Topical Collection "Mathematical Analysis and Applications"

Editor

Prof. Dr. Hari Mohan Srivastava
grade Website SciProfiles
Collection Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization
Special Issues and Collections in MDPI journals

Topical Collection Information

Dear Colleagues,

Investigations involving the theory and applications of the mathematical analytic tools and techniques are remarkably widespread in many diverse areas of the mathematical, physical, chemical, engineering, and statistical sciences. In this Collection, we invite and welcome review, expository, and original research articles dealing with the recent advances in mathematical analysis and its multidisciplinary applications.

Looking forward to your contribution to this Topical Collection,

Cordially yours,

Prof. Dr. H. M. Srivastava
Collection Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Mathematical (or higher transcendental) functions and their applications
  • Fractional calculus and its applications
  • q-series and q-polynomials
  • Analytic number theory
  • Special functions of mathematical physics and applied mathematics
  • Geometric function theory of complex analysis

Related Special Issues

Published Papers (43 papers)

2020

Jump to: 2019, 2018

Open AccessArticle
Convergence of Weak*-Scalarly Integrable Functions
Axioms 2020, 9(3), 112; https://doi.org/10.3390/axioms9030112 - 22 Sep 2020
Abstract
Let (Ω,F,μ) be a complete probability space, E a separable Banach space and E the topological dual vector space of E. We present some compactness results in LE1E, the Banach [...] Read more.
Let (Ω,F,μ) be a complete probability space, E a separable Banach space and E the topological dual vector space of E. We present some compactness results in LE1E, the Banach space of weak*-scalarly integrable E-valued functions. As well we extend the classical theorem of Komlós to the bounded sequences in LE1E. Full article
Open AccessArticle
The Existence and Uniqueness of an Entropy Solution to Unilateral Orlicz Anisotropic Equations in an Unbounded Domain
Axioms 2020, 9(3), 109; https://doi.org/10.3390/axioms9030109 - 17 Sep 2020
Abstract
The purpose of this work is to prove the existence and uniqueness of a class of nonlinear unilateral elliptic problem (P) in an arbitrary domain, managed by a low-order term and non-polynomial growth described by an N-uplet of N-function [...] Read more.
The purpose of this work is to prove the existence and uniqueness of a class of nonlinear unilateral elliptic problem (P) in an arbitrary domain, managed by a low-order term and non-polynomial growth described by an N-uplet of N-function satisfying the Δ2-condition. The source term is merely integrable. Full article
Open AccessArticle
Star-Shapedness of \({\mathcal N}\)-Structures in Euclidean Spaces
Axioms 2020, 9(3), 107; https://doi.org/10.3390/axioms9030107 - 12 Sep 2020
Abstract
The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a [...] Read more.
The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a quasi (or, pseudo) star-shaped set to be a star-shaped set are provided. Full article
Open AccessArticle
A Proof of Komlós Theorem for Super-Reflexive Valued Random Variables
Axioms 2020, 9(3), 106; https://doi.org/10.3390/axioms9030106 - 11 Sep 2020
Abstract
We give a geometrical proof of Komlós’ theorem for sequences of random variables with values in super-reflexive Banach space. Our approach is inspired by the elementary proof given by Guessous in 1996 for the Hilbert case and uses some geometric properties of smooth [...] Read more.
We give a geometrical proof of Komlós’ theorem for sequences of random variables with values in super-reflexive Banach space. Our approach is inspired by the elementary proof given by Guessous in 1996 for the Hilbert case and uses some geometric properties of smooth spaces. Full article
Open AccessArticle
A New Common Fixed Point Theorem for Three Commuting Mappings
Axioms 2020, 9(3), 105; https://doi.org/10.3390/axioms9030105 - 11 Sep 2020
Abstract
In the present paper, we propose a common fixed point theorem for three commuting mappings via a new contractive condition which generalizes fixed point theorems of Darbo, Hajji and Aghajani et al. An application is also given to illustrate our main result. Moreover, [...] Read more.
In the present paper, we propose a common fixed point theorem for three commuting mappings via a new contractive condition which generalizes fixed point theorems of Darbo, Hajji and Aghajani et al. An application is also given to illustrate our main result. Moreover, several consequences are derived, which are generalizations of Darbo’s fixed point theorem and a Hajji’s result. Full article
Open AccessArticle
Global Optimization and Common Best Proximity Points for Some Multivalued Contractive Pairs of Mappings
Axioms 2020, 9(3), 102; https://doi.org/10.3390/axioms9030102 - 07 Sep 2020
Abstract
In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. [...] Read more.
