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 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 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.

On a necessary and sufficient condition for the negativeness of the Green’s function of a two-point boundary value problem for a functional differential equation

2021 ◽  
pp. 382-393
Author(s):  
Sergey M. Labovskiy
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Value ◽  
Sufficient Condition ◽  
Point Boundary ◽  
Necessary And Sufficient Condition ◽  
Functional Differential ◽  
Necessary And Sufficient

Conditions of negativity for the Green’s function of a two-point boundary value problem L_λ u≔u^((n) )-λ∫_0^l▒〖u(s) d_s r(x,s)=f(x), x∈[0,l], B^k (u)=α,〗 where B^k (u)=(u(0),…,u^((n-k-1) ) (0),u(l),-u^'(l) ,…,(-1)^((k-1) ) u^((k-1) ) (0) ), n≥3, 0<k<n, k is odd, are considered. The function r(x,s) is assumed to be non-decreasing in the second argument. A necessary and sufficient condition for the nonnegativity of the solution of this boundary value problem on the set E of functions satisfying the conditions u(0)=⋯=u^((n-k-2) ) (0)=0, u(l)=⋯=u^((k-2) ) (l)=0, u^((n-k-1) ) (0)≥0, u^((k-1) ) (l)≥0, f(x)≤0 is obtained. This condition lies in the subcriticality of boundary value problems with vector functionals B^(k-1) and B^(k+1). Let k be even and λ^k be the smallest positive value of λ for which the problem L_λ u=0, B^k u=0 has a nontrivial solution. Then the pair of conditions λ<λ^(k-1) and λ<λ^(k+1) is necessary and sufficient for positivity of the solution of the problem.

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