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Positive Solutions of Second-order Difference Equation with Variable Coefficient on the Infinite Interval
  
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KeyWord:Positive solution  Green's function  Compact  Infinite interval
Author NameAffiliation
Yanqiong Lu Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China 
Rui Wang Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China 
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Abstract:
      In this paper, based on the one-signed Green's function and the compact results on the infinite interval, we obtain the existence and multiplicity of positive solutions for the boundary value problems \begin{align*} \left\{\begin{array}{ll} \Delta^2x(n-1)-p(n)\Delta x(n-1)-q(n)x(n-1)+f(n,x(n))=0, \ n\in\mathbb{N}, \\[0ex] \alpha x(0)-\beta\Delta x(0)=0, \quad \ \ \lim\limits_{n\rightarrow\infty}x(n)=0 \end{array} \right. \end{align*} by the fixed point theorem in cones. The main results extend some results in the previous literature.