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21 pages, 380 KiB

Open AccessArticle

Approximation of Fixed Points of Multivalued Generalized (α,β)-Nonexpansive Mappings in an Ordered CAT(0) Space

by Mujahid Abbas, Hira Iqbal, Manuel De la Sen and Khushdil Ahmad

Mathematics 2021, 9(16), 1945; https://doi.org/10.3390/math9161945 - 15 Aug 2021

Cited by 1 | Viewed by 1507

Abstract

The purpose of this article is to initiate the notion of monotone multivalued generalized (

α

,

β

)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S [...] Read more.

The purpose of this article is to initiate the notion of monotone multivalued generalized (

α

,

β

)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S-iteration algorithm in CAT(0) space to prove some convergence results. Moreover, some examples and useful results related to the proposed mapping are provided. Numerical experiments are also provided to illustrate and compare the convergence of the iteration scheme. Finally, an application of the iterative scheme has been presented in finding the solutions of integral differential equation. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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12 pages, 292 KiB

Open AccessArticle

Nadler’s Theorem on Incomplete Modular Space

by Fatemeh Lael, Naeem Saleem, Liliana Guran and Monica Felicia Bota

Mathematics 2021, 9(16), 1927; https://doi.org/10.3390/math9161927 - 13 Aug 2021

Cited by 3 | Viewed by 2108

Abstract

This manuscript is focused on the role of convexity of the modular, and some fixed point results for contractive correspondence and single-valued mappings are presented. Further, we prove Nadler’s Theorem and some fixed point results on orthogonal modular spaces. We present an application [...] Read more.

This manuscript is focused on the role of convexity of the modular, and some fixed point results for contractive correspondence and single-valued mappings are presented. Further, we prove Nadler’s Theorem and some fixed point results on orthogonal modular spaces. We present an application to a particular form of integral inclusion to support our proposed version of Nadler’s theorem. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

18 pages, 317 KiB

Open AccessArticle

Inertial Extragradient Methods for Solving Split Equilibrium Problems

by Suthep Suantai, Narin Petrot and Manatchanok Khonchaliew

Mathematics 2021, 9(16), 1884; https://doi.org/10.3390/math9161884 - 08 Aug 2021

Cited by 2 | Viewed by 1420

Abstract

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. [...] Read more.

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

8 pages, 254 KiB

Open AccessArticle

Approximation of Endpoints for α—Reich–Suzuki Nonexpansive Mappings in Hyperbolic Metric Spaces

by Izhar Uddin, Sajan Aggarwal and Afrah A. N. Abdou

Mathematics 2021, 9(14), 1692; https://doi.org/10.3390/math9141692 - 19 Jul 2021

Cited by 3 | Viewed by 1685

Abstract

The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving

α

—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong [...] Read more.

The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving

α

—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong and

Δ

—convergence theorems in a hyperbolic metric space. Thus, our results generalize and improve many existing results. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

20 pages, 664 KiB

Open AccessArticle

A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications

by Dawan Chumpungam, Panitarn Sarnmeta and Suthep Suantai

Mathematics 2021, 9(13), 1562; https://doi.org/10.3390/math9131562 - 02 Jul 2021

Cited by 2 | Viewed by 1727

Abstract

For the past few decades, various algorithms have been proposed to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. The convergence of these algorithms was guaranteed under the L-Lipschitz condition on the gradient of [...] Read more.

For the past few decades, various algorithms have been proposed to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. The convergence of these algorithms was guaranteed under the L-Lipschitz condition on the gradient of the objective function. In recent years, an inertial technique has been widely used to accelerate the convergence behavior of an algorithm. In this work, we introduce a new forward–backward splitting algorithm using a new line search and inertial technique to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. A weak convergence of our proposed method is established without assuming the L-Lipschitz continuity of the gradient of the objective function. Moreover, a complexity theorem is also given. As applications, we employed our algorithm to solve data classification and image restoration by conducting some experiments on these problems. The performance of our algorithm was evaluated using various evaluation tools. Furthermore, we compared its performance with other algorithms. Based on the experiments, we found that the proposed algorithm performed better than other algorithms mentioned in the literature. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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14 pages, 308 KiB

Open AccessArticle

Some Fixed Point Results of Weak-Fuzzy Graphical Contraction Mappings with Application to Integral Equations

by Shamoona Jabeen, Zhiming Zheng, Mutti-Ur Rehman, Wei Wei and Jehad Alzabut

Mathematics 2021, 9(5), 541; https://doi.org/10.3390/math9050541 - 04 Mar 2021

Cited by 3 | Viewed by 1443

Abstract

The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some fixed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of [...] Read more.

The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some fixed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of mappings involved, our results enrich and generalize some well-known results in fixed point theory. With the help of new lemmas, our proofs are straight forward. We furnish the validity of our findings with appropriate examples. This approach is completely new and will be beneficial for the future aspects of the related study. We provide an application of integral equations to illustrate the usability of our theory. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

25 pages, 455 KiB

Open AccessFeature PaperArticle

Discrete Group Actions on Digital Objects and Fixed Point Sets by Iso_k(·)-Actions

by Sang-Eon Han

Mathematics 2021, 9(3), 290; https://doi.org/10.3390/math9030290 - 01 Feb 2021

Viewed by 1942

Abstract

Given a digital image (or digital object)

(X, k), X \subset Z^{n}

, this paper initially establishes a group structure of the set of self-k-isomorphisms of

(X, k)

with the function composition, denoted [...] Read more.

