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Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects. (English) Zbl 1281.34134
Summary: We present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects $$\aligned & u^{\Delta^2}(t)+A(\sigma(t))u(\sigma(t))+\nabla F(\sigma(t),u(\sigma(t)))=0, \quad \Delta \text{-a.e.} \ t\in[0,T]^\kappa_\mathbb T; \\ & u(0)-u(T)=u^\Delta(0)-u^\Delta(T)=0,\\ & (u^i)^\Delta(t^+_j)-(u^i)^\Delta(t^-_j)=I_{ij}(u^i(t_j)), \ i=1,2,\dots,N,\,j=1,2,\dots,p, \endaligned$$ where $t_0=0<t_1<t_2<\dots<t_p<t_{p+1}=T$, $t_j\in [0,T]_\mathbb T \ ( j=0,1,2,\dots,p+1)$, $u(t)=(u^1(t),u^2(t),\dots,u^N(t))\in \mathbb R^N$, $A(t)=[d_{lm}(t)]$ is a symmetric $N\times N$ matrix-valued function defined on $[0,T]_\mathbb T$ with $d_{lm}\in L^\infty([0,T]_\mathbb T, \mathbb R)$ for all $l,m=1,2,\dots, N$, $I_{ij}:\mathbb R\to \mathbb R \ (i=1,2,\dots,N,\, j=1,2,\dots,p)$ are continuous and $F:[0,T]_\mathbb T\times \mathbb R^N\to \mathbb R$. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.
34N05Dynamic equations on time scales or measure chains
34B37Boundary value problems for ODE with impulses
58E30Variational principles on infinite-dimensional spaces
58E50Applications of variational methods in infinite-dimensional spaces
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