Sahand Communications in Mathematical Analysis
Volume:20 Issue: 1, Winter 2023

A new class of retro Banach frames called retro bi-Banach frame has been introduced and studied with illustrative examples. Relationships of a retro bi-Banach frame with various existing classes of Banach frame are presented. In the sequel, we deal with characterizations of the near-exact retro Banach frame and discuss the invariance of near-exact retro Banach frames under block perturbation. Finally, applications regarding the rank of a matrix and eigenvalue problems have been demonstrated.

In this paper, we determine the upper and lower bounds for the norm of lower triangular matrix operators on Ces\`{a}ro weighted $(p,v)-$fractional difference sequence spaces of modulus functions. We consider the matrix operators acting between $\ell_{p}(w)$ and $C_{p}(v,\omega,\\\Delta^{(\eta,\ell)},\mathcal{F})$ and identify their bounds and vice-versa. We also investigate the same characteristics for N\"{o}rlund and weighted mean matrix operators.

The purpose of this article is to introduce the triple sequences and its convergence over instuitionistic fuzzy metric space (\textbf{IFMS}). The article also discusses ideal convergence of triple sequences, the uniqueness of ideal limits, the relationship between Pringsheim's limit and ideal limits, the ideal Cauchy sequences, and various specific spaces of triple sequences with respect to IFMS.

In this paper, polynomial-based superconvergent degenerate kernel and {Nyström} methods for solving {Fredholm} integral equations of the second kind with  the smooth kernel are studied. By using an interpolatory projection based on Legendre polynomials of degree $\leq n,$ we analyze the convergence of these methods and we establish superconvergence results for their iterated versions. Two numerical examples are given to illustrate the theoretical estimates.

In this paper, the concept of a pseudosymmetric space which is a natural generalization of the concept of a symmetric space is defined. All basic concepts such as the Luxemburg representation theorem, the Boyd indices, the fundamental function and its properties, Calderon's theorem, etc., is transferred over the pseudosymmetric case. Examples are given for pseudosymmetric spaces. The quasi-symmetric spaces expand the scope of the application of symmetric space results.

Initial-boundary value problems including space-time fractional PDEs have been used to model a wide range of problems in physics and engineering fields. In this paper, a non-self adjoint initial boundary value problem containing a third order fractional differential equation is considered. First, a spectral problem for this problem is presented. Then the eigenvalues and eigenfunctions of the main spectral problem are calculated. In order to calculate the roots of their characteristic equation, the asymptotic expansion of the roots is used. Finally, by suitable choice of these asymptotic expansions, related eigenfunctions and Mittag-Lefler functions, the analytic and numerical solutions to the main initial-boundary value problem are given.

We study the convergence of the Krasnoselskii sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ for non-self mappings on closed intervals. We show that if $g$ satisfies $g^{'}\ge -1$ along with some other conditions, this sequence converges to a fixed point of $g$. We extend this fixed-point result to a novel and efficient root-finding method. We present concrete examples at the end. In these examples, we make a comparison between Newton-Raphson and our method. These examples also reveal how our method can be applied efficiently to find the fixed points of a real-valued function.

In the present investigation, our main aim is to introduce a certain subclass of analytic univalent functions related to the Error function. We discuss the implications of our main results, including the coefficient bound, extreme points, weighted mean, convolution, convexity, and radii properties,  as well as any other related properties.

For a non-zero normed linear space $A$, we consider $ C^b\left(K\right) $, the complex-valued, bounded and continuous functions space on $K$ with $ \left\| \cdot \right\|_\infty $, where $ K = \overline{B^{\left(0\right)}_1} $ (the closed unit ball of $A$). Also for a non-zero element $\varphi \in A^*$ with $ \left\| \varphi \right\| \leq 1 $, we consider the space $ C^{b\varphi}\left(K\right) $ as the linear space $ C^b\left(K\right) $ with the new norm $ \left\| f \right\|_\varphi = \left\| f\varphi \right\|_\infty $ for all $ f \in C^b\left(K\right) $. Some basic properties such as, proximinality, E-proximinality, strongly proximinality and quasi Chebyshev for certain subsets of $ C^b\left(K\right) $ are characterized with the norms $ \left\| \cdot \right\|_\varphi $ and $ \left\|\cdot \right\|_\infty $. Some examples for illustration and for comparison between the norms $ \left\| \cdot \right\|_\varphi $ and $ \left\| \cdot \right\|_\infty $ on $ C^b\left(K\right) $ are presented.

In this paper, contractive mappings of \CM{} type in fuzzy metric spaces are studied. A class $\Psi_1$ of gauge functions $\psi:(0,1]\to(0,1]$ such that, for any $r\in(0,1)$, there exists $\rho\in(r,1)$ such that $1-r> \tau >1-\rho$ implies $\psi(\tau)\geq 1-r$, is introduced, and it is shown that fuzzy $\psi$-contractive mappings are fuzzy contractive mappings of \CM{} type. A characterization of Cauchy sequences in fuzzy metric spaces is presented, and it is utilized to establish fixed point theorems. Examples are given to support the results. Our results cover those of Mihet (Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. 159(2008) 739-744), Wardowski (Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 222(2013) 108-114) and others.