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Currently displaying 1 – 2 of 2

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Existence of solutions for a class of damped vibration problems on time scales.

Li, Yongkun; Zhou, Jianwen — 2010

Advances in Difference Equations [electronic only]

Existence and multiplicity of solutions for a class of damped vibration problems with impulsive effects

Jianwen Zhou; Yongkun Li — 2011

Annales Polonici Mathematici

Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects ⎧ u”(t) + g(t)u’(t) + f(t,u(t)) = 0, a.e. t ∈ [0,T ⎨ u(0) = u(T) = 0 ⎩ $Δ u^{'} (t_{j}) = u^{'} (t ⁺_{j} - u^{'} (t ¯_{j}) = I_{j} (u (t_{j}))$ , j = 1,...,p, are established, where $t ₀ = 0 < t ₁ < \dots < t_{p} < t_{p + 1} = T$ , g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and $I_{j} : ℝ \to ℝ$ , j = 1,...,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness of our results....

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