Axioms

11 pages, 258 KiB

Open AccessArticle

Generalized Hyers–Ulam Stability of the Additive Functional Equation

by Yang-Hi Lee and Gwang Hui Kim

Axioms 2019, 8(2), 76; https://doi.org/10.3390/axioms8020076 - 25 Jun 2019

Cited by 4 | Viewed by 2905

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x₁ + y₁, x₂ + y₂, …, x_n + y_n) = f(x₁, x [...] Read more.

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x₁ + y₁, x₂ + y₂, …, x_n + y_n) = f(x₁, x₂, … x_n) + f(y₁, y₂, …, y_n). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

22 pages, 324 KiB

Open AccessArticle

Riemann–Liouville Operator in Weighted L_p Spaces via the Jacobi Series Expansion

by Maksim V. Kukushkin

Axioms 2019, 8(2), 75; https://doi.org/10.3390/axioms8020075 - 23 Jun 2019

Cited by 6 | Viewed by 2616

Abstract

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously [...] Read more.

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces. Full article

(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)

18 pages, 1418 KiB

Open AccessArticle

Sumudu Decomposition Method for Solving Fuzzy Integro-Differential Equations

by Shin Min Kang, Zain Iqbal, Mustafa Habib and Waqas Nazeer

Axioms 2019, 8(2), 74; https://doi.org/10.3390/axioms8020074 - 20 Jun 2019

Cited by 11 | Viewed by 2998

Abstract

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of [...] Read more.

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of linear fuzzy Volterra integro-differential equations (FVIDEs) are more generalized and have better applications. In this report, the Sumudu decomposition method (SDM) was used to find the solution to some linear and nonlinear fuzzy integro-differential equations (FIDEs). Some examples are given to show the validity of the presented method. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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11 pages, 257 KiB

Open AccessArticle

New Entropy-Based Similarity Measure between Interval-Valued Intuitionstic Fuzzy Sets

by Saida S. Mohamed, Areeg Abdalla and Robert I. John

Axioms 2019, 8(2), 73; https://doi.org/10.3390/axioms8020073 - 18 Jun 2019

Cited by 2 | Viewed by 2644

Abstract

In this paper, we propose a new approach to constructing similarity measures using the entropy measure for Interval-Valued Intuitionistic Fuzzy Sets. In addition, we provide several illustrative examples to demonstrate the practicality and effectiveness of the proposed formula. Finally, we use the new [...] Read more.

In this paper, we propose a new approach to constructing similarity measures using the entropy measure for Interval-Valued Intuitionistic Fuzzy Sets. In addition, we provide several illustrative examples to demonstrate the practicality and effectiveness of the proposed formula. Finally, we use the new proposed similarity measure to develop a new approach for solving problems of pattern recognition and multi-criteria fuzzy decision-making. Full article

(This article belongs to the Special Issue Softcomputing: Theories and Applications)

36 pages, 432 KiB

Open AccessArticle

Recent Advances on the Results for Nonunique Fixed in Various Spaces

by Erdal Karapınar

Axioms 2019, 8(2), 72; https://doi.org/10.3390/axioms8020072 - 05 Jun 2019

Cited by 9 | Viewed by 2688

Abstract

In this short survey, we aim to underline the importance of the non-unique fixed point results in various abstract spaces. We recall a brief background on the topic and we combine, collect and unify several existing non-unique fixed points in the literature. Some [...] Read more.

In this short survey, we aim to underline the importance of the non-unique fixed point results in various abstract spaces. We recall a brief background on the topic and we combine, collect and unify several existing non-unique fixed points in the literature. Some interesting examples are considered. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

19 pages, 307 KiB

Open AccessArticle

Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

by Olga Tsekhan

Axioms 2019, 8(2), 71; https://doi.org/10.3390/axioms8020071 - 03 Jun 2019

Cited by 8 | Viewed by 2672

Abstract

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the [...] Read more.

