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13 pages, 285 KB

Open AccessArticle

Generalized Local Morrey Spaces Associated with Ball Banach Function Spaces and Their Application

by Feiyang Zhang and Jiang Zhou

Axioms 2025, 14(12), 894; https://doi.org/10.3390/axioms14120894 - 1 Dec 2025

Viewed by 161

This paper is devoted to the analysis of boundedness for fractional integral operators, Calderón–Zygmund singular integral operators, and their corresponding commutators on generalized local Morrey spaces associated with ball Banach function spaces. These foundational results are then applied to establish the local regularity [...] Read more.

This paper is devoted to the analysis of boundedness for fractional integral operators, Calderón–Zygmund singular integral operators, and their corresponding commutators on generalized local Morrey spaces associated with ball Banach function spaces. These foundational results are then applied to establish the local regularity within the

L M_{X}^{φ}

Morrey spaces for the solution gradients of second-order elliptic equations expressed in divergence form. Full article

(This article belongs to the Special Issue Applications in Harmonic Analysis)

26 pages, 382 KB

Open AccessArticle

Some Realisation of the Banach Space of All Continuous Linear Functionals on ℓ₁ Approximated by Weakly Symmetric Continuous Linear Functionals

by Mykhailo Varvariuk and Taras Vasylyshyn

Symmetry 2025, 17(11), 1896; https://doi.org/10.3390/sym17111896 - 6 Nov 2025

Viewed by 207

Abstract

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is

ℓ_{1}

(i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation [...] Read more.

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is

ℓ_{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

ℓ_{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}{\underset{︸}{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

ℓ_{1}

that can be approximated by weakly symmetric functions. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis: 3rd Edition)

22 pages, 351 KB

Open AccessArticle

On the Multiplication Operators from the Natural μ-Bloch-Type Space into Another Natural ω-Bloch-Type Space

by Xiaoman Liu and Yongmin Liu

Mathematics 2025, 13(20), 3302; https://doi.org/10.3390/math13203302 - 16 Oct 2025

Viewed by 255

Abstract

This paper investigates the boundedness of multiplication operators

M_{ψ}

between natural

μ

-Bloch-type spaces

B_{μ, nat} (B_{X})

(or their little

μ

-Bloch counterparts) and natural

ω

-Bloch-type spaces

B_{ω, nat} (B_{X})

on [...] Read more.

This paper investigates the boundedness of multiplication operators

M_{ψ}

between natural

μ

-Bloch-type spaces

B_{μ, nat} (B_{X})

(or their little

μ

-Bloch counterparts) and natural

ω

-Bloch-type spaces

B_{ω, nat} (B_{X})

on the unit ball

B_{X}

of a complex Banach space X. We establish complete characterizations for the boundedness of

M_{ψ}

under varying conditions on the weight functions

μ

and

ω

, including specific cases such as logarithmic and power-weighted Bloch spaces. The results extend classical operator theory to infinite-dimensional settings, unifying prior work on finite-dimensional domains. Full article

19 pages, 300 KB

Open AccessArticle

Certain Novel Best Proximity Theorems with Applications to Complex Function Theory and Integral Equations

by Moosa Gabeleh

Axioms 2025, 14(9), 657; https://doi.org/10.3390/axioms14090657 - 27 Aug 2025

Viewed by 658

Abstract

Let

E

and

F

be nonempty disjoint subsets of a metric space

(M, d)

. For a non-self-mapping

φ : E \to F

, which is fixed-point free, a point

ϰ \in E

is said to be a best proximity [...] Read more.

Let

E

and

F

be nonempty disjoint subsets of a metric space

(M, d)

. For a non-self-mapping

φ : E \to F

, which is fixed-point free, a point

ϰ \in E

is said to be a best proximity point for the mapping

φ

whenever the distance of the point

ϰ

to its image under

φ

is equal to the distance between the sets,

E

and

F

. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

16 pages, 304 KB

Open AccessArticle

On the Characterizations of Some Strongly Bounded Operators on C(K, X) Spaces

by Ioana Ghenciu

Axioms 2025, 14(8), 558; https://doi.org/10.3390/axioms14080558 - 23 Jul 2025

Viewed by 477

Abstract

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and

C (K, X)

is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators [...] Read more.

