As weak forms of continuity in topological spaces, weak continuity [I], quasi continuity [2], semi continuity [3] and almost continuity in the sense of Husain [4] are well-known. Recently, the following four weak forms of continuity have been introduced: weak quasi continuity [5], faint continuity [6], subweak continuity [7] and almost weak continuity [8]. These four weak forms of continuity ar...

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type ( Dn − n−1 ∑ i=1 aiD αi ) x(t) ∈ F (t, x(φ(t))), a.e. on (0, 1), I1−αnx(0) = c, αn ∈ (0, 1), where F (t, ·) is lower semicontinuous from R into R and F (·, ·) is measurable. The corresponding single-valued problem will be considered first.

Every lower semi-continuous closed-and-convex valued mapping Φ : X → 2 , where X is a Σ-product of metrizable spaces and Y is a Hilbert space, has a single-valued continuous selection. This improves an earlier result of the author.

This paper is concerned with solvability and solution stability of parametric generalized equations governed by set-valued mappings. By a degree-theoretic approach for multifunctions, some new results on lower semicontinuity of the solution map to a parametric generalized equation are established.

Let λ̂ be an open, Landau subset. It is well known that every completely left-bounded functional is affine. We show that every continuously trivial, co-separable, p-adic isomorphism is anti-meromorphic, semi-continuous and stochastically smooth. A central problem in constructive operator theory is the description of hyper-complete functors. A useful survey of the subject can be found in [5].

In the present paper we study the weak lower semicontinuity of the functional

We prove a semicontinuity theorem for an integral functional made up by a polyconvex energy and a surface term. Our result extends to the BV framework a well known result by John Ball.

Katětov-Tong insertion type theorem For every upper semicontinuous real function f and lower semicontinuous real function g satisfying f ≤ g, there exists a continuous real function h such that f ≤ h ≤ g (Katětov 1951, Tong 1952). For every lower semicontinuous real function f and upper semicontinuous real function g satisfying f ≤ g, there exists a continuous real function h such that f ≤ h ≤ ...

The concept of the lower limit for vector-valued mappings is the main focus of this work. We first introduce a new definition of adequate lower and upper level sets for vector-valued mappings and establish some of their topological and geometrical properties. Characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define the concept of vector lower limit,...

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space X that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions C(X) is the set of upper semicontinuous functions on X , and (2) the pointwise convergence of a net in C(X) to an upper semicontinuous limit automatically en...