Space of Functions with Growths Tempered by a Modulus of Continuity
The principal aim of this chapter is to consider the function space consisting of functions defined on a compact metric space with growths tempered by a given modulus of continuity and its connection with the measures of noncompactness. The chapter is ins
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Space of Functions with Growths Tempered by a Modulus of Continuity I.J. Cabrera, J. Harjani, B. López and K.B. Sadarangani
Abstract The principal aim of this chapter is to consider the function space consisting of functions defined on a compact metric space with growths tempered by a given modulus of continuity and its connection with the measures of noncompactness. The chapter is inspired in the papers [1, 2].
4.1 Space of Functions with Growths Tempered by a Modulus of Continuity The content of this section appears in [1]. By (X, d), we denote a compact metric space and put R+ = [0, ∞). Definition 4.1 A function w : R+ → R+ is said to be a modulus of continuity if w(0) = 0, w(ε) > 0 for ε > 0 and w is nondecreasing on R+ . Examples of modulus of continuity are w(ε) = ε, w(ε) = ln(1 + ε), w(ε) = εα with α > 0, w(ε) = eε − 1 and w(ε) = ε/(1 + ε). In the sequel, we assume that w is a modulus of continuity which is continuous at the point ε = 0, i.e., w(ε) → 0 as ε → 0. By C(X ), we denote the space of the functions x : X → R being continuous on X . In C(X ), we consider the classical supremum norm · ∞ given by x∞ = sup{|x(u)| : u ∈ X }. I.J. Cabrera (B) · J. Harjani · B. López · K.B. Sadarangani Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain e-mail: [email protected] J. Harjani e-mail: [email protected] B. López e-mail: [email protected] K.B. Sadarangani e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 J. Bana´s et al. (eds.), Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, DOI 10.1007/978-981-10-3722-1_4
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It is well known that (C(X ), · ∞ ) is a Banach space. In the sequel, we take as w = w(ε) a fixed modulus of continuity which is continuous at the point ε = 0. By Cw (X ), we denote the space of all real functions defined on X such that its growth is controlled by the modulus of continuity w, i.e., x : X → R belongs to Cw (X ) if there exists a constant K x > 0 such that for any u, v ∈ X , |x(u) − x(v)| ≤ K x w(d(u, v)). It is clear that x ∈ Cw (X ) and only if sup
|x(u) − x(v)| : u, v ∈ X, u = v < ∞. w(d(u, v))
If we take u 0 ∈ X an arbitrary fixed element of X then the quantity defined by xw = |x(u 0 )| + sup
|x(u) − x(v)| : u, v ∈ X, u = v , w(d(u, v))
for x ∈ Cw (X ) defines a norm and (C(X ), · w ) is a Banach space. Notice that if w is continuous at ε = 0 then Cw (X ) ⊂ C(X ). Indeed, for (u n ) ⊂ X and u ∈ X with u n → u and x ∈ Cw (X ), since |x(u n ) − x(u)| ≤ K x w(d(u n , u)) from u n → u, and the continuity of w at ε = 0 it follows that x(u n ) → x(u) and, consequently, x ∈ C(X ). Remark 4.1 If w is not continuous at ε = 0 then the inclusion Cw (X ) ⊂ C(X ) can be false. For example, if we take as w : R+ → R+ the function defined by w(ε) =
0, 1,
ε = 0, ε > 0,
then w is a modulus of continuity which is not continuous at ε = 0. It is easily seen that in this case Cw (X ) is the space of the
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