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13 pages, 1144 KB

Open AccessArticle

Self-Adaptive Limited-Memory Quasi-Newton Method with Function Value Information for Large-Scale Unconstrained Optimization

by Jiangwen Ju, Weixin Lin and Hao Liu

Mathematics 2026, 14(10), 1750; https://doi.org/10.3390/math14101750 - 19 May 2026

Viewed by 234

Abstract

We extend the modified BFGS algorithm to a limited-memory framework, and propose a self-adaptive limited-memory quasi-Newton method, denoted as LADBFGS, for large-scale unconstrained optimization. The proposed method fully exploits function value information to improve the curvature approximation of the objective function, while enabling [...] Read more.

We extend the modified BFGS algorithm to a limited-memory framework, and propose a self-adaptive limited-memory quasi-Newton method, denoted as LADBFGS, for large-scale unconstrained optimization. The proposed method fully exploits function value information to improve the curvature approximation of the objective function, while enabling dynamic and adaptive adjustment of parameters. We establish the global R-linear convergence of the proposed algorithm for uniformly convex problems. Numerical experiments on 102 standard unconstrained test functions with dimensions of no less than 1000 show that the proposed LADBFGS method outperforms the standard limited-memory BFGS method in terms of iteration count, number of function and gradient evaluations, and computational time, and also achieves a higher success rate for solving the test problems. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

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13 pages, 353 KB

Open AccessArticle

On Uniformly δ-Geometric Convex Functions

by Yamin Sayyari, Hasan Barsam and Loredana Ciurdariu

Fractal Fract. 2026, 10(5), 289; https://doi.org/10.3390/fractalfract10050289 - 24 Apr 2026

Viewed by 454

Abstract

In this paper, we give some new Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities for uniformly

δ

-geometric convex functions. In addition, some limit bounds for Caputo–Fabrizio fractional integral operators are established as an application in the case of uniformly

δ

-geometric convex functions. Some [...] Read more.

In this paper, we give some new Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities for uniformly

δ

-geometric convex functions. In addition, some limit bounds for Caputo–Fabrizio fractional integral operators are established as an application in the case of uniformly

δ

-geometric convex functions. Some new examples and graphical representations are provided in order to illustrate the validity of our results. Full article

(This article belongs to the Section General Mathematics, Analysis)

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31 pages, 430 KB

Open AccessArticle

A Length Preserving Geodesic Curvature Difference Flow in the Hyperbolic Plane

by Qian Liu, Zhizhong Zheng, Fang Yang and Xinxin Pan

Mathematics 2026, 14(7), 1096; https://doi.org/10.3390/math14071096 - 24 Mar 2026

Viewed by 417

Abstract

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane

H^{2}

. Given an initial curve

γ_{1}

and a target curve

γ_{2}

of the same hyperbolic [...] Read more.

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane

H^{2}

. Given an initial curve

γ_{1}

and a target curve

γ_{2}

of the same hyperbolic length, we evolve

γ_{1}

by a normal speed given by the difference of the reciprocals of geodesic curvatures evaluated at points with the same outward unit normal, together with a time-dependent scalar term

Γ (t)

chosen to preserve the hyperbolic length. Using Leichtweiβ’s hyperbolic support function and Howe’s curvature formula, the flow is reformulated as a quasilinear uniformly parabolic equation on

S^{1}

with a nonlocal term

Γ (t)

. We prove short-time existence, uniqueness, and preservation of strict horocyclic convexity. Linearizing the support function equation at the target support function yields a uniformly elliptic operator whose kernel contains the infinitesimal isometry directions. Under a spectral gap assumption on a normalized slice transverse to the isometry orbit, we prove global existence and exponential convergence for initial data sufficiently close to the target curve. In the last section, this assumption is verified explicitly when the target curve is a geodesic circle. Full article

15 pages, 280 KB

Open AccessArticle

Locally Nearly Uniformly Convex Points in Orlicz Spaces Equipped with the Luxemburg Norm

by Yunan Cui, Xiaoxia Wang and Yaoming Niu

Axioms 2026, 15(1), 74; https://doi.org/10.3390/axioms15010074 - 20 Jan 2026

Viewed by 333

Abstract

This research explores two novel geometric concepts—nearly convex points and locally nearly uniformly convex points within the frameworks of Banach spaces and Orlicz spaces equipped with the Luxemburg norm. First, we establish the general characterization criteria for nearly convex points in Banach spaces. [...] Read more.

This research explores two novel geometric concepts—nearly convex points and locally nearly uniformly convex points within the frameworks of Banach spaces and Orlicz spaces equipped with the Luxemburg norm. First, we establish the general characterization criteria for nearly convex points in Banach spaces. Then, we analyze the intrinsic connection between locally nearly uniformly convex points and nearly extreme points in Banach spaces. Additionally, we provide comprehensive characterizations of locally nearly uniformly convex points in both Orlicz function spaces and Orlicz sequence spaces under the Luxemburg norm. These findings enrich the geometric theory system of Banach and Orlicz spaces, offering new theoretical support for related research directions. Full article

23 pages, 6989 KB

Open AccessFeature PaperArticle

Simulation Teaching of Adaptive Fault-Tolerant Containment Control for Nonlinear Multi-Agent Systems

by Shangkun Liu, Wangjin Zhang, Jingli Huang and Jie Huang

Mathematics 2025, 13(21), 3475; https://doi.org/10.3390/math13213475 - 31 Oct 2025

Viewed by 659

Abstract

An adaptive fault-tolerant containment control approach is developed for nonlinear multi-agent systems to address issues related to both communication link and actuator faults. This approach achieves fault-tolerant containment control through the introduction of a convex hull signal estimator and a fault compensation mechanism. [...] Read more.

