The nonlinear eigenvalue problem Lu+f(x,u)=λu in (a,b) , with u(a)=u(b)=0 , where Lu=−(p(x)u ′ ) ′ +q(x)u is the usual differential operator for Sturm-Liouville problems, f is continuous and such that f(x,0)=0 for all x , and λ is a real parameter, is considered in this paper. Here λ is called an eigenvalue if there is a nontrivial (nonzero) solution for the above problem. It is proved that if f is odd, such that 0≤f(x,s)≤αs 2 for some α>0 and all x and s , and satisfies additional assumptions, then all eigenvalues are contained in the intervals [λ 0 n ,λ 0 n +α] , where λ 0 n are the eigenvalues for the linear problem Lu=λu . The proof uses spectral properties of the associated linear operator. Some complementary results concerning families of eigenvalues are also included.
Chiappinelli, R. (1989). Estimates on the eigenvalues for some nonlinear ordinary differential operators. COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE, 30(3), 441-446.
Estimates on the eigenvalues for some nonlinear ordinary differential operators
CHIAPPINELLI, RAFFAELE
1989-01-01
Abstract
The nonlinear eigenvalue problem Lu+f(x,u)=λu in (a,b) , with u(a)=u(b)=0 , where Lu=−(p(x)u ′ ) ′ +q(x)u is the usual differential operator for Sturm-Liouville problems, f is continuous and such that f(x,0)=0 for all x , and λ is a real parameter, is considered in this paper. Here λ is called an eigenvalue if there is a nontrivial (nonzero) solution for the above problem. It is proved that if f is odd, such that 0≤f(x,s)≤αs 2 for some α>0 and all x and s , and satisfies additional assumptions, then all eigenvalues are contained in the intervals [λ 0 n ,λ 0 n +α] , where λ 0 n are the eigenvalues for the linear problem Lu=λu . The proof uses spectral properties of the associated linear operator. Some complementary results concerning families of eigenvalues are also included.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/11365/33468
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