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and

C (K, X)

is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators

T : C (K, X) \to Y

with representing measures

m : Σ \to L (X, Y)

, where

L (X, Y)

is the Banach space of all operators

T : X \to Y

and

Σ

is the

σ

-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

13 pages, 281 KB

Open AccessArticle

m-Isometric Operators with Null Symbol and Elementary Operator Entries

by Bhagwati Prashad Duggal

Axioms 2025, 14(7), 503; https://doi.org/10.3390/axioms14070503 - 27 Jun 2025

Viewed by 412

Abstract

A pair

(A, B)

of Banach space operators is strict

(m, X)

-isometric for a Banach space operator

X \in B (X)

and a positive integer m if [...] Read more.

A pair

(A, B)

of Banach space operators is strict

(m, X)

-isometric for a Banach space operator

X \in B (X)

and a positive integer m if

▵_{A, B}^{m} (X) = (\sum_{j = 0}^{m} (\begin{matrix} m \\ j \end{matrix}) L_{A}^{j} R_{B}^{j}) (X) = 0

and

▵_{A, B}^{m - 1} (X) \neq 0

, where

L_{A}

and

R_{B} \in B (B (X))

are, respectively, the operators of left multiplication by A and right multiplication by B. Define operators

E_{A, B}

and

E_{A, B} (X)

by

E_{A, B} = L_{A} R_{B}

and

{(E_{A, B} (X))}^{n} = E_{A, B}^{n} (X)

for all non-negative integers n. Using little more than an algebraic argument, the following generalised version of a result relating

(m, X)

-isometric properties of pairs

(A_{1}, A_{2})

and

(B_{1}, B_{2})

to pairs

(E_{A_{1}, A_{2}} (S_{1}), E_{B_{1}, B_{2}} (S_{2}))

and

(E_{A_{1}, A_{2}}, E_{B_{1}, B_{2}})

is proved: if

A_{i}, B_{i}, S_{i}, X

are operators in

B (X)

,

1 \leq i \leq 2

and X a quasi-affinity, then the pair

(E_{A_{1}, A_{2}} (S_{1}), E_{B_{1}, B_{2}} (S_{2}))

(resp., the pair

(E_{A_{1}, A_{2}}, E_{B_{1}, B_{2}})

) is strict

(m, X)

-isometric for all

X \in B (X)

if and only if there exist positive integers

m_{i} \leq m

,

1 \leq i \leq 2

and

m = m_{1} + m_{2} - 1

, and a non-zero scalar

β

such that

I - E_{β A_{1}, A_{2}} (S_{1})

is (strict)

m_{1}

-nilpotent and

I - E_{\frac{1}{β} B_{1}, B_{2}} (S_{2})

is (strict)

m_{2}

-nilpotent (resp.,

(β A_{1}, B_{1})

is strict

(m_{1}, I)

-isometric and

(\frac{1}{β} B_{2}, A_{2})

is strict

(m_{2}, I)

-isometric). Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

26 pages, 513 KB

Open AccessArticle

Stability of Weak Rescaled Pure Greedy Algorithms

by Wan Li, Man Lu, Peixin Ye and Wenhui Zhang

Axioms 2025, 14(6), 446; https://doi.org/10.3390/axioms14060446 - 6 Jun 2025

Viewed by 641

Abstract

We study the stability of Weak Rescaled Pure Greedy Algorithms for convex optimization, WRPGA(co), in general Banach spaces. We obtain the convergence rates of WRPGA(co) with noise and errors under a weaker assumption for the modulus of smoothness of the objective function. The [...] Read more.

We study the stability of Weak Rescaled Pure Greedy Algorithms for convex optimization, WRPGA(co), in general Banach spaces. We obtain the convergence rates of WRPGA(co) with noise and errors under a weaker assumption for the modulus of smoothness of the objective function. The results show that the rate is almost the same as that of WRPGA(co) without noise and errors, which is optimal and independent of the spatial dimension. This makes WRPGA(co) more practically applicable and scalable for high-dimensional data. Furthermore, we apply WRPGA(co) with errors to the problem of m-term approximation and derive the optimal convergence rate. This indicates the flexibility of WRPGA(co) and its wide utility across machine learning and signal processing. Our numerical experiments verify the stability of WRPGA(co). Thus, WRPGA(co) is a desirable choice for practical implementation. Full article

(This article belongs to the Section Mathematical Analysis)

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12 pages, 279 KB

Open AccessArticle

Construction of ε-Nets for the Space of Planar Convex Bodies Endowed with the Banach–Mazur Metric

by Yanmei Chen, Yunfang Lyu, Shenghua Gao and Senlin Wu

Mathematics 2025, 13(8), 1358; https://doi.org/10.3390/math13081358 - 21 Apr 2025

Viewed by 530

Abstract

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et [...] Read more.

