The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem

summary:In this paper we prove an existence theorem for the Cauchy problem $$x^{\prime }(t) = f(t, x(t)), \quad x(0) = x_0, \quad t \in I_{\alpha } = [0, \alpha ]$$ using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function $f$ are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function $f$ satisfies some conditions expressed in terms of measures of weak noncompactness

On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given

Complexity Bounds for Ordinal-Based Termination

`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program running times. Our main tool for this are length function theorems, which provide complexity bounds on the use of well quasi orders. We illustrate how to prove such theorems in the simple yet until now untreated case of ordinals. We show how to apply this new theorem to derive complexity bounds on programs when they are proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability Problems (RP 2014, 22-24 September 2014, Oxford

A Girsanov Result through Birkhoff Integral

A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one

Effects of Interspecific Coexistence on Laying Date and Clutch Size in Two Closely Related Species of Hole-nesting Birds

Coexistence between great tits Parus major and blue tits Cyanistes caeruleus, but also other hole-nesting taxa, constitutes a classic example of species co-occurrence resulting in potential interference and exploitation competition for food and for breeding and roosting sites. However, the spatial and temporal variations in coexistence and its consequences for competition remain poorly understood. We used an extensive database on reproduction in nest boxes by great and blue tits based on 87 study plots across Europe and Northern Africa during 1957–2012 for a total of 19,075 great tit and 16,729 blue tit clutches to assess correlative evidence for a relationship between laying date and clutch size, respectively, and density consistent with effects of intraspecific and interspecific competition. In an initial set of analyses, we statistically controlled for a suite of site-specific variables. We found evidence for an effect of intraspecific competition on blue tit laying date (later laying at higher density) and clutch size (smaller clutch size at higher density), but no evidence of significant effects of intraspecific competition in great tits, nor effects of interspecific competition for either species. To further control for site-specific variation caused by a range of potentially confounding variables, we compared means and variances in laying date and clutch size of great and blue tits among three categories of difference in density between the two species. We exploited the fact that means and variances are generally positively correlated. If interspecific competition occurs, we predicted a reduction in mean and an increase in variance in clutch size in great tit and blue tit when density of heterospecifics is higher than the density of conspecifics, and for intraspecific competition, this reduction would occur when density of conspecifics is higher than the density of heterospecifics. Such comparisons of temporal patterns of means and variances revealed evidence, for both species, consistent with intraspecific competition and to a smaller extent with interspecific competition. These findings suggest that competition associated with reproductive behaviour between blue and great tits is widespread, but also varies across large spatial and temporal scales. © 2018 The Authors. Journal of Animal EcologyAcademy of Finland 26585