Volume 29, Issue 1

The uniform closure of the neural networks with Heaviside activation is specified.

When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study the possible behaviors of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the infinite-dimensional inverse problem – the underlying motivations being the stability or instability of infinite-dimensional Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an infinite-dimensional inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem.

This article focuses on the creation of an existence theorem for a fully nonlinear Hadamard fractional boundary value problem subject to special three-point boundary conditions. By making use of the coincidence degree theory, it is proved that our governing problem makes resonance, that is, the linear part of the differential operator is non-invertible (equally, the corresponding linear problem has at least one nontrivial solution). Constructing some hypotheses on the linear part of the differential operator, nonlinearities and boundary conditions, we give an existence criterion for at least one solution of the fractional-order resonant boundary value problem under study. At the end, a numerical example is presented to illustrate the obtained theoretical results.

In this paper, we discuss the existence and uniqueness of a positive solution for a p -Laplacian differential equation containing left and right Caputo derivatives. By the help of the Guo–Krasnoselskii theorem, we prove the existence of at least one positive solution. The existence of a unique positive solution is established under the assumption that the corresponding operator is α-concave and increasing. Numerical examples are given to check the obtained results.

We apply the averaging theory of first and second order to a class of generalized polynomial Kukles differential systems, which can bifurcate from the periodic orbits of the linear center x ˙ = y {\dot{x}=y} , y ˙ = - x {\dot{y}=-x} , in order to study the maximum number of limit cycles of these systems.

We are concerned with the stability of a 1-D coupled Rayleigh beam-string transmission system. We obtain the polynomial decay rate t - 1 {t^{-1}} or the exponential decay rate for the given transmission system whether the frictional damping is only effective in the beam part or the string part, respectively. This paper generalizes the recent result in [Y.-F. Li, Z.-J. Han and G.-Q. Xu, Explicit decay rate for coupled string-beam system with localized frictional damping, Appl. Math. Lett. 78 2018, 51–58]. The main ingredient of the proof is some careful analysis for the Rayleigh beam and string transmission system.

In this article, we use a particular case of convolution as an operator to discuss a number of problems concerning multiplier results between function spaces such as Hardy and B p {B^{p}} -spaces. As a consequence, we extend certain well-known results on fractional derivatives and fractional integrals. Also, we find condition on the parameters b , c {b,c} such that 𝒫 b , c {{\mathcal{P}}^{b,c}} in B p {B^{p}} .

In this paper we prove the Belitskii–Lyubich conjecture for triangular and gradient mappings. These results resemble those obtained for the discrete Markus–Yamabe conjecture. However, the proofs are quite different, which sheds new light on the subject. We complete the picture by showing that the general versions of the two conjectures can be turned into theorems at little cost, simply by relaxing their spectral condition.

Complex uncertain variables are measurable functions from an uncertain space to the set of complex numbers and are used to model complex uncertain quantities. In this paper, we introduce the deferred-statistically convergence concepts of complex uncertain sequences, deferred-statistically convergence almost surely, deferred-statistically convergence in measure, deferred-statistically convergence in mean and some relationship among them are discussed.

We prove via a Galerkin approximation the time local existence of regular solutions to a Landau–Lifshitz–Bloch equation with applied current in a bounded domain. The uniqueness of the solution is also established. Moreover, we show the global in time existence of a regular solution in dimension two.

Theorems which are converse to the Ohlin lemma for convex and strongly convex functions are proved. New proofs of probabilistic characterizations of convex and strongly convex functions are presented.

In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces. We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and the Mönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 2008, 3, 536–546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.

We provide new proofs of two intrinsic properties of prox-regular sets in Hilbert spaces.

In this paper the right upper semicontinuity at p = 1 {p=1} and continuity at p = ∞ {p=\infty} of the set-valued map p → B Ω , 𝒳 , p ⁢ ( r ) {p\rightarrow B_{\Omega,\mathcal{X},p}(r)} , p ∈ [ 1 , ∞ ] {p\in[1,\infty]} , are studied where B Ω , 𝒳 , p ⁢ ( r ) {B_{\Omega,\mathcal{X},p}(r)} is the closed ball of the space L p ⁢ ( Ω , Σ , μ ; 𝒳 ) {L_{p}(\Omega,\Sigma,\mu;\mathcal{X})} centered at the origin with radius r , ( Ω , Σ , μ ) {(\Omega,\Sigma,\mu)} is a finite and positive measure space, 𝒳 {\mathcal{X}} is a separable Banach space. It is proved that the considered set-valued map is right upper semicontinuous at p = 1 {p=1} and continuous at p = ∞ {p=\infty} . An application of the obtained results to the set of integrable outputs of the input-output system described by the Urysohn-type integral operator is discussed.

We establish L 𝔮 {L^{\mathfrak{q}}} convergence for Hamiltonian Monte Carlo (HMC) algorithms. More specifically, under mild conditions for the associated Hamiltonian motion, we show that the outputs of the algorithms converge (strongly for 2 ≤ 𝔮 < ∞ {2\leq\mathfrak{q}<\infty} and weakly for 1 < 𝔮 < 2 {1<\mathfrak{q}<2} ) to the desired target distribution. In addition, we establish a general convergence rate for an L 𝔮 {L^{\mathfrak{q}}} convergence given a convergence rate at a specific q * {q^{*}} , and apply this result to conclude geometric convergence in the Euclidean space for HMC with uniformly strongly logarithmic concave target and auxiliary distributions. We also present the results of experiments to illustrate convergence in L 𝔮 {L^{\mathfrak{q}}} .

The main purpose of this study is to determine the radii of starlikeness and convexity of the generalized k -Bessel functions for three different kinds of normalization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre–Pólya class plays a significant role in this paper. Moreover, the interlacing properties of the zeros of the k -Bessel function and its derivative is also useful in the proof of the main results. By making use of the Euler–Rayleigh inequalities for the real zeros of the generalized k -Bessel function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero.

The purpose of this paper is to introduce an iterative algorithm for approximating the solution of the split equality monotone variational inclusion problem (SEMVIP) for monotone operators, which is also a solution of the split equality fixed point problem (SEFPP) for strictly pseudocontractive maps in real Hilbert spaces. We establish the strong convergence of the sequence generated by our iterative algorithm. Our result complements and extends some related results in literature.