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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....

Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales

Chao Wang;  Yongkun Li — 2013

Annales Polonici Mathematici

We propose a concept of weighted pseudo almost automorphic functions on almost periodic time scales and study some important properties of weighted pseudo almost automorphic functions on almost periodic time scales. As applications, we obtain the conditions for the existence of weighted pseudo almost automorphic mild solutions to a class of semilinear dynamic equations on almost periodic time scales.

Three periodic solutions for a class of higher-dimensional functional differential equations with impulses

Yongkun Li;  Changzhao Li;  Juan Zhang — 2010

Annales Polonici Mathematici

By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), t ≠ t j , j ∈ ℤ, ⎨ ⎩ y ( t ⁺ j ) = y ( t ¯ j ) + I j ( y ( t j ) ) , where A ( t ) = ( a i j ( t ) ) n × n is a nonsingular matrix with continuous real-valued entries.

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