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.

scholarly journals A note on resonance problems with nonlinearity bounded in one direction

1995 ◽  
Vol 52 (2) ◽  
pp. 183-188 ◽  
Author(s):  
To Fu Ma ◽  
Luís Sanchez
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
First Eigenvalue ◽  
Existence Of A Solution ◽  
The First Eigenvalue ◽  
At Resonance ◽  
Growth Restrictions ◽  
Problem At Resonance ◽  
Bounded Below

We prove the existence of a solution for a semilinear boundary value problem at resonance in the first eigenvalue. The nonlinearity is assumed to be bounded below or above; no further growth restrictions are assumed.

scholarly journals Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Chenghua Gao
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Sufficient Condition ◽  
First Eigenvalue ◽  
Necessary And Sufficient Condition ◽  
The First Eigenvalue ◽  
Necessary And Sufficient

This paper is concerned with the existence of solutions for the discrete second-order boundary value problemΔ2u(t-1)+λ1u(t)+g(Δu(t))=f(t),t∈{1,2,…,T},u(0)=u(T+1)=0, whereT>1is an integer,f:{1,…,T}→R,g:R→Ris bounded and continuous, andλ1is the first eigenvalue of the eigenvalue problemΔ2u(t-1)+λu(t)=0,t∈T,u(0)=u(T+1)=0.

scholarly journals On the existence threshold for positive solutions of p-Laplacian equations with a concave–convex nonlinearity

2015 ◽  
Vol 17 (06) ◽  
pp. 1450044 ◽  
Author(s):  
Fernando Charro ◽  
Enea Parini
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Boundary Value ◽  
First Eigenvalue ◽  
Limit Case ◽  
The First Eigenvalue ◽  
Laplacian Equations

We study the following boundary value problem with a concave–convex nonlinearity: [Formula: see text] Here Ω ⊂ ℝnis a bounded domain and 1 < q < p < r < p*. It is well known that there exists a number Λq, r> 0 such that the problem admits at least two positive solutions for 0 < Λ < Λq, r, at least one positive solution for Λ = Λq, r, and no positive solution for Λ > Λq, r. We show that [Formula: see text] where λ1(p) is the first eigenvalue of the p-Laplacian. It is worth noticing that λ1(p) is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case q = p.

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