scholarly journals Derivation of a certain inequality for the first eigenvalue of two boundary value problems

1960 ◽  
Vol 10 (1) ◽  
pp. 68-82
Author(s):  
Ludvík Janoš
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
First Eigenvalue ◽  
The First Eigenvalue

THE REGION OF SOLVABILITY OF A PARAMETERIZED BOUNDARY VALUE PROBLEM CAN BE DISCONNECTED

2004 ◽  
Vol 06 (06) ◽  
pp. 901-912
Author(s):  
ANTONIO J. UREÑA
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Second Eigenvalue

A celebrated result by Amann, Ambrosetti and Mancini [1] implies the connectedness of the region of existence for some parameter-depending boundary value problems which are resonant at the first eigenvalue. The analogous thing does not hold for problems which are resonant at the second eigenvalue.

Existence theorems for equations in normed spaces and boundary value problems for nonlinear vector ordinary differential equations

1984 ◽  
Vol 98 (1-2) ◽  
pp. 1-11 ◽  
Author(s):  
A. Cañada ◽  
R. Ortega
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Part ◽  
Sufficient Conditions ◽  
First Eigenvalue ◽  
Normed Spaces ◽  
Nonlinear Vector

SynopsisThe existence of solutions to equations in normed spaces is proved when the nonlinear part of the equation satisfies growth and asymptotic conditions, whether the linear part is invertible or not. For this, we use the coincidence degree theory developed by Mawhin. We apply our abstract results to boundary value problems for nonlinear vector ordinary differential equations. In particular, we consider the Picard boundary value problem at the first eigenvalue and the periodic boundary value problem at resonance. In both cases, the nonlinear term can be of superlinear type. Also, necessary and sufficient conditions of Landesman-Lazer type are obtained.

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