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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