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Abstract
We are concerned with solutions to the difference equation Py(t - k) = f(t,y(t)) where
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Here, k and n are fixed integers with 0 (LESSTHEQ) k < n, and the coefficients (alpha)(,i)(t) are defined on I + k where I is an interval of integers of the form a,b or a,(INFIN)). Py(t - k) = 0 is said to be disconjugate on J provided no nontrivial solution has n zeros on J. We will be interested in right and left (j,n - j)-disconjugacy as defined by Peterson. We obtain a partial factorization for P if P is right (j,n - j)-disconjugate and results relating right and left (j,n-j)-disconjugacy are given. An adjoint to Py(t) = 0 and disconjugacy properties for the adjoint equation are discussed.
Next, the equation Ly(t) + p(t)y(t) = 0 is considered, where L is a disconjugate operator. This work is motivated by results of Elias for the corresponding differential equation. A theorem which bounds the number of certain types of zeros for solutions on an interval is obtained. Using this, sign conditions on p(t) are determined that will guarantee that Ly(t) + p(t)y(t) = 0 is right (k,n - k)-disconjugate, and a uniqueness theorem for solutions to certain types of bound- ary value problems is given. Several results give some properties of solutions to this difference equation. Furthermore, a classification of solutions is obtained based on their behavior in a neighborhood of infinity.
Finally, we consider the nonlinear equation Py(t - k) = f(t,y(t)). A comparison theorem for solutions of related linear inequalities is obtained. This leads to some disconjugacy results. The Brouwer invariance of domain theorem is used to establish some results on the (j,n - j)-problem and its related variational equation. Then we show that under suitable conditions on f and certain related linear equations that the (n - 2,2)-boundary value problem has a unique solution.
