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37 pages, 417 KiB

Open AccessFeature PaperArticle

On the Properties of Iterations Generated with Composition Maps of Cyclic Contractive Self-Mappings and Strict Contractions in Metric Spaces

by Manuel De la Sen

Mathematics 2025, 13(14), 2224; https://doi.org/10.3390/math13142224 - 8 Jul 2025

Viewed by 127

Abstract

This paper studies the convergence of distances between sequences of points and that of sequences of points in metric spaces. This investigation is focused on the iterative processes built with composed self-mappings of a cyclic contraction, which can involve more than two nonempty [...] Read more.

This paper studies the convergence of distances between sequences of points and that of sequences of points in metric spaces. This investigation is focused on the iterative processes built with composed self-mappings of a cyclic contraction, which can involve more than two nonempty closed subsets in a metric space, which are combined with compositions of a strict contraction with itself, which operates in each of the individual subsets, in any order and any number of mutual compositions. It is admitted, in the most general case, the involvement of any number of repeated compositions of both self-maps with themselves. It is basically seen that, if one of the best-proximity points in the cyclic disposal is unique in a boundedly compact subset of the metric space is sufficient to achieve unique asymptotic cycles formed by a best-proximity point per each adjacent subset. The same property is achievable if such a subset is strictly convex and the metric space is a uniformly convex Banach space. Furthermore, all the sequences with arbitrary initial points in the union of all the subsets of the cyclic disposal converge to such a limit cycle. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 2nd Edition)

22 pages, 386 KiB

Open AccessArticle

Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains

by Abdelkader Moumen, Sabri T. M. Thabet, Hussien Albala, Khaled Aldwoah, Hicham Saber, Eltigani I. Hassan and Alawia Adam

Fractal Fract. 2025, 9(7), 442; https://doi.org/10.3390/fractalfract9070442 - 3 Jul 2025

Viewed by 303

Abstract

This article concerns a novel coupled implicit differential system under

φ

–Riemann–Liouville (

RL

) fractional derivatives with

p

-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains [...] Read more.

This article concerns a novel coupled implicit differential system under

φ

–Riemann–Liouville (

RL

) fractional derivatives with

p

-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains

[c, \infty)

. The explicit iterative solution’s existence and uniqueness (

E a U

) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias (

UHR

) stabilities are investigated. Ultimately, we provide a numerical application of a coupled

φ

-

RL

fractional turbulent flow model to illustrate and test the effectiveness of our outcomes. Full article

(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)

14 pages, 294 KiB

Open AccessArticle

Stability of a General Functional Equation

by Anna Bahyrycz

Symmetry 2025, 17(7), 1017; https://doi.org/10.3390/sym17071017 - 27 Jun 2025

Viewed by 145

Abstract

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations [...] Read more.

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. We also show, using examples, how some additional assumptions imposed on the function when examining the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation. The functional equation studied in this paper has symmetric coefficients (with precision up to the sign), and it is a generalization of an equation characterizing n-quadratic functions, as well as many other functional equations with symmetric coefficients: for example, the multi-Cauchy equation and the multi-Jensen equation. Our results generalize many known outcomes. Full article

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

13 pages, 281 KiB

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 142

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)

36 pages, 544 KiB

Open AccessArticle

Well-Posedness of Cauchy-Type Problems for Nonlinear Implicit Hilfer Fractional Differential Equations with General Order in Weighted Spaces

by Jakgrit Sompong, Samten Choden, Ekkarath Thailert and Sotiris K. Ntouyas

Symmetry 2025, 17(7), 986; https://doi.org/10.3390/sym17070986 - 22 Jun 2025

Viewed by 164

Abstract

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness [...] Read more.

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)

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12 pages, 1132 KiB

Open AccessFeature PaperArticle

On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries

by Isabelle Chalendar, Lucas Oger and Jonathan R. Partington

Mathematics 2025, 13(13), 2053; https://doi.org/10.3390/math13132053 - 20 Jun 2025

Viewed by 189

Abstract

A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to [...] Read more.

A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach–Stone theorem to surjective and nonsurjective cases. Full article

(This article belongs to the Section C4: Complex Analysis)

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12 pages, 248 KiB

Open AccessArticle

On Structral Properties of Some Banach Space-Valued Schröder Sequence Spaces

by Yılmaz Yılmaz, A. Nihal Tuncer and Seçkin Yalçın

Symmetry 2025, 17(7), 977; https://doi.org/10.3390/sym17070977 - 20 Jun 2025

Viewed by 200

Abstract

Some properties on Banach spaces, such as the Radon–Riesz, Dunford–Pettis and approximation properties, allow us to better understand the naive details about the structure of space and the robust inhomogeneities and symmetries in space. In this work we try to examine such properties [...] Read more.

