In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

Abstract. We develop continuous first order logic, a variant of the logic described in [CK66]. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson’s logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.

Abstract. We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin [Poi06]. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are ℵ0-categorical and ℵ0-stable. In particular they are stable.

Abstract. We develop continuous first order logic, a variant of the logic described in [CK66]. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson’s logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.

Abstract: In this paper, we prove common fixed point theorems of Gregus type for three discontinuous and weak compatible mappings in convex metric spaces. We improve, extend and generalize some well known results by many authors Key words: common fixed point, compatible mapping, convex metric space, W-affine mapping. 1.