chapter  11
49 Pages

Chapter 11

Boundary Value Problems Higher Order Difference Equations

Results stated in Chapter 10 play a fundamental role in the study of various higher order boundary value problems induding those discussed in Section 1.5. Using these results we provide easily verifiable sets of mx:cssary and suffident <:onditions so that each of these boundary value problems has at lea.o;;t one solution. Sufficient conditions ensuring the uniqueness of these solutions arc also induded. This is followed by the mnvcrgenee of the constructive methods: Picard's method, the approximate Picard's method, quasilincarization, and the approximate quasilinearization. The results obtained herein arc more explicit than those discussed in Chapter 9 for the systems of difference equations. The monotonic <:onvcrgcnee of the Picard's iterative method is analyzed in Section 11.4. Next, we shall show that the initial value methods discussed in Chapters 8 and 9 for constructing the solutions of boundary value problems can also be used to prove the existence and uniqueness theorems for the higher order discrete boundary value problems. In Section 10.9, we have noticed that the uniqueness of the solutions of the linear boundary value problems implies the existence of the solutions. The argument employed in proving this assertion is algebraic and is based on the linear structure of the fundamental system of solutions of the difference equations and the linearity of the boundary mnditions. In Section 11 .6 sufficient conditions which guarantee this property for the nonlinear boundary value problems arc provided.