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12 pages, 257 KB

Open AccessArticle

Efficient Solution of DC-Type Vector Optimization via Abstract Convex Analysis

by Ruimin Gao and Chaoli Yao

Mathematics 2026, 14(6), 1070; https://doi.org/10.3390/math14061070 - 22 Mar 2026

Viewed by 257

The concept of vector topical functions, which take values in a partially ordered Banach space endowed with a complete lattice structure, was introduced in our preceding study. This structure enabled the development of an abstract convexity theory for vector topical functions, utilizing the [...] Read more.

The concept of vector topical functions, which take values in a partially ordered Banach space endowed with a complete lattice structure, was introduced in our preceding study. This structure enabled the development of an abstract convexity theory for vector topical functions, utilizing the notion of vector support. In this paper, applying these abstract convex theories, a DC-type vector optimization is investigated. Using the idea of a slack, the support set of a vector-valued map can be fully characterized by the subdifferential in abstract convex sense. Then, with the aid of this result, a sufficient condition to detect the efficient solutions for the DC-type vector optimization is obtained. In addition, a dual problem for the DC optimization is proposed, for which some strong dual results are established. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

19 pages, 375 KB

Open AccessArticle

Fixed Point of Polynomial F-Contraction with an Application

by Amjad E. Hamza, Hayel N. Saleh, Bakri Younis, Khaled Aldwoah, Osman Osman, Hicham Saber and Alawia Adam

Mathematics 2026, 14(4), 589; https://doi.org/10.3390/math14040589 - 8 Feb 2026

Viewed by 529

Abstract

This paper introduces polynomial F-contractions, a novel category of contractive mappings within metric spaces. This concept synthesizes two powerful generalizations of the Banach contraction principle: the F-contractions originally developed by Wardowski and the polynomial-type contractions studied very recently by Jleli et [...] Read more.

This paper introduces polynomial F-contractions, a novel category of contractive mappings within metric spaces. This concept synthesizes two powerful generalizations of the Banach contraction principle: the F-contractions originally developed by Wardowski and the polynomial-type contractions studied very recently by Jleli et al. We formulate fixed point theorems for this new class of mappings in complete metric spaces, which extends and unifies several established theorems in fixed point theory. We first prove our main result for continuous mappings and then extend it to a broader class of mappings that are not necessarily continuous but satisfy the Picard continuity condition. The significance and novelty of our results are highlighted through illustrative examples and further supported by applications to a fractional boundary value problem. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

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13 pages, 310 KB

Open AccessArticle

A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces

by Changchi Huang, Jigen Peng, Liqian Qin and Yuchao Tang

Mathematics 2026, 14(2), 245; https://doi.org/10.3390/math14020245 - 8 Jan 2026

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Abstract

The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality [...] Read more.

The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality mapping. In this paper, we propose and analyze an RFBS algorithm in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. To the best of our knowledge, this work presents the first strong (R-linear) convergence result for the RFBS method in such Banach spaces, achieved under a newly adapted notion of strong monotonicity. Our results thus establish a foundational theoretical guarantee for RFBS in Banach spaces under strengthened monotonicity conditions, while highlighting the open problem of proving weak convergence for the general monotone case. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

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16 pages, 1409 KB

Open AccessFeature PaperArticle

Solving Variational Inclusion Problems with Inertial S*Forward-Backward Algorithm and Application to Stroke Prediction Data Classification

by Wipawinee Chaiwino, Payakorn Saksuriya and Raweerote Suparatulatorn

Mathematics 2026, 14(1), 101; https://doi.org/10.3390/math14010101 - 26 Dec 2025

Viewed by 566

Abstract

This article introduces an iterative algorithm that is created by integrating the

S^{*}

-iteration process with the inertial forward-backward algorithm. The algorithm is designed to solve optimization problems formulated as variational inclusions in a real Hilbert space. We establish the weak convergence [...] Read more.

This article introduces an iterative algorithm that is created by integrating the

S^{*}

-iteration process with the inertial forward-backward algorithm. The algorithm is designed to solve optimization problems formulated as variational inclusions in a real Hilbert space. We establish the weak convergence of the algorithm under conventional assumptions. One of the applications of the algorithm is to solve the extreme learning machine, which can be transformed into the variational inclusion problem. Different algorithms, with all parameters set to be identical, are employed to solve the stroke classification problem in order to evaluate the algorithm’s performance. The results indicate that our algorithm converges faster than others and achieves a precision of 93.90%, a recall of 100%, and an F1-score of 96.58%. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

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15 pages, 586 KB

Open AccessArticle

On Probabilistic Convergence Rates of Symmetric Stochastic Bernstein Polynomials

by Shenggang Zhang, Qinjiao Gao and Chungang Zhu

Mathematics 2025, 13(20), 3281; https://doi.org/10.3390/math13203281 - 14 Oct 2025

Cited by 1 | Viewed by 719

Abstract

This paper analyzes the exponential convergence properties of Symmetric Stochastic Bernstein Polynomials (SSBPs), a novel approximation framework that combines the deterministic precision of classical Bernstein polynomials (BPs) with the adaptive node flexibility of Stochastic Bernstein Polynomials (SBPs). Through innovative applications of order statistics [...] Read more.

This paper analyzes the exponential convergence properties of Symmetric Stochastic Bernstein Polynomials (SSBPs), a novel approximation framework that combines the deterministic precision of classical Bernstein polynomials (BPs) with the adaptive node flexibility of Stochastic Bernstein Polynomials (SBPs). Through innovative applications of order statistics concentration inequalities and modulus of smoothness analysis, we derive the first probabilistic convergence rates for SSBPs across all

L_{p}

(

1 \leq p \leq \infty

) norms and in pointwise approximation. Numerical experiments demonstrate dual advantages: (1) SSBPs achieve comparable

L_{\infty}

errors to BPs in approximating fundamental stochastic functions (uniform distribution and normal density), while significantly outperforming SBPs; (2) empirical convergence curves validate exponential decay of approximation errors. These results position SSBPs as a principal solution for stochastic approximation problems requiring both mathematical rigor and computational adaptability. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

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12 pages, 269 KB

Open AccessArticle

On a p(x)-Biharmonic Kirchhoff Problem with Logarithmic Nonlinearity

by Dongyun Pan and Changmu Chu

Mathematics 2025, 13(18), 3054; https://doi.org/10.3390/math13183054 - 22 Sep 2025

Viewed by 774

Abstract

This paper is devoted to the study of a class of the

p (x)

-biharmonic Kirchhoff problem with logarithmic nonlinearity. With the help of the mountain pass theorem, the existence of a nontrivial weak solution to this problem is obtained. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

10 pages, 223 KB

Open AccessArticle

Nash’s Existence Theorem for Non-Compact Strategy Sets

by Xinyu Zhang, Chunyan Yang, Renjie Han and Shiqing Zhang

Mathematics 2024, 12(13), 2017; https://doi.org/10.3390/math12132017 - 28 Jun 2024

Viewed by 1370

Abstract

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, [...] Read more.

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

31 pages, 724 KB

Open AccessArticle

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

by Chein-Shan Liu, Chung-Lun Kuo and Chih-Wen Chang

Mathematics 2024, 12(11), 1761; https://doi.org/10.3390/math12111761 - 5 Jun 2024

Viewed by 2593

Abstract

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting [...] Read more.

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an

m \times m

positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m). Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

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