Research

11 pages, 302 KiB

Open AccessArticle

Interpolative Reich–Rus–Ćirić and Hardy–Rogers Contraction on Quasi-Partial b-Metric Space and Related Fixed Point Results

by Vishnu Narayan Mishra, Luis Manuel Sánchez Ruiz, Pragati Gautam and Swapnil Verma

Mathematics 2020, 8(9), 1598; https://doi.org/10.3390/math8091598 - 17 Sep 2020

Cited by 26 | Viewed by 2085

Abstract

The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction [...] Read more.

The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction in the framework of quasi-partial b-metric space and proved the corresponding common fixed point theorem by adopting the notion of interpolation. The results are further validated with the application based on them. Full article

(This article belongs to the Special Issue Variational Inequality)

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15 pages, 1102 KiB

Open AccessArticle

Strong Convergent Theorems Governed by Pseudo-Monotone Mappings

by Liya Liu, Xiaolong Qin and Jen-Chih Yao

Mathematics 2020, 8(8), 1256; https://doi.org/10.3390/math8081256 - 31 Jul 2020

Viewed by 1549

Abstract

The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are [...] Read more.

The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are established in Hilbert spaces under mild assumption that the associated mapping is Lipschitz continuous, pseudo-monotone and sequentially weakly continuous. Numerical experiments are performed to illustrate the behaviors of our proposed methods, as well as comparing them with the existing one in literature. Full article

(This article belongs to the Special Issue Variational Inequality)

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13 pages, 790 KiB

Open AccessArticle

The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces

by Eskandar Naraghirad, Luoyi Shi and Ngai-Ching Wong

Mathematics 2020, 8(6), 1022; https://doi.org/10.3390/math8061022 - 22 Jun 2020

Cited by 1 | Viewed by 1951

Abstract

The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like [...] Read more.

The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like maps in the general Banach space setting. In this paper, after introducing the notion of Bregman generalized hybrid sequences in a reflexive Banach space, we prove (with using the Bregman–Opial property instead of the Opial property) convergence theorems for such sequences. We also provide new fixed point theorems for Bregman generalized hybrid maps defined on an arbitrary but not necessarily convex subset of a reflexive Banach space. We end this paper with a brief discussion of the existence of Bregman absolute fixed points of such maps. Full article

(This article belongs to the Special Issue Variational Inequality)

15 pages, 280 KiB

Open AccessArticle

A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space

by Wataru Takahashi

Mathematics 2020, 8(3), 435; https://doi.org/10.3390/math8030435 - 17 Mar 2020

Cited by 4 | Viewed by 2164

Abstract

In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common [...] Read more.

In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we get well-known and new strong convergence theorems in a Hilbert space. Full article

(This article belongs to the Special Issue Variational Inequality)

7 pages, 206 KiB

Open AccessFeature PaperArticle

Existence of a Unique Fixed Point for Nonlinear Contractive Mappings

by Simeon Reich and Alexander J. Zaslavski

Mathematics 2020, 8(1), 55; https://doi.org/10.3390/math8010055 - 01 Jan 2020

Cited by 12 | Viewed by 1870

Abstract

In a recent work, we established the existence of a unique fixed point for nonlinear contractive self-mappings of a bounded and closed set in a Banach space. In the present paper we extend this result to the case of unbounded sets. Full article

(This article belongs to the Special Issue Variational Inequality)

18 pages, 793 KiB

Open AccessArticle

Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems

by Lu-Chuan Ceng and Meijuan Shang

Mathematics 2019, 7(10), 933; https://doi.org/10.3390/math7100933 - 10 Oct 2019

Cited by 7 | Viewed by 1496

Abstract

In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where

1 < q \leq 2

. We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, [...] Read more.

In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where

1 < q \leq 2

. We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonexpansive mappings in X. The generalized Mann-like viscosity implicit rule investigated in this work is based on the Korpelevich’s extragradient technique, the implicit viscosity iterative method and the Mann’s iteration method. We show that the iterative sequences governed by our generalized Mann-like viscosity implicit rule converges strongly to a solution of the general optimization system. Full article

(This article belongs to the Special Issue Variational Inequality)

13 pages, 292 KiB

Open AccessArticle

Convergence of Two Splitting Projection Algorithms in Hilbert Spaces

by Marwan A. Kutbi, Abdul Latif and Xiaolong Qin

Mathematics 2019, 7(10), 922; https://doi.org/10.3390/math7100922 - 03 Oct 2019

Viewed by 1401

Abstract

The aim of this present paper is to study zero points of the sum of two maximally monotone mappings and fixed points of a non-expansive mapping. Two splitting projection algorithms are introduced and investigated for treating the zero and fixed point problems. Possible [...] Read more.

