1,720,974 research outputs found
Unbounded Solutions to Systems of Differential Equations at Resonance
Full text linkWe deal with a weakly coupled system of ODEs of the type xj′′+nj2xj+hj(x1,...,xd)=pj(t),j=1,...,d,with hj locally Lipschitz continuous and bounded, pj continuous and 2 π-periodic, nj∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h1, ... , hd are assumed
Periodic solutions of asymptotically linear second order equations with changing sign weight
No full textIn this paper we study the ordinary differential equation x'' + q(t)g(x) = 0, where g is a locally Lipschitz continuous function that satisfies g(x)x > 0 for all non zero x and is asymptotically linear, while q is a continuous, π-periodic and changing sign weight. By the application of a recent result on the existence and multiplicity of fixed points of planar maps, we give conditions on q and on the behavior of the ratio g(x)/x near zero and near infinity in order to obtain multiple periodic solutions with the prescribed number of zeros in the intervals of positivity and negativity of q, as well as multiple subharmonics of any order and uncountably many bounded solutions
Periodic solutions to a forced kepler problem in the plane
Full text linkGiven a smooth function U(t, x), T-periodic in the first variable and satisfying U(t, x) = O(vertical bar x vertical bar(alpha)) for some alpha is an element of (0, 2) as vertical bar x vertical bar -> infinity, we prove that the forced Kepler problem(sic) = -x/vertical bar x vertical bar(3) + del U-x(t, x), x is an element of R-2,has a generalized T-periodic solution, according to the definition given in the paper by A. Boscaggin, R. Ortega, and L. Zhao [Trans. Amer. Math. Soc. 372 (2019), 677-703]. The proof relies on variational arguments
Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential
Full text linkWe consider the Lorentz force equation in the physically relevant case of a singular electric field E. Assuming that E and B are T-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T-periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem
Periodic perturbations of central force problems and an application to a restricted 3-body problem
Full text linkWe consider a perturbation of a central force problem of the form x ̈=V′(|x|)[Formula presented]+ε∇xU(t,x),x∈R2∖{0}, where ε∈R is a small parameter, V:(0,+∞)→R and U:R×(R2∖{0})→R are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε=0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at ε=0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V(r)=κ/rα for α∈(−∞,2)∖{−2,0,1}). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction
Linear and nonlinear eigenvalue problems for Dirac systems in unbounded domains
No full textWe first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solution by means of the rotation number. We then give a global bifurcation result for a planar nonlinear Dirac system in the open half-line. As an application,we provide a global continuum of solutions of the nonlinear Dirac equation which have a special form
A global bifurcation result for a second order singular equation
Full text linkDedicated, with gratefulness and friendship, to Professor Fabio Zanolin on the occasion of his 60th birthday Abstract. We deal with a boundary value problem associated to a second order singular equation in the open interval (0, 1]. We first study the eigenvalue problem in the linear case and discuss the nodal properties of the eigenfunctions. We then give a global bifurcation result for nonlinear problems
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