An adaptive fault-tolerant containment control approach is developed for nonlinear multi-agent systems to address issues related to both communication link and actuator faults. This approach achieves fault-tolerant containment control through the introduction of a convex hull signal estimator and a fault compensation mechanism. First, a leader–follower network model with communication link faults is constructed, and distributed containment errors are established. The proposed framework involves three key components: the design of an adaptive backstepping control law, the introduction of a nonlinear filter for boundary error elimination, and the application of a radial basis function neural network (RBFNN) for the approximation of unknown nonlinear terms. Meanwhile, an adaptive convex hull estimator is designed to estimate the signals formed by the leaders, and an actuator fault estimator is constructed to compensate for fault signals online. Additionally, Lyapunov stability analysis demonstrates that all containment errors remain uniformly bounded. To support simulation teaching and validation, numerical simulations and autonomous underwater vehicle (AUV) simulations are used to not only to confirm the efficacy of the presented control technique but also to provide illustrative cases for educational purposes. Full article

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24 pages, 1596 KB

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 755

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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18 pages, 369 KB

Open AccessArticle

Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint

by Yanrong Lu, Jize Li and Yonghui Zhou

Mathematics 2025, 13(8), 1327; https://doi.org/10.3390/math13081327 - 18 Apr 2025

Viewed by 1020

Abstract

This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic [...] Read more.

This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. Full article

(This article belongs to the Special Issue Stochastic Optimal Control, Game Theory, and Related Applications)

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19 pages, 370 KB

Open AccessArticle

On Quantum Hermite-Hadamard-Fejer Type Integral Inequalities via Uniformly Convex Functions

by Hasan Barsam, Somayeh Mirzadeh, Yamin Sayyari and Loredana Ciurdariu

Fractal Fract. 2025, 9(2), 108; https://doi.org/10.3390/fractalfract9020108 - 12 Feb 2025

Cited by 3 | Viewed by 1809

Abstract

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s [...] Read more.

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s inequality are used in demonstrations. Some particular functions are chosen to illustrate the investigated results by two examples analyzed and the result obtained have been graphically visualized. Full article

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31 pages, 364 KB

Open AccessArticle

An Efficient Iterative Scheme for Approximating the Fixed Point of a Function Endowed with Condition (B_γ,μ) Applied for Solving Infectious Disease Models

by Godwin Amechi Okeke, Akanimo Victor Udo and Rubayyi T. Alqahtani

Mathematics 2025, 13(4), 562; https://doi.org/10.3390/math13040562 - 8 Feb 2025

Cited by 4 | Viewed by 1748

Abstract

The purpose of this paper is to construct a new fixed-point iterative scheme, called the Picard-like iterative scheme, for approximating the fixed point of a mapping that satisfies condition

(B_{γ, μ})

in the setting of a uniformly convex Banach [...] Read more.

The purpose of this paper is to construct a new fixed-point iterative scheme, called the Picard-like iterative scheme, for approximating the fixed point of a mapping that satisfies condition

(B_{γ, μ})

in the setting of a uniformly convex Banach space. We prove that this novel iterative scheme converges faster than some existing iterative schemes in the literature. Moreover,

G

-stability and almost

G

-stability results are proven. Furthermore, we apply our results for approximating the solution of an integral equation that models the spread of some infectious diseases. Similarly, we also applied the results for approximating the solution of the boundary value problem by embedding Green’s function in our novel method. Our results extend and generalize other existing results in the literature. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)

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17 pages, 304 KB

Open AccessArticle

Quasi-Lower C² Functions and Their Application to Nonconvex Variational Problems

by Messaoud Bounkhel

Axioms 2024, 13(12), 870; https://doi.org/10.3390/axioms13120870 - 13 Dec 2024

Viewed by 1129

Abstract

This study presents a novel category of nonconvex functions in Banach spaces, referred to as quasi-lower

C^{2}

functions on nonempty closed sets. We establish the existence of solutions for nonconvex variational problems involving quasi-lower

C^{2}

functions defined in Banach spaces. To [...] Read more.

