Definition. Let X be a metric space with metric d. (a) A collection {G α } α∈A of open sets is called an open cover of X if every x ∈ X belongs to at least one of the G α , α ∈ A. An open cover is finite if the index set A is finite. (b) X is compact if every open cover of X contains a finite subcover. Definition. Let X be a metric space with metric d and let A ⊂ X. We say that A is a compact s...

The aim of this paper is to define and emphasize a strong form D-compact sets in generalized metricspaces, namely D-precompact sets. Also with other sets, we shall study the relationships. Furthermore, wegive notions sequentially

For n ≥ 3, let Ω ⊂ R be a bounded smooth domain and N ⊂ R be a compact smooth Riemannian submanifold without boundary. Suppose that {un} ⊂ W (Ω, N) are weak solutions to the perturbed n-harmonic map equation (1.2), satisfying (1.3), and uk → u weakly in W (Ω, N). Then u is an n-harmonic map. In particular, the space of n-harmonic maps is sequentially compact for the weak-W 1,n topology. §

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to...

A subset of a locally convex space E is called (relatively) sequentially complete if every Cauchy sequence $$\left\{ x_{n}\right\} _{n=1}^{\infty }$$ in contained converges to point $$x\in A$$ (a E$$ ). Asanov and Velichko proved that X countably compact, functionally bounded set $$C_{p}\left( X\right) $$ relatively Baturov showed Lindelöf $$\Sigma -space, each compact (so bounded) C_{p}\left( ...

This paper presents new fixed point results for weakly sequentially upper semicontinuous maps defined on locally convex Hausdorff topological spaces which are angelic when furnished with the weak topology. Moreover, we establish an applicable Leray-Schauder alternative (Theorem 2.12) for a certain subclass of these maps. Our alternative combines the advantages of the strong topology (i.e., the ...