Some properties on Banach spaces, such as the Radon–Riesz, Dunford–Pettis and approximation properties, allow us to better understand the naive details about the structure of space and the robust inhomogeneities and symmetries in space. In this work we try to examine such properties of vector-valued Schröder sequence spaces. Further, we show that these sequence spaces have a kind of Schauder basis. We also prove that

ℓ_{1} (S, V)

possesses the Dunford–Pettis property and demonstrate that

ℓ_{p} (S, V)

satisfies the approximation property for

1 \leq p < \infty

under certain conditions and

ℓ_{\infty} (S, V)

has the Hahn–Banach extension property. Finally, we show that

ℓ_{2} (S, V)

has the Radon–Riesz property whenever V has it. Full article

(This article belongs to the Section Mathematics)

24 pages, 1596 KiB

Open AccessArticle

Convergence and ω²-Stability Analysis of a Hybrid-Type Iterative Scheme with Application to Functional Delay Differential Equations

by Safeer Hussain Khan, Hina Dilawer, Hira Iqbal and Mujahid Abbas

Axioms 2025, 14(6), 475; https://doi.org/10.3390/axioms14060475 - 19 Jun 2025

Viewed by 165

Abstract

The purpose of this article is to analyze a hybrid-type iterative algorithm for a class of generalized non-expansive mappings satisfying the Garcia-Falset property in uniformly convex Banach spaces. Some existing results for such mappings have been obtained using the given algorithm. The [...] Read more.

The purpose of this article is to analyze a hybrid-type iterative algorithm for a class of generalized non-expansive mappings satisfying the Garcia-Falset property in uniformly convex Banach spaces. Some existing results for such mappings have been obtained using the given algorithm. The

ω^{2}

-stability of the iterative process is also studied. Using some examples, numerical experiments are conducted by comparing this iterative algorithm with different well-known iterative schemes. It is concluded that this iterative algorithm converges faster to the fixed point and is preferable over the previously known iterative schemes using the Garcia-Falset property. A weak solution of the Volterra–Stieltjes-type delay functional differential equation is presented to demonstrate the significance of the proposed results. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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24 pages, 1508 KiB

Open AccessArticle

The Stochastic Evolution of Financial Asset Prices

by Ioannis Paraskevopoulos and Alvaro Santos

Mathematics 2025, 13(12), 2002; https://doi.org/10.3390/math13122002 - 17 Jun 2025

Viewed by 192

Abstract

This paper examines the relationship between dependence and independence alternatives in general stochastic processes and explores the duality between the true (yet unknown) stochastic process and the functional representation that fits the observed data. We demonstrate that the solution depends on its historic [...] Read more.

This paper examines the relationship between dependence and independence alternatives in general stochastic processes and explores the duality between the true (yet unknown) stochastic process and the functional representation that fits the observed data. We demonstrate that the solution depends on its historic realizations, challenging existing theoretical frameworks that assume independence between the solution and the history of the true process. Under orthogonality conditions, we investigate parameter spaces within data-generating processes and establish conditions under which data exhibit mean-reverting, random, cyclical, history-dependent, or explosive behaviors. We validate our theoretical framework through empirical analysis of an extensive dataset comprising daily prices from the S&P500, 10-year US Treasury bonds, the EUR/USD exchange rate, Brent oil, and Bitcoin from 1 January 2002 to 1 February 2024. Our out-of-sample predictions, covering the period from 17 February 2019 to 1 February 2024, demonstrate the model’s exceptional forecasting capability, yielding correct predictions with between 73% and 92% accuracy, significantly outperforming naïve and moving average models, which only achieved 47% to 54% accuracy. Full article

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24 pages, 434 KiB

Open AccessArticle

Three-Step Iterative Methodology for the Solution of Extended Ordered XOR-Inclusion Problems Incorporating Generalized Cayley–Yosida Operators

by Doaa Filali, Imran Ali, Montaser Saudi Ali, Nidal H. E. Eljaneid, Esmail Alshaban and Faizan Ahmad Khan

Mathematics 2025, 13(12), 1969; https://doi.org/10.3390/math13121969 - 14 Jun 2025

Viewed by 283

Abstract

The system of extended ordered XOR-inclusion problems (in short, SEOXORIP) involving generalized Cayley and Yosida operators is introduced and studied in this paper. The solution is obtained in a real ordered Banach space using a fixed-point approach. First, we develop the fixed-point lemma [...] Read more.

