Сompactness of one class of fractional integrating operator with variable upper limit

The paper establishes a characterization of the compactness for fractional operators of a general class, including the Riemann-Liouville, Hadamard and Erdelyi-Kober operators. The paper considers an integral fractional integration operator of Hardy type with nonnegative kernels and a variable limit of integration (a function as the upper limit of integration) and under certain conditions on the kernel, a criterion of the compactness in weighted Lebesgue spaces is obtained for this operator, when the parameters of the spaces satisfy the conditions Moreover, more general results are obtained for the weighted differential inequality of Hardy type on the set of locally absolutely continuous functions that vanish and infinity at the ends of the interval, covering the previously known results, and more precise estimates for the best constant are given. The localization method, Schauder’s theorem, the Kantorovich test, and the theorem on the uniform limit of compact operators were used in the proof of the main theorem. The obtained results of the study the compactness of fractional integration operators can be used in the estimation of solutions of differential equations that model various processes in mathematics. In particular, these results yield new results in the theory of Hardy-type inequalities.

Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(0,4) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index α∈(0,4); and no restriction on the rate of ρn is necessary.

(F, L) Contractions on Complete Weak Partial Metric Spaces for a Commuting Family of Self Mappings

The aim of this paper is to clarify the choice of the self map T : X → X in Kaya et als (F,L) weak contractions by choosing a family Tn, n ∈ N of (F,L) contractions. Motivated by the fact that the uniform limit T of the family of self maps is a better approximation, we are guaranteed the choice of the self map. By this, the choice of T is no longer arbitrary. Again, for any nite family T1, T2, T3, · · · , TN of (F,L) contractions their composition is an (F,L) contraction. This concept generalizes and improves on several results especially Theorems 3.1 and 3.2 of [10].

Entire functions with prescribed singular values

We introduce a new class of entire functions [Formula: see text] which consists of all [Formula: see text] for which there exists a sequence [Formula: see text] and a sequence [Formula: see text] satisfying [Formula: see text] for all [Formula: see text]. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set [Formula: see text] containing the origin and at least one more point is the set of singular values of some locally univalent function in [Formula: see text], hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set [Formula: see text] is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class [Formula: see text] contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

On finite sums of periodic functions

It is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

Quasi-uniform and uniform convergence of Riemann and Riemann-type integrable functions with values in a Banach space

In this article, we study quasi-uniform and uniform convergence of nets and sequences of different types of functions defined on a topological space, in particular, on a closed bounded interval of R, with values in a metric space and in some cases in a Banach space. We show that boundedness and continuity are inherited to the quasi-uniform limit, and integrability is inherited to the uniform limit of a net of functions. Given a sequence of functions, we construct functions with values in a sequence space and consequently we infer some important properties of such functions. Finally, we study convergence of partially equi-regulated* nets of functions which is shown to be a generalized notion of exhaustiveness.

On Partitions and Periodic Sequences

In the first part we associate a periodic sequence with a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved.