Mathematics

Research

13 pages, 282 KiB

Open AccessArticle

Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems

by Yuanheng Wang, Mingyue Yuan and Bingnan Jiang

Mathematics 2021, 9(13), 1548; https://doi.org/10.3390/math9131548 - 1 Jul 2021

Cited by 6 | Viewed by 1782

Abstract

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong [...] Read more.

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems. Full article

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

21 pages, 812 KiB

Open AccessArticle

Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

by Li Wei, Xin-Wang Shen and Ravi P. Agarwal

Mathematics 2021, 9(13), 1504; https://doi.org/10.3390/math9131504 - 27 Jun 2021

Cited by 1 | Viewed by 1598

Abstract

Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under [...] Read more.

Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under mild restrictions, which extend and complement some already existing work. Full article

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

12 pages, 281 KiB

Open AccessArticle

A New Approach of Some Contractive Mappings on Metric Spaces

by Ion Marian Olaru and Nicolae Adrian Secelean

Mathematics 2021, 9(12), 1433; https://doi.org/10.3390/math9121433 - 19 Jun 2021

Cited by 7 | Viewed by 1976

Abstract

In this paper, we introduce a new contraction-type mapping and provide a fixed-point theorem which generalizes and improves some existing results in the literature. Thus, we prove that the Boyd and Wong theorem (1969) and, more recently, the fixed-point results due to Wardowski [...] Read more.

In this paper, we introduce a new contraction-type mapping and provide a fixed-point theorem which generalizes and improves some existing results in the literature. Thus, we prove that the Boyd and Wong theorem (1969) and, more recently, the fixed-point results due to Wardowski (2012), Turinici (2012), Piri and Kumam (2016), Secelean (2016), Proinov (2020), and others are consequences of our main result. An application in integral equations and some illustrative examples are indicated. Full article

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

24 pages, 1106 KiB

Open AccessArticle

Halpern-Subgradient Extragradient Method for Solving Equilibrium and Common Fixed Point Problems in Reflexive Banach Spaces

by Annel Thembinkosi Bokodisa, Lateef Olakunle Jolaoso and Maggie Aphane

Mathematics 2021, 9(7), 743; https://doi.org/10.3390/math9070743 - 31 Mar 2021

Viewed by 2041

Abstract

In this paper, using the concept of Bregman distance, we introduce a new Bregman subgradient extragradient method for solving equilibrium and common fixed point problems in a real reflexive Banach space. The algorithm is designed, such that the stepsize is chosen without prior [...] Read more.

In this paper, using the concept of Bregman distance, we introduce a new Bregman subgradient extragradient method for solving equilibrium and common fixed point problems in a real reflexive Banach space. The algorithm is designed, such that the stepsize is chosen without prior knowledge of the Lipschitz constants. We also prove a strong convergence result for the sequence that is generated by our algorithm under mild conditions. We apply our result to solving variational inequality problems, and finally, we give some numerical examples to illustrate the efficiency and accuracy of the algorithm. Full article

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

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19 pages, 827 KiB

Open AccessArticle

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

by Nishu Gupta, Mihai Postolache, Ashish Nandal and Renu Chugh

Mathematics 2021, 9(4), 372; https://doi.org/10.3390/math9040372 - 13 Feb 2021

Cited by 19 | Viewed by 1988

Abstract

The aim of this paper is to formulate and analyze a cyclic iterative algorithm in real Hilbert spaces which converges strongly to a common solution of fixed point problem and multiple-sets split common fixed point problem involving demicontractive operators without prior knowledge of [...] Read more.

The aim of this paper is to formulate and analyze a cyclic iterative algorithm in real Hilbert spaces which converges strongly to a common solution of fixed point problem and multiple-sets split common fixed point problem involving demicontractive operators without prior knowledge of operator norm. Significance and range of applicability of our algorithm has been shown by solving the problem of multiple-sets split common null point, multiple-sets split feasibility, multiple-sets split variational inequality, multiple-sets split equilibrium and multiple-sets split monotone variational inclusion. Full article

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

11 pages, 259 KiB

Open AccessArticle

On An Open Question in Controlled Rectangular b-Metric Spaces

by Reny George, Abdelkader Belhenniche, Sfya Benahmed, Zoran D. Mitrović, Nabil Mlaiki and Liliana Guran

Mathematics 2020, 8(12), 2239; https://doi.org/10.3390/math8122239 - 18 Dec 2020

Cited by 4 | Viewed by 2557

Abstract

In this paper, we give an affirmative answer to an open question posed recently by Mlaiki et al. As a consequence of our results, we get some known results in the literature. We also give an application of our results to the existence [...] Read more.

In this paper, we give an affirmative answer to an open question posed recently by Mlaiki et al. As a consequence of our results, we get some known results in the literature. We also give an application of our results to the existence of a solution of nonlinear fractional differential equations. Full article

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

17 pages, 279 KiB

Open AccessArticle

Set-Valued Symmetric Generalized Strong Vector Quasi-Equilibrium Problems with Variable Ordering Structures

by Jing-Nan Li, San-Hua Wang and Yu-Ping Xu

Mathematics 2020, 8(9), 1604; https://doi.org/10.3390/math8091604 - 17 Sep 2020

Cited by 2 | Viewed by 1756

Abstract

In this paper, two types of set-valued symmetric generalized strong vector quasi-equilibrium problems with variable ordering structures are discussed. By using the concept of cosmically upper continuity rather than the one of upper semicontinuity for cone-valued mapping, some existence theorems of solutions are [...] Read more.

