Nonlinear Resonance: a Comparison Between Landesman-Lazer and Ahmad-Lazer-Paul Conditions

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Alessandro Fonda ◽  
Maurizio Garrione
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Nonlinear Resonance ◽  
Elliptic Operators ◽  
Scalar Case ◽  
Asymmetric Oscillator ◽  
Existence Conditions

AbstractWe show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

scholarly journals Nonresonance conditions for fourth order nonlinear boundary value problems

1994 ◽  
Vol 17 (4) ◽  
pp. 725-740 ◽  
Author(s):  
C. De Coster ◽  
C. Fabry ◽  
F. Munyamarere
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Linear Problem ◽  
Bounded Function ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Existence Conditions

This paper is devoted to the study of the problemu(4)=f(t,u,u′,u″,u‴),u(0)=u(2π),   u′(0)=u′(2π),   u″(0)=u″(2π),   u‴(0)=u‴(2π).We assume thatfcan be written under the formf(t,u,u′,u″,u‴)=f2(t,u,u′,u″,u‴)u″+f1(t,u,u′,u″,u‴)u′+f0(t,u,u′,u″,u‴)u+r(t,u,u′,u″,u‴)whereris a bounded function. We obtain existence conditions related to uniqueness conditions for the solution of the linear problemu(4)=au+bu″,u(0)=u(2π),   u′(0)=u′(2π),   u″(0)=u″(2π),   u‴(0)=u‴(2π).

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