Mathematics

Research

15 pages, 788 KiB

Open AccessArticle

Fundamental Questions and New Counterexamples for b-Metric Spaces and Fatou Property

by Ning Lu, Fei He and Wei-Shih Du

Mathematics 2019, 7(11), 1107; https://doi.org/10.3390/math7111107 - 14 Nov 2019

Cited by 7 | Viewed by 3309

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction [...] Read more.

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction with contraction constant

λ \in [\frac{1}{s}, 1)

in a complete b-metric space with

s > 1

have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

20 pages, 282 KiB

Open AccessArticle

Best Proximity Results with Applications to Nonlinear Dynamical Systems

by Hamed H Al-Sulami, Nawab Hussain and Jamshaid Ahmad

Mathematics 2019, 7(10), 900; https://doi.org/10.3390/math7100900 - 26 Sep 2019

Cited by 9 | Viewed by 1934

Abstract

Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution

ω

that is optimal in the sense that the error

σ (ω, J ω)

assumes the global minimum value [...] Read more.

Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution

ω

that is optimal in the sense that the error

σ (ω, J ω)

assumes the global minimum value

σ (θ, ϑ)

. The aim of this paper is to define the notion of Suzuki

α

-

Θ

-proximal multivalued contraction and prove the existence of best proximity points

ω

satisfying

σ (ω, J ω) = σ (θ, ϑ),

where

J

is assumed to be continuous or the space

M

is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

11 pages, 266 KiB

Open AccessArticle

Existence and Uniqueness of Zeros for Vector-Valued Functions with K-Adjustability Convexity and Their Applications

by Wei-Shih Du

Mathematics 2019, 7(9), 809; https://doi.org/10.3390/math7090809 - 2 Sep 2019

Cited by 2 | Viewed by 1822

Abstract

In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued [...] Read more.

In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued functions with K-adjustability convexity. As their applications, we obtain existence theorems for the minimization problem and fixed point problem which are original and quite different from the known results in the existing literature. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

12 pages, 785 KiB

Open AccessArticle

A Fixed Point Result with a Contractive Iterate at a Point

by Badr Alqahtani, Andreea Fulga and Erdal Karapınar

Mathematics 2019, 7(7), 606; https://doi.org/10.3390/math7070606 - 7 Jul 2019

Cited by 6 | Viewed by 2449

Abstract

In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition [...] Read more.

In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition for the considered mapping is not required. Second, the contraction inequality contains all possible geometrical distances. Third, the contraction inequality is formulated for some iteration of the considered operator, instead of the dealing with the given operator. Fourth and last, the iteration number may vary for each point in the domain of the operator for which we look for a fixed point. Consequently, the proved results generalize the acknowledged results in the field, including the well-known theorems of Seghal, Kincses-Totik, and Banach-Caccioppoli. We present two illustrative examples to support our results. As an application, we consider an Ulam-stability of one of our results. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

7 pages, 221 KiB

Open AccessArticle

On Bilateral Contractions

by Chi-Ming Chen, Gholamreza Heidary Joonaghany, Erdal Karapınar and Farshid Khojasteh

Mathematics 2019, 7(6), 538; https://doi.org/10.3390/math7060538 - 12 Jun 2019

Cited by 7 | Viewed by 2609

Abstract

In this manuscript, we introduce a new type of contraction, bilateral contraction which merges two significant approaches in the fixed point theory: Caristi type contractions and Jaggi type contractions. The principal aim of the main result is to enrich the literature by combining [...] Read more.

In this manuscript, we introduce a new type of contraction, bilateral contraction which merges two significant approaches in the fixed point theory: Caristi type contractions and Jaggi type contractions. The principal aim of the main result is to enrich the literature by combining the techniques of the mentioned two celebrated results that belong to Jaggi and Caristi. We consider an example to indicate the validity and genuine nature of the main result. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

12 pages, 326 KiB

Open AccessArticle

On (G, G′)-Prešić–Ćirić Operators in Graphical Metric Spaces

by Satish Shukla, Nabil Mlaiki and Hassen Aydi

Mathematics 2019, 7(5), 445; https://doi.org/10.3390/math7050445 - 18 May 2019

Cited by 9 | Viewed by 2213

Abstract

The purpose of this paper is to introduce a new type of operators in graphical metric spaces and to prove some fixed point results for these operators. Several known results are generalized and extended in this new setting of graphical metric spaces. The [...] Read more.