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

12 pages, 276 KiB

Open AccessArticle

On Almost b-Metric Spaces and Related Fixed Point Results

by Nabil Mlaiki, Katarina Kukić, Milanka Gardašević-Filipović and Hassen Aydi

Axioms 2019, 8(2), 70; https://doi.org/10.3390/axioms8020070 - 01 Jun 2019

Cited by 9 | Viewed by 3264

Abstract

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b [...] Read more.

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b-metrics to the corresponding b-metrics. Later, we show that this approach can not work for all kinds of contractions. To confirm this, we present a proof in which the contraction condition is such that it cannot be reduced to corresponding b-metrics. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

9 pages, 264 KiB

Open AccessArticle

Fixed Point Theorems via α-ϱ-Fuzzy Contraction

by Badshah-e- Rome, Muhammad Sarwar and Poom Kumam

Axioms 2019, 8(2), 69; https://doi.org/10.3390/axioms8020069 - 31 May 2019

Cited by 1 | Viewed by 2771

Abstract

Some well known results from the existing literature are extended and generalized via new contractive type mappings in fuzzy metric spaces. A non trivial supporting example is also provided to demonstrate the validity of the obtained results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

10 pages, 774 KiB

Open AccessArticle

(p, q)-Hermite–Hadamard Inequalities for Double Integral and (p, q)-Differentiable Convex Functions

by Julalak Prabseang, Kamsing Nonlaopon and Jessada Tariboon

Axioms 2019, 8(2), 68; https://doi.org/10.3390/axioms8020068 - 28 May 2019

Cited by 19 | Viewed by 3087

Abstract

The aim of this paper is to establish some new

(p, q)

-calculus of Hermite–Hadamard inequalities for the double integral and refinements of the Hermite–Hadamard inequality for

(p, q)

-differentiable convex functions. [...] Read more.

The aim of this paper is to establish some new

(p, q)

-calculus of Hermite–Hadamard inequalities for the double integral and refinements of the Hermite–Hadamard inequality for

(p, q)

-differentiable convex functions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

10 pages, 247 KiB

Open AccessArticle

Best Approximation Results in Various Frameworks

by Taoufik Sabar, Abdelhafid Bassou and Mohamed Aamri

Axioms 2019, 8(2), 67; https://doi.org/10.3390/axioms8020067 - 27 May 2019

Cited by 1 | Viewed by 2262

Abstract

We first provide a best proximity point result for quasi-noncyclic relatively nonexpansive mappings in the setting of dualistic partial metric spaces. Then, those spaces will be endowed with convexity and a result for a cyclic mapping will be obtained. Afterwards, we prove a [...] Read more.

We first provide a best proximity point result for quasi-noncyclic relatively nonexpansive mappings in the setting of dualistic partial metric spaces. Then, those spaces will be endowed with convexity and a result for a cyclic mapping will be obtained. Afterwards, we prove a best proximity point result for tricyclic mappings in the framework of the newly introduced extended partial

S_{b}

-metric spaces. In this way, we obtain extensions of some results in the literature. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

6 pages, 273 KiB

Open AccessArticle

Generic Homeomorphisms with Shadowing of One-Dimensional Continua

by Alfonso Artigue and Gonzalo Cousillas

Axioms 2019, 8(2), 66; https://doi.org/10.3390/axioms8020066 - 26 May 2019

Cited by 1 | Viewed by 2253

Abstract

In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property. Full article

(This article belongs to the Special Issue Shadowing in Dynamical Systems)

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11 pages, 1150 KiB

Open AccessArticle

An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

by Deepak Kumar, Janak Raj Sharma and Clemente Cesarano

Axioms 2019, 8(2), 65; https://doi.org/10.3390/axioms8020065 - 25 May 2019

Cited by 8 | Viewed by 2896

Abstract

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that [...] Read more.

