Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, P. O. Box 13185/768, Tehran, Iran.

In this paper, we introduce various definitions of R-duals, to be called R-duals of type I, II, which leads to a generalization of the duality principle in Banach spaces. A basic problem of interest in connection with the study of R-duals in Banach spaces is that of characterizing those R-duals which can essentially be regarded as M-basis. We give some conditions under which an R-dual sequence to be an M-basis for .

Adv Differ Equ 2021 9;2021(1):40. Epub 2021 Jan 9.

Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba R7A 6A9 Canada.

The goal of this paper is to study the uniqueness of solutions of several Hadamard-type integral equations and a related coupled system in Banach spaces. The results obtained are new and based on Babenko's approach and Banach's contraction principle. We also present several examples for illustration of the main theorems. Read More

Adv Differ Equ 2021 7;2021(1):35. Epub 2021 Jan 7.

Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain.

We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. Read More

Philos Trans A Math Phys Eng Sci 2021 Feb 4;379(2191):20190379. Epub 2021 Jan 4.

Section de mathématiques Station 8, EPFL, Lausanne, CH 1015, Switzerland.

Consider [Formula: see text] such that (, 0) = 0 for all [Formula: see text], where and are Banach spaces. Bifurcation from the line [Formula: see text] of trivial solutions is investigated in cases where (, · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for to be a bifurcation point and yield global information about the connected component of [Formula: see text] containing (, 0). Read More

J Am Stat Assoc 2020 11;115(529):307-317. Epub 2019 Apr 11.

Department of Statistical Science, School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China

Technological advances in science and engineering have led to the routine collection of large and complex data objects, where the dependence structure among those objects is often of great interest. Those complex objects (e.g, different brain subcortical structures) often reside in some Banach spaces, and hence their relationship cannot be well characterized by most of the existing measures of dependence such as correlation coefficients developed in Hilbert spaces. Read More

Banach J Math Anal 2021 19;15(1):14. Epub 2020 Oct 19.

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz n. 1, 1090 Wien, Austria.

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions. Read More