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Axioms, Volume 9, Issue 4 (December 2020) – 32 articles

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Open AccessArticle
Unification Theories: Means and Generalized Euler Formulas
Axioms 2020, 9(4), 144; https://doi.org/10.3390/axioms9040144 - 20 Dec 2020
Viewed by 437
Abstract
The main concepts in this paper are the means and Euler type formulas; the generalized mean which incorporates the harmonic mean, the geometric mean, the arithmetic mean, and the quadratic mean can be further generalized. Results on the Euler’s formula, the (modified) Yang–Baxter [...] Read more.
The main concepts in this paper are the means and Euler type formulas; the generalized mean which incorporates the harmonic mean, the geometric mean, the arithmetic mean, and the quadratic mean can be further generalized. Results on the Euler’s formula, the (modified) Yang–Baxter equation, coalgebra structures, and non-associative structures are also included in the current paper. Full article
Open AccessArticle
On a Viscosity Iterative Method for Solving Variational Inequality Problems in Hadamard Spaces
Axioms 2020, 9(4), 143; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040143 - 16 Dec 2020
Viewed by 401
Abstract
In this paper, we propose and study an iterative algorithm that comprises of a finite family of inverse strongly monotone mappings and a finite family of Lipschitz demicontractive mappings in an Hadamard space. We establish that the proposed algorithm converges strongly to a [...] Read more.
In this paper, we propose and study an iterative algorithm that comprises of a finite family of inverse strongly monotone mappings and a finite family of Lipschitz demicontractive mappings in an Hadamard space. We establish that the proposed algorithm converges strongly to a common solution of a finite family of variational inequality problems, which is also a common fixed point of the demicontractive mappings. Furthermore, we provide a numerical experiment to demonstrate the applicability of our results. Our results generalize some recent results in literature. Full article
(This article belongs to the Special Issue Theory and Application of Fixed Point)
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Open AccessArticle
Gently Paraconsistent Calculi
Axioms 2020, 9(4), 142; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040142 - 12 Dec 2020
Viewed by 556
Abstract
In this paper, we consider some paraconsistent calculi in a Hilbert-style formulation with the rule of detachment as the sole rule of interference. Each calculus will be expected to contain all axiom schemas of the positive fragment of classical propositional calculus and respect [...] Read more.
In this paper, we consider some paraconsistent calculi in a Hilbert-style formulation with the rule of detachment as the sole rule of interference. Each calculus will be expected to contain all axiom schemas of the positive fragment of classical propositional calculus and respect the principle of gentle explosion. Full article
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Open AccessArticle
New Results on Start-Points for Multi-Valued Maps
Axioms 2020, 9(4), 141; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040141 - 03 Dec 2020
Cited by 1 | Viewed by 412
Abstract
In this manuscript we investigate the existence of start-points for the generalized weakly contractive multi-valued mappings in the setting of left K-complete quasi-pseudo metric space. We provide an example to support the given result. Full article
(This article belongs to the Special Issue Theory and Application of Fixed Point)
Open AccessArticle
A Self-Adaptive Shrinking Projection Method with an Inertial Technique for Split Common Null Point Problems in Banach Spaces
Axioms 2020, 9(4), 140; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040140 - 02 Dec 2020
Viewed by 424
Abstract
In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm [...] Read more.
In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm of the bounded operator and a strong convergence result is proved for the sequence generated by our algorithm under mild conditions. Moreover, we give some applications of our result to split convex minimization and split equilibrium problems in real Banach spaces. This result improves and extends several other results in this direction in the literature. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
Open AccessArticle
Bitcoin Analysis and Forecasting through Fuzzy Transform
Axioms 2020, 9(4), 139; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040139 - 28 Nov 2020
Viewed by 380
Abstract
Sentiment analysis to characterize the properties of Bitcoin prices and their forecasting is here developed thanks to the capability of the Fuzzy Transform (F-transform for short) to capture stylized facts and mutual connections between time series with different natures. The recently proposed L [...] Read more.
Sentiment analysis to characterize the properties of Bitcoin prices and their forecasting is here developed thanks to the capability of the Fuzzy Transform (F-transform for short) to capture stylized facts and mutual connections between time series with different natures. The recently proposed Lp-norm F-transform is a powerful and flexible methodology for data analysis, non-parametric smoothing and for fitting and forecasting. Its capabilities are illustrated by empirical analyses concerning Bitcoin prices and Google Trend scores (six years of daily data): we apply the (inverse) F-transform to both time series and, using clustering techniques, we identify stylized facts for Bitcoin prices, based on (local) smoothing and fitting F-transform, and we study their time evolution in terms of a transition matrix. Finally, we examine the dependence of Bitcoin prices on Google Trend scores and we estimate short-term forecasting models; the Diebold–Mariano (DM) test statistics, applied for their significance, shows that sentiment analysis is useful in short-term forecasting of Bitcoin cryptocurrency. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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Open AccessArticle
Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point
Axioms 2020, 9(4), 138; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040138 - 27 Nov 2020
Cited by 1 | Viewed by 351
Abstract
The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. [...] Read more.
The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. The problem with a “simple” turning point is considered in the case, when the eigenvalue vanishes at t=0 and has the form tm/na(t). The asymptotic convergence of the regularized series is proved. Full article
Open AccessArticle
Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces
Axioms 2020, 9(4), 137; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040137 - 26 Nov 2020
Viewed by 397
Abstract
A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is [...] Read more.
A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results. Full article
(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)
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Open AccessArticle
Oscillation of Emden–Fowler-Type Neutral Delay Differential Equations
Axioms 2020, 9(4), 136; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040136 - 20 Nov 2020
Cited by 1 | Viewed by 419
Abstract
In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results. Full article
Open AccessEditorial
Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of His 80th Birthday
Axioms 2020, 9(4), 135; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040135 - 18 Nov 2020
Viewