Asymptotic Behaviour of the Iterates of Positive Linear Operators

and Applied Analysis 3 then lim k Uf Lf, ∀f ∈ C X . 3.2 Moreover, if X is a compact metric space, then the convergence is uniform. Proof. Let f ∈ C X . The case when Lf f is trivial. Indeed, in this case, since Lf ∈ V andU preserves the elements of V, we haveUf ULf Lf , and hence Uf Lf , k 1, 2, . . . . If Lf / f , for sufficiently small ε > 0, the inverse image of the open set −ε, ε under the continuous function Lf − f is an open set G, YL ⊆ G/ X. It follows that ∣Lf − f∣ < ε, on G. 3.3 Since X is compact andG is open, it follows thatX \G ⊆ X \YL is a nonempty compact subset of X, and we obtain mε : inf x∈X\G ( Uφ x − φ x ) > 0. 3.4 Consequently, the following decisive inequality ∣ ∣Lf − f∣ < ε ∥f − Lf∥ mε ( Uφ − φ 3.5 is satisfied. By applying the positive operatorU to 3.5 , we get ∣∣ ∣Lf −Uf ∣∣ ∣ < ε ∥f − Lf∥ mε ∣∣ ∣U 1φ −Uφ ∣∣ ∣. 3.6 Since Uφ ≥ φ, we obtain φ ≤ Uφ ≤ U 1φ ≤ ∥φ∥, k 1, 2, . . . . 3.7 The sequence Uφ k≥1 is monotone and bounded. It follows that it is pointwise convergent. Since ε was chosen arbitrarily, by using 3.6 we deduce that Uf pointwise −−−−−−−→ Lf . In the particular case when X is a compact metric space, since Uφ pointwise −−−−−−−→ Lφ ∈ C X , 3.8 by Dini’s Theorem, we obtain that Uφ uniformly −−−−−−−−→ Lφ. From the inequalities ∥ ∥∥Lf −Ukf ∥ ∥∥ < ε ∥ ∥f − Lf∥ mε ∥ ∥∥Uk 1φ −Ukφ ∥ ∥∥, 3.9 we deduce that Uf uniformly −−−−−−−−→ Lf . 4 Abstract and Applied Analysis In the following we give more information on the limit operator L. Theorem 3.2. The limit interpolation operator L is unique, positive, and satisfies the equalities