Search Results (77)

In this work, we develop and investigate statistical extensions of gauge integrability and gauge summability for double sequences of functions of two real variables, formulated within the framework of deferred weighted means. We begin by establishing several fundamental limit theorems that serve to connect these generalized notions and provide a rigorous theoretical foundation. Based on these results, we establish Korovkin-type approximation theorems using the classical test function set $1,s,t,s^2+t^2$ in the Banach space $\mathcal{C}\left({[0,1]}^2\right)$. To demonstrate the applicability of the proposed framework, we further present an example involving families of positive linear operators associated with the Meyer-König and Zeller (MKZ) operators. These findings not only extend classical Korovkin-type theorems to the setting of statistical deferred gauge integrability and summability but also underscore their robustness in addressing double sequences and the approximation of two-variable functions. Full article

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is ${\ell }_1$ (i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation is a periodic sequence. We consider the completion of the space of all such linear continuous functionals on ${\ell }_1$ with periods of Riesz representations equal to powers of 2. It is known that this completion is a Banach space with a Schauder basis. In this work, we construct a sequence Banach space with the standard Schauder basis $\{e_m=(\underset{m-1}{\underbrace{0,\dots ,0}},1,0,\dots ){\}}_{m=1}^{\infty }$ that is isometrically isomorphic to this completion. Results of the work can be used to describe spectra of topological algebras of analytic functions on ${\ell }_1$ that can be approximated by weakly symmetric functions. Full article

We develop a convergence framework for Grünwald–Letnikov (GL) fractional and classical integer difference operators acting on sequences in fuzzy-paranormed (fp) spaces, motivated by data that are imprecise and contain sporadic outliers. Fuzzy paranorms provide a resolution-dependent notion of proximity, while statistical and lacunary statistical convergence downweight sparse deviations by natural density; together, they yield robust criteria for difference-filtered signals. Within this setting, we establish uniqueness of fp–${\mathsf{\Delta }}^m$ statistical limits; an equivalence between fp-statistical convergence of ${\mathsf{\Delta }}^m$ (and its GL extension ${\mathsf{\Delta }}^{\alpha }$) and fp-strong p-Cesàro summability; an equivalence between lacunary fp-${\mathsf{\Delta }}^m$ statistical convergence and blockwise strong p-Cesàro summability; and a density-based decomposition into a classically convergent part plus an fp-null remainder. We also show that GL binomial weights act as an ${\ell }^1$ convolution, ensuring continuity of ${\mathsf{\Delta }}^{\alpha }$ in the fp topology, and that nabla/delta forms are transferred by the discrete Q–operator. The usefulness of the criteria is illustrated on simple engineering-style examples (e.g., relaxation with memory, damped oscillations with bursts), where the fp-Cesàro decay of difference residuals serves as a practical diagnostic for Cesàro compliance. Beyond illustrative mathematics, we report engineering-style diagnostics where the fuzzy Cesàro residual index correlates with measurable quantities (e.g., vibration amplitude and energy surrogates) under impulsive disturbances and missing data. We also calibrate a global decision threshold ${\tau }_{\mathrm{glob}}$ via sensitivity analysis across $(\alpha ,p,m)$, where $m\in \mathbb{N}$ is the integer difference order, $\alpha >0$ is the fractional order, and $p\ge 1$ is the Cesàro exponent, and provide quantitative baselines (median/M-estimators, ${\ell }_1$ trend filtering, Gaussian Kalman filtering, and an $\alpha$-stable filtering structure) to show complementary gains under bursty regimes. The results are stated for integer m and lifted to fractional orders $\alpha >0$ through the same binomial structure and duality. Full article

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set $\{1,x,x^2\}$ propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data. Full article

In the present paper, we use the proximal point method with remotest set control for find an approximate common zero of a finite collection of maximal monotone maps in a real Hilbert space under the presence of computational errors. We prove that the inexact proximal point method generates an approximate solution if these errors are summable. Also, we show that if the computational errors are small enough, then the inexact proximal point method generates approximate solutions Full article

