Mathematics

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18 pages, 361 KiB

Open AccessArticle

Digital Topological Properties of an Alignment of Fixed Point Sets

by Sang-Eon Han

Mathematics 2020, 8(6), 921; https://doi.org/10.3390/math8060921 - 05 Jun 2020

Cited by 2 | Viewed by 1945

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image [...] Read more.

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image

(X, k)

, let

F (X)

be the set of cardinalities of the fixed point sets of all k-continuous self-maps of

(X, k)

(see Definition 4). In this paper we call it an alignment of fixed point sets of

(X, k)

. Then we have the following unsolved problem. How many components are there in

F (X)

up to 2-connectedness? In particular, let

C_{k}^{n, l}

be a simple closed k-curve with l elements in

Z^{n}

and

X : = C_{k}^{n, l_{1}} \lor C_{k}^{n, l_{2}}

be a digital wedge of

C_{k}^{n, l_{1}}

and

C_{k}^{n, l_{2}}

in

Z^{n}

. Then we need to explore both the number of components of

F (X)

up to digital 2-connectivity (see Definition 4) and perfectness of

F (X)

(see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models

C_{2 n}^{n, 4}

,

C_{3^{n} - 1}^{n, 4}

, and

C_{k}^{n, 6}

play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that

X^{♯} \geq 2

. Full article

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

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21 pages, 383 KiB

Open AccessArticle

Modified Inertial-Type Multi-Choice CQ-Algorithm for Countable Weakly Relatively Non-Expansive Mappings in a Banach Space, Applications and Numerical Experiments

by Li Wei, Yibin Xin, Ruilan Zhang and Ravi P. Agarwal

Mathematics 2020, 8(4), 613; https://doi.org/10.3390/math8040613 - 16 Apr 2020

Viewed by 1447

Abstract

In this paper, some new modified inertial-type multi-choice CQ-algorithms for approximating common fixed point of countable weakly relatively non-expansive mappings are presented in a real uniformly convex and uniformly smooth Banach space. New proof techniques are used to prove strong convergence theorems, which [...] Read more.

In this paper, some new modified inertial-type multi-choice CQ-algorithms for approximating common fixed point of countable weakly relatively non-expansive mappings are presented in a real uniformly convex and uniformly smooth Banach space. New proof techniques are used to prove strong convergence theorems, which extend some previous work. The connection and application to maximal monotone operators are demonstrated. Numerical experiments are conducted to illustrate that the rate of convergence is accelerated compared to some previous ones for some special cases. Full article

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

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11 pages, 331 KiB

Open AccessArticle

The Fixed Point Property of the Infinite M-Sphere

by Sang-Eon Han and Selma Özçağ

Mathematics 2020, 8(4), 599; https://doi.org/10.3390/math8040599 - 15 Apr 2020

Cited by 3 | Viewed by 1650

Abstract

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space

(Z^{2}, γ)

. This compactification is called the infinite M-topological sphere and denoted by [...] Read more.

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space

(Z^{2}, γ)

. This compactification is called the infinite M-topological sphere and denoted by

({(Z^{2})}^{*}, γ^{*})

, where

{(Z^{2})}^{*} : = Z^{2} \cup {*}, * \notin Z^{2}

and

γ^{*}

is the topology for

{(Z^{2})}^{*}

induced by the topology

γ

on

Z^{2}

. With the topological space

({(Z^{2})}^{*}, γ^{*})

, since any open set containing the point

“ * ”

has the cardinality

ℵ_{0}

, we call

({(Z^{2})}^{*}, γ^{*})

the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does

({(Z^{2})}^{*}, γ^{*})

have the fixed point property (FPP, for short)? The present paper proves that

({(Z^{2})}^{*}, γ^{*})

has the FPP in the category

M o p (γ^{*})

whose object is the only

({(Z^{2})}^{*}, γ^{*})

and morphisms are all continuous self-maps g of

({(Z^{2})}^{*}, γ^{*})

such that

| g ({(Z^{2})}^{*}) | = ℵ_{0}

with

* \in g ({(Z^{2})}^{*})

or

g ({(Z^{2})}^{*})

is a singleton. Since

({(Z^{2})}^{*}, γ^{*})

can be a model for a digital sphere derived from the M-topological space

(Z^{2}, γ)

, it can play a crucial role in topology, digital geometry and applied sciences. Full article

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

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14 pages, 289 KiB

Open AccessArticle

Fixed Point Theorems for Generalized (αβ-ψ)-Contractions in F -Metric Spaces with Applications

by Saleh Abdullah Al-Mezel, Jamshaid Ahmad and Giuseppe Marino

Mathematics 2020, 8(4), 584; https://doi.org/10.3390/math8040584 - 14 Apr 2020

Cited by 12 | Viewed by 1714

Abstract

The purpose of this paper is to define generalized (

α β

-

ψ)

-contraction in the context of

F

-metric space and obtain some new fixed point results. As applications, we solve a nonlinear neutral differential equation with an unbounded delay [...] Read more.