In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. Next, we put forward the concept of multivalued Kannan-type contractive pair and also the concept of weak Δ-property to determine the existence of common best proximity point for such a pair of maps. Full article
Open AccessArticle
Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature
Axioms 2020, 9(3), 91; https://doi.org/10.3390/axioms9030091 - 29 Jul 2020
Abstract
A novel method for generating and providing quadrature solutions to families of linear, second-order, ordinary differential equations is presented in this paper. It is based upon a comparison of control system feedback diagrams—one representing the system and equation under study and a second [...] Read more.
A novel method for generating and providing quadrature solutions to families of linear, second-order, ordinary differential equations is presented in this paper. It is based upon a comparison of control system feedback diagrams—one representing the system and equation under study and a second equalized to it and providing solutions. The resulting Riccati equation connection between them is utilized to generate and solve groups of equations parameterized by arbitrary functions and constants. This method also leads to a formal solution mechanism for all second-order linear differential equations involving an infinite series of integrals of each equation’s Schwarzian derivative. The practicality of this mechanism is strongly dependent on the series rates of and allowed regions for convergence. The feedback diagram method developed is shown to be equivalent to a comparable method based on the differential equation’s normal form and another relying upon the grouping of terms for a reduction of the equation order, but augmenting their results. Applications are also made to the Helmholtz equation. Full article
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Open AccessArticle
Mindfulness Model Using Polariton Oscillation in Plasmonic Circuit for Human Performance Management
Axioms 2020, 9(3), 76; https://doi.org/10.3390/axioms9030076 - 08 Jul 2020
Abstract
We have proposed that human life is formed on a space and time function relationship basis, which is distorted after fertilization in the ovum, from which growth is generated by a space–time distortion against the universe’s gravity. A space–time distortion’s reduction can be [...] Read more.
We have proposed that human life is formed on a space and time function relationship basis, which is distorted after fertilization in the ovum, from which growth is generated by a space–time distortion against the universe’s gravity. A space–time distortion’s reduction can be managed by space and time separation, which is known as mindfulness. A space–time distortion in human cells is configured by a polariton traveling in a gold grating film, which can be employed to investigate mindfulness characteristics. Mindfulness is the steady state of the time function of energy after the separation. Energy levels of mindfulness based on polariton aspects are categorized by a quantum number (n), which can be reduced to be a two-level system called Rabi oscillation by a successive filtering method. We have assumed a cell space–time distortion can reduce to reach the original state, which is the stopping state. Mindfulness with a certain frequency energy level of n = 2 was achieved. Several techniques in the practice of mindfulness based on successive filtering called meditation are given and explained, where the required levels of the mindfulness state can be achieved. The criteria of the proposed method are a low energy level (n) and high frequency (f) outputs, which can apply to having a working performance improvement. Full article
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Open AccessArticle
Fixed Point Results under Generalized c-Distance in Cone b-Metric Spaces Over Banach Algebras
Axioms 2020, 9(1), 31; https://doi.org/10.3390/axioms9010031 - 21 Mar 2020
Abstract
In this work, we define the concept of a generalized c-distance in cone b-metric spaces over a Banach algebra and introduce some its properties. Then, we prove the existence and uniqueness of fixed points for mappings satisfying weak contractive conditions such [...] Read more.
In this work, we define the concept of a generalized c-distance in cone b-metric spaces over a Banach algebra and introduce some its properties. Then, we prove the existence and uniqueness of fixed points for mappings satisfying weak contractive conditions such as Han–Xu-type contraction and Cho-type contraction with respect to this distance. Our assertions are useful, since we remove the continuity condition of the mapping and the normality condition for the cone. Several examples are given to support the main results. Full article
Open AccessEditorial
Mathematical Analysis and Applications II
Axioms 2020, 9(1), 16; https://doi.org/10.3390/axioms9010016 - 06 Feb 2020
Abstract
Web Site: http://www [...] Full article