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques. Full article

(This article belongs to the Special Issue Special Functions and Their Applications)

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3 pages, 168 KiB

Open AccessBook Review

Review of “The Significance of the New Logic” Willard Van Orman Quine. Edited and Translated by Walter Carnielli, Frederique Janssen-Lauret, and William Pickering. Cambridge University Press, Cambridge, UK, 2018, pp. 1–200. ISBN-10: 1107179025 ISBN-13: 978-1107179028

by Alfredo Roque Freire

Axioms 2019, 8(2), 64; https://doi.org/10.3390/axioms8020064 - 22 May 2019

Cited by 1 | Viewed by 2696

Abstract

In this review, I will discuss the historical importance of “The Significance of the New Logic” by Quine. This is a translation of the original “O Sentido da Nova Lógica” in Portuguese by Carnielli, Janssen-Lauret, and Pickering. The American philosopher wrote this book [...] Read more.

In this review, I will discuss the historical importance of “The Significance of the New Logic” by Quine. This is a translation of the original “O Sentido da Nova Lógica” in Portuguese by Carnielli, Janssen-Lauret, and Pickering. The American philosopher wrote this book in the beginning of the 1940s, before a major shift in his philosophy. Thus, I will argue that the reader must see this book as an introduction to an important period in his thinking. I will provide a brief summary of the chapters, remarking on valuable features in each of them and positions Quine abandoned in his later work. Full article

(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)

12 pages, 276 KiB

Open AccessArticle

Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions

by Rekha Srivastava, Humera Naaz, Sabeena Kazi and Asifa Tassaddiq

Axioms 2019, 8(2), 63; https://doi.org/10.3390/axioms8020063 - 21 May 2019

Cited by 2 | Viewed by 2904

Abstract

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from [...] Read more.

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from

(0 < ℜ (s) < 1)

to

(0 < ℜ (s) < μ) .

This leads to fresh insights for a new generalization of the Riemann zeta function. The results are validated by obtaining the classical series representation of the polylogarithm and Hurwitz–Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose–Einstein and Fermi–Dirac functions with Apostol–Euler–Nörlund polynomials are established to prove new identities. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

10 pages, 262 KiB

Open AccessArticle

A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

by Maxie D. Schmidt

Axioms 2019, 8(2), 62; https://doi.org/10.3390/axioms8020062 - 19 May 2019

Cited by 1 | Viewed by 3901

Abstract

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and [...] Read more.

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

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7 pages, 253 KiB

Open AccessArticle

Oscillation of Fourth-Order Functional Differential Equations with Distributed Delay

by Clemente Cesarano and Omar Bazighifan

Axioms 2019, 8(2), 61; https://doi.org/10.3390/axioms8020061 - 18 May 2019

Cited by 34 | Viewed by 3209

Abstract

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential [...] Read more.

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples. Full article

(This article belongs to the Special Issue Special Functions and Their Applications)

5 pages, 215 KiB

Open AccessArticle

Unification Theories: New Results and Examples

by Florin F. Nichita

Axioms 2019, 8(2), 60; https://doi.org/10.3390/axioms8020060 - 18 May 2019

Cited by 12 | Viewed by 2515

Abstract

This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We [...] Read more.

This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We also explain that derivations and co-derivations can be unified. Finally, we consider a “modified” Yang–Baxter type equation, which unifies several problems in mathematics. Full article

(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)

6 pages, 582 KiB

Open AccessCorrection

Correction to “On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension” [Axioms 7(3), 58, 2018]

by Francesca Mazzia and Alessandra Sestini

Axioms 2019, 8(2), 59; https://doi.org/10.3390/axioms8020059 - 16 May 2019

Viewed by 2165

Abstract

The authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order

p + r,

where

r = 2

and p is the order of the method. This generalization [...] Read more.

The authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order

p + r,

where

r = 2

and p is the order of the method. This generalization of conjugate-symplecticity states that the methods conserve quadratic first integrals and the Hamiltonian function over time intervals of length

O (h^{- r})

. Theorem 1 of the above mentioned paper is then replaced by a new one. All the other results in the paper do not change. Two new figures related to the already considered Kepler problem are also added. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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16 pages, 7224 KiB

Open AccessArticle

Visual-Servoing Based Global Path Planning Using Interval Type-2 Fuzzy Logic Control

by Mahmut Dirik, Oscar Castillo and Adnan Fatih Kocamaz

Axioms 2019, 8(2), 58; https://doi.org/10.3390/axioms8020058 - 10 May 2019

Cited by 28 | Viewed by 4908

Abstract

Mobile robot motion planning in an unstructured, static, and dynamic environment is faced with a large amount of uncertainties. In an uncertain working area, a method should be selected to address the existing uncertainties in order to plan a collision-free path between the [...] Read more.