The system of extended ordered XOR-inclusion problems (in short, SEOXORIP) involving generalized Cayley and Yosida operators is introduced and studied in this paper. The solution is obtained in a real ordered Banach space using a fixed-point approach. First, we develop the fixed-point lemma for the solution of SEOXORIP. By using the fixed-point lemma, we develop a three-step iterative scheme for obtaining the approximate solution of SEOXORIP. Under the Lipschitz continuous assumptions of the cost mappings, the strong convergence of the scheme is demonstrated. Lastly, we provide a numerical example with a convergence graph generated using MATLAB 2018a to verify the convergence of the sequence generated by the proposed scheme. Full article

(This article belongs to the Special Issue Advances in Mathematical Analysis and Inequalities)

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13 pages, 330 KiB

Open AccessArticle

Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative

by Mahir Almatarneh, Sonuc Zorlu and Nazim I. Mahmudov

Fractal Fract. 2025, 9(6), 374; https://doi.org/10.3390/fractalfract9060374 - 12 Jun 2025

Viewed by 482

Abstract

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to [...] Read more.

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results. Full article

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67 pages, 3117 KiB

Open AccessArticle

Stability Analysis and Local Convergence of a New Fourth-Order Optimal Jarratt-Type Iterative Scheme

by Eulalia Martínez, José A. Reyes, Alicia Cordero and Juan R. Torregrosa

Computation 2025, 13(6), 142; https://doi.org/10.3390/computation13060142 - 9 Jun 2025

Viewed by 1340

Abstract

In this work, using the weight function technique, we introduce a new family of fourth-order iterative methods optimal in the sense of Kung and Traub for scalar equations, generalizing Jarratt’s method. Through Taylor series expansions, we confirm that all members of this family [...] Read more.

In this work, using the weight function technique, we introduce a new family of fourth-order iterative methods optimal in the sense of Kung and Traub for scalar equations, generalizing Jarratt’s method. Through Taylor series expansions, we confirm that all members of this family achieve fourth-order convergence when derivatives up to the fourth order are bounded. Additionally, a stability analysis is performed on quadratic polynomials using complex discrete dynamics, enabling differentiation among the methods based on their stability. To demonstrate practical applicability, a numerical example illustrates the effectiveness of the proposed family. Extending our findings to Banach spaces, we conduct local convergence analyses on a specific subfamily containing Jarratt’s method, requiring only boundedness of the first derivative. This significantly broadens the method’s applicability to more general spaces and reduces constraints on higher-order derivatives. Finally, additional examples validate the existence and uniqueness of approximate solutions in Banach spaces, provided the initial estimate lies within the locally determined convergence radius obtained using majorizing functions. Full article

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1 pages, 121 KiB

Open AccessCorrection

Correction: El-hady et al. On Approximate Multi-Cubic Mappings in 2-Banach Spaces. Symmetry 2025, 17, 475

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

Symmetry 2025, 17(6), 909; https://doi.org/10.3390/sym17060909 - 9 Jun 2025

Viewed by 215

Abstract

In the published publication [...] Full article

26 pages, 513 KiB

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 233

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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19 pages, 332 KiB

Open AccessArticle

Analytical Approach to Convergence, Stability, and Data-Dependence of Jungck-KF Iterative Scheme with Applications in Dynamic Market Equilibrium Analysis

by Khushdil Ahmad, Khurram Shabbir, Faraz Ali, Monica-Felicia Bota and Liliana Guran

Symmetry 2025, 17(6), 885; https://doi.org/10.3390/sym17060885 - 5 Jun 2025

Viewed by 355

Abstract

In this work, we employ a more thorough contractive condition to examine the stability and convergence behavior of an Jungck-type iterative scheme for a pair of non-self mappings in a Banach space. Our results show that this iterative scheme has a better rate [...] Read more.

In this work, we employ a more thorough contractive condition to examine the stability and convergence behavior of an Jungck-type iterative scheme for a pair of non-self mappings in a Banach space. Our results show that this iterative scheme has a better rate of convergence as compared to all existing Jungck-type iterative schemes. The norm of a Banach space is symmetric with respect to the origin. Symmetry can significantly influence both the theoretical underpinnings and practical convergence behavior of iterative schemes. Furthermore, we show the convergence behaviour of various Jungck-type iterative schemes with an Jungck-KF iterative scheme through an example. We also prove the data-dependence result for our proposed iterative scheme for non-self-mapping. Additionally, we provide an application of the Jungck-KF iterative scheme related to Dynamic Market Equilibrium. Full article

(This article belongs to the Special Issue Application of Symmetry in Equations)

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