2019

Jump to: 2020, 2018

Open AccessArticle
Repeated Derivatives of Hyperbolic Trigonometric Functions and Associated Polynomials
Axioms 2019, 8(4), 138; https://doi.org/10.3390/axioms8040138 - 06 Dec 2019
Cited by 1
Abstract
Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by [...] Read more.
Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of sec ( . ) , tan ( . ) and for their hyperbolic counterparts. Full article
Open AccessArticle
Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators
Axioms 2019, 8(4), 111; https://doi.org/10.3390/axioms8040111 - 10 Oct 2019
Cited by 3
Abstract
Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation [...] Read more.
Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Szász-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Szász-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Szász-Mirakjan Beta-type operators. Full article
Open AccessArticle
Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction
Axioms 2019, 8(3), 88; https://doi.org/10.3390/axioms8030088 - 26 Jul 2019
Cited by 4
Abstract
In this paper, for a given direction b C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t [...] Read more.
In this paper, for a given direction b C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t C } for any z 0 C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n R + . Full article
Open AccessArticle
On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions
Axioms 2019, 8(3), 87; https://doi.org/10.3390/axioms8030087 - 26 Jul 2019
Cited by 4
Abstract
In the theory of generalized hypergeometric functions, classical summation theorems for the series 2 F 1 , 3 F 2 , 4 F 3 , 5 F 4 and 7 F 6 play a key role. Very recently, Masjed-Jamei and Koepf established generalizations [...] Read more.
In the theory of generalized hypergeometric functions, classical summation theorems for the series 2 F 1 , 3 F 2 , 4 F 3 , 5 F 4 and 7 F 6 play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions p F p for p = 2 , 3 , 4 , 5 and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings. Full article
Open AccessArticle
Generalized Hyers–Ulam Stability of the Additive Functional Equation
Axioms 2019, 8(2), 76; https://doi.org/10.3390/axioms8020076 - 25 Jun 2019
Cited by 1
Abstract
We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, …, xn + yn) = 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, …, xn + yn) = f(x1, x2, … xn) + f(y1, y2, …, yn). 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
Open AccessArticle
On Almost b-Metric Spaces and Related Fixed Point Results
Axioms 2019, 8(2), 70; https://doi.org/10.3390/axioms8020070 - 01 Jun 2019
Cited by 6
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
Open AccessArticle
(p, q)-Hermite–Hadamard Inequalities for Double Integral and (p, q)-Differentiable Convex Functions
Axioms 2019, 8(2), 68; https://doi.org/10.3390/axioms8020068 - 28 May 2019
Cited by 2
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
Open AccessArticle
Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions
Axioms 2019, 8(2), 63; https://doi.org/10.3390/axioms8020063 - 21 May 2019
Cited by 1
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 < ( s ) < 1 ) to ( 0 < ( 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
Open AccessArticle
A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions
Axioms 2019, 8(2), 62; https://doi.org/10.3390/axioms8020062 - 19 May 2019
Cited by 1
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
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Open AccessArticle
Fixed Point Theorems through Modified ω-Distance and Application to Nontrivial Equations
Axioms 2019, 8(2), 57; https://doi.org/10.3390/axioms8020057 - 08 May 2019
Cited by 5
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
Open AccessReview
Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey
Axioms 2019, 8(2), 50; https://doi.org/10.3390/axioms8020050 - 25 Apr 2019
Cited by 1
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
Open AccessArticle
A Note on the Displacement Problem of Elastostatics with Singular Boundary Values
Axioms 2019, 8(2), 46; https://doi.org/10.3390/axioms8020046 - 19 Apr 2019
Cited by 2
Abstract
The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum a is considered. Precisely, if the elastic body is represented by a domain of class C k ( k 2 ) of R 3 and a [...] Read more.
The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum a is considered. Precisely, if the elastic body is represented by a domain of class C k ( k 2 ) of R 3 and a W 2 k 1 / q , q ( Ω ) , q ( 1 , + ) , then it is proved that there exists a solution which is of class C in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class. Full article
Open AccessArticle
Fixed Point Results in Partial Symmetric Spaces with an Application
Axioms 2019, 8(1), 13; https://doi.org/10.3390/axioms8010013 - 22 Jan 2019
Cited by 5
Abstract
In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of [...] Read more.
In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of Fredholm integral equations. Furthermore, we introduce an analogue of the Hausdorff metric in the context of partial symmetric spaces and utilize the same to prove an analogue of the Nadler contraction principle in such spaces. Our results extend and improve many results in the existing literature. We also give some examples exhibiting the utility of our newly established results. Full article
Open AccessArticle
A New Identity for Generalized Hypergeometric Functions and Applications
Axioms 2019, 8(1), 12; https://doi.org/10.3390/axioms8010012 - 18 Jan 2019
Cited by 1
Abstract
We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized [...] Read more.
We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38). Full article