Mobile robot motion planning in an unstructured, static, and dynamic environment is faced with a large amount of uncertainties. In an uncertain working area, a method should be selected to address the existing uncertainties in order to plan a collision-free path between the desired two points. In this paper, we propose a mobile robot path planning method in the visualize plane using an overhead camera based on interval type-2 fuzzy logic (IT2FIS). We deal with a visual-servoing based technique for obstacle-free path planning. It is necessary to determine the location of a mobile robot in an environment surrounding the robot. To reach the target and for avoiding obstacles efficiently under different shapes of obstacle in an environment, an IT2FIS is designed to generate a path. A simulation of the path planning technique compared with other methods is performed. We tested the algorithm within various scenarios. Experiment results showed the efficiency of the generated path using an overhead camera for a mobile robot. Full article

(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)

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12 pages, 260 KiB

Open AccessArticle

Fixed Point Theorems through Modified ω-Distance and Application to Nontrivial Equations

by Tariq Qawasmeh, Abdalla Tallafha and Wasfi Shatanawi

Axioms 2019, 8(2), 57; https://doi.org/10.3390/axioms8020057 - 08 May 2019

Cited by 17 | Viewed by 2773

Abstract

In this manuscript, we utilize the concept of modified

ω

-distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified

ω

-distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, [...] Read more.

In this manuscript, we utilize the concept of modified

ω

-distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified

ω

-distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, 203, 120–129] in 2016 to introduce the notions of

(ω, φ)

-Suzuki contraction and generalized

(ω, φ)

-Suzuki contraction. We employ these notions to prove some fixed point results. Moreover, we introduce an example to show the novelty of our results. Furthermore, we introduce some applications for our results. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

10 pages, 284 KiB

Open AccessArticle

Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case

by Galina Kurina

Axioms 2019, 8(2), 56; https://doi.org/10.3390/axioms8020056 - 08 May 2019

Cited by 5 | Viewed by 2698

Abstract

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with [...] Read more.

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix

A (t)

standing in front of the unknown function. In the present paper, the orthogonal projectors onto

k e r A (t)

and

k e r A {(t)}^{'}

(the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

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14 pages, 1334 KiB

Open AccessEditor’s ChoiceArticle

Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability

by Francisco I. Chicharro, Alicia Cordero, Neus Garrido and Juan R. Torregrosa

Axioms 2019, 8(2), 55; https://doi.org/10.3390/axioms8020055 - 06 May 2019

Cited by 10 | Viewed by 3181

Abstract

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are [...] Read more.

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)

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4 pages, 192 KiB

Open AccessArticle

A Note on Anosov Homeomorphisms

by Mauricio Achigar

Axioms 2019, 8(2), 54; https://doi.org/10.3390/axioms8020054 - 01 May 2019

Cited by 5 | Viewed by 2932

Abstract

For an

α

-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the

α

-shadowing property for one-jump [...] Read more.

For an

α

-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the

α

-shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system. Full article

(This article belongs to the Special Issue Shadowing in Dynamical Systems)

19 pages, 5187 KiB

Open AccessArticle

Comparative Study of the Conventional Mathematical and Fuzzy Logic Controllers for Velocity Regulation

by Fevrier Valdez, Oscar Castillo, Camilo Caraveo and Cinthia Peraza

Axioms 2019, 8(2), 53; https://doi.org/10.3390/axioms8020053 - 01 May 2019

Cited by 7 | Viewed by 3548

Abstract

Currently, we are in the digital era, where robotics, with the help of the Internet of Things (IoT), is exponentially advancing, and in the technology market we can find multiple devices for achieving these systems, such as the Raspberry Pi, Arduino, and so [...] Read more.