The aim of this present paper is to study zero points of the sum of two maximally monotone mappings and fixed points of a non-expansive mapping. Two splitting projection algorithms are introduced and investigated for treating the zero and fixed point problems. Possible computational errors are taken into account. Two convergence theorems are obtained and applications are also considered in Hilbert spaces Full article

(This article belongs to the Special Issue Variational Inequality)

16 pages, 305 KiB

Open AccessArticle

Fixed Point Results for Generalized ℱ-Contractions in Modular b-Metric Spaces with Applications

by Vahid Parvaneh, Nawab Hussain, Maryam Khorshidi, Nabil Mlaiki and Hassen Aydi

Mathematics 2019, 7(10), 887; https://doi.org/10.3390/math7100887 - 23 Sep 2019

Cited by 13 | Viewed by 2842

Abstract

The aim of this paper is to generalize the

F

-contractive condition in the framework of

α - ν

-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications [...] Read more.

The aim of this paper is to generalize the

F

-contractive condition in the framework of

α - ν

-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications are presented to support the obtained results. Full article

(This article belongs to the Special Issue Variational Inequality)

19 pages, 294 KiB

Open AccessFeature PaperArticle

Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

by Lu-Chuan Ceng, Xiaolong Qin, Yekini Shehu and Jen-Chih Yao

Mathematics 2019, 7(10), 881; https://doi.org/10.3390/math7100881 - 22 Sep 2019

Cited by 14 | Viewed by 1899

Abstract

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms [...] Read more.

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI). Full article

(This article belongs to the Special Issue Variational Inequality)

14 pages, 279 KiB

Open AccessArticle

Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces

by Mujahid Abbas, Yusuf Ibrahim, Abdul Rahim Khan and Manuel De la Sen

Mathematics 2019, 7(8), 749; https://doi.org/10.3390/math7080749 - 16 Aug 2019

Cited by 8 | Viewed by 2326

Abstract

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in [...] Read more.

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in

C A T (0)

spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of

C A T (0)

spaces. Full article

(This article belongs to the Special Issue Variational Inequality)

15 pages, 306 KiB

Open AccessArticle

New Hybrid Contractions on b-Metric Spaces

by Erdal Karapınar and Andreea Fulga

Mathematics 2019, 7(7), 578; https://doi.org/10.3390/math7070578 - 28 Jun 2019

Cited by 26 | Viewed by 2181

Abstract

In this manuscript, we introduce the notion of b-hybrid contraction in the setting of b-metric space. We investigate the existence and uniqueness of a fixed point for this contraction. Our results combine and merge several existing results in the corresponding literature [...] Read more.

In this manuscript, we introduce the notion of b-hybrid contraction in the setting of b-metric space. We investigate the existence and uniqueness of a fixed point for this contraction. Our results combine and merge several existing results in the corresponding literature and we list some of them as corollaries. Finally, we consider an Ulam’s type stability for an application. Full article

(This article belongs to the Special Issue Variational Inequality)

16 pages, 291 KiB

Open AccessArticle

Strong Convergence Theorems for Variational Inequalities and Common Fixed-Point Problems Using Relaxed Mann Implicit Iteration Methods

by Lu-Chuan Ceng and Meijuan Shang

Mathematics 2019, 7(5), 424; https://doi.org/10.3390/math7050424 - 11 May 2019

Cited by 3 | Viewed by 1840

Abstract

Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a [...] Read more.

Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a countable family of uniformly Lipschitzian pseudocontractive mappings. More convergence theorems are proved under some suitable weak condition in both 2-uniformly smooth and uniformly convex Banach spaces. Full article

(This article belongs to the Special Issue Variational Inequality)

14 pages, 283 KiB

Open AccessArticle

Generalized Implicit Set-Valued Variational Inclusion Problem with ⊕ Operation

by Rais Ahmad, Imran Ali, Saddam Husain, A. Latif and Ching-Feng Wen

Mathematics 2019, 7(5), 421; https://doi.org/10.3390/math7050421 - 10 May 2019

Cited by 2 | Viewed by 1692

Abstract

In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence [...] Read more.

In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence result is proven for a generalized implicit set-valued variational inclusion problem with ⊕ operation. Some special cases of a generalized implicit set-valued variational inclusion problem with ⊕ operation are discussed. An example is constructed to illustrate some of the concepts used in this paper. Full article

(This article belongs to the Special Issue Variational Inequality)

20 pages, 296 KiB

Open AccessArticle

Systems of Variational Inequalities with Nonlinear Operators

by Lu-Chuan Ceng and Qing Yuan

Mathematics 2019, 7(4), 338; https://doi.org/10.3390/math7040338 - 09 Apr 2019

Cited by 4 | Viewed by 2188

Abstract

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly [...] Read more.

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces. Full article

(This article belongs to the Special Issue Variational Inequality)

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