 
Summary: EVEN ORDER SELF ADJOINT TIME SCALE PROBLEMS
DOUGLAS R. ANDERSON AND JOAN HOFFACKER
Abstract. Even order self adjoint differential time scale expressions are introduced, to
gether with associated self adjoint boundary conditions; the result is established by in
duction. Several fourthorder nabladelta deltanabla examples are given for select self
adjoint boundary conditions, together with the specific corresponding Green's functions
over common time scales. One derived Green's function is shown directly to be symmetric.
Some self adjoint boundary value problems (BVPs) for second order differential equa
tions on time scales were constructed and studied earlier in [1] by making use of both delta
and nabla derivatives. Next, certain BVPs for higher order equations on time scales were
investigated in [2, 3, 4] where, however, the considered BVPs turned out, in general, nonself
adjoint because their Green's function were found nonsymmetric. Therefore it remained
unclear as to how to place the successive delta and nabla derivatives for higher order to
get selfadjoint differential expressions that can yield symmetric Green's functions. Gu
seinov [5] offered a possible resolution of this problem; in this paper we offer a direct proof
by mathematical induction of his conjecture, in the case where we stack nabla derivatives
and one delta derivative on the inside first, followed by stacked deltas and one nabla on
the outside (see below). In a subsequent, closely related sequel [6], a more abstract but
comprehensive approach is used to establish self adjoint deltanabla equations and bound
ary conditions, using quasiderivative notation to consolidate (though unfortunately also