In this paper, two types of set-valued symmetric generalized strong vector quasi-equilibrium problems with variable ordering structures are discussed. By using the concept of cosmically upper continuity rather than the one of upper semicontinuity for cone-valued mapping, some existence theorems of solutions are established under suitable assumptions of cone-continuity and cone-convexity for the equilibrium mappings. Moreover, the results of compactness for solution sets are proven. As applications, some existence results of strong saddle points are obtained. The main results obtained in this paper unify and improve some recent works in the literature. Full article

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

17 pages, 296 KiB

Open AccessArticle

Minirobots Moving at Different Partial Speeds

by Constantin Udrişte and Ionel Ţevy

Mathematics 2020, 8(6), 1036; https://doi.org/10.3390/math8061036 - 24 Jun 2020

Cited by 2 | Viewed by 2318

Abstract

In this paper, we present the mathematical point of view of our research group regarding the multi-robot systems evolving in a multi-temporal way. We solve the minimum multi-time volume problem as optimal control problem for a group of planar micro-robots moving in the [...] Read more.

In this paper, we present the mathematical point of view of our research group regarding the multi-robot systems evolving in a multi-temporal way. We solve the minimum multi-time volume problem as optimal control problem for a group of planar micro-robots moving in the same direction at different partial speeds. We are motivated to solve this problem because a similar minimum-time optimal control problem is now in vogue for micro-scale and nano-scale robotic systems. Applying the (weak and strong) multi-time maximum principle, we obtain necessary conditions for optimality and that are used to guess a candidate control policy. The complexity of finding this policy for arbitrary initial conditions is dominated by the computation of a planar convex hull. We pointed this idea by applying the technique of multi-time Hamilton-Jacobi-Bellman PDE. Our results can be extended to consider obstacle avoidance by explicit parameterization of all possible optimal control policies. Full article

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

11 pages, 263 KiB

Open AccessArticle

Strong Convergence of Mann’s Iteration Process in Banach Spaces

by Hong-Kun Xu, Najla Altwaijry and Souhail Chebbi

Mathematics 2020, 8(6), 954; https://doi.org/10.3390/math8060954 - 11 Jun 2020

Cited by 5 | Viewed by 2603

Abstract

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but [...] Read more.

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants. Full article

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

8 pages, 232 KiB

Open AccessArticle

Δ-Convergence of Products of Operators in p-Uniformly Convex Metric Spaces

by Byoung Jin Choi

Mathematics 2020, 8(5), 741; https://doi.org/10.3390/math8050741 - 8 May 2020

Cited by 3 | Viewed by 2078

Abstract

In this paper, we first introduce the new notion of p-strongly quasi-nonexpansive maps on p-uniformly convex metric spaces, and then we study the

Δ

(weak)-convergence of products of p-strongly quasi-nonexpansive maps on p-uniformly convex metric spaces. Furthermore, using the [...] Read more.

In this paper, we first introduce the new notion of p-strongly quasi-nonexpansive maps on p-uniformly convex metric spaces, and then we study the

Δ

(weak)-convergence of products of p-strongly quasi-nonexpansive maps on p-uniformly convex metric spaces. Furthermore, using the result, we prove the

Δ

-convergence of the weighted averaged method for projection operators. Full article

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

13 pages, 276 KiB

Open AccessArticle

On Pata–Suzuki-Type Contractions

by Erdal Karapınar and V. M. L. Hima Bindu

Mathematics 2020, 8(3), 389; https://doi.org/10.3390/math8030389 - 10 Mar 2020

Cited by 5 | Viewed by 2084

Abstract

In this manuscript, we introduce two notions, Pata–Suzuki

Z

-contraction and Pata

Z

-contraction for the pair of self-mapping

g, f

in the context of metric spaces. For such types of contractions, both the existence and uniqueness of a common fixed point [...] Read more.

In this manuscript, we introduce two notions, Pata–Suzuki

Z

-contraction and Pata

Z

-contraction for the pair of self-mapping

g, f

in the context of metric spaces. For such types of contractions, both the existence and uniqueness of a common fixed point are examined. We provide examples to illustrate the validity of the given results. Further, we consider ordinary differential equations to apply our obtained results. Full article

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

6 pages, 262 KiB

Open AccessArticle

Ekeland Variational Principle in the Variable Exponent Sequence Spaces ℓ_p(·)

by Monther R. Alfuraidan and Mohamed A. Khamsi

Mathematics 2020, 8(3), 375; https://doi.org/10.3390/math8030375 - 7 Mar 2020

Viewed by 2348

Abstract

In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces

ℓ_{p (\cdot)}

. The core obstacle in the development of a modular version of the EVP is the [...] Read more.

In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces

ℓ_{p (\cdot)}

. The core obstacle in the development of a modular version of the EVP is the failure of the triangle inequality for the module. It is the lack of this inequality, which is indispensable in the establishment of the classical EVP, that has hitherto prevented a successful treatment of the modular case. As an application, we establish a modular version of Caristi’s fixed point theorem in

ℓ_{p (\cdot)}

. Full article

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

34 pages, 435 KiB

Open AccessArticle

Output-Space Branch-and-Bound Reduction Algorithm for a Class of Linear Multiplicative Programs

by Bo Zhang, Yuelin Gao, Xia Liu and Xiaoli Huang

Mathematics 2020, 8(3), 315; https://doi.org/10.3390/math8030315 - 1 Mar 2020

Cited by 24 | Viewed by 2494

Abstract

In this paper, a new relaxation bounding method is proposed for a class of linear multiplicative programs. Although the

2 p - 1

variable is introduced in the construction of equivalence problem, the branch process of the algorithm is only carried out in [...] Read more.

