1,721,027 research outputs found
Periodic solutions to planar Hamiltonian systems: high multiplicity and chaotic dynamics
Full text linkMultiple positive solutions for nonlinear problems with indefinite weight: an overview of Fabio Zanolin's contributions
No full textWe revisit some results obtained by the authors together with Fabio Zanolin about the existence and multiplicity of positive solutions of boundary value problems associated with second-order equations of the type u''+q(t)g(u)=0, where q is a sign-changing weight and g is a non-negative nonlinear function
Planar Hamiltonian systems at resonance: The Ahmad-Lazer-Paul condition
Full text linkWe consider the planar Hamiltonian system Ju′ = ∇F(u) + ∇uR(t,u), t ∈ [0,T], u ∈ R2, with F(u) positive and positively 2-homogeneous and ∇uR(t, u) sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman-Lazer condition is analyzed, as well. © 2012 Springer Basel AG
Resonant Sturm-Liouville boundary value problems for differential systems in the plane
Full text linkWe study the Sturm-Liouville boundary value problem associated with the planar differential system Jz'= ∇V (z) + R(t, z), where V (z) is positive and positively 2-homogeneous and R(t, z) is bounded. Assuming Landesman-Lazer type conditions, we obtain the existence of a solution in the resonant case. The proofs are performed via a shooting argument. Some applications to boundary value problems associated with scalar second order asymmetric equations are discussed
Positive periodic solutions to an indefinite Minkowski-curvature equation
No full textWe investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., T-periodic) and subharmonic (i.e., kT-periodic for some integer k≥2) to the equation (u'/sqrt(1-|u'|^2))'+λa(t)g(u)=0, where λ>0 is a parameter, a(t) is a T-periodic sign-changing weight function and g: [0,+∞[→[0,+∞[ is a continuous function having superlinear growth at zero. In particular, we prove that for both g(u)=u^p, with p>1, and g(u)=u^p/(1+u^{p−q}), with 0≤q≤1, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a "small" one and a "large" one) for λ large enough. Moreover, in both cases the "small" T-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of T-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincaré–Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞
Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities
Full text linkWe study the Neumann boundary value problem for the second order ODE (Formula presented.),t∈[0,T],where (Formula presented.) is a bounded function of constant sign, (Formula presented.) are the positive/negative part of a sign-changing weight a(t) and μ>0 is a real parameter. Depending on the sign of (Formula presented.) at infinity, we find existence/multiplicity of solutions for μ in a “small” interval near the value (Formula presented.).The proof exploits a change of variables, transforming the sign-indefinite Eq. (1) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for (Formula presented.) and (Formula presented.) are given, as well
A note on a linear spectral theorem for a class of first order systems in
Full text linkAlong the lines of Atkinson, a spectral theorem is proved for the boundary value problem
where is real-valued and are symmetric matrices, with positive definite. A suitable rotation index associated to the system is used to highlight the connections between the eigenvalues and the nodal properties of the corresponding eigenfunctions
Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball
Full text link By using a shooting technique, we prove that the quasilinear boundary value problem [Formula: see text] where [Formula: see text] is a ball and [Formula: see text], has more and more pairs of nodal solutions on growing of the parameter [Formula: see text]. The radial Neumann problem and the periodic problem for the corresponding one-dimensional equation are considered, as well. </jats:p
Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line
No full textA counterexample to a priori bounds under the Ahmad-Lazer-Paul condition
Full text linkIn the context of scalar second order ODEs at resonance,we construct a counterexample showing that, in general, the Ahmad-Lazer-Paul condition does not imply a priori bounds for T-periodic solutions
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