### Antiperiodic boundary value problems for finite dimensional differential systems.

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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 ⎩ $\Delta {u}^{\text{'}}\left({t}_{j}\right)={u}^{\text{'}}(t{\u207a}_{j}-{u}^{\text{'}}\left(t{\xaf}_{j}\right)={I}_{j}\left(u\left({t}_{j}\right)\right)$, j = 1,...,p, are established, where $t\u2080=0<t\u2081<\cdots <{t}_{p}<{t}_{p+1}=T$, g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and ${I}_{j}:\mathbb{R}\to \mathbb{R}$, 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....

This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure...

In this paper, a class of damped vibration problems with impulsive effects is considered. An existence result is obtained by using the variational method and the critical point theorem due to Brezis and Nirenberg. The obtained result is also valid and new for the corresponding second-order impulsive Hamiltonian system. Finally, an example is presented to illustrate the feasibility and effectiveness of the result.

The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(\right|\dot{u}\left(t\right){|}^{p-2}\dot{u}\left(t\right))=\nabla F(t,u\left(t\right)),\phantom{\rule{1.0em}{0ex}}\text{a.e.}\phantom{\rule{4pt}{0ex}}t\in [0,T],\\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0,\\ \Delta {\dot{u}}^{i}\left({t}_{j}\right)={\dot{u}}^{i}\left({t}_{j}^{+}\right)-{\dot{u}}^{i}\left({t}_{j}^{-}\right)={I}_{ij}\left({u}^{i}\left({t}_{j}\right)\right),\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,N;\phantom{\rule{4pt}{0ex}}j=1,2,\cdots ,m.\end{array}$$ By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.

A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point...

In this paper we investigate the existence of solutions to impulsive problems with a $p\left(t\right)$-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence...

In the paper we consider the impulsive periodic boundary value problem with a general linear left hand side. The results are based on the topological degree theorems for the corresponding operator equation $(I-F)u=0$ on a certain set $\Omega $ that is established using properties of strict lower and upper functions of the boundary value problem.