In this paper, a new relaxation bounding method is proposed for a class of linear multiplicative programs. Although the

2 p - 1

variable is introduced in the construction of equivalence problem, the branch process of the algorithm is only carried out in

p -

dimensional space. In addition, a super-rectangular reduction technique is also given to greatly improve the convergence rate. Furthermore, we construct an output-space branch-and-bound reduction algorithm based on solving a series of linear programming sub-problems, and prove the convergence and computational complexity of the algorithm. Finally, to verify the feasibility and effectiveness of the algorithm, we carried out a series of numerical experiments and analyzed the advantages and disadvantages of the algorithm by numerical results. Full article

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

12 pages, 247 KiB

Open AccessArticle

Common Fixed Point and Endpoint Theorems for a Countable Family of Multi-Valued Mappings

by Hüseyin Işık, Babak Mohammadi, Choonkil Park and Vahid Parvaneh

Mathematics 2020, 8(2), 292; https://doi.org/10.3390/math8020292 - 21 Feb 2020

Cited by 3 | Viewed by 2244

Abstract

We prove some common fixed point and endpoint theorems for a countable infinite family of multi-valued mappings, as well as Allahyari et al. (2015) did for self-mappings. An example and an application to a system of integral equations are given to show the [...] Read more.

We prove some common fixed point and endpoint theorems for a countable infinite family of multi-valued mappings, as well as Allahyari et al. (2015) did for self-mappings. An example and an application to a system of integral equations are given to show the usability of the results. Full article

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

7 pages, 249 KiB

Open AccessArticle

New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings

by Hüseyin Işık, Mohammad Reza Haddadi, Vahid Parvaneh, Choonkil Park and Somayeh Kornokar

Mathematics 2020, 8(2), 284; https://doi.org/10.3390/math8020284 - 20 Feb 2020

Viewed by 1926

Abstract

In recent works, many authors generated strongly relatively nonexpansive sequences of mappings by the sequences of firmly nonexpansive-like mappings. In this paper, we introduce a new method for construction of strongly relatively nonexpansive sequences from firmly nonexpansive-like mappings. Full article

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

11 pages, 765 KiB

Open AccessArticle

Quasi (s,r)-Contractive Multi-Valued Operators and Related Fixed Point Theorems

by Ei Ei Nyein, Dianlu Tian and Aung Khaing Zaw

Mathematics 2020, 8(1), 64; https://doi.org/10.3390/math8010064 - 2 Jan 2020

Cited by 1 | Viewed by 2188

Abstract

This paper gives the new concepts of quasi

(s, r)

-contractive multi-valued operators and establishes some related fixed point results for such operators. In addition, an application to certain functional equations arising from dynamic programming is given to illustrate the [...] Read more.

This paper gives the new concepts of quasi

(s, r)

-contractive multi-valued operators and establishes some related fixed point results for such operators. In addition, an application to certain functional equations arising from dynamic programming is given to illustrate the usage of the obtained results. Full article

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

14 pages, 789 KiB

Open AccessFeature PaperArticle

On a Periodic Boundary Value Problem for a Fractional–Order Semilinear Functional Differential Inclusions in a Banach Space

by Mikhail Kamenski, Valeri Obukhovskii, Garik Petrosyan and Jen-Chih Yao

Mathematics 2019, 7(12), 1146; https://doi.org/10.3390/math7121146 - 23 Nov 2019

Cited by 11 | Viewed by 2202

Abstract

We consider the periodic boundary value problem (PBVP) for a semilinear fractional-order delayed functional differential inclusion in a Banach space. We introduce and study a multivalued integral operator whose fixed points coincide with mild solutions of our problem. On that base, we prove [...] Read more.

We consider the periodic boundary value problem (PBVP) for a semilinear fractional-order delayed functional differential inclusion in a Banach space. We introduce and study a multivalued integral operator whose fixed points coincide with mild solutions of our problem. On that base, we prove the main existence result (Theorem 4). We present an example dealing with existence of a trajectory for a time-fractional diffusion type feedback control system with a delay satisfying periodic boundary value condition. Full article

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

12 pages, 262 KiB

Open AccessArticle

Common Fixed Point Results for Generalized Wardowski Type Contractive Multi-Valued Mappings

by Hüseyin Işık, Vahid Parvaneh, Babak Mohammadi and Ishak Altun

Mathematics 2019, 7(11), 1130; https://doi.org/10.3390/math7111130 - 18 Nov 2019

Cited by 14 | Viewed by 2300

Abstract

In this paper, we introduce generalized Wardowski type quasi-contractions called

α

-

(φ, Ω)

-contractions for a pair of multi-valued mappings and prove the existence of the common fixed point for such mappings. An illustrative example and an application are [...] Read more.

