Modular Locally Constant Mappings in Vector Ultrametric Spaces

and Applied Analysis 3 Example 1.2. The vector spaceC1 0, 1 consisting of all continuously differentiable real-valued functions on 0, 1 equipped with the modular ρ defined by ρ x max t∈ 0,1 |x t | max t∈ 0,1 ∣ ∣x′ t ∣ ∣, ( x ∈ C1 0, 1 ) 1.7 constitutes a complete modular space. The subset P { x ∈ C1 0, 1 : x t ≥ 0, ∀t ∈ 0, 1 } 1.8 is a unital cone in C1 0, 1 with unit 1. The cone P is not normal since, for example, x t t 1, for n ≥ 1 does not imply that ρ x ≤ ρ 1 . Throughout this note, we suppose that P is a cone in complete modular spaceAρ, and is the partial order induced by P. Definition 1.3. A vector ultrametric on a nonempty set X is a mapping d : X × X → Aρ satisfying the conditions: CUM1 d x, y 0 for all x, y ∈ X and d x, y 0 if and only if x y; CUM2 d x, y d y, x for all x, y ∈ X; CUM3 If d x, z p and d y, z p, then d x, y p, for any x, y, z ∈ X, and p ∈ P. Then the triple X, d,P is called a vector ultrametric space. If P is unital and normal, then X, d,P is called a unital-normal vector ultrametric space. For unital-normal vector ultrametric space X, d,P , since d ( x, y ) ρdx, ye, dy, z ρdy, ze, 1.9 from CUM3 we get d x, z maxρdx, y, ρdy, ze, 1.10 and therefore ρ d x, z ≤ maxρdx, y, ρdy, z. 1.11 Let X, d,P be a unital-normal vector ultrametric space. If x ∈ X and p ∈ P \ {0}, the ball B x, p centered at x with radius p is defined as B ( x, p ) : { y ∈ X : ρdx, y ≤ ρp. 1.12 The unital-normal vector ultrametric space X, d,P is called spherically complete if every chain of balls with respect to inclusion has a nonempty intersection. 4 Abstract and Applied Analysis The following lemma may be easily obtained. Lemma 1.4. Let X, d,P be a unital-normal vector ultrametric space. 1 If a, b ∈ X, 0 p and b ∈ B a, p , then B a, p B b, p . 2 If a, b ∈ X, 0 ≺ p q, then either B a, p B b, q ∅ or B a, p ⊆ B b, q . Definition 1.5. Let X, d,P be a unital-normal vector ultrametric space. A mapping f : X → P\{0} is said to bemodular locally constant provided that for any x ∈ X and any y ∈ B x, f x one has ρ f x ρ f y . 2. Main Theorem Theorem 2.1. Let X, d,P be a spherically complete unital-normal vector ultrametric space and T : X → X be a mapping such that for every x, y ∈ X, x / y, either ρ ( d ( Tx, Ty )) < max { ρ d x, Tx , ρ ( d ( y, Ty ))} 2.1