We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low type-level witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence of a continuous inverse to a monotonically increasi...

A mapping f : X → X from a set X to itself has a fixed point if there is an x ∈ X such that f(x) = x. The simplest fixed point theorem is that a continuous function f : [a, b] → [a, b] has at least one fixed point. This is a consequence of the intermediate value theorem from calculus, as follows. Since f(a) ≥ a and f(b) ≤ b, we have f(b) − b ≤ 0 ≤ f(a) − a. The difference f(x)−x is continuous, ...

A procedure is described that gives values to set variables in automatic theorem proving. The result is that a theorem is thereby reduced to first order logic, which is often much easier to prove. This procedure handles a part of higher order logic, a small but important part. It is not as general as the methods of Huet, Andrews, Pietrzykowski, and Haynes and Henschen, but it seems to be much f...

It is well known that different kinds of expressions of modulus of convexity are essentially based on two geometrical propositions. For one of the propositions we first present a new proof by the Hahn–Banach theorem and intermediate value theorem, then give some corollaries to it which lead to some new expressions of modulus of convexity.

We prove an intermediate value theorem of an arithmetical flavor, involving the consecutive averages {x̄n}n≥1 of sequences with terms in a given finite set {a1, ..., ar}. For every such set we completely characterize the numbers Π (”intermediate values”) with the property that the consecutive averages {x̄n} of every sequence {xn}n≥1 with terms in {a1, ..., ar} cannot increase from a value x̄k < Π ...

this paper studies a fractional boundary value problem of nonlineardifferential equations of arbitrary orders. new existence and uniquenessresults are established using banach contraction principle. other existenceresults are obtained using schaefer and krasnoselskii fixed point theorems.in order to clarify our results, some illustrative examples are alsopresented.