In this paper, we introduce generalized Wardowski type quasi-contractions called

α

-

(φ, Ω)

-contractions for a pair of multi-valued mappings and prove the existence of the common fixed point for such mappings. An illustrative example and an application are given to show the usability of our results. Full article

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

26 pages, 923 KiB

Open AccessArticle

Modified Halpern Iterative Method for Solving Hierarchical Problem and Split Combination of Variational Inclusion Problem in Hilbert Space

by Bunyawee Chaloemyotphong and Atid Kangtunyakarn

Mathematics 2019, 7(11), 1037; https://doi.org/10.3390/math7111037 - 3 Nov 2019

Cited by 1 | Viewed by 2108

Abstract

The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern [...] Read more.

The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern iterative method, we prove the strong convergence theorem for finding a common solution for the hierarchical fixed point problem and the split combination of variational inclusion problem. The result presented in this paper demonstrates the corresponding result for the split zero point problem and the split combination of variation inequality problem. Moreover, we discuss a numerical example for supporting our result and the numerical example shows that our result is not true if some conditions fail. Full article

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

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7 pages, 224 KiB

Open AccessArticle

An Investigation of the Common Solutions for Coupled Systems of Functional Equations Arising in Dynamic Programming

by Hüseyin Işık and Wutiphol Sintunavarat

Mathematics 2019, 7(10), 977; https://doi.org/10.3390/math7100977 - 16 Oct 2019

Cited by 3 | Viewed by 2248

Abstract

The purpose of this paper is to introduce the new notion of a specific point in the space of the bounded real-valued functions on a given non-empty set and present a result based on the existence and uniqueness of such points. As a [...] Read more.

The purpose of this paper is to introduce the new notion of a specific point in the space of the bounded real-valued functions on a given non-empty set and present a result based on the existence and uniqueness of such points. As a consequence of our results, we discuss the existence of a unique common solution to coupled systems of functional equations arising in dynamic programming. Full article

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

10 pages, 278 KiB

Open AccessArticle

Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods

by Xin Xu and Yang Dong Xu

Mathematics 2019, 7(10), 947; https://doi.org/10.3390/math7100947 - 11 Oct 2019

Viewed by 2221

Abstract

The connectedness and path connectedness of the solution sets to vector optimization problems is an important and interesting study in optimization theories and applications. Most papers involving the direction established the connectedness and connectedness for the solution sets of vector optimization problems or [...] Read more.

The connectedness and path connectedness of the solution sets to vector optimization problems is an important and interesting study in optimization theories and applications. Most papers involving the direction established the connectedness and connectedness for the solution sets of vector optimization problems or vector equilibrium problems by means of the linear scalarization method rather than the nonlinear scalarization method. The aim of the paper is to deal with the connectedness and the path connectedness for the weak efficient solution set to a vector optimization problem by using the nonlinear scalarization method. Firstly, the union relationship between the weak efficient solution set to the vector optimization problem and the solution sets to a series of parametric scalar minimization problems, is established. Then, some properties of the solution sets of scalar minimization problems are investigated. Finally, by using the union relationship, the connectedness and the path connectedness for the weak efficient solution set of the vector optimization problem are obtained. Full article

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

10 pages, 251 KiB

Open AccessArticle

Gradient Methods with Selection Technique for the Multiple-Sets Split Equality Problem

by Dianlu Tian, Lining Jiang and Luoyi Shi

Mathematics 2019, 7(10), 928; https://doi.org/10.3390/math7100928 - 8 Oct 2019

Cited by 4 | Viewed by 1973

Abstract

The inverse problem is one of the four major problems in computational mathematics. There is an inverse problem in medical image reconstruction and radiotherapy that is called the multiple-sets split equality problem. The multiple-sets split equality problem is a unified form of the [...] Read more.

The inverse problem is one of the four major problems in computational mathematics. There is an inverse problem in medical image reconstruction and radiotherapy that is called the multiple-sets split equality problem. The multiple-sets split equality problem is a unified form of the split feasibility problem, split equality problem, and split common fixed point problem. In this paper, we present two iterative algorithms for solving it. The suggested algorithms are based on the gradient method with a selection technique. Based on this technique, we only need to calculate one projection in each iteration. Full article

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

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18 pages, 838 KiB

Open AccessArticle

Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis

by Anchalee Sripattanet and Atid Kangtunyakarn

Mathematics 2019, 7(10), 916; https://doi.org/10.3390/math7100916 - 2 Oct 2019

Cited by 2 | Viewed by 2755

Abstract

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational [...] Read more.

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented. Full article

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

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9 pages, 256 KiB

Open AccessArticle

Fixed Points for a Pair of F-Dominated Contractive Mappings in Rectangular b-Metric Spaces with Graph

by Tahair Rasham, Giuseppe Marino and Abdullah Shoaib

Mathematics 2019, 7(10), 884; https://doi.org/10.3390/math7100884 - 23 Sep 2019

Cited by 6 | Viewed by 2500

Abstract

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of [...] Read more.

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results. Full article

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

21 pages, 823 KiB

Open AccessArticle

A New Global Optimization Algorithm for a Class of Linear Fractional Programming

by X. Liu, Y.L. Gao, B. Zhang and F.P. Tian

Mathematics 2019, 7(9), 867; https://doi.org/10.3390/math7090867 - 19 Sep 2019

Cited by 29 | Viewed by 3026

Abstract

In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the [...] Read more.

