A functional interpretation for nonstandard arithmetic

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Goedel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of extensional Heyting and Peano arithmetic in all finite types, strengthening earlier results by Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. To conclude the paper, we will point out some open problems and directions for future research and mention some initial results on saturation principles

Rate of convergence under weak contractiveness conditions

We introduce a new class of selfmaps T of metric spaces, which generalizes the weakly Zamfirescu maps (and therefore weakly contraction maps, weakly Kannan maps, weakly Chatterjea maps and quasi-contraction maps with constant h < 1 / 2). We give an explicit Cauchy rate for the Picard iteration sequences {T nx0}n∈N for this type of maps, and show that if the space is complete, then all Picard iteration sequences converge to the unique fixed point of T. Our Cauchy rate depends on the space (X, d), the map T, and the starting point x0 ∈ X only through an upper bound b ≥ d(x0, T x0) and certain moduli θ, µ for the map, but is otherwise fully uniform. As a step on the way to proving our fixed point result we also calculate a modulus of uniqueness for this type of maps.Junta de AndalucíaResearch Council of NorwayDirección General de Enseñanza Superio

A rate of convergence for asymptotic contractions

AbstractIn [P. Gerhardy, A quantitative version of Kirk's fixed point theorem for asymptotic contractions, J. Math. Anal. Appl. 316 (2006) 339–345], P. Gerhardy gives a quantitative version of Kirk's fixed point theorem for asymptotic contractions. This involves modifying the definition of an asymptotic contraction, subsuming the old definition under the new one, and giving a bound, expressed in the relevant moduli and a bound on the Picard iteration sequence, on how far one must go in the iteration sequence to at least once get close to the fixed point. However, since the convergence to the fixed point needs not be monotone, this theorem does not provide a full rate of convergence. We here give an explicit rate of convergence for the iteration sequence, expressed in the relevant moduli and a bound on the sequence. We furthermore give a characterization of asymptotic contractions on bounded, complete metric spaces, showing that they are exactly the mappings for which every Picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point

On Rates of Convergence in Metric Fixed Point Theory

This thesis investigates some effective and quantitative aspects of metric fixed point theory in the light of methods from proof theory. The thesis consists of contributions to the program of proof mining, as developed by Kohlenbach and various collaborators since the early 1990s (but with roots back to Kreisel's program "unwinding of proofs" from the 1950s). The contributions involve both case studies - studying given prima facie ineffective proofs of certain fixed point theorems to extract "hidden" effective information like explicit bounds and rates of convergence for iteration sequences, and also developing further the use of the logical machinery involved. The main theoretical tools involve Gödel's functional ("Dialectica") interpretation combined with negative translation and a variant of Howard's majorizability relation - and specically the logical metatheorems of Kohlenbach and Gerhardy, where the reach of these techniques is extended to formal systems for analysis with various abstract spaces added as new ground types. The main contributions of the thesis are twofold: (1) We construct explicit and effective full rates of convergence for the Picard iteration sequences for two classes of selfmaps on metric spaces. One of these are Kirk's asymptotic contractions, and as a byproduct of the logical analysis we obtain a string of results concerning this class of mappings, including a characterization on nonempty, bounded, complete metric spaces as exactly the mappings for which there exists a point to which all Picard iteration sequences converge with a rate of convergence which is uniform in the starting point. This shows that in the setting of bounded metric spaces the asymptotic contractions in the sense of Kirk in some sense are the most general mappings which still exhibit convergence of the Picard iteration sequences of "Banach type" - to the same point and with strong uniformity with respect to the starting point. The other class of mappings for which we construct explicit rates of convergence are the so-called uniformly continuous uniformly generalized p-contractive mappings. Logical analysis of the concepts involved - using monotone functional interpretation - allows us to develop an extension of a related fixed point theorem from the case where the space is compact to arbitrary metric spaces. This is possible because monotone functional interpretation automatically leads us to consider the "right" uniform version of the corresponding contractive type condition - whereas in the proof of the original theorem the compactness of the space "secretly" upgrades the generalized contractive condition in question to this uniform version. Also in this case we were able to give an effective and highly uniform rate of convergence for the Picard iteration sequences, and by the uniformity features of the resulting rate of convergence it follows that the mappings under consideration are asymptotic contractions in the sense of Kirk. (2) We develop a method for finding, under general conditions, explicit and highly uniform rates of convergence for the Picard iteration sequences for selfmaps on bounded metric spaces from ineffective proofs of convergence to a unique fixed point. We are able to extract full rates of convergence by extending the use of a logical metatheorem due to Kohlenbach. Our novel method provides an explanation in logical terms for the fact that we in the case studies mentioned above could find such explicit rates of convergence. This amounts, loosely speaking, to general conditions under which we in this specic setting can transform a $\forall\exists\forall$-sentence into a $\forall\exists$-sentence via an argument involving product spaces. This reduction in logical complexity allows us to use the existing machinery to extract quantitative bounds of the sort we need