Theorem: Let and satisfy the said inequalities and satisfy .

In addition, let and be so smooth that , the solution of the given BVP has a continuous sixth derivative on . Then (10.8) holds with , the solution of the difference system, and , the solution of the BVP

(10.9)

where

.

Proof: We first note that the BVP (10.9) has a unique solution. Now define the net function by

using the given BVP, difference system, and the T.E. we obtain

.

Now the T.E. can be written as

(using the continuity of ).

Thus we have