We show that the Arzelà–Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly’s theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.

In this paper, we investigate a class of semilinear fractional evolution equations with nonlocal initial conditions given by (1) ⎧⎨ ⎩ dqu(t) dtq = Au(t)+(Fu)(t), t ∈ I, u(0)+g(u) = u0, where 0 < q< 1 , I is a compact interval. Sufficient conditions for the existence of mild solutions for the equation (1) are derived. The main tools include Laplace transform, Arzela-Ascoli’s Theorem, Schauder’s ...

in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

*Correspondence: [email protected] School of Science, Guilin University of Technology, Guilin, Guangxi 541004, P.R. China Abstract In this paper, we build a multispecies predator-prey model with mutual interference and time delays. By means of the comparison theorem, Ascoli theorem and Lebesgue dominated convergence theorem, we establish the sufficient conditions of permanence and investigate ...

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are ...

Abstract We establish inter alia a compactness criterion in metric spaces involving sequence of completely continuous mappings, which is continuously convergent, the sense H. Hahn, to identity mapping. For Banach spaces, linear version that result coincides with theorem due Mazur. also present new proof Ascoli–Arzelà theorem, we use above applied Bernstein operators.