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing

ε

-nets for

(K^{n}, d_{B M})

based on finite subsets of

Z^{n}

theoretically. In this work, we present an algorithm to construct

ε

-nets for

(K^{2}, d_{B M})

and a

(1 / 4)

-net for

(C^{2}, d_{B M})

is constructed. To the best of our knowledge, this is the first concrete

ε

-net for

(C^{2}, d_{B M})

for such a small

ε

. Full article

(This article belongs to the Section B: Geometry and Topology)

21 pages, 391 KB

Open AccessArticle

Universal Covering System and Borsuk’s Problem in Finite Dimensional Banach Spaces

by Xincong Qi, Xinling Zhang, Yunfang Lyu and Senlin Wu

Axioms 2025, 14(4), 277; https://doi.org/10.3390/axioms14040277 - 6 Apr 2025

Viewed by 628

Abstract

For each n-dimensional real Banach space X and each positive integer m, let

β (X, m)

be the infimum of

δ \in (0, 1]

such that each set

A \subseteq X

having diameter 1 can [...] Read more.

For each n-dimensional real Banach space X and each positive integer m, let

β (X, m)

be the infimum of

δ \in (0, 1]

such that each set

A \subseteq X

having diameter 1 can be represented as the union of m subsets of A, whose diameters are not greater than

δ

. Providing accurate estimations of

β (X, m)

for specific choices of X and m is crucial for addressing the extension of the classical Borsuk’s problem. A general framework for estimating

β (X, m)

via constructing and refining universal covering systems is presented. As an example, a universal covering system is constructed in

ℓ_{1}^{3}

and it is shown that

β (ℓ_{1}^{3}, 8) \leq 11 / 12

by a feasible partitioning of members in this universal covering system. Full article

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17 pages, 312 KB

Open AccessArticle

On Approximate Multi-Cubic Mappings in 2-Banach Spaces

by El-sayed El-hady, Ghazyiah Alsahli, Abasalt Bodaghi and Mehdi Dehghanian

Symmetry 2025, 17(4), 475; https://doi.org/10.3390/sym17040475 - 21 Mar 2025

Cited by 1 | Viewed by 566 | Correction

Abstract

The present article presents a system of symmetric equations defining multi-cubic mappings (M-CMs). Next, we describe how these mappings are structured and obtain an equation for describing them. Moreover, we Address the Hyers-Ulam stability (H-UStab) in the sense of Găvruţa for a symmetric [...] Read more.

The present article presents a system of symmetric equations defining multi-cubic mappings (M-CMs). Next, we describe how these mappings are structured and obtain an equation for describing them. Moreover, we Address the Hyers-Ulam stability (H-UStab) in the sense of Găvruţa for a symmetric multi-cubic equation through the application of the so-called Hyers (direct) method in the setting of 2-Banach spaces. For a typical case, by means of a norm, induced from a 2-norm of

R^{d}

, we examine the stability and hyperstability of a mapping

f : R^{d n} ⟶ R^{d}

by using a fixed point (FP) result. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities, 2nd Edition)

16 pages, 301 KB

Open AccessArticle

On Ulam Stability of the Davison Functional Equation in m-Banach Spaces

by El-sayed El-hady and Janusz Brzdęk

Axioms 2025, 14(2), 107; https://doi.org/10.3390/axioms14020107 - 30 Jan 2025

Viewed by 934

Abstract

We prove new Ulam stability results for the Davison functional equation, in the class of mappings h from a ring F into an m-Banach space. In this way, we complement several earlier outcomes, by extending them to the case of m-normed [...] Read more.