In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the same time, p new nonlinear equality constraints are added to the original problem. By classifying the coefficient symbols of all linear functions in the objective function of the original problem, four sets are obtained, which are

I_{i}^{+}

,

I_{i}^{-}

,

J_{i}^{+}

and

J_{i}^{-}

. Combined with the multiplication rule of real number operation, the objective function and constraint conditions of the equivalent problem are linearized into a lower bound linear relaxation programming problem. Our lower bound determination method only needs

e_{i}^{T} x + f_{i} \neq 0

, and there is no need to convert molecules to non-negative forms in advance for some special problems. A output-space branch and bound algorithm based on solving the linear programming problem is proposed and the convergence of the algorithm is proved. Finally, in order to illustrate the feasibility and effectiveness of the algorithm, we have done a series of numerical experiments, and show the advantages and disadvantages of our algorithm by the numerical results. Full article

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

8 pages, 239 KiB

Open AccessArticle

On a New Generalization of Banach Contraction Principle with Application

by Hüseyin Işık, Babak Mohammadi, Mohammad Reza Haddadi and Vahid Parvaneh

Mathematics 2019, 7(9), 862; https://doi.org/10.3390/math7090862 - 18 Sep 2019

Cited by 9 | Viewed by 2817

Abstract

The main purpose of the current work is to present firstly a new generalization of Caristi’s fixed point result and secondly the Banach contraction principle. An example and an application is given to show the usability of our results. Full article

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

13 pages, 280 KiB

Open AccessArticle

Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption

by Jinzuo Chen, Mihai Postolache and Li-Jun Zhu

Mathematics 2019, 7(9), 777; https://doi.org/10.3390/math7090777 - 23 Aug 2019

Cited by 6 | Viewed by 2352

Abstract

Two iterative algorithms are suggested for approximating a solution of the split common fixed point problem involved in pseudo-contractive operators without Lipschitz assumption. We prove that the sequence generated by the first algorithm converges weakly to a solution of the split common fixed [...] Read more.

Two iterative algorithms are suggested for approximating a solution of the split common fixed point problem involved in pseudo-contractive operators without Lipschitz assumption. We prove that the sequence generated by the first algorithm converges weakly to a solution of the split common fixed point problem and the second one converges strongly. Moreover, the sequence

{x_{n}}

generated by Algorithm 3 strongly converges to

z = {proj}_{S} 0,

which is the minimum-norm solution of problem (1). Numerical examples are included. Full article

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

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24 pages, 368 KiB

Open AccessArticle

Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

by Mihai Postolache, Ashish Nandal and Renu Chugh

Mathematics 2019, 7(9), 773; https://doi.org/10.3390/math7090773 - 22 Aug 2019

Cited by 12 | Viewed by 2525

Abstract

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity [...] Read more.

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward-backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem. Full article

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

15 pages, 873 KiB

Open AccessArticle

Three-Step Projective Methods for Solving the Split Feasibility Problems

by Suthep Suantai, Nontawat Eiamniran, Nattawut Pholasa and Prasit Cholamjiak

Mathematics 2019, 7(8), 712; https://doi.org/10.3390/math7080712 - 6 Aug 2019

Cited by 10 | Viewed by 3773

Abstract

In this paper, we focus on studying the split feasibility problem (SFP) in Hilbert spaces. Based on the CQ algorithm involving the self-adaptive technique, we introduce a three-step iteration process for approximating the solution of SFP. Then, the convergence results are established under [...] Read more.

In this paper, we focus on studying the split feasibility problem (SFP) in Hilbert spaces. Based on the CQ algorithm involving the self-adaptive technique, we introduce a three-step iteration process for approximating the solution of SFP. Then, the convergence results are established under mild conditions. Numerical experiments are provided to show the efficiency in signal processing. Some comparisons to various methods are also provided in this paper. Full article

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

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19 pages, 337 KiB

Open AccessArticle

On Fixed Point Results in G_b-Metric Spaces

by Hassen Aydi, Dušan Rakić, Asadolah Aghajani, Tatjana Došenović, Mohd Salmi Md Noorani and Haitham Qawaqneh

Mathematics 2019, 7(7), 617; https://doi.org/10.3390/math7070617 - 11 Jul 2019

Cited by 20 | Viewed by 4029

Abstract

The purpose of this paper is to consider various results in the context of G

_{b}

-metric spaces that have been recently published after the paper (Aghajani, A.; Abbas, M.; Roshan, J.R. Common fixed point of generalized weak contractive mappings in partially ordered [...] Read more.

The purpose of this paper is to consider various results in the context of G

_{b}

-metric spaces that have been recently published after the paper (Aghajani, A.; Abbas, M.; Roshan, J.R. Common fixed point of generalized weak contractive mappings in partially ordered G

_{b}

-metric spaces. Filomat 2014, 28, 1087–1101). Our new results improve, complement, unify, enrich and generalize already well known results on G

_{b}

-metric spaces. Moreover, some coupled and tripled coincidence point results have been provided. Full article

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

14 pages, 288 KiB

Open AccessArticle

Fixed Point Results for Multivalued Prešić Type Weakly Contractive Mappings

by Abdul Latif, Talat Nazir and Mujahid Abbas

Mathematics 2019, 7(7), 601; https://doi.org/10.3390/math7070601 - 6 Jul 2019

Cited by 4 | Viewed by 2472

Abstract

We present fixed points results of multivalued Prešić type k-step iterative mappings satisfying generalized weakly contraction conditions in metric spaces. An example is presented to support the main result proved herein. The stability of fixed point sets of multivalued Prešić type weakly [...] Read more.