The purpose of this paper is to introduce a new type of operators in graphical metric spaces and to prove some fixed point results for these operators. Several known results are generalized and extended in this new setting of graphical metric spaces. The results are illustrated and justified with examples. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

11 pages, 246 KiB

Open AccessArticle

Best Proximity Point Theorems for Two Weak Cyclic Contractions on Metric-Like Spaces

by Erdal Karapınar, Chi-Ming Chen and Chih-Te Lee

Mathematics 2019, 7(4), 349; https://doi.org/10.3390/math7040349 - 14 Apr 2019

Cited by 12 | Viewed by 2900

Abstract

In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic

φ

-contraction via the

MT

-function. We express some examples to [...] Read more.

In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic

φ

-contraction via the

MT

-function. We express some examples to indicate the validity of the presented results. Our results unify and generalize a number of best proximity point results on the topic in the corresponding recent literature. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

10 pages, 277 KiB

Open AccessArticle

Some Remarks and New Results in Ordered Partial b-Metric Spaces

by Jelena Vujaković, Hassen Aydi, Stojan Radenović and Aiman Mukheimer

Mathematics 2019, 7(4), 334; https://doi.org/10.3390/math7040334 - 6 Apr 2019

Cited by 24 | Viewed by 3082

Abstract

We improve and generalize some new results in fixed point theory in the context of partial b-metric spaces. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

19 pages, 309 KiB

Open AccessArticle

Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems

by Lu-Chuan Ceng and Qing Yuan

Mathematics 2019, 7(2), 187; https://doi.org/10.3390/math7020187 - 16 Feb 2019

Cited by 1 | Viewed by 2543

Abstract

In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem [...] Read more.

In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

12 pages, 741 KiB

Open AccessArticle

Strong Convergence of a New Iterative Algorithm for Split Monotone Variational Inclusion Problems

by Lu-Chuan Ceng and Qing Yuan

Mathematics 2019, 7(2), 123; https://doi.org/10.3390/math7020123 - 24 Jan 2019

Cited by 1 | Viewed by 2661

Abstract

The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of [...] Read more.

The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of infinite dimensional Hilbert spaces. Convergence theorem of the sequences generated in our proposed implicit algorithm is obtained under some weak assumptions. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

7 pages, 231 KiB

Open AccessArticle

Interpolative Ćirić-Reich-Rus Type Contractions via the Branciari Distance

by Hassen Aydi, Chi-Ming Chen and Erdal Karapınar

Mathematics 2019, 7(1), 84; https://doi.org/10.3390/math7010084 - 15 Jan 2019

Cited by 105 | Viewed by 3554

Abstract

In this paper, we initiate the concept of interpolative Ćirić-Reich-Rus type contractions via the Branciari distance and prove some related fixed points results for such mappings. Moreover, an example is provided to show the useability of our obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

8 pages, 255 KiB

Open AccessArticle

Global Optimization for Quasi-Noncyclic Relatively Nonexpansive Mappings with Application to Analytic Complex Functions

by Poom Kumam and Chirasak Mongkolkeha

Mathematics 2019, 7(1), 46; https://doi.org/10.3390/math7010046 - 4 Jan 2019

Cited by 2 | Viewed by 2849

Abstract

The purpose of this article is to resolve a global optimization problem for quasi-noncyclic relatively nonexpansive mappings by giving an algorithm that determines an optimal approximate solution of the following minimization problem, [...] Read more.

The purpose of this article is to resolve a global optimization problem for quasi-noncyclic relatively nonexpansive mappings by giving an algorithm that determines an optimal approximate solution of the following minimization problem,

min_{x \in A} d (x, T x), min_{y \in B} d (y, T y) and min_{(x, y) \in A \times B} d (x, y);

also, we provide some illustrative examples to support our results. As an application, the existence of a solution of the analytic complex function is discussed. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

10 pages, 269 KiB

Open AccessArticle

Double Controlled Metric Type Spaces and Some Fixed Point Results

by Thabet Abdeljawad, Nabil Mlaiki, Hassen Aydi and Nizar Souayah

Mathematics 2018, 6(12), 320; https://doi.org/10.3390/math6120320 - 12 Dec 2018

Cited by 119 | Viewed by 6332

Abstract

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions

α (x, y)

and

μ (x, y)

on the right-hand side of the

b -

triangle [...] Read more.

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions

α (x, y)

and

μ (x, y)

on the right-hand side of the

b -

triangle inequality, that is,

d (x, y) \leq α (x, z) d (x, z) + μ (z, y) d (z, y), for all x, y, z \in X .

Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and

ϕ

-nonlinear type contractions in the setting of double controlled metric type spaces. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

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