A Landesman–Lazer-type condition for asymptotically linear second-order equations with a singularity

2012 ◽  
Vol 142 (6) ◽  
pp. 1263-1277 ◽  
Author(s):  
Alessandro Fonda ◽  
Maurizio Garrione
Keyword(s):  
Periodic Problem ◽  
Second Order ◽  
Type Condition ◽  
Asymptotically Linear ◽  
Existence Of A Solution ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Linear Behaviour ◽  
Image Position ◽  
Second Order Equations

We consider the T-periodic problemwhere g: [0,T]×]0,+∞[→ℝ exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behaviour at infinity. In particular, for large x, g(t, x) is controlled from both sides by two consecutive asymptotes of the T-periodic Fučik spectrum, with possible equality on one side. Using a suitable Landesman–Lazer-type condition, we prove the existence of a solution.

Multiplicity results for systems of asymptotically linear second order equations

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Anna Capietto ◽  
Francesca Dalbono
Keyword(s):  
Lower Bounds ◽  
Second Order ◽  
Upper And Lower Bounds ◽  
Asymptotically Linear ◽  
Multiplicity Results ◽  
Number Of Zeros ◽  
Existence And Multiplicity ◽  
Abstract Continuation Theorem ◽  
Nodal Properties ◽  
Second Order Equations

AbstractWe 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.

scholarly journals ON SOME FUČÍK TYPE PROBLEM WITH CUBIC NONLINEARITY

2011 ◽  
Vol 16 (1) ◽  
pp. 52-61
Author(s):  
N. Sergejeva
Keyword(s):  
Boundary Conditions ◽  
Second Order ◽  
Type Problem ◽  
Cubic Nonlinearity ◽  
Normalization Condition ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Fucik Spectrum

We study the second order Fučík type problem with cubic nonlinearity and construct the Fučík spectrum for this problem. The spectrum obtained under normalization condition (otherwise problem may have continuous spectra) structurally is similar to Fučík spectra for the problem with the same boundary conditions.

On a structure of the set of positive solutions to second-order equations with a super-linear non-linearity

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiří Šremr
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Periodic Problem ◽  
Second Order ◽  
Solution Set ◽  
Existence And Multiplicity ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions ◽  
Lower And Upper Functions ◽  
Second Order Equations

Abstract We study the existence and multiplicity of positive solutions to the periodic problem u ′′ = p ⁢ ( t ) ⁢ u - q ⁢ ( t , u ) ⁢ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\quad u^{\prime}(0)=u^{\prime}(\omega), where p , f ∈ L ⁢ ( [ 0 , ω ] ) p,f\in L([0,\omega]) and q : [ 0 , ω ] × R → R q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R} is a Carathéodory function. By using the method of lower and upper functions, we show some properties of the solution set of the considered problem and, in particular, the existence of a minimal positive solution.

Multiplicity of solutions of Dirichlet problems associated with second-order equations in ℝ2

2009 ◽  
Vol 52 (3) ◽  
pp. 569-581 ◽  
Author(s):  
F. Dalbono ◽  
C. Rebelo
Keyword(s):  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Second Order ◽  
Point Boundary ◽  
Dirichlet Problems ◽  
Asymptotically Linear ◽  
Planar System ◽  
Shooting Technique ◽  
Sign Conditions ◽  
Second Order Equations

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.

scholarly journals Fučik Spectrum for the Second Order BVP with Nonlocal Boundary Condition

2007 ◽  
Vol 12 (3) ◽  
pp. 419-429 ◽  
Author(s):  
N. Sergejeva
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Order Differential Equation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Fucik Spectrum

We construct the Fučik spectrum for some second order boundary value problem with nonlocal boundary condition. This spectrum differs essentially from the known Fučik spectra. We apply this result to the second order differential equation x'' + g(x) = f(t, x, x') with the conditions x(a) = 0, ∫ab x(s)ds = 0.

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