The purpose of this paper is to define generalized (

α β

-

ψ)

-contraction in the context of

F

-metric space and obtain some new fixed point results. As applications, we solve a nonlinear neutral differential equation with an unbounded delay

ϑ^{/} (ι) = - ρ_{1} (ι) ϑ (ι) + ρ_{2} (ι) L (ϑ (ι - ς (ι))) + ρ_{3} (ι) ϑ^{/} (ι - ς (ι)),

where

ρ_{1} (ι)

,

ρ_{2} (ι)

are continuous,

ρ_{3} (ι)

is continuously differentiable and

ς (ι) > 0,

for all

ι \in R

and is twice continuously differentiable. Full article

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

10 pages, 773 KiB

Open AccessArticle

Stability of Unbounded Differential Equations in Menger k-Normed Spaces: A Fixed Point Technique

by Masoumeh Madadi, Reza Saadati and Manuel De la Sen

Mathematics 2020, 8(3), 400; https://doi.org/10.3390/math8030400 - 11 Mar 2020

Cited by 8 | Viewed by 1671

Abstract

We attempt to solve differential equations

υ^{'} (ν) = Γ (ν, υ (ν))

and use the fixed point technique to prove its Hyers–Ulam–Rassias stability in Menger k-normed spaces. [...] Read more.

We attempt to solve differential equations

υ^{'} (ν) = Γ (ν, υ (ν))

and use the fixed point technique to prove its Hyers–Ulam–Rassias stability in Menger k-normed spaces. Full article

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

23 pages, 690 KiB

Open AccessArticle

Digital k-Contractibility of an n-Times Iterated Connected Sum of Simple Closed k-Surfaces and Almost Fixed Point Property

by Sang-Eon Han

Mathematics 2020, 8(3), 345; https://doi.org/10.3390/math8030345 - 04 Mar 2020

Cited by 9 | Viewed by 1970

Abstract

The paper firstly establishes the so-called n-times iterated connected sum of a simple closed k-surface in

Z^{3}

, denoted by

C_{k}^{n}

,

k \in {6, 18, 26}

. Secondly, for a simple closed 18-surface [...] Read more.

The paper firstly establishes the so-called n-times iterated connected sum of a simple closed k-surface in

Z^{3}

, denoted by

C_{k}^{n}

,

k \in {6, 18, 26}

. Secondly, for a simple closed 18-surface

M S S_{18}

, we prove that there are only two types of connected sums of it up to 18-isomorphism. Besides, given a simple closed 6-surface

M S S_{6}

, we prove that only one type of

M S S_{6} ♯ M S S_{6}

exists up to 6-isomorphism, where ♯ means the digital connected sum operator. Thirdly, we prove the digital k-contractibility of

C_{k}^{n} : = {\overset{︷}{M S S_{k} ♯ \dots ♯ M S S_{k}}}^{n - times}, k \in {18, 26}

, which leads to the simply k-connectedness of

C_{k}^{n}

,

k \in {18, 26}

,

n \in N

. Fourthly, we prove that

C_{6}^{2}

and

C_{k}^{n}

do not have the almost fixed point property (AFPP, for short),

k \in {18, 26}

. Finally, assume a closed k-surface

S_{k} (\subset Z^{3})

which is

(k, \bar{k})

-isomorphic to

(X, k)

in the picture

(Z^{3}, k, \bar{k}, X)

and the set X is symmetric according to each of

x y

-,

y z

-, and

x z

-planes of

R^{3}

. Then we prove that

S_{k}

does not have the AFPP. In this paper given a digital image

(X, k)

is assumed to be k-connected and its cardinality

| X | \geq 2

. Full article

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

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16 pages, 261 KiB

Open AccessArticle

Several Fixed Point Theorems in Convex b-Metric Spaces and Applications

by Lili Chen, Chaobo Li, Radoslaw Kaczmarek and Yanfeng Zhao

Mathematics 2020, 8(2), 242; https://doi.org/10.3390/math8020242 - 14 Feb 2020

Cited by 20 | Viewed by 3104

Abstract

Our paper is devoted to indicating a way of generalizing Mann’s iteration algorithm and a series of fixed point results in the framework of b-metric spaces. First, the concept of a convex b-metric space by means of a convex structure is [...] Read more.

Our paper is devoted to indicating a way of generalizing Mann’s iteration algorithm and a series of fixed point results in the framework of b-metric spaces. First, the concept of a convex b-metric space by means of a convex structure is introduced and Mann’s iteration algorithm is extended to this space. Next, by the help of Mann’s iteration scheme, strong convergence theorems for two types of contraction mappings in convex b-metric spaces are obtained. Some examples supporting our main results are also presented. Moreover, the problem of the T-stability of Mann’s iteration procedure for the above mappings in complete convex b-metric spaces is considered. As an application, we apply our main result to approximating the solution of the Fredholm linear integral equation. Full article

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

16 pages, 420 KiB

Open AccessFeature PaperArticle

Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

by Sang-Eon Han

Mathematics 2020, 8(1), 18; https://doi.org/10.3390/math8010018 - 19 Dec 2019

Cited by 1 | Viewed by 2145

Abstract

Given a Khalimsky (for short, K-) topological space X, the present paper examines if there are some relationships between the contractibility of X and the existence of the fixed point property of X. Based on a K-homotopy for K [...] Read more.