This study presents a novel category of nonconvex functions in Banach spaces, referred to as quasi-lower

C^{2}

functions on nonempty closed sets. We establish the existence of solutions for nonconvex variational problems involving quasi-lower

C^{2}

functions defined in Banach spaces. To illustrate the applicability of our findings, an example is provided in

L^{p}

spaces. Full article

(This article belongs to the Special Issue New Developments in Analysis of Variational Inequalities and Related Fields)

10 pages, 261 KB

Open AccessArticle

Relations of Harmonic Starlike Function Subclasses with Mittag–Leffler Function

by Naci Taşar, Fethiye Müge Sakar, Seher Melike Aydoğan and Georgia Irina Oros

Axioms 2024, 13(12), 826; https://doi.org/10.3390/axioms13120826 - 26 Nov 2024

Viewed by 920

Abstract

In this study, the connection between certain subfamilies of harmonic univalent functions is established by utilizing a convolution operator involving the Mittag–Leffler function. The investigation reveals inclusion relations concerning harmonic

γ

-uniformly starlike mappings in the open unit disc, harmonic starlike functions and [...] Read more.

In this study, the connection between certain subfamilies of harmonic univalent functions is established by utilizing a convolution operator involving the Mittag–Leffler function. The investigation reveals inclusion relations concerning harmonic

γ

-uniformly starlike mappings in the open unit disc, harmonic starlike functions and harmonic convex functions, highlighting the improvements given by the results presented here on previously published outcomes. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

16 pages, 276 KB

Open AccessArticle

Monotonicities of Quasi-Normed Orlicz Spaces

by Dong Ji and Yunan Cui

Axioms 2024, 13(10), 696; https://doi.org/10.3390/axioms13100696 - 7 Oct 2024

Cited by 2 | Viewed by 1320

Abstract

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces

L^{Φ}

generated by the b-Orlicz function

Φ

equipped with a Luxemburg quasi-norm contain both classical spaces [...] Read more.

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces

L^{Φ}

generated by the b-Orlicz function

Φ

equipped with a Luxemburg quasi-norm contain both classical spaces

L^{p} (p \geq 1)

and

L^{p} (0 < p < 1)

. The Orlicz spaces

L^{Φ}

are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given. Full article

12 pages, 5661 KB

Open AccessEditor’s ChoiceArticle

An Adaptive Sliding Mode Control Using a Novel Adaptive Law Based on Quasi-Convex Functions and Average Sliding Variables for Robot Manipulators

by Dong Hee Seo, Jin Woong Lee, Hyuk Mo An and Seok Young Lee

Electronics 2024, 13(19), 3940; https://doi.org/10.3390/electronics13193940 - 5 Oct 2024

Cited by 2 | Viewed by 2531

Abstract

This paper proposes a novel adaptive law that uses a quasi-convex function and a novel sliding variable in an adaptive sliding mode control (ASMC) scheme for robot manipulators. Since the dynamic equations of robot manipulators inevitably include model uncertainties and disturbances, time-delay estimation [...] Read more.

This paper proposes a novel adaptive law that uses a quasi-convex function and a novel sliding variable in an adaptive sliding mode control (ASMC) scheme for robot manipulators. Since the dynamic equations of robot manipulators inevitably include model uncertainties and disturbances, time-delay estimation (TDE) errors occur when using the time-delay control (TDC) approach. Further, the ASMC method used to compensate for TDE errors naturally causes a chattering phenomenon. To improve tracking performance while reducing or maintaining chattering, this paper proposes an adaptive law based on a quasi-convex function that is convex at the origin and concave at the gain switching point, respectively. We also adopt a novel sliding variable that uses previously sampled tracking errors and their time derivatives. Further, this paper proves that the sliding variable of the robot manipulator controlled by the proposed ASMC satisfies uniformly ultimately bounded stability. The simulation and experimental results illustrate the effectiveness of the proposed methods in terms of tracking performance. Full article

(This article belongs to the Special Issue Intelligence Control and Applications of Intelligence Robotics)

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16 pages, 818 KB

Open AccessArticle

Starlikeness and Convexity of Generalized Bessel-Maitland Function

by Muhammad Umar Nawaz, Daniel Breaz, Mohsan Raza and Luminiţa-Ioana Cotîrlă

Axioms 2024, 13(10), 691; https://doi.org/10.3390/axioms13100691 - 4 Oct 2024

Cited by 4 | Viewed by 1408

Abstract

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order

δ

, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of [...] Read more.

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order

δ

, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies. Full article

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17 pages, 378 KB

Open AccessArticle

Generalized Quasilinearization Method for Caputo Fractional Differential Equations with Initial Conditions with Applications

by Aghalaya S. Vatsala and Govinda Pageni

Foundations 2024, 4(3), 345-361; https://doi.org/10.3390/foundations4030023 - 25 Jul 2024

Cited by 1 | Viewed by 1462

Abstract

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using

q,

which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the [...] Read more.

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using

q,

which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the data. The numerical methods available in the literature provide only the local existence of the solution. However, the interval of existence is known and guaranteed by the natural upper and lower solutions of the nonlinear differential equations. In this work, we develop monotone iterates, together with lower and upper solutions that converge uniformly, monotonically, and quadratically to the unique solution of the Caputo nonlinear fractional differential equation over its entire interval of existence. The nonlinear function is assumed to be the sum of convex and concave functions. The method is referred to as the generalized quasilinearization method. We provide a Caputo fractional logistic equation as an example whose interval of existence is

[0, \infty) .

Full article

(This article belongs to the Section Mathematical Sciences)

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