scholarly journals Multiple solutions for a fourth-order difference boundary value problem with parameter via variational approach

2012 ◽  
Vol 36 (9) ◽  
pp. 4385-4392 ◽  
Author(s):  
Guo-Dong Zhang ◽  
Hong-Rui Sun
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Variational Approach ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Order Difference

scholarly journals Uniqueness and Multiplicity of Solutions for a Second-Order Discrete Boundary Value Problem with a Parameter

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Xi-Lan Liu ◽  
Jian-Hua Wu
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Order Difference ◽  
Multiplicity Of Solutions ◽  
Discrete Boundary Value Problem ◽  
Second Order Difference Equation ◽  
Order Difference Equation

This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.

scholarly journals Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zuoshi Xie ◽  
Yuanfeng Jin ◽  
Chengmin Hou
Keyword(s):  
Boundary Value Problem ◽  
Critical Point Theory ◽  
Variational Approach ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Point Theory ◽  
Linking Theorem ◽  
Fractional Difference ◽  
Variational Framework

By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

scholarly journals Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Bo Zheng ◽  
Huafeng Xiao
Keyword(s):  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Local Minimizer ◽  
Second Order ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Difference Problem ◽  
Order Difference

This paper studies the existence of multiple solutions of the second-order difference boundary value problemΔ2u(n−1)+V′(u(n))=0,n∈ℤ(1,T),u(0)=0=u(T+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalueλk  (k≥2)of linear difference problemΔ2u(n−1)+λu(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1)near infinity and the trivial solution of the first equation is a local minimizer under some assumptions onV.

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