ABSTRACT
The heart of this work utilizes continuous dependence of solutions of (18.1) to find differences with respect boundary points and derivatives with respect to boundary values. After imposing a few disconjugacy-type conditions on (18.1), we will see that derivatives of a solution, w(t), to (18.1) relate to the following linear nth order difference equation
z(t+ n) =
∂f
∂di (t, w(t), . . . , w(t+ n− 1))z(t+ i− 1) (18.3)
called the variational equation along w(t). The motivation for the research conducted in this work on the relationship between a
solution to a differential or difference equation and the associated variational equation can trace its origin to 1964 when Hartman proved a theorem he attributed to Peano about initial value problems for differential equations [155]. Since then, Henderson and several others have extended and redefined these results in various ways including boundary value problems for both differential and difference equations. For differential equations results, we point the reader to [110, 159, 161, 207]. For results on difference equations, we reference [45,92,93,156,239]. Also, interest in multipoint and nonlocal boundary value problems has grown significantly in recent years as can be seen in [12, 30, 113,186] which is why we have incorporated nonlocality into our boundary conditions. Also, related to the results of this chapter is the recent paper by Hopkins et al. [172].