Given a Khalimsky (for short, K-) topological space X, the present paper examines if there are some relationships between the contractibility of X and the existence of the fixed point property of X. Based on a K-homotopy for K-topological spaces, we firstly prove that a K-homeomorphism preserves a K-homotopy between two K-continuous maps. Thus, we obtain that a K-homeomorphism preserves K-contractibility. Besides, the present paper proves that every simple closed K-curve in the n-dimensional K-topological space,

S C_{K}^{n, l}, n \geq 2, l \geq 4

, is not K-contractible. This feature plays an important role in fixed point theory for K-topological spaces. In addition, given a K-topological space X, after developing the notion of K-contractibility relative to each singleton

{x} (\subset X)

, we firstly compare it with the concept of K-contractibility of X. Finally, we prove that the K-contractibility does not imply the K-contractibility relative to each singleton

{x_{0}} (\subset X)

. Furthermore, we deal with certain conjectures involving the (almost) fixed point property in the categories KTC and KAC, where KTC (see Section 3) (resp. KAC (see Section 5)) denotes the category of K-topological (resp. KA-) spaces, KA-) spaces are subgraphs of the connectedness graphs of the K-topology on

Z^{n}

. Full article

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

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19 pages, 590 KiB

Open AccessArticle

Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

by Sang-Eon Han

Mathematics 2019, 7(12), 1244; https://doi.org/10.3390/math7121244 - 16 Dec 2019

Cited by 2 | Viewed by 2002

Abstract

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and [...] Read more.

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in

Z^{3}

, say

S_{k}

, we can see that both the minimal 26-adjacency neighborhood of a point

x \in S_{k}

, denoted by

M_{k} (x)

, and the geometric realization of it in

R^{3}

, denoted by

D_{k} (x)

, play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces

M S S_{18}

and

M S S_{18}^{'}

do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al. Full article

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

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10 pages, 265 KiB

Open AccessArticle

Approximations of Fixed Points in the Hadamard Metric Space CAT_p(0)

by Mostafa Bachar and Mohamed Amine Khamsi

Mathematics 2019, 7(11), 1088; https://doi.org/10.3390/math7111088 - 11 Nov 2019

Cited by 3 | Viewed by 2081

Abstract

In this paper, we consider the recently introduced

C A T_{p} (0)

, where the comparison triangles belong to

ℓ_{p}

, for

p \geq 2

. We first establish an inequality in these nonlinear metric spaces. Then, we use [...] Read more.

In this paper, we consider the recently introduced

C A T_{p} (0)

, where the comparison triangles belong to

ℓ_{p}

, for

p \geq 2

. We first establish an inequality in these nonlinear metric spaces. Then, we use it to prove the existence of fixed points of asymptotically nonexpansive mappings defined in

C A T_{p} (0)

. Moreover, we discuss the behavior of the successive iteration introduced by Schu for these mappings in Banach spaces. In particular, we prove that the successive iteration generates an approximate fixed point sequence. Full article

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

14 pages, 326 KiB

Open AccessArticle

Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

by Sang-Eon Han

Mathematics 2019, 7(10), 954; https://doi.org/10.3390/math7100954 - 12 Oct 2019

Cited by 7 | Viewed by 1528

Abstract

This paper explores a certain relationship between the almost fixed point property (AFPP for short) of a compact and n-dimensional Euclidean space and that of its digitized space. Based on several types of digitizations, we prove that the AFPP of a [...] Read more.

This paper explores a certain relationship between the almost fixed point property (AFPP for short) of a compact and n-dimensional Euclidean space and that of its digitized space. Based on several types of digitizations, we prove that the AFPP of a compact and n-dimensional Euclidean cube is preserved by each of the

U (k)

, the

L (k)

and the Khalimsky digitizations,

k \in 3^{n} - 1, n \in N

. Full article

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

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13 pages, 297 KiB

Open AccessArticle

Hybrid Methods for a Countable Family of G-Nonexpansive Mappings in Hilbert Spaces Endowed with Graphs

by Suthep Suantai, Mana Donganont and Watcharaporn Cholamjiak

Mathematics 2019, 7(10), 936; https://doi.org/10.3390/math7100936 - 10 Oct 2019

Cited by 18 | Viewed by 2015

Abstract

In this paper, we introduce the iterative scheme for finding a common fixed point of a countable family of G-nonexpansive mappings by the shrinking projection method which generalizes Takahashi Takeuchi and Kubota’s theorem in a Hilbert space with a directed graph. Simultaneously, we [...] Read more.

In this paper, we introduce the iterative scheme for finding a common fixed point of a countable family of G-nonexpansive mappings by the shrinking projection method which generalizes Takahashi Takeuchi and Kubota’s theorem in a Hilbert space with a directed graph. Simultaneously, we give examples and numerical results for supporting our main theorems and compare the rate of convergence of some examples under the same conditions. Full article

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

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12 pages, 311 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

The Fixed Point Property of Non-Retractable Topological Spaces

by Jeong Min Kang, Sang-Eon Han and Sik Lee

Mathematics 2019, 7(10), 879; https://doi.org/10.3390/math7100879 - 21 Sep 2019

Cited by 8 | Viewed by 2613

Abstract

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for [...] Read more.

