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 a Fourth-Order Boundary Value Problem at Resonance

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Man Xu ◽  
Ruyun Ma
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Elastic Beam ◽  
First Eigenvalue ◽  
At Resonance ◽  
Problem At Resonance ◽  
Spectrum Structure

We investigate the spectrum structure of the eigenvalue problem u4x=λux,  x∈0,1;  u0=u1=u′0=u′1=0. As for the application of the spectrum structure, we show the existence of solutions of the fourth-order boundary value problem at resonance -u4x+λ1ux+gx,ux=hx,  x∈0,1;  u0=u1=u′0=u′1=0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded.

scholarly journals On Removable Sets of the First Boundary-Value Problem for Degenerated Elliptic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-20
Author(s):  
Tair S. Gadjiev ◽  
Asghar I. Shariffar ◽  
Rafig A. Rasulov
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Removable Sets ◽  
Necessary And Sufficient

In the paper, the necessary and sufficient condition of compact removability is obtained.

scholarly journals Positive Solutions of Two-Point Boundary Value Problems for Monge-Ampère Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Baoqiang Yan ◽  
Meng Zhang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Existence Of Positive Solutions ◽  
Necessary And Sufficient ◽  
Monge Ampere

This paper considers the following boundary value problem:((-u'(t))n)'=ntn-1f(u(t)),  0<t<1,  u'(0)=0,  u(1)=0, wheren>1is odd. We establish the method of lower and upper solutions for some boundary value problems which generalizes the above equations and using this method we present a necessary and sufficient condition for the existence of positive solutions to the above boundary value problem and some sufficient conditions for the existence of positive solutions.

scholarly journals A necessary and sufficient condition for the existence of a unique solution of a discrete boundary value problem

2003 ◽  
Vol 2003 (39) ◽  
pp. 2455-2463 ◽  
Author(s):  
Raghib Abu-Saris ◽  
Wajdi Ahmad
Keyword(s):  
Boundary Value Problem ◽  
Unique Solution ◽  
Boundary Value ◽  
Linear Difference Equation ◽  
Fundamental Question ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Constant Coefficients ◽  
Necessary And Sufficient

Akth-order linear difference equation with constant coefficients subject to boundary conditions is considered. A necessary and sufficient condition for the existence of a unique solution for such a boundary value problem is established. The condition established answers a fundamental question for well-posedness and can be easily applied using a simple and computationally tractable algorithm that does not require finding the roots of the associated characteristic equation.

scholarly journals On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Rossitza Semerdjieva
Keyword(s):  
Free Boundary ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Condition ◽  
Free Boundary Value Problem ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Differentiability ◽  
Differential Condition

AbstractWe consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on Cm-smoothness of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.

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