This paper introduces and analyzes the concept of $\lambda$-statistical convergence in fuzzy paranormed spaces, demonstrating its relevance to supply chain inventory management under demand shocks. We establish key relationships between generalized convergence methods and fuzzy convex analysis, showing how these results extend classical summability theory to uncertain demand environments. By exploring $\lambda$-statistical Cauchy sequences and $(V,\lambda )$-summability in fuzzy paranormed spaces, we provide new insights applicable to adaptive inventory optimization and decision-making in supply chains. Our findings bridge theoretical aspects of fuzzy convexity with practical convergence tools, advancing the robust modeling of demand uncertainty. Full article

The Eulerian iterative method of the summation of divergent series, invented in Institutiones Calculi Differentialis, is studied. We demonstrate that the method is equivalent to the Karamata–Lototsky–Jakimovski summability method, introduced in the 1950s. We prove a new theorem on the Euler iterative summability of the series of alternating factorials. Ensuing summability corollaries are discussed. Full article

In this paper, we introduce and investigate the concept of statistical derivatives within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Using this approach, we establish a novel Korovkin-type theorem for a specific set of exponential test functions, namely 1, $e^{-\upsilon }$ and $e^{-2\upsilon }$, which are defined on the Banach space $\mathcal{C}[0,\infty )$. Our results significantly extend several well-known Korovkin-type theorems. Additionally, we analyze the rate of convergence associated with the statistical derivatives under deferred Cesàro summability. To support our theoretical findings, we provide compelling numerical examples, followed by graphical representations generated using MATLAB software, to visually illustrate and enhance the understanding of the convergence behavior of the operators. Full article

This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable norms. The focus is to determine the conditions under which the Kirk iteration sequence converges strongly or weakly to a fixed point. Finally, some examples are given in this article to demonstrate the advantages of the preferred iteration method and to verify the results obtained. Full article

We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also considered. Overdetermination conditions are the integrals over the boundary of a solution’s domain with weights. We focus on existence and uniqueness theorems and stability estimates for solutions to these inverse problems. An operator equation to which the problem is reduced is studied with the use of the contraction mapping principle. A solution belongs to some Sobolev space and has all generalized derivatives occurring into the equation summable to some power. The method of the proof is constructive, and it can be used for developing new numerical algorithms for solving the problem. Full article

This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks. Full article

In this paper, we prove existence and regularity results for a nonlinear elliptic problem of p-Laplacian type with a singular potential like ${\displaystyle \frac{f}{u^{\gamma }}}$ and a lower order term $bu$, where u is the solution and b and f are only assumed to be summable functions. We show that, despite the lack of regularity of the data, for suitable choices of the function b in the lower order term, a strong regularizing effect appears. In particular we exhibit the existence of bounded solutions. Worth notice is that this result fails if $b\equiv 0$, i.e., in absence of the lower order term. Moreover, we show that, if the singularity is “not too large” (i.e., $\gamma \le 1$), such a regular solution is also unique. Full article

In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is $-{\Delta }_{p}u=\beta {|\nabla u|}^q+f \mathrm{i}\mathrm{n} \Omega , u=0 \mathrm{o}\mathrm{n} \partial \Omega ,$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$, $N\ge 2$, $1<p<N$, ${\Delta }_{p}u=\mathrm{div}\left({|\nabla u|}^{p-2}\nabla u\right)$, $p-1<q<p$, $\beta$ is a positive constant and f is a measurable function satisfying suitable summability conditions depending on q and a smallness condition. Full article

In this paper, the Fibonacci sequence, renowned for its significance across various fields, its ability to illuminate numerical concepts, and its role in uncovering patterns in mathematics and nature, forms the foundation of this research. This study introduces innovative concepts of weighted density, weighted statistical summability, weighted statistical convergence, and weighted statistical Cauchy, uniquely defined via the Fibonacci sequence and modulus functions. Key theorems, relationships, examples, and properties substantiate these novel principles, advancing our comprehension of sequence behavior. Additionally, we extend the notion of statistical cluster points within a broader framework, surpassing traditional definitions and offering deeper insights into convergence in a statistical context. Our findings in this paper open avenues for new applications and further exploration in various mathematical fields. Full article