Solutions of second order homogeneous Dirichlet systems

Nontrivial solutions of second-order singular Dirichlet systems

where q ∈ C((, ),R), e = (e, . . . , eN )T ∈ C((, ),RN ), N ≥ , and the nonlinear term f (t,u) ∈ C((, )×RN \ {},RN ). We are mainly motivated by the recent excellent works [–], in which singular periodic systems were extensively studied. Let R+ denote the set of vectors of RN with positive components. For a fixed vector v ∈R+ , we say that system (.) presents a singularity at the o...

On Second Order Homogeneous Linear Differential Equations with Liouvillian Solutions

We determine all minimal polynomials for second order homogeneous linear diierential equations with algebraic solutions decomposed into in-variants and we show how easily one can recover the known conditions on diierential Galois groups 12,19,25] using invariant theory. Applying these conditions and the diierential invariants of a diierential equation we deduce an alternative method to the algo...

Periodic Solutions of Second Order Hamiltonian Systems

We provide sufficient conditions for the existence of periodic solutions of the second order Hamiltonian system −x′′ − λx = εV ′ x (t, x) , where ε is a small parameter, x ∈ R and V (t, x) is 2π-periodic in t. Moreover we provide two applications.

Stability Analysis of Second-Order Switched Homogeneous Systems

Periodic solutions of second order non-autonomous singular dynamical systems

In this paper, we establish two different existence results of positive periodic solutions for second order non-autonomous singular dynamical systems. The first one is based on a nonlinear alternative principle of Leray–Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder’s fixed point theorem an...