2018

Jump to: 2020, 2019

Open AccessArticle
Extended Partial Sb-Metric Spaces
Axioms 2018, 7(4), 87; https://doi.org/10.3390/axioms7040087 - 21 Nov 2018
Cited by 3
Abstract
In this paper, we introduce the concept of extended partial S b -metric spaces, which is a generalization of the extended S b -metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric [...] Read more.
In this paper, we introduce the concept of extended partial S b -metric spaces, which is a generalization of the extended S b -metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric spaces generalize many results in the literature. Moreover, we prove some fixed point theorems under some different contractions, and some examples are given to illustrate our results. Full article
Open AccessEditorial
Mathematical Analysis and Applications
Axioms 2018, 7(4), 82; https://doi.org/10.3390/axioms7040082 - 12 Nov 2018
Abstract
Website: http://www.math.uvic.ca/faculty/harimsri/ [...] Full article
Open AccessArticle
On the Fixed-Circle Problem and Khan Type Contractions
Axioms 2018, 7(4), 80; https://doi.org/10.3390/axioms7040080 - 08 Nov 2018
Cited by 6
Abstract
In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of F C -Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the [...] Read more.
In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of F C -Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the common fixed-circle problem. Full article
Open AccessArticle
Common Fixed Point Theorems for Generalized Geraghty (α,ψ,ϕ)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces
Axioms 2018, 7(4), 74; https://doi.org/10.3390/axioms7040074 - 25 Oct 2018
Cited by 12
Abstract
The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty ( α , ψ , ϕ ) -quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show [...] Read more.
The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty ( α , ψ , ϕ ) -quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via α admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces. Full article
Open AccessArticle
New Bell–Sheffer Polynomial Sets
Axioms 2018, 7(4), 71; https://doi.org/10.3390/axioms7040071 - 08 Oct 2018
Cited by 3
Abstract
In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results [...] Read more.
In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear. Full article
Open AccessArticle
Periodically Forced Nonlinear Oscillatory Acoustic Vacuum
Axioms 2018, 7(4), 69; https://doi.org/10.3390/axioms7040069 - 22 Sep 2018
Cited by 1
Abstract
In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system. Full article
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Open AccessArticle
Solutions to Abel’s Integral Equations in Distributions
Axioms 2018, 7(3), 66; https://doi.org/10.3390/axioms7030066 - 02 Sep 2018
Cited by 2
Abstract
The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space D ( R + ) based on inverse convolutional operators and Babenko’s approach. Furthermore, we [...] Read more.
The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space D ( R + ) based on inverse convolutional operators and Babenko’s approach. Furthermore, we provide interesting applications of Abel’s integral equations in viscoelastic systems, as well as solving other integral equations, such as θ π / 2 y ( φ ) cos β φ ( cos θ cos φ ) α d φ = f ( θ ) , and 0 x 1 / 2 g ( x ) y ( x + t ) d x = f ( t ) . Full article
Open AccessArticle
Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions
Axioms 2018, 7(3), 65; https://doi.org/10.3390/axioms7030065 - 01 Sep 2018
Cited by 3
Abstract
The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation [...] Read more.
The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative. Full article
Open AccessArticle
Umbral Methods and Harmonic Numbers
Axioms 2018, 7(3), 62; https://doi.org/10.3390/axioms7030062 - 01 Sep 2018
Cited by 2
Abstract
The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals. Full article
Open AccessArticle
Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution
Axioms 2018, 7(3), 61; https://doi.org/10.3390/axioms7030061 - 23 Aug 2018
Cited by 1
Abstract
A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control [...] Read more.
A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control variable is discretized into piece-wise constant intervals. The problem is solved by Differential Evolution (DE), which is one of the evolutionary algorithm developed for optimization. Two scenarios, namely four time horizons and eight time horizons, are compared and discussed. The simulations show that models with controls lead to decreasing the number of patients as well as epidemic period length. From the optimal solution, vector elimination is the prioritized strategy for disease control. Full article