We prove new Ulam stability results for the Davison functional equation, in the class of mappings h from a ring F into an m-Banach space. In this way, we complement several earlier outcomes, by extending them to the case of m-normed spaces. Our proofs are based on an earlier Ulam stability result obtained for some functional equation in a single variable. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

14 pages, 273 KB

Open AccessArticle

Partitioning Functional of a Class of Convex Bodies

by Xinling Zhang

Axioms 2025, 14(1), 48; https://doi.org/10.3390/axioms14010048 - 9 Jan 2025

Cited by 1 | Viewed by 808

Abstract

For each n-dimensional real Banach space X, each positive integer m, and each bounded set

A \subseteq X

with diameter greater than 0, let

β_{X} (A, m)

be the infimum of [...] Read more.

For each n-dimensional real Banach space X, each positive integer m, and each bounded set

A \subseteq X

with diameter greater than 0, let

β_{X} (A, m)

be the infimum of

δ \in (0, 1]

such that

A \subseteq X

can be represented as the union of m subsets of A, whose diameters are not greater than

δ

times the diameter of A. Estimating

β_{X} (A, m)

is an important part of Chuanming Zong’s quantitative program for attacking Borsuk’s problem. However, estimating the partitioning functionals of general convex bodies in finite dimensional Banach spaces is challenging, so we will begin with the estimation of partitioning functionals for special convex bodies. In this paper, we prove a series of inequalities about partitioning functionals of convex cones. Several estimations of partitioning functionals of the convex hull of

(A + u) \cup (A - u)

and

(A + u) \cup (- A - u)

are also presented, where

A \subseteq R^{n - 1} \times {0}

is a convex body with the origin o in its interior, and

u \in R^{n} ∖ (R^{n - 1} \times {0})

. These results contribute to the study of Borsuk’s problem through Zong’s program. Full article

(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)

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21 pages, 343 KB

Open AccessArticle

Fixed-Point Results for Krasnoselskii, Meir–Keeler, and Boyd–Wong-Type Mappings with Applications to Dynamic Market Equilibrium

by Lifang Guo, Rabia Bibi, Abeer Alshejari, Ekrem Savas, Tayyab Kamran and Umar Ishtiaq

Axioms 2024, 13(12), 867; https://doi.org/10.3390/axioms13120867 (registering DOI) - 12 Dec 2024

Cited by 1 | Viewed by 1196

Abstract

This paper introduces the idea of a cone m-hemi metric space, which extends the idea of an m-hemi metric space. By presenting non-trivial examples, we demonstrate the superiority of cone m-hemi metric spaces over m-hemi metric spaces. Further, we [...] Read more.

This paper introduces the idea of a cone m-hemi metric space, which extends the idea of an m-hemi metric space. By presenting non-trivial examples, we demonstrate the superiority of cone m-hemi metric spaces over m-hemi metric spaces. Further, we extend the Banach contraction principle and Krasnoselskii, Meir–Keeler, Boyd–Wong, and some other fixed-point results in the setting of complete and compact cone m-hemi metric spaces. Furthermore, we provide several non-trivial examples and applications to the Fredholm integral equation and dynamic market equilibrium to demonstrate the validity of the main results. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

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20 pages, 326 KB

Open AccessArticle

Noncommutative Multi-Parameter Subsequential Wiener–Wintner-Type Ergodic Theorem

by Mu Sun and Yinmei Zhang

Axioms 2024, 13(9), 595; https://doi.org/10.3390/axioms13090595 - 31 Aug 2024

Viewed by 1234

Abstract

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class

Δ

instead of the weight class case. In [...] Read more.

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class

Δ

instead of the weight class case. In our subsequence class, every term of

\underset{̲}{k} \in Δ

is one of the three kinds of nonzero density subsequences we consider. As key ingredients, we give the maximal ergodic inequalities of multi-parameter subsequential averages and obtain a noncommutative subsequential analogue of the Banach principle. Then, by combining the critical result of the uniform convergence for a dense subset of the noncommutative

L_{p} (M)

space and the noncommutative Orlicz space, we immediately obtain the main theorem. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

11 pages, 248 KB

Open AccessArticle

Stability and Instability of an Apollonius-Type Functional Equation

by Ponmana Selvan Arumugam, Won-Gil Park and Jaiok Roh

Mathematics 2024, 12(14), 2274; https://doi.org/10.3390/math12142274 - 21 Jul 2024

Viewed by 990

Abstract

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this [...] Read more.

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers’ direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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