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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.