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space

Y ∖ {p}

. Full article

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

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9 pages, 267 KiB

Open AccessArticle

Analytical Solution of Urysohn Integral Equations by Fixed Point Technique in Complex Valued Metric Spaces

by Hasanen A. Hammad and Manuel De la Sen

Mathematics 2019, 7(9), 852; https://doi.org/10.3390/math7090852 - 15 Sep 2019

Cited by 19 | Viewed by 2560

Abstract

The purpose of this article is to introduce a fixed point result for a general contractive condition in the context of complex valued metric spaces. Also, some important corollaries under this contractive condition are obtained. As an application, we find a unique solution [...] Read more.

The purpose of this article is to introduce a fixed point result for a general contractive condition in the context of complex valued metric spaces. Also, some important corollaries under this contractive condition are obtained. As an application, we find a unique solution for Urysohn integral equations, and some illustrative examples are given to support our obtaining results. Our results extend and generalize the results of Azam et al. Previous known related results in the literarure and some other known results in the literature. Full article

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

7 pages, 258 KiB

Open AccessArticle

Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in b-Metric Spaces

by Pradip Debnath and Manuel de La Sen

Mathematics 2019, 7(9), 849; https://doi.org/10.3390/math7090849 - 14 Sep 2019

Cited by 29 | Viewed by 2847

Abstract

In this paper, using an interpolative approach, we investigate two fixed point theorems in the framework of a b-metric space whose all closed and bounded subsets are compact. One of the theorems is for set-valued Hardy–Rogers-type and the other one is for [...] Read more.

In this paper, using an interpolative approach, we investigate two fixed point theorems in the framework of a b-metric space whose all closed and bounded subsets are compact. One of the theorems is for set-valued Hardy–Rogers-type and the other one is for set-valued Reich–Rus–Ćirić-type contractions. Examples are provided to validate the results. Full article

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

10 pages, 750 KiB

Open AccessArticle

Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings

by Yaqin Wang, Xiaoli Fang and Tae-Hwa Kim

Mathematics 2019, 7(9), 844; https://doi.org/10.3390/math7090844 - 12 Sep 2019

Cited by 5 | Viewed by 2110

Abstract

We, first, propose a new method for solving split common fixed point problems for demicontractive mappings in Hilbert spaces, and then establish the strong convergence of such an algorithm, which extends the Halpern type algorithm studied by Wang and Xu to a viscosity [...] Read more.

We, first, propose a new method for solving split common fixed point problems for demicontractive mappings in Hilbert spaces, and then establish the strong convergence of such an algorithm, which extends the Halpern type algorithm studied by Wang and Xu to a viscosity iteration. Above all, the step sizes in this algorithm are chosen without a priori knowledge of the operator norms. Full article

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

15 pages, 632 KiB

Open AccessArticle

A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

by Suthep Suantai, Suparat Kesornprom and Prasit Cholamjiak

Mathematics 2019, 7(9), 789; https://doi.org/10.3390/math7090789 - 27 Aug 2019

Cited by 18 | Viewed by 2701

Abstract

In this paper, we focus on studying the split feasibility problem (SFP), which has many applications in signal processing and image reconstruction. A popular technique is to employ the iterative method which is so called the relaxed CQ algorithm. However, the speed of [...] Read more.

In this paper, we focus on studying the split feasibility problem (SFP), which has many applications in signal processing and image reconstruction. A popular technique is to employ the iterative method which is so called the relaxed CQ algorithm. However, the speed of convergence usually depends on the way of selecting the step size of such algorithms. We aim to suggest a new hybrid CQ algorithm for the SFP by using the self adaptive and the line-search techniques. There is no computation on the inverse and the spectral radius of a matrix. We then prove the weak convergence theorem under mild conditions. Numerical experiments are included to illustrate its performance in compressed sensing. Some comparisons are also given to show the efficiency with other CQ methods in the literature. Full article

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

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19 pages, 330 KiB

Open AccessArticle

Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph

by Reny George, Hossam A. Nabwey, Rajagopalan Ramaswamy and Stojan Radenović

Mathematics 2019, 7(8), 754; https://doi.org/10.3390/math7080754 - 18 Aug 2019

Cited by 8 | Viewed by 2455

Abstract

We have introduced the new notions of R-weakly graph preserving and R-weakly

α

-admissible pair of multivalued mappings which includes the class of graph preserving mappings, weak graph preserving mappings as well as

α

-admissible mappings of type S, [...] Read more.

