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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 < ⋯ < 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....

Existence and multiplicity results for nonlinear eigenvalue problems with discontinuities

Nikolaos Papageorgiou, Francesca Papalini (2000)

Annales Polonici Mathematici

We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations. In order to study the existence of solutions we replace these two problems with their multivalued approximations and, for the first problem, we estabilish an existence result while for the second problem we prove the existence of multiple nontrivial solutions. The approach used is variational.

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