ABSTRACT
We are given a seminormed linear space (V, ‖·‖) on which a sequence of operators T˜ =T1,T2, . . . is defined. We may take norms to be a special case of seminorms, the theory of this chapter generally applying in the samemanner to each. We usually expect each operator to have a common fixed point V∗ =TkV∗, or an equivalence class of fixed points. At the very least, interest is in evaluating or approximating some fixed point V∗ using the iteration algorithm:
V0 = v0 Vk = TkVk−1, k=1, 2, . . . , (10.1)
for a given initial solution v0 ∈V. We always assume ‖v0‖<∞ and ‖V∗‖<∞. The intention is that Vk ∈V for all k≥1, and that the sequence converges to V∗ in the given seminorm, that is, limk ‖Vk − V∗‖=0. In many cases, the algorithm will be homogenous in the sense that Tk =T, so that Vk =Tkv0, but it turns out that results obtainable for homogenous algorithms are extendible to the nonhomgenous case with little loss of generality, and much gain in applicability.
