The Fučík spectrum for nonlocal BVP with Sturm–Liouville boundary condition

2014 ◽  
Vol 19 (3) ◽  
pp. 503-516 ◽  
Author(s):  
Natalija Sergejeva ◽  
Keyword(s):  
Boundary Condition ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Sturm Liouville ◽  
Nonlocal Bvp ◽  
Fucik Spectrum

On the periodic Fučik spectrum and a superlinear Sturm–Liouville equation

1993 ◽  
Vol 123 (1) ◽  
pp. 95-107 ◽  
Author(s):  
Djairo G. Figueiredo ◽  
Bernhard Ruf
Keyword(s):  
Liouville Equation ◽  
Exponential Nonlinearity ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Sturm Liouville ◽  
Fucik Spectrum

SynopsisIn the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the Fučik spectrum.

On a variational characterization of the Fučík spectrum of the Laplacian and a superlinear Sturm–Liouville equation

2004 ◽  
Vol 134 (3) ◽  
pp. 557-577 ◽  
Author(s):  
Eugenio Massa
Keyword(s):  
Finite Number ◽  
Liouville Equation ◽  
Neumann Boundary ◽  
Linking Theorem ◽  
Variational Characterization ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Sturm Liouville ◽  
Fucik Spectrum

In the first part of this paper, a variational characterization of parts of the Fučík spectrum for the Laplacian in a bounded domain Ω is given. The proof uses a linking theorem on sets obtained through a suitable deformation of subspaces of H1 (Ω). In the second part, a nonlinear Sturm–Liouville equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but a finite number of eigenvalues. It is proved that, under certain conditions, this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values related to the Fucík spectrum.

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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