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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

$\begin{array}{cc}& {u}^{\left(iv\right)}\left(t\right)+A{u}^{\text{'}\text{'}}\left(t\right)+Bu\left(t\right)=f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}\text{a.e.}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}t\in \left[0,T\right],\hfill \\ & {\Delta }\left({u}^{\text{'}\text{'}}\left({t}_{j}\right)\right)={I}_{1j}\left({u}^{\text{'}}\left({t}_{j}\right)\right),\phantom{\rule{1.em}{0ex}}j=1,\cdots ,l,\hfill \\ & {\Delta }\left({u}^{\text{'}\text{'}\text{'}}\left({t}_{j}\right)\right)={I}_{2j}\left(u\left({t}_{j}\right)\right),\phantom{\rule{1.em}{0ex}}j=1,\cdots ,l,\hfill \\ & u\left(0\right)=u\left(T\right)={u}^{\text{'}\text{'}}\left({0}^{+}\right)={u}^{\text{'}\text{'}}\left({T}^{-}\right)=0\hfill \end{array}$

is considered, where $A$ and $B$ are real constants, $f:\left[0,T\right]×ℝ\to ℝ$ is continuous, ${I}_{1j},{I}_{2j}\in C\left(ℝ,ℝ\right)$ for $1\le j\le l,$ $0={t}_{0}<{t}_{1}<...<{t}_{l}<{t}_{l+1}=T,$ ${\Delta }\left({u}^{\text{'}\text{'}}\left({t}_{j}\right)\right)={u}^{\text{'}\text{'}}\left({t}_{j}^{+}\right)-{u}^{\text{'}\text{'}}\left({t}_{j}^{-}\right),$ ${\Delta }\left({u}^{\text{'}\text{'}\text{'}}\left({t}_{j}\right)\right)={u}^{\text{'}\text{'}\text{'}}\left({t}_{j}^{+}\right)-{u}^{\text{'}\text{'}\text{'}}\left({t}_{j}^{-}\right)·$ 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.

##### MSC:
 34B37 Boundary value problems for ODE with impulses 34B15 Nonlinear boundary value problems for ODE 58E30 Variational principles on infinite-dimensional spaces 58E05 Abstract critical point theory
##### References:
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