We present fixed points results of multivalued Prešić type k-step iterative mappings satisfying generalized weakly contraction conditions in metric spaces. An example is presented to support the main result proved herein. The stability of fixed point sets of multivalued Prešić type weakly contractive mappings are also established. Global attractivity result for the class of matrix difference equations is derived as application of the result presented herein. These results generalize and extend various comparable results in the existing literature. Full article

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

13 pages, 297 KiB

Open AccessArticle

F -Metric, F-Contraction and Common Fixed-Point Theorems with Applications

by Awais Asif, Muhammad Nazam, Muhammad Arshad and Sang Og Kim

Mathematics 2019, 7(7), 586; https://doi.org/10.3390/math7070586 - 30 Jun 2019

Cited by 21 | Viewed by 5908

Abstract

In this paper, we noticed that the existence of fixed points of F-contractions, in

F

-metric space, can be ensured without the third condition (F3) imposed on the Wardowski function

F : (0, \infty) \to R

. We obtain [...] Read more.

In this paper, we noticed that the existence of fixed points of F-contractions, in

F

-metric space, can be ensured without the third condition (F3) imposed on the Wardowski function

F : (0, \infty) \to R

. We obtain fixed points as well as common fixed-point results for Reich-type F-contractions for both single and set-valued mappings in

F

-metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented. The application shows the role of fixed-point theorems in dynamic programming, which is widely used in computer programming and optimization. Our results extend and generalize the previous results in the existing literature. Full article

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

11 pages, 268 KiB

Open AccessArticle

Extended Mizoguchi-Takahashi Type Fixed Point Theorems and Their Application

by Babak Mohammadi, Vahid Parvaneh, Hassen Aydi and Hüseyin Işık

Mathematics 2019, 7(7), 575; https://doi.org/10.3390/math7070575 - 27 Jun 2019

Cited by 5 | Viewed by 2605

Abstract

The aim of this work is to extend the Mizoguchi-Takahashi fixed point result motivated by the approach of Wardowski (2012) and provide some related fixed point results in (ordered) metric spaces. An example is given to support the main results. Moreover, we provide [...] Read more.

The aim of this work is to extend the Mizoguchi-Takahashi fixed point result motivated by the approach of Wardowski (2012) and provide some related fixed point results in (ordered) metric spaces. An example is given to support the main results. Moreover, we provide an application on nonlinear differential equations. Full article

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

14 pages, 296 KiB

Open AccessArticle

Bounded Perturbation Resilience and Superiorization of Proximal Scaled Gradient Algorithm with Multi-Parameters

by Yanni Guo and Xiaozhi Zhao

Mathematics 2019, 7(6), 535; https://doi.org/10.3390/math7060535 - 12 Jun 2019

Cited by 2 | Viewed by 2573

Abstract

In this paper, a multi-parameter proximal scaled gradient algorithm with outer perturbations is presented in real Hilbert space. The strong convergence of the generated sequence is proved. The bounded perturbation resilience and the superiorized version of the original algorithm are also discussed. The [...] Read more.

In this paper, a multi-parameter proximal scaled gradient algorithm with outer perturbations is presented in real Hilbert space. The strong convergence of the generated sequence is proved. The bounded perturbation resilience and the superiorized version of the original algorithm are also discussed. The validity and the comparison with the use or not of superiorization of the proposed algorithms were illustrated by solving the

l^{1} - l^{2}

problem. Full article

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

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11 pages, 256 KiB

Open AccessArticle

Approximation of Fixed Points for Suzuki’s Generalized Non-Expansive Mappings

by Javid Ali, Faeem Ali and Puneet Kumar

Mathematics 2019, 7(6), 522; https://doi.org/10.3390/math7060522 - 6 Jun 2019

Cited by 17 | Viewed by 3308

Abstract

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki’s generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that the considered iterative [...] Read more.

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki’s generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that the considered iterative scheme converges faster than some other known iterations for Suzuki’s generalized non-expansive mappings. To support our claim, we give an illustrative numerical example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature. Full article

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

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16 pages, 1397 KiB

Open AccessArticle

The Generalized Viscosity Implicit Midpoint Rule for Nonexpansive Mappings in Banach Space

by Huancheng Zhang, Yunhua Qu and Yongfu Su

Mathematics 2019, 7(6), 512; https://doi.org/10.3390/math7060512 - 4 Jun 2019

Cited by 1 | Viewed by 2284

Abstract

This paper constructs the generalized viscosity implicit midpoint rule for nonexpansive mappings in Banach space. It obtains strong convergence conclusions for the proposed algorithm and promotes the related results in this field. Moreover, this paper gives some applications. Finally, the paper gives six [...] Read more.

This paper constructs the generalized viscosity implicit midpoint rule for nonexpansive mappings in Banach space. It obtains strong convergence conclusions for the proposed algorithm and promotes the related results in this field. Moreover, this paper gives some applications. Finally, the paper gives six numerical examples to support the main results. Full article

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

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12 pages, 264 KiB

Open AccessArticle

Prešić Type Nonself Operators and Related Best Proximity Results

by Muhammad Usman Ali, Misbah Farheen, Tayyab Kamran and Georgeta Maniu

Mathematics 2019, 7(5), 394; https://doi.org/10.3390/math7050394 - 30 Apr 2019

Cited by 6 | Viewed by 2372

Abstract

The purpose of this article is to discuss the existence of best proximity points for Pre

\overset{ˇ}{s}

i

\overset{´}{c}

-type nonself operators, say

T : A^{k} \to B

. We also give several examples to support our results. As a [...] Read more.

