Search results for "asymptotically linear"
showing 8 items of 8 documents
Multiplicity results for asymmetric boundary value problems with indefinite weights
2004
We prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the formu″+f(t,u)=0,u(0)=u(T)=0. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues.
Multiplicity results for systems of asymptotically linear second order equations
2002
Abstract We prove the existence and multiplicity of solutions, with prescribed nodal properties, for a BVP associated with a system of asymptotically linear second order equations. The applicability of an abstract continuation theorem is ensured by upper and lower bounds on the number of zeros of each component of a solution.
Multiplicity of Solutions for Second Order Two-Point Boundary Value Problems with Asymptotically Asymmetric Nonlinearities at Resonance
2007
Abstract Estimations of the number of solutions are given for various resonant cases of the boundary value problem 𝑥″ + 𝑔(𝑡, 𝑥) = 𝑓(𝑡, 𝑥, 𝑥′), 𝑥(𝑎) cos α – 𝑥′(𝑎) sin α = 0, 𝑥(𝑏) cos β – 𝑥′(𝑏) sin β = 0, where 𝑔(𝑡, 𝑥) is an asymptotically linear nonlinearity, and 𝑓 is a sublinear one. We assume that there exists at least one solution to the BVP.
Multiplicity results for asymptotically linear equations, using the rotation number approach
2007
By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties.
A Multiplicity result for a class of strongly indefinite asymptotically linear second order systems
2010
We prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the Maslov index of the linear Hamiltonian systems associated to the asymptotic limits of the given nonlinearity.
Multiplicity of solutions for asymptotically linear $n$-th order boundary value problems
2007
In this paper we investigate existence and multiplicity of solutions, with prescribed nodal properties, to a two-point boundary value problem of asymptotically linear $n$-th order equations. The proof follows a shooting approach and it is based on the weighted eigenvalue theory for linear $n$-th order boundary value problems
Multiplicity of solutions of Dirichlet problems associated with second-order equations in ℝ2
2009
AbstractWe study the existence of multiple solutions for a two-point boundary-value problem associated with a planar system of second-order ordinary differential equations by using a shooting technique. We consider asymptotically linear nonlinearities satisfying suitable sign conditions. Multiplicity is ensured by assumptions involving the Morse indices of the linearizations at zero and at infinity.
Sign-preserving solutions for a class of asymptotically linear systems of second-order ordinary differential equations
2022
We study multiplicity of solutions to an asymptotically linear Dirichlet problem associated with a planar system of second order ordinary differential equations. The existence of two sign-preserving component-wise solutions is guaranteed when the Morse indexes of the linearizations at zero and at infinity do not coincide, and one of the asymptotic problems has zero-index. The proof is developed in the framework of topological and shooting methods and it is based on a detailed analysis and characterization of the phase angles in a two-dimensional setting.