We have introduced the new notions of R-weakly graph preserving and R-weakly

α

-admissible pair of multivalued mappings which includes the class of graph preserving mappings, weak graph preserving mappings as well as

α

-admissible mappings of type S,

α_{*}

-admissible mappings of type S and

α_{*}

- orbital admissible mappings of type S respectively. Some generalized contraction and rational contraction classes are also introduced for a pair of multivalued mappings and common fixed point theorems are proved in a b-metric space endowed with a graph. We have also applied our results to obtain common fixed point theorems for R-weakly

α

-admissible pair of multivalued mappings in a b-metric space which are the proper extension and generalization of many known results. Proper examples are provided in support of our results. Our main results and its consequences improve, generalize and extend many known fixed point results existing in literature. Full article

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

11 pages, 248 KiB

Open AccessArticle

Some General Theorems for Compact Acyclic Multifunctions

by Donal O’Regan

Mathematics 2019, 7(8), 682; https://doi.org/10.3390/math7080682 - 31 Jul 2019

Cited by 1 | Viewed by 1588

Abstract

We present general Leray-Schauder type theorems for compact acyclic Multifunctions, using the topological transversality theorem by the author. Full article

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

19 pages, 318 KiB

Open AccessArticle

(C , Ψ * , G ) Class of Contractions and Fixed Points in a Metric Space Endowed with a Graph

by Reny George, Ekta Tamrakar, Jelena Vujaković, Hemant Kumar Pathak and Selvavinayagam Velusamy

Mathematics 2019, 7(5), 482; https://doi.org/10.3390/math7050482 - 27 May 2019

Cited by 2 | Viewed by 2284

Abstract

In this paper, we introduce the

(C, Ψ^{*}, G)

class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and [...] Read more.

In this paper, we introduce the

(C, Ψ^{*}, G)

class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and prove common fixed point theorems for such mappings in a metric space endowed with a graph. Our results unify and generalize many important fixed point results existing in literature. As an application of our main result, we have derived fixed point theorems for a pair of

α

-admissible set valued mappings in a metric space. Full article

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

19 pages, 326 KiB

Open AccessArticle

New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

by Li Wei, Yingzi Shang and Ravi P. Agarwal

Mathematics 2019, 7(5), 466; https://doi.org/10.3390/math7050466 - 24 May 2019

Cited by 3 | Viewed by 2010

Abstract

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed [...] Read more.

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified. Full article

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

10 pages, 257 KiB

Open AccessArticle

Reich, Jungck, and Berinde Common Fixed Point Results on ℱ-Metric Spaces and an Application

by Zoran D. Mitrović, Hassen Aydi, Nawab Hussain and Aiman Mukheimer

Mathematics 2019, 7(5), 387; https://doi.org/10.3390/math7050387 - 28 Apr 2019

Cited by 21 | Viewed by 2547

Abstract

Jleli and Samet (2018) introduced a new concept, named an

F

-metric space, as a generalization of the notion of a metric space. In this paper, we prove certain common fixed point theorems in

F

-metric spaces. As consequences of our results, we [...] Read more.

Jleli and Samet (2018) introduced a new concept, named an

F

-metric space, as a generalization of the notion of a metric space. In this paper, we prove certain common fixed point theorems in

F

-metric spaces. As consequences of our results, we obtain results of Banach, Jungck, Reich, and Berinde in these spaces. An application in dynamic programming is also given. Full article

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

25 pages, 1310 KiB

Open AccessArticle

Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

by Yan Tang and Yeol Je Cho

Mathematics 2019, 7(3), 255; https://doi.org/10.3390/math7030255 - 12 Mar 2019

Cited by 1 | Viewed by 2342

Abstract

In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian [...] Read more.

In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian and Fakhri, a new self-adaptive step size algorithm is proposed to find a common element of the solution set of the problems SVIP and EP. Convergence theorems are established under suitable conditions for the algorithm and application to the common solution of the fixed point problem, and the split convex optimization problem is considered. Finally, the performances and computational experiments are presented and a comparison with the related algorithms is provided to illustrate the efficiency and applicability of our new algorithms. Full article

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

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15 pages, 286 KiB

Open AccessArticle

Common Fixed Point Theorems of Generalized Multivalued (ψ,ϕ)-Contractions in Complete Metric Spaces with Application

by Eskandar Ameer, Muhammad Arshad, Dong Yun Shin and Sungsik Yun

Mathematics 2019, 7(2), 194; https://doi.org/10.3390/math7020194 - 18 Feb 2019

Cited by 3 | Viewed by 2875

Abstract

The purpose of this paper is to introduce the notion of generalized multivalued

(ψ, ϕ)

-type contractions and generalized multivalued

(ψ, ϕ)

-type Suzuki contractions and establish some new common fixed point theorems for such multivalued mappings in complete metric spaces. [...] Read more.

The purpose of this paper is to introduce the notion of generalized multivalued

(ψ, ϕ)

-type contractions and generalized multivalued

(ψ, ϕ)

-type Suzuki contractions and establish some new common fixed point theorems for such multivalued mappings in complete metric spaces. Our results are extension and improvement of the Suzuki and Nadler contraction theorems, Jleli and Samet, Piri and Kumam, Mizoguchi and Takahashi, and Liu et al. fixed point theorems. We provide an example for supporting our new results. Moreover, an application of our main result to the existence of solution of system of functional equations is also presented. Full article

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

19 pages, 306 KiB

Open AccessArticle

Weak and Strong Convergence Theorems for the Inclusion Problem and the Fixed-Point Problem of Nonexpansive Mappings

by Prasit Cholamjiak, Suparat Kesornprom and Nattawut Pholasa

Mathematics 2019, 7(2), 167; https://doi.org/10.3390/math7020167 - 13 Feb 2019

Cited by 7 | Viewed by 2823

Abstract

In this work, we study the inclusion problem of the sum of two monotone operators and the fixed-point problem of nonexpansive mappings in Hilbert spaces. We prove the weak and strong convergence theorems under some weakened conditions. Some numerical experiments are also given [...] Read more.

