1,721,104 research outputs found
On the topological degree of planar maps avoiding normal cones
Full text linkThe classical Poincaré–Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones. We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than 1
Non-well-ordered lower and upper solutions for semilinear systems of PDEs
Full text linkWe prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type
Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces
Full text linkWe prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations
Non-well-ordered lower and upper solutions for semilinear systems of PDEs
Full text linkWe prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type
A Poincaré–Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems
No full textWe provide a new version of the Poincaré–Birkhoff theorem for possibly multivalued successor maps associated with planar non-autonomous Hamiltonian systems. As an application, we prove the existence of periodic and subharmonic solutions of the scalar second order equation x ̈+λg(t,x)=0, for λ>0 sufficiently small, with g(t, x) having a superlinear growth at infinity, without requiring the existence of an equilibrium point
Two-point boundary value problems for planar systems: A lower and upper solutions approach
Full text linkWe extend the theory of lower and upper solutions to planar systems of ordinary differential equations with separated boundary conditions, both in the well-ordered and in the non-well-ordered cases. We are able to deal with general Sturm–Liouville boundary conditions in the well-ordered case, and we analyze the Dirichlet problem in the non-well-ordered case. Our results apply in particular to scalar second order differential equations, including those driven by the mean curvature operator. Higher dimensional systems are also treated, with the same approach
Well-Ordered and Non-Well-Ordered Lower and Upper Solutions for Periodic Planar Systems
Full text linkThe aim of this paper is to extend the theory of lower and upper solutions to the periodic problem associated with planar systems of differential equations. We generalize previously given definitions and we are able to treat both the well-ordered case and the non-well-ordered case. The proofs involve topological degree arguments, together with a detailed analysis of the solutions in the phase plane
Multiplicity of Periodic Solutions for Nearly Resonant Hamiltonian Systems
No full textWe prove a multiplicity result for the periodic problem associated with
a Hamiltonian system whose Hamiltonian function has a twisting part and a
nonresonant part. The possible approach to resonance together with some kind of
Landesman–Lazer conditions is also analyzed. We propose a new version of this
condition, and we also treat the so-called double resonance situation
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