Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Earlier work showed all of these implications to be strict. Here we re-prove one of the strictness results, using very different arguments. The technique used is a mixture of realizability, forcing in the gu...

Based on the G-H-KKM selections, some nonempty intersection theorems and their applications to minimax inequalities are presented.

Introduction. This paper deals with locales and their spaces of points in intuitionistic analysis or, if you like, in (Grothendieck) toposes. One of the important aspects of the problem whether a certain locale has enough points is that it is directly related to the (constructive) completeness of a geometric theory. A useful exposition of this relationship may be found in [1], and we will assum...

In this paper, we prove that if G is 3-connected noncomplete graph of order n satisfying min{max{d(u), d(v)} : d(u, v) = 2} = μ, then for each edge e, G has a cycle containing e of length at least min{n, 2μ}, unless G is a spanning subgraph of Kμ +K n−μ or K3 + (lKμ−2 ∪Ks), where n = l(μ− 2) + s+ 3, 1 ≤ s ≤ μ− 2.

In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer’s Fan Theorem for detachable bars on the binary fan is equivalent in Bishop’s sense to various principles including a version WKL! of Weak König’s Lemma with a strong effective uniqueness hypothesis. Schwichtenberg [9] proved the equivalence directly and formalized his proof in M...

The bipotential theory is based on an extension of Fenchel’s inequality (see section 1 and example 1 in section 2). Despite several powerful applications (frictional contact [6], non-associated Drucker-Prager model [1], or Lemaitre plastic ductile damage law [2], to cite a few), the bipotentials don’t have yet a complete mathematical treatment. This is a second paper on the mathematics of the b...

In this paper, the solvability of a class of strong vector quasi-equilibrium problems(for short, SVQEP) are studied in real topological vector space. By using the Kakutani-Fan-Glicksberg fixed point theorem, some existence theorems for SVQEP are obtained without any monotonicity assumption.

We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees. This is then applied to prove several separation results involving variants of the Fan Theorem.