In this work, we study the inclusion problem of the sum of two monotone operators and the fixed-point problem of nonexpansive mappings in Hilbert spaces. We prove the weak and strong convergence theorems under some weakened conditions. Some numerical experiments are also given to support our main theorem. Full article

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

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15 pages, 837 KiB

Open AccessArticle

Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications

by Pongsakorn Sunthrayuth, Nuttapol Pakkaranang, Poom Kumam, Phatiphat Thounthong and Prasit Cholamjiak

Mathematics 2019, 7(2), 161; https://doi.org/10.3390/math7020161 - 11 Feb 2019

Cited by 5 | Viewed by 2770

Abstract

In this paper, we introduce a generalized viscosity explicit method (GVEM) for nonexpansive mappings in the setting of Banach spaces and, under some new techniques and mild assumptions on the control conditions, prove some strong convergence theorems for the proposed method, which converge [...] Read more.

In this paper, we introduce a generalized viscosity explicit method (GVEM) for nonexpansive mappings in the setting of Banach spaces and, under some new techniques and mild assumptions on the control conditions, prove some strong convergence theorems for the proposed method, which converge to a fixed point of the given mapping and a solution of the variational inequality. As applications, we apply our main results to show the existence of fixed points of strict pseudo-contractions and periodic solutions of nonlinear evolution equations and Fredholm integral equations. Finally, we give some numerical examples to illustrate the efficiency and implementation of our method. Full article

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

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13 pages, 772 KiB

Open AccessArticle

Fixed Point Results for Multi-Valued Contractions in b−Metric Spaces and an Application

by Haitham Qawaqneh, Mohd Salmi Md Noorani, Wasfi Shatanawi, Hassen Aydi and Habes Alsamir

Mathematics 2019, 7(2), 132; https://doi.org/10.3390/math7020132 - 01 Feb 2019

Cited by 73 | Viewed by 3297

Abstract

In this paper, by characterizing a weak contractive condition based on using

C -

functions and

α -

admissible multi-valued mapping of type S, we present some fixed point results for

(α, F) -

admissible multi-valued mappings in the [...] Read more.

In this paper, by characterizing a weak contractive condition based on using

C -

functions and

α -

admissible multi-valued mapping of type S, we present some fixed point results for

(α, F) -

admissible multi-valued mappings in the setting of

b -

metric spaces. Some examples and an application are added in order to show the reliability of our obtained results. Our results amend, unify, and generalize some existing results in the literature. The scientific novelty of our main results is to take new contraction self-mappings in

b -

metric spaces for multi-valued mappings. Full article

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

16 pages, 288 KiB

Open AccessArticle

Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces

by Yixuan Yang, Yuchao Tang and Chuanxi Zhu

Mathematics 2019, 7(2), 131; https://doi.org/10.3390/math7020131 - 01 Feb 2019

Cited by 3 | Viewed by 2686

Abstract

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking [...] Read more.

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions. Full article

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

17 pages, 295 KiB

Open AccessArticle

Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

by Xinglong Wang, Jing Zhao and Dingfang Hou

Mathematics 2019, 7(2), 119; https://doi.org/10.3390/math7020119 - 23 Jan 2019

Cited by 2 | Viewed by 2181

Abstract

The split feasibility problem models inverse problems arising from phase retrievals problems and intensity-modulated radiation therapy. For solving the split feasibility problem, Xu proposed a relaxed CQ algorithm that only involves projections onto half-spaces. In this paper, we use the dual variable to [...] Read more.

The split feasibility problem models inverse problems arising from phase retrievals problems and intensity-modulated radiation therapy. For solving the split feasibility problem, Xu proposed a relaxed CQ algorithm that only involves projections onto half-spaces. In this paper, we use the dual variable to propose a new relaxed CQ iterative algorithm that generalizes Xu’s relaxed CQ algorithm in real Hilbert spaces. By using projections onto half-spaces instead of those onto closed convex sets, the proposed algorithm is implementable. Moreover, we present modified relaxed CQ algorithm with viscosity approximation method. Under suitable conditions, global weak and strong convergence of the proposed algorithms are proved. Some numerical experiments are also presented to illustrate the effectiveness of the proposed algorithms. Our results improve and extend the corresponding results of Xu and some others. Full article

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

11 pages, 264 KiB

Open AccessArticle

Extension of Extragradient Techniques for Variational Inequalities

by Yonghong Yao, Ke Wang, Xiaowei Qin and Li-Jun Zhu

Mathematics 2019, 7(2), 111; https://doi.org/10.3390/math7020111 - 22 Jan 2019

Cited by 1 | Viewed by 2020

Abstract

An extragradient type method for finding the common solutions of two variational inequalities has been proposed. The convergence result of the algorithm is given under mild conditions on the algorithm parameters. Full article

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

11 pages, 259 KiB

Open AccessArticle

Fixed Point Results for the Family of Multivalued F-Contractive Mappings on Closed Ball in Complete Dislocated b-Metric Spaces

by Qasim Mahmood, Abdullah Shoaib, Tahair Rasham and Muhammad Arshad

Mathematics 2019, 7(1), 56; https://doi.org/10.3390/math7010056 - 07 Jan 2019

Cited by 23 | Viewed by 3208

Abstract

The purpose of this paper is to find out fixed point results for the family of multivalued mappings fulfilling a generalized rational type F-contractive conditions on a closed ball in complete dislocated b-metric space. An application to the system of integral [...] Read more.

