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

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Zhou J., Li Y.

Annales Polonici Mathematici

2011

tom 100

nr 1

87-98

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 ⎩ $Δu'(t_{j}) = u'(t⁺_{j} - u'(t¯_{j}) = I_{j}(u(t_{j}))$, j = 1,...,p, are established, where $t₀ = 0 < t₁ < ⋯ < t_{p} < t_{p+1} = T$, g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and $I_{j}: ℝ → ℝ$, 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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Li Y.

1 Zhou J.

1 2011

Wyniki wyszukiwania

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