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Some Identities for Euler and Bernoulli Polynomials and Their Zeros
Axioms 2018, 7(3), 56; https://doi.org/10.3390/axioms7030056 - 14 Aug 2018
Cited by 16Correction
Abstract
In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer. Full article
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A New Type of Generalization on W—Asymptotically J λ—Statistical Equivalence with the Number of α
Axioms 2018, 7(3), 54; https://doi.org/10.3390/axioms7030054 - 02 Aug 2018
Cited by 1
Abstract
In our paper, by using the concept of Wasymptotically J statistical equivalence of order α which has been previously defined, we present the definitions of Wasymptotically Jλstatistical equivalence of order α, Wstrongly [...] Read more.
In our paper, by using the concept of Wasymptotically J statistical equivalence of order α which has been previously defined, we present the definitions of Wasymptotically Jλstatistical equivalence of order α, Wstrongly asymptotically Jλstatistical equivalence of order α, and Wstrongly Cesáro asymptotically Jstatistical equivalence of order α where 0<α1. We also extend these notions with a sequence of positive real numbers, p=(pk), and we investigate how our results change if p is constant. Full article
Open AccessArticle
Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative
Axioms 2018, 7(3), 48; https://doi.org/10.3390/axioms7030048 - 18 Jul 2018
Cited by 4
Abstract
One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed [...] Read more.
One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed. Full article
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Some Summation Theorems for Generalized Hypergeometric Functions
Axioms 2018, 7(2), 38; https://doi.org/10.3390/axioms7020038 - 08 Jun 2018
Cited by 5
Abstract
Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in [...] Read more.
Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in order to extend some classical summation theorems of hypergeometric functions such as Gauss, Kummer, Dixon, Watson, Whipple, Pfaff–Saalschütz and Dougall formulas and also obtain some new summation theorems in the sequel. Full article
Open AccessArticle
Pre-Metric Spaces Along with Different Types of Triangle Inequalities
Axioms 2018, 7(2), 34; https://doi.org/10.3390/axioms7020034 - 24 May 2018
Cited by 1
Abstract
The T 1 -spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies. [...] Read more.
The T 1 -spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies. Full article
Open AccessArticle
Yukawa Potential, Panharmonic Measure and Brownian Motion
Axioms 2018, 7(2), 28; https://doi.org/10.3390/axioms7020028 - 01 May 2018
Cited by 3
Abstract
This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind [...] Read more.
This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind the algorithm for the case of the Yukawa PDE. We study the panharmonic measure, which is a generalization of the harmonic measure for the Yukawa PDE. We show that there are natural stochastic definitions for the panharmonic measure in terms of the Brownian motion and that the harmonic and the panharmonic measures are all mutually equivalent. Furthermore, we calculate their Radon–Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm. Full article
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Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator
Axioms 2018, 7(2), 27; https://doi.org/10.3390/axioms7020027 - 24 Apr 2018
Cited by 2
Abstract
Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved. Full article
Open AccessArticle
New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences
Axioms 2018, 7(2), 24; https://doi.org/10.3390/axioms7020024 - 13 Apr 2018
Cited by 2
Abstract
In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical [...] Read more.
In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical convergence and strongly A I -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S θ A I , F and N θ A I , F , respectively. We give some inclusion relations between S A I , F , S θ A I , F and N θ A I , F . We also investigate Cesáro summability for A I and we obtain some basic results between A I -Cesáro summability, strongly A I -Cesáro summability and the spaces mentioned above. Full article
Open AccessArticle
Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods
Axioms 2018, 7(2), 22; https://doi.org/10.3390/axioms7020022 - 01 Apr 2018
Cited by 2
Abstract
In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order [...] Read more.
In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 n , k ; λ . Finally, we make some remarks and observations regarding these identities and relations. Full article
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