The purpose of this paper is to find out fixed point results for the family of multivalued mappings fulfilling a generalized rational type F-contractive conditions on a closed ball in complete dislocated b-metric space. An application to the system of integral equations is presented to show the novelty of our results. Our results extend several comparable results in the existing literature. Full article

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

16 pages, 281 KiB

Open AccessArticle

PPF-Dependent Fixed Point Results for New Multi-Valued Generalized F-Contraction in the Razumikhin Class with an Application

by Hasanen A. Hammad and Manuel De la Sen

Mathematics 2019, 7(1), 52; https://doi.org/10.3390/math7010052 - 07 Jan 2019

Cited by 2 | Viewed by 2355

Abstract

In this paper, a new multi-valued generalized F-contraction mapping is given. Using it, the existence of PPF-dependent fixed point for such mappings in the Razumikhin class is obtained. Moreover, an application for nonlinear integral equations with delay is presented here to illustrate [...] Read more.

In this paper, a new multi-valued generalized F-contraction mapping is given. Using it, the existence of PPF-dependent fixed point for such mappings in the Razumikhin class is obtained. Moreover, an application for nonlinear integral equations with delay is presented here to illustrate the usability of the obtained results. Full article

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

20 pages, 961 KiB

Open AccessArticle

Some Common Fixed Point Theorems for Generalized F-Contraction Involving w-Distance with Some Applications to Differential Equations

by Chirasak Mongkolkeha and Dhananjay Gopal

Mathematics 2019, 7(1), 32; https://doi.org/10.3390/math7010032 - 30 Dec 2018

Cited by 7 | Viewed by 3386

Abstract

In this paper, we introduce the Ćirić type generalized F-contraction and establish certain common fixed point results for such F-contraction in metric spaces with the w-distances. In addition, we give some examples to support our results. Finally, we apply our [...] Read more.

In this paper, we introduce the Ćirić type generalized F-contraction and establish certain common fixed point results for such F-contraction in metric spaces with the w-distances. In addition, we give some examples to support our results. Finally, we apply our results to show the existence of solutions of the second order differential equation. Full article

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

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14 pages, 291 KiB

Open AccessArticle

Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems

by Peichao Duan, Xubang Zheng and Jing Zhao

Mathematics 2019, 7(1), 14; https://doi.org/10.3390/math7010014 - 24 Dec 2018

Cited by 2 | Viewed by 2431

Abstract

In this paper, we propose a viscosity approximation method to solve the split common fixed point problem and consider the bounded perturbation resilience of the proposed method in general Hilbert spaces. Under some mild conditions, we prove that our algorithms strongly converge to [...] Read more.

In this paper, we propose a viscosity approximation method to solve the split common fixed point problem and consider the bounded perturbation resilience of the proposed method in general Hilbert spaces. Under some mild conditions, we prove that our algorithms strongly converge to a solution of the split common fixed point problem, which is also the unique solution of the variational inequality problem. Finally, we show the convergence and effectiveness of the algorithms by two numerical examples. Full article

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

14 pages, 301 KiB

Open AccessArticle

Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

by Xiangkai Sun, Hongyong Fu and Jing Zeng

Mathematics 2019, 7(1), 12; https://doi.org/10.3390/math7010012 - 23 Dec 2018

Cited by 17 | Viewed by 2885

Abstract

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using [...] Read more.

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints. Full article

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

14 pages, 2904 KiB

Open AccessArticle

New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations

by Sujitra Sanhan, Winate Sanhan and Chirasak Mongkolkeha

Mathematics 2018, 6(12), 324; https://doi.org/10.3390/math6120324 - 12 Dec 2018

Cited by 4 | Viewed by 2518

Abstract

The purpose of this article is to prove some existences of fixed point theorems for generalized

F

-contraction mapping in metric spaces by using the concept of generalized pseudodistance. In addition, we give some examples to illustrate our main results. As the application, [...] Read more.

The purpose of this article is to prove some existences of fixed point theorems for generalized

F

-contraction mapping in metric spaces by using the concept of generalized pseudodistance. In addition, we give some examples to illustrate our main results. As the application, the existence of the solution of the second order differential equation is given. Full article

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

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19 pages, 280 KiB

Open AccessArticle

Some PPF Dependent Fixed Point Theorems for Generalized α-F-Contractions in Banach Spaces and Applications

by Yeol Je Cho, Shin Min Kang and Peyman Salimi

Mathematics 2018, 6(11), 267; https://doi.org/10.3390/math6110267 - 19 Nov 2018

Cited by 1 | Viewed by 2444

Abstract

In this paper, we introduce the concepts of an

α

-admissible nonself-mapping, an

α

-F-contractive nonself-mapping, a weak

α

-F-contractive nonself-mapping, and a generalized

α

-F-contractive nonself-mapping and prove some

P P F

(past-present-future)-dependent fixed point theorems [...] Read more.

