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

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.

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On the first eigenvalue of non-uniformly elliptic boundary value problems

Mathematische Zeitschrift ◽

10.1007/bf01161411 ◽

1980 ◽

Vol 174 (3) ◽

pp. 227-232 ◽

Cited By ~ 3

Author(s):

Neil S. Trudinger

Keyword(s):

Boundary Value Problems ◽

Elliptic Boundary ◽

Boundary Value ◽

Elliptic Boundary Value Problems ◽

First Eigenvalue ◽

Elliptic Boundary Value ◽

The First Eigenvalue

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A Unified Approach to Estimation of Lower Bounds for the First Eigenvalue of Several Elliptic Boundary Value Problems

Mathematische Nachrichten ◽

10.1002/mana.19871310117 ◽

1987 ◽

Vol 131 (1) ◽

pp. 183-199 ◽

Cited By ~ 6

Author(s):

Klaus Gürlebeck ◽

Wolfgang Sprössig

Keyword(s):

Boundary Value Problems ◽

Lower Bounds ◽

Elliptic Boundary ◽

Boundary Value ◽

Elliptic Boundary Value Problems ◽

First Eigenvalue ◽

Unified Approach ◽

Elliptic Boundary Value ◽

The First Eigenvalue

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The monotonicity of the change in the first eigenvalue for a class of nonselfadjoint boundary value problems in the theory of hydrodynamical stability

Seven Papers in Applied Mathematics - American Mathematical Society Translations: Series 2 ◽

10.1090/trans2/125/03 ◽

1985 ◽

pp. 33-43

Author(s):

V. G. Babskiĭ ◽

A. D. Myshkis

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

First Eigenvalue ◽

The First Eigenvalue ◽

Hydrodynamical Stability

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Nonuniform nonresonance at the first eigenvalue for singular boundary value problems with sign changing nonlinearities

Journal of Mathematical Analysis and Applications ◽

10.1016/s0022-247x(02)00340-2 ◽

2002 ◽

Vol 274 (1) ◽

pp. 404-423 ◽

Cited By ~ 4

Author(s):

R.P. Agarwal ◽

Donal O'Regan ◽

V. Lakshmikantham

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Singular Boundary ◽

First Eigenvalue ◽

Singular Boundary Value Problems ◽

The First Eigenvalue

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A Nonlinear Boundary Value Problem with Potential Oscillating Around the First Eigenvalue

Journal of Differential Equations ◽

10.1006/jdeq.1995.1060 ◽

1995 ◽

Vol 117 (2) ◽

pp. 428-445 ◽

Cited By ~ 11

Author(s):

P. Habets ◽

R. Manasevich ◽

F. Zanolin

Keyword(s):

Boundary Value Problem ◽

Nonlinear Boundary ◽

Boundary Value ◽

Nonlinear Boundary Value Problem ◽

First Eigenvalue ◽

The First Eigenvalue ◽

Nonlinear Boundary Value

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500025518 ◽

1984 ◽

Vol 98 (1-2) ◽

pp. 1-11 ◽

Cited By ~ 5

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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