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Green's functions, Cauchy functions and cone theoretic eigenvalue results for difference equations
Abstract
Initially, we present eigenvalue results for conjugate and focal boundary value problems involving the second order vector difference operator $\rm Ly(t)={-\Delta}\lbrack P(t - 1)\Delta y(t - 1)\rbrack$ on the discrete interval $\rm \lbrack a+1,b+1\rbrack{\equiv}\{a+1,a+2,\cdots,b+1\}.$ Here, P(t) is an n $\times$ n real matrix function on (a,b + 1). Green's matrix functions are derived for the conjugate and focal cases, and used to define summation operators on appropriate Banach spaces. Results from the theory of cones in a Banach space are then used to show the existence of least positive eigenvalues with corresponding eigenfunctions that satisfy certain sign conditions, find bounds on these least positive eigenvalues, and prove comparison theorems. We next consider the conjugate boundary value problem on the infinite discrete interval $\rm \lbrack a+1,\infty){\equiv}\{a+1,a+2,\cdots\}.$ Sufficient conditions are given which allow the methods applied to the finite interval case to be applied to the infinite interval case. Given these conditions, we present eigenvalue results for the infinite interval problem. We conclude by considering the scalar n$\sp{\rm th}$ order linear difference equation $\rm \sum\sbsp{i=0}{n}\ p\sb{i}(t)y(t+i)=0.$ In the constant coefficients case, we derive expressions for the Cauchy functions for equations having various combinations of characteristic roots.
Subject Area
Mathematics
Recommended Citation
Schneider, John Martin, "Green's functions, Cauchy functions and cone theoretic eigenvalue results for difference equations" (1992). ETD collection for University of Nebraska-Lincoln. AAI9314437.
https://digitalcommons.unl.edu/dissertations/AAI9314437