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Variational methods to fourth-order impulsive differential equations. (English) Zbl 1218.34029
The fourth-order boundary value problem for impulsive differential equations $$\aligned &u^{(iv)}(t)+Au''(t)+Bu(t)=f(t,u(t)), \quad \text{a.e.}\,\,\, t\in [0,T],\\ &\Delta(u''(t_j))=I_{1j}(u'(t_j)), \quad j=1,\dots,l,\\ &\Delta(u'''(t_j))=I_{2j}(u(t_j)), \quad j=1,\dots,l,\\ &u(0)=u(T)=u''(0^+)=u''(T^-)=0\endaligned$$ is considered, where $A$ and $B$ are real constants, $f: [0,T]\times {\Bbb R}\to {\Bbb R}$ is continuous, $I_{1j}, I_{2j}\in C({\Bbb R},{\Bbb R})$ for $1\le j\le l,$ $0=t_0<t_1<\ldots<t_l<t_{l+1}=T,$ $\Delta(u''(t_j))=u''(t_j^+)-u''(t_j^-),$ $\Delta(u'''(t_j))=u'''(t_j^+)-u'''(t_j^-).$ By using variational methods and critical point theory, the authors prove the existence of at least one classical solution and infinitely many distinct classical solutions. Some examples to illustrate the results are also presented.

34B37Boundary value problems for ODE with impulses
34B15Nonlinear boundary value problems for ODE
58E30Variational principles on infinite-dimensional spaces
58E05Abstract critical point theory
Full Text: DOI
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