In this paper, we introduce the concepts of an

α

-admissible nonself-mapping, an

α

-F-contractive nonself-mapping, a weak

α

-F-contractive nonself-mapping, and a generalized

α

-F-contractive nonself-mapping and prove some

P P F

(past-present-future)-dependent fixed point theorems for the proposed contractive nonself-mappings in certain Razumikhin classes. By using our results, we derive some

P P F

-dependent fixed point theorems for an

α

-F-contractive nonself-mapping endowed with a graph or a partial order. Finally, we give some applications to illustrate the main results. Full article

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

12 pages, 771 KiB

Open AccessArticle

Random Best Proximity Points for α-Admissible Mappings via Simulation Functions

by Chayut Kongban, Poom Kumam and Juan Martínez-Moreno

Mathematics 2018, 6(11), 262; https://doi.org/10.3390/math6110262 - 18 Nov 2018

Cited by 1 | Viewed by 3011

Abstract

In this paper, we introduce a new concept of random

α

-proximal admissible and random

α

-

Z

-contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces. [...] Read more.

In this paper, we introduce a new concept of random

α

-proximal admissible and random

α

-

Z

-contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces. Full article

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

19 pages, 1085 KiB

Open AccessArticle

Strong Convergence Theorems for Fixed Point Problems for Nonexpansive Mappings and Zero Point Problems for Accretive Operators Using Viscosity Implicit Midpoint Rules in Banach Spaces

by Huancheng Zhang, Yunhua Qu and Yongfu Su

Mathematics 2018, 6(11), 257; https://doi.org/10.3390/math6110257 - 16 Nov 2018

Cited by 3 | Viewed by 2260

Abstract

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results [...] Read more.

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results of the proposed algorithm and improves the relevant results of researchers in this field. In the end, this paper gives numerical examples to support the main results. Full article

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

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7 pages, 237 KiB

Open AccessArticle

Interpolative Reich–Rus–Ćirić Type Contractions on Partial Metric Spaces

by Erdal Karapinar, Ravi Agarwal and Hassen Aydi

Mathematics 2018, 6(11), 256; https://doi.org/10.3390/math6110256 - 16 Nov 2018

Cited by 134 | Viewed by 4590

Abstract

By giving a counter-example, we point out a gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85–87) where the given fixed point may be not unique and we present the corrected version. We also state the celebrated [...] Read more.

By giving a counter-example, we point out a gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85–87) where the given fixed point may be not unique and we present the corrected version. We also state the celebrated fixed point theorem of Reich–Rus–Ćirić in the framework of complete partial metric spaces, by taking the interpolation theory into account. Some examples are provided where the main result in papers by Reich (Can. Math. Bull. 1971, 14, 121–124; Boll. Unione Mat. Ital. 1972, 4, 26–42 and Boll. Unione Mat. Ital. 1971, 4, 1–11.) is not applicable. Full article

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

13 pages, 337 KiB

Open AccessArticle

The Combination Projection Method for Solving Convex Feasibility Problems

by Songnian He and Qiao-Li Dong

Mathematics 2018, 6(11), 249; https://doi.org/10.3390/math6110249 - 12 Nov 2018

Cited by 2 | Viewed by 2625

Abstract

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some [...] Read more.

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some

x^{*} \in C : = \cap_{i = 1}^{m} {x \in H | c_{i} (x) \leq 0}

, where m is a positive integer,

H

is a real Hilbert space, and

{c_{i}}_{i = 1}^{m}

are convex functions defined as

H

. The key of the CPM is that, for the current iterate

x^{k}

, the CPM firstly constructs a new level set

H_{k}

through a convex combination of some of

{c_{i}}_{i = 1}^{m}

in an appropriate way, and then updates the new iterate

x^{k + 1}

only by using the projection

P_{H_{k}}

. We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods. Full article

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

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17 pages, 301 KiB

Open AccessArticle

Best Proximity Point Results in b-Metric Space and Application to Nonlinear Fractional Differential Equation

by Azhar Hussain, Tanzeela Kanwal, Muhammad Adeel and Stojan Radenovic

Mathematics 2018, 6(11), 221; https://doi.org/10.3390/math6110221 - 28 Oct 2018

Cited by 10 | Viewed by 2792

Abstract

Based on the concepts of contractive conditions due to Suzuki (Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, 2008, 136, 1861–1869) and Jleli (Jleli, M., Samet, B., A new generalization of the Banach contraction [...] Read more.

Based on the concepts of contractive conditions due to Suzuki (Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, 2008, 136, 1861–1869) and Jleli (Jleli, M., Samet, B., A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014, 2014, 38), our aim is to combine the aforementioned concepts in more general way for set valued and single valued mappings and to prove the existence of best proximity point results in the context of b-metric spaces. Endowing the concept of graph with b-metric space, we present some best proximity point results. Some concrete examples are presented to illustrate the obtained results. Moreover, we prove the existence of the solution of nonlinear fractional differential equation involving Caputo derivative. Presented results not only unify but also generalize several existing results on the topic in the corresponding literature. Full article

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

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