The purpose of this article is to discuss the existence of best proximity points for Pre

\overset{ˇ}{s}

i

\overset{´}{c}

-type nonself operators, say

T : A^{k} \to B

. We also give several examples to support our results. As a consequence of our results, we have provided some interesting formulations of Pre

\overset{ˇ}{s}

i

\overset{´}{c}

fixed point results. Full article

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

13 pages, 292 KiB

Open AccessFeature PaperArticle

Numerical Reckoning Fixed Points of (ρE)-Type Mappings in Modular Vector Spaces

by Wissam Kassab and Teodor Ţurcanu

Mathematics 2019, 7(5), 390; https://doi.org/10.3390/math7050390 - 29 Apr 2019

Cited by 16 | Viewed by 2931

Abstract

In this paper, we study an iteration process introduced by Thakur et al. for Suzuki mappings in Banach spaces, in the new context of modular vector spaces. We establish existence results for a more recent version of Suzuki generalized non-expansive mappings. The stability [...] Read more.

In this paper, we study an iteration process introduced by Thakur et al. for Suzuki mappings in Banach spaces, in the new context of modular vector spaces. We establish existence results for a more recent version of Suzuki generalized non-expansive mappings. The stability and data dependence of the scheme for

ρ

-contractions is studied as well. Full article

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

13 pages, 290 KiB

Open AccessArticle

Generalized Viscosity Implicit Iterative Process for Asymptotically Non-Expansive Mappings in Banach Spaces

by Chanjuan Pan and Yuanheng Wang

Mathematics 2019, 7(5), 379; https://doi.org/10.3390/math7050379 - 26 Apr 2019

Cited by 7 | Viewed by 2458

Abstract

In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium [...] Read more.

In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium problems. Further, a numerical example is given to illustrate our convergence analysis. The results generalize and improve corresponding results in the literature. Full article

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

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12 pages, 242 KiB

Open AccessArticle

Cayley Inclusion Problem Involving XOR-Operation

by Imran Ali, Rais Ahmad and Ching-Feng Wen

Mathematics 2019, 7(3), 302; https://doi.org/10.3390/math7030302 - 25 Mar 2019

Cited by 11 | Viewed by 3523

Abstract

In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point [...] Read more.

In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work. Full article

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

20 pages, 337 KiB

Open AccessFeature PaperArticle

Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions

by Lu-Chuan Ceng, Mihai Postolache, Ching-Feng Wen and Yonghong Yao

Mathematics 2019, 7(3), 270; https://doi.org/10.3390/math7030270 - 16 Mar 2019

Cited by 10 | Viewed by 2239

Abstract

Multistep composite implicit and explicit extragradient-like schemes are presented for solving the minimization problem with the constraints of variational inclusions and generalized mixed equilibrium problems. Strong convergence results of introduced schemes are given under suitable control conditions. Full article

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

20 pages, 590 KiB

Open AccessArticle

A General Algorithm for the Split Common Fixed Point Problem with Its Applications to Signal Processing

by Wachirapong Jirakitpuwapat, Poom Kumam, Yeol Je Cho and Kanokwan Sitthithakerngkiet

Mathematics 2019, 7(3), 226; https://doi.org/10.3390/math7030226 - 28 Feb 2019

Cited by 19 | Viewed by 3786

Abstract

In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s [...] Read more.

In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s motivation, in 2015, Boikanyo constructed also a new algorithm for demicontractive operators and obtained some strong convergence theorems for this problem without using the operator norm. In this paper, we consider a viscosity iterative algorithm in Boikanyo’s algorithm to approximate to a solution of this problem and prove some strong convergence theorems of our proposed algorithm to a solution of this problem. Finally, we apply our main results to some applications, signal processing and others and compare our algorithm with five algorithms such as Cui and Wang’s algorithm, Boikanyo’s algorithm, forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm (FISTA). Full article

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

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20 pages, 280 KiB

Open AccessArticle

Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems

by Lu-Chuan Ceng and Xiaoye Yang

Mathematics 2019, 7(3), 218; https://doi.org/10.3390/math7030218 - 26 Feb 2019

Cited by 1 | Viewed by 2332

Abstract

This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive [...] Read more.

This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI), and introduces some Mann-type implicit iteration methods for solving it. Norm convergence of the proposed methods of the iteration methods is guaranteed under some suitable assumptions. Full article

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

13 pages, 257 KiB

Open AccessArticle

Extension and Application of the Yamada Iteration Algorithm in Hilbert Spaces

by Ming Tian and Meng-Ying Tong

Mathematics 2019, 7(3), 215; https://doi.org/10.3390/math7030215 - 26 Feb 2019

Cited by 1 | Viewed by 2247

Abstract

In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence [...] Read more.

In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence theorem in Hilbert space. In particular, the set of zero points of an inverse strongly-monotone mapping can be transformed into the solution set of the variational inequality problem. Further, based on this result, we also obtain some new weak convergence theorems which are used to solve the equilibrium problem and the split feasibility problem. Full article

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

12 pages, 253 KiB

Open AccessArticle

Convergence and Best Proximity Points for Generalized Contraction Pairs

by Slah Sahmim, Abdelbasset Felhi and Hassen Aydi

Mathematics 2019, 7(2), 176; https://doi.org/10.3390/math7020176 - 15 Feb 2019

Cited by 6 | Viewed by 2335

Abstract

This paper is devoted to studying the existence of best proximity points and convergence for a class of generalized contraction pairs by using the concept of proximally-complete pairs and proximally-complete semi-sharp proximinal pairs. The obtained results are generalizations of the result of Sadiq [...] Read more.

