On a measure of noncompactness in the Holder space Ck,γ(Ω) and its application

Existence of a solution in the Holder space for a nonlinear functional integral equation

“…Theorem 1 [7] Suppose S is a nonempty, bounded, convex and closed subset of a Banach space Y , and let φ : S → S is a continuous mapping such that ν(φX) ≤ λν(X) where λ ∈ [0, 1) and X is an arbitrary nonempty subset of S. Then φ has a fixed point in the set X. Now, we recall the notion of the measure of noncompactness on C k,γ (Ω) given in [10]. To define such a measure, the author used the help of papers [2] and [4].…”

Existence of solutions for some classes of integro-differential equations on C1(R+) by applying a measure of noncompactness

“…To create a measure of noncompactness on a Banach space, we need to separate the precompact sets from the other subsets. For example, see [4,3,6,22]. If X is a Banach space, let us denote all nonempty bounded subsets of X with MX and all nonempty precompact sets of X, with NX .…”

Compactness in Lipschitz spaces and around

“…This was already known to Johnson in the real-valued setting (see [19,Theorem 3.2]) and to García-Lirola et al in a slight different context of little Lipschitz functions that are continuous with respect to a topology which needs not to come from the metric of the underlying space (see [14,Lemma 2.7]). Finally, based on the ideas of Banaś and Nalepa, in [25] Saiedinezhad used the uniform local flatness to provide a sufficient condition for relative compactness of a non-empty subset of the space C k,α (X, R) of real-valued multivariate functions that are k-times continuously differentiable and whose partial derivative of order k is H ölder continuous with exponent α; here X is a compact subset of R n . Clearly, this result cannot be necessary.…”