Currently, we are in the digital era, where robotics, with the help of the Internet of Things (IoT), is exponentially advancing, and in the technology market we can find multiple devices for achieving these systems, such as the Raspberry Pi, Arduino, and so on. The use of these devices makes our work easier regarding processing information or controlling physical mechanisms, as some of these devices have microcontrollers or microprocessors. One of the main challenges in speed control applications is to make the decision to use a fuzzy logic control (FLC) system instead of a conventional controller system, such as a proportional integral (PI) or a proportional integral-derivative (PID). The main contribution of this paper is the design, integration, and comparative study of the use of these three types of controllers—FLC, PI, and PID—for the speed control of a robot built using the Lego Mindstorms EV3 kit. The root mean square error (RMSE) and the settling time were used as metrics to validate the performance of the speed control obtained with the controllers proposed in this paper. Full article

(This article belongs to the Special Issue Softcomputing: Theories and Applications)

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22 pages, 430 KiB

Open AccessArticle

PIP-Space Valued Reproducing Pairs of Measurable Functions

by Jean-Pierre Antoine and Camillo Trapani

Axioms 2019, 8(2), 52; https://doi.org/10.3390/axioms8020052 - 30 Apr 2019

Cited by 4 | Viewed by 2151

Abstract

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space [...] Read more.

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space

Z : = Z (X, μ)

, where

(X, μ)

is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

15 pages, 1336 KiB

Open AccessArticle

Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

by Alicia Cordero, Javier G. Maimó, Juan R. Torregrosa and María P. Vassileva

Axioms 2019, 8(2), 51; https://doi.org/10.3390/axioms8020051 - 25 Apr 2019

Viewed by 2625

Abstract

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The [...] Read more.

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)

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15 pages, 238 KiB

Open AccessEditor’s ChoiceReview

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

by Paolo Emilio Ricci

Axioms 2019, 8(2), 50; https://doi.org/10.3390/axioms8020050 - 25 Apr 2019

Cited by 2 | Viewed by 2636

Abstract

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

20 pages, 330 KiB

Open AccessArticle

Relation Theoretic Common Fixed Point Results for Generalized Weak Nonlinear Contractions with an Application

by Atiya Perveen, Idrees A. Khan and Mohammad Imdad

Axioms 2019, 8(2), 49; https://doi.org/10.3390/axioms8020049 - 24 Apr 2019

Cited by 3 | Viewed by 2820

Abstract

In this paper, by introducing the concept of generalized Ćirić-type weak

(ϕ_{g}, R)

-contraction, we prove some common fixed point results in partial metric spaces endowed with binary relation

R

. We also deduce some useful consequences showing the [...] Read more.

In this paper, by introducing the concept of generalized Ćirić-type weak

(ϕ_{g}, R)

-contraction, we prove some common fixed point results in partial metric spaces endowed with binary relation

R

. We also deduce some useful consequences showing the usability of our results. Finally, we present an application to establish the solution of a system of integral equations. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

30 pages, 1942 KiB

Open AccessEditor’s ChoiceArticle

A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets

by Luciano Stefanini and Barnabas Bede

Axioms 2019, 8(2), 48; https://doi.org/10.3390/axioms8020048 - 24 Apr 2019

Cited by 12 | Viewed by 4229

Abstract

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space [...] Read more.

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space of compact convex sets in terms of a pseudo-norm, i.e., the magnitude of the difference set. A computational procedure for two dimensional sets is outlined and some examples of the new difference are given. Full article

(This article belongs to the Special Issue Softcomputing: Theories and Applications)

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11 pages, 252 KiB

Open AccessArticle

On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

by Mama Foupouagnigni and Salifou Mboutngam

Axioms 2019, 8(2), 47; https://doi.org/10.3390/axioms8020047 - 21 Apr 2019

Cited by 3 | Viewed by 2299

Abstract

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, [...] Read more.

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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Axioms, Volume 8, Issue 2 (June 2019) – 40 articles

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