This paper is devoted to studying the existence of best proximity points and convergence for a class of generalized contraction pairs by using the concept of proximally-complete pairs and proximally-complete semi-sharp proximinal pairs. The obtained results are generalizations of the result of Sadiq Basha (Basha, S., Best proximity points: global optimal approximate solutions, J. Glob. Optim. 2011, 49, 15–21) As an application, we give a result for nonexpansive mappings in normed vector spaces. Full article

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

12 pages, 305 KiB

Open AccessArticle

Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

by Chanjuan Pan and Yuanheng Wang

Mathematics 2019, 7(2), 156; https://doi.org/10.3390/math7020156 - 8 Feb 2019

Cited by 7 | Viewed by 2437

Abstract

In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we [...] Read more.

In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method. Full article

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

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16 pages, 299 KiB

Open AccessArticle

Sehgal Type Contractions on Dislocated Spaces

by Badr Alqahtani, Andreea Fulga, Erdal Karapınar and Panda Sumati Kumari

Mathematics 2019, 7(2), 153; https://doi.org/10.3390/math7020153 - 6 Feb 2019

Cited by 24 | Viewed by 2547

Abstract

In this paper, we investigate the contractive type inequalities for the iteration of the mapping at a given point in the setting of dislocated metric space. We consider an example to illustrate the validity of the given result. Further, as an application, we [...] Read more.

In this paper, we investigate the contractive type inequalities for the iteration of the mapping at a given point in the setting of dislocated metric space. We consider an example to illustrate the validity of the given result. Further, as an application, we propose a solution for a boundary value problem of the second order differential equation. Full article

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

9 pages, 230 KiB

Open AccessArticle

An Iterative Approach to the Solutions of Proximal Split Feasibility Problems

by Li-Jun Zhu and Yonghong Yao

Mathematics 2019, 7(2), 145; https://doi.org/10.3390/math7020145 - 3 Feb 2019

Viewed by 2328

Abstract

The proximal split feasibility problem is investigated in Hilbert spaces. An iterative procedure is introduced for finding the solution of the proximal split feasibility problem. Strong convergence analysis of the presented algorithm is proved. Full article

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

16 pages, 303 KiB

Open AccessArticle

ON H⁺Type Multivalued Contractions and Applications in Symmetric and Probabilistic Spaces

by Pradip Patle, Deepesh Patel, Hassen Aydi and Stojan Radenović

Mathematics 2019, 7(2), 144; https://doi.org/10.3390/math7020144 - 3 Feb 2019

Cited by 57 | Viewed by 2798

Abstract

The main idea in this article is to establish some fixed and common fixed point results for multivalued

H^{+}

-type contraction mappings in symmetric spaces. New results are accompanied with illustrative examples. An application of the obtained results to probabilistic spaces is [...] Read more.

The main idea in this article is to establish some fixed and common fixed point results for multivalued

H^{+}

-type contraction mappings in symmetric spaces. New results are accompanied with illustrative examples. An application of the obtained results to probabilistic spaces is presented. Full article

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

24 pages, 844 KiB

Open AccessArticle

Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

by Lu-Chuan Ceng and Qing Yuan

Mathematics 2019, 7(2), 142; https://doi.org/10.3390/math7020142 - 2 Feb 2019

Cited by 5 | Viewed by 2315

Abstract

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common [...] Read more.

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces. Full article

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

15 pages, 793 KiB

Open AccessEditor’s ChoiceArticle

An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems

by Yonghong Yao, Mihai Postolache and Jen-Chih Yao

Mathematics 2019, 7(1), 61; https://doi.org/10.3390/math7010061 - 8 Jan 2019

Cited by 98 | Viewed by 5585

Abstract

In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested [...] Read more.

In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested algorithm is demonstrated. Full article

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

13 pages, 272 KiB

Open AccessArticle

A Class of Nonlinear Fuzzy Variational Inequality Problems

by Cunlin Li, Zhifu Jia and Yeong-Cheng Liou

Mathematics 2019, 7(1), 54; https://doi.org/10.3390/math7010054 - 7 Jan 2019

Cited by 3 | Viewed by 2432

Abstract

In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed [...] Read more.

In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed the

S C^{1}

property of the FERM model. Furthermore, results of convergence analysis were obtained based on an approximation model of the FERM model. The convergence of global optimal solutions and the convergence of stationary points were analysed. Full article

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

14 pages, 292 KiB

Open AccessArticle

System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces

by Lu-Chuan Ceng, Mihai Postolache, Xiaolong Qin and Yonghong Yao

Mathematics 2019, 7(1), 5; https://doi.org/10.3390/math7010005 - 20 Dec 2018

Cited by 2 | Viewed by 2412

Abstract

The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point [...] Read more.

The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point problem, we construct an implicit multiple-viscosity approximation method for solving the GSVI. Under very mild conditions, we prove the strong convergence of the proposed method to a solution of the GSVI with the HVI constraint, for infinitely many pseudocontractions. Full article

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

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