Given a digital image (or digital object)

(X, k), X \subset Z^{n}

, this paper initially establishes a group structure of the set of self-k-isomorphisms of

(X, k)

with the function composition, denoted by

I s o_{k} (X)

or

A u t_{k} (X)

. In particular, let

C_{k}^{n, l}

be a simple closed k-curve with l elements in

Z^{n}

. Then, the group

I s o_{k} (C_{k}^{n, l})

is proved to be isomorphic to the standard dihedral group

D_{l}

with order l. The calculation of this quantity

I s o_{k} (C_{k}^{n, l})

is a key step for obtaining many new results. Indeed, it is essential for exploring many features of

I s o_{k} (X)

. Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an

I s o_{k} (X)

-action on

(X, k)

, we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images

(X, k)

whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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10 pages, 1447 KiB

Open AccessArticle

Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

by Grzegorz Sroka and Mariusz Oszust

Mathematics 2021, 9(3), 264; https://doi.org/10.3390/math9030264 - 29 Jan 2021

Cited by 1 | Viewed by 1579

Abstract

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and [...] Read more.

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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25 pages, 700 KiB

Open AccessArticle

Fixed Point Sets of Digital Curves and Digital Surfaces

by Sang-Eon Han

Mathematics 2020, 8(11), 1896; https://doi.org/10.3390/math8111896 - 31 Oct 2020

Cited by 2 | Viewed by 1431

Abstract

Given a digital image (or digital object)

(X, k)

, we address some unsolved problems related to the study of fixed point sets of k-continuous self-maps of

(X, k)

from the viewpoints of digital curve and [...] Read more.

Given a digital image (or digital object)

(X, k)

, we address some unsolved problems related to the study of fixed point sets of k-continuous self-maps of

(X, k)

from the viewpoints of digital curve and digital surface theory. Consider two simple closed k-curves with

l_{i}

elements in

Z^{n}

,

i \in {1, 2}, l_{1} ⪈ l_{2} \geq 4

. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers

l_{i}, i \in {1, 2}

, instead of the k-adjacency. Furthermore, given digital k-surfaces, we also study an alignment of fixed point sets of digital k-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image

(X, k)

is assumed to be k-connected and

X^{♯} \geq 2

unless stated otherwise. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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21 pages, 523 KiB

Open AccessArticle

The Most Refined Axiom for a Digital Covering Space and Its Utilities

by Sang-Eon Han

Mathematics 2020, 8(11), 1868; https://doi.org/10.3390/math8111868 - 27 Oct 2020

Cited by 4 | Viewed by 1763

Abstract

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local [...] Read more.

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local

(k_{0}, k_{1})

-isomorphisms and use the most simplified local

(k_{0}, k_{1})

-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local

(k_{0}, k_{1})

-isomorphism is proved to be a

(k_{0}, k_{1})

-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed

k : = k (t, n)

-curve with five elements in

Z^{n}

, denoted by

S C_{k}^{n, 5}

. After introducing the notion of digital topological imbedding, we investigate some properties of

S C_{k}^{n, 5}

, where

k : = k (t, n), 3 \leq t \leq n

. Since

S C_{k}^{n, 5}

is the minimal and simple closed k-curve with odd elements in

Z^{n}

which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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10 pages, 232 KiB

Open AccessArticle

Local Sharp Vector Variational Type Inequality and Optimization Problems

by Jong Kyu Kim and Salahuddin

Mathematics 2020, 8(10), 1844; https://doi.org/10.3390/math8101844 - 20 Oct 2020

Cited by 3 | Viewed by 1637

Abstract

In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, [...] Read more.

In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate

η

-convexity conditions for locally Lipschitzian functions. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

12 pages, 923 KiB

Open AccessArticle

Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem

by Yaqian Jiang, Rudong Chen and Luoyi Shi

Mathematics 2020, 8(10), 1836; https://doi.org/10.3390/math8101836 - 19 Oct 2020

Viewed by 1363

Abstract

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in [...] Read more.

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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26 pages, 473 KiB

Open AccessArticle

Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

by Sang-Eon Han

Mathematics 2020, 8(9), 1617; https://doi.org/10.3390/math8091617 - 18 Sep 2020

Cited by 2 | Viewed by 1732

Abstract

Let

C_{k}^{n, l}

be a simple closed k-curves with l elements in

Z^{n}

and

W : = \overset{m - times}{\overset{︷}{C_{k}^{n, l} \lor \dots \lor C_{k}^{n, l}}}

be an m-iterated [...] Read more.

Let

C_{k}^{n, l}

be a simple closed k-curves with l elements in

Z^{n}

and

W : = \overset{m - times}{\overset{︷}{C_{k}^{n, l} \lor \dots \lor C_{k}^{n, l}}}

be an m-iterated digital wedges of

C_{k}^{n, l}

, and

F (C o n_{k} (W))

be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of

F (C o n_{k} (W))

. Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of

C_{k}^{n, l}

, denoted by

F (C o n_{k} (C_{k}^{n, l}))

, where

l (\geq 7)

is an odd natural number and

k \neq 2 n

. Secondly, given a digital image

(X, k)

with

X^{♯} = n

, we find a certain condition that supports

n - 1, n - 2 \in F (C o n_{k} (X))

. Thirdly, after finding some features of

F (C o n_{k} (W))

, we develop a method of making

F (C o n_{k} (W))

perfect according to the (even or odd) number l of

C_{k}^{n, l}

. Finally, we prove that the perfectness of

F (C o n_{k} (W))

is equivalent to that of

F (C o n_{k} (C_{k}^{n, l}))

. This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images

(X, k)

such that

X^{♯} \geq 2

. Full article

(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)

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