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24 pages, 1140 KB

Open AccessArticle

Flexible and Efficient Iterative Solutions for General Variational Inequalities in Real Hilbert Spaces

by Emirhan Hacıoğlu, Müzeyyen Ertürk, Faik Gürsoy and Gradimir V. Milovanović

Axioms 2025, 14(4), 288; https://doi.org/10.3390/axioms14040288 - 11 Apr 2025

Viewed by 493

This paper introduces a novel Picard-type iterative algorithm for solving general variational inequalities in real Hilbert spaces. The proposed algorithm enhances both the theoretical framework and practical applicability of iterative algorithms by relaxing restrictive conditions on parametric sequences, thereby expanding their scope of use. We establish convergence results, including a convergence equivalence with a previous algorithm, highlighting the theoretical relationship while demonstrating the increased flexibility and efficiency of the new approach. The paper also addresses gaps in the existing literature by offering new theoretical insights into the transformations associated with variational inequalities and the continuity of their solutions, thus paving the way for future research. The theoretical advancements are complemented by practical applications, such as the adaptation of the algorithm to convex optimization problems and its use in real-world contexts like machine learning. Numerical experiments confirm the proposed algorithm’s versatility and efficiency, showing superior performance and faster convergence compared to an existing method. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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17 pages, 588 KB

Open AccessArticle

Fractals of Interpolative Kannan Mappings

by Xiangting Shi, Umar Ishtiaq, Muhammad Din and Mohammad Akram

Fractal Fract. 2024, 8(8), 493; https://doi.org/10.3390/fractalfract8080493 - 21 Aug 2024

Cited by 37 | Viewed by 1491

Abstract

In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept by introducing the IKC-iterated function/multi-function system using interpolative Kannan operators, including a broader area of mappings. Moreover, we establish the Collage Theorem endowed with the iterated function system (IFS) based on the IKC, and show the well-posedness of the IKC-IFS. Interpolative Kannan contractions are meaningful due to their applications in fractals, offering a more versatile framework for creating intricate geometric structures with potentially fewer constraints compared to classical approaches. Full article

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21 pages, 336 KB

Open AccessArticle

The Generalized Iterated Function System and Common Attractors of Generalized Hutchinson Operators in Dislocated Metric Spaces

by Talat Nazir and Sergei Silvestrov

Fractal Fract. 2023, 7(12), 832; https://doi.org/10.3390/fractalfract7120832 - 23 Nov 2023

Cited by 1 | Viewed by 1566

Abstract

In this paper, we present the generalized iterated function system for the construction of common fractals of generalized contractive mappings in the setup of dislocated metric spaces. The well-posedness of attractors’ problems of rational contraction maps in the framework of dislocated metric spaces [...] Read more.

10 pages, 291 KB

Open AccessArticle

Fractals via Controlled Fisher Iterated Function System

by C. Thangaraj and D. Easwaramoorthy

Fractal Fract. 2022, 6(12), 746; https://doi.org/10.3390/fractalfract6120746 - 19 Dec 2022

Cited by 4 | Viewed by 2337

Abstract

This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the controlled metric space is used in this paper to generate a new type of fractal set called controlled Fisher fractals (CF-Fractals) by constructing a system named the controlled Fisher iterated function system (CF-IFS). Furthermore, the interesting results and consequences of the controlled Fisher iterated function system and controlled Fisher fractals are demonstrated. In addition, the collage theorem on controlled Fisher fractals is established as well. The newly developing IFS and fractal set in the controlled metric space can provide the novel directions in the fractal theory. Full article

(This article belongs to the Special Issue Stability Analysis and Control of Fractional-Order Markovian Jump Systems)

26 pages, 15315 KB

Open AccessArticle

IFS-Based Image Reconstruction of Binary Images with Functional Networks

by Akemi Gálvez, Iztok Fister, Andrés Iglesias, Iztok Fister, Valentín Gómez-Jauregui, Cristina Manchado and César Otero

Mathematics 2022, 10(7), 1107; https://doi.org/10.3390/math10071107 - 29 Mar 2022

Cited by 1 | Viewed by 2049

Abstract

This work addresses the IFS-based image reconstruction problem for binary images. Given a binary image as the input, the goal is to obtain all the parameters of an iterated function system whose attractor approximates the input image accurately; the quality of this approximation is measured according to a similarity function between the original and the reconstructed images. This paper introduces a new method to tackle this issue. The method is based on functional networks, a powerful extension of neural networks that uses functions instead of the scalar weights typically found in standard neural networks. The method relies on an artificial network comprised of several functional networks, one for each of the contractive affine maps forming the IFS. The method is applied to an illustrative and challenging example of a fractal binary image exhibiting a complicated shape. The graphical and numerical results show that the method performs very well and is able to reconstruct the input image using IFS with high accuracy. The results also show that the method is not yet optimal and offers room for further improvement. Full article

(This article belongs to the Special Issue Computer Graphics, Image Processing and Artificial Intelligence)

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17 pages, 291 KB

Open AccessArticle

Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications

by Sudesh Kumari, Renu Chugh, Jinde Cao and Chuangxia Huang

Mathematics 2019, 7(10), 967; https://doi.org/10.3390/math7100967 - 14 Oct 2019

Cited by 16 | Viewed by 3182

Abstract

In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results obtained by Chifu et al. (2014) and N.A. Secelean (2015) and generalize the results of Nazir et al. (2016) by using the assumptions imposed by Dung et al. (2017) to the case of ciric type generalized multi-iterated function system (CGMIFS) composed of ciric type generalized multivalued G-contractions defined on multifractal space

C (U)

in the framework of a Hausdorff b-metric space, where

U = U_{1} \times U_{2} \times \dots \times U_{N}

, N being a finite natural number. As an application of our study, we derive collage theorem which can be used to construct general fractals and to solve inverse problem in Hausdorff b-metric spaces which are more general spaces than Hausdorff metric spaces. Full article

(This article belongs to the Special Issue Impulsive Control Systems and Complexity)

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