Existence of Solutions for Equations Involving Iterated Functional Series

This section collects the standard terminology and results used in the sequel (see [5]). Let I = [a,b] be an interval of real numbers. C1(I ,I), the set of all continuously differentiable functions from I into I , is a closed subset of the Banach Space C1(I ,R) of all continuously differentiable functions from I into R with the norm ‖ · ‖c1 defined by ‖φ‖c1 = ‖φ‖c0 +‖φ′‖c0 , φ∈ C1(I ,R) where ‖φ‖c0 =maxx∈I |φ(x)| and φ′ is the derivative of φ. Following Zhang [5], for given constants M ≥ 0, M∗ ≥ 0, and δ > 0, we define the families of functions 1 ( I ,M,M∗ )= {φ∈ C(I ,I) : φ(a)= a, φ(b)= b, 0≤ φ′(x)≤M ∀x ∈ I , ∣∣φ′(x1)−φ′(x2)∣∣≤M∗∣∣x1− x2∣∣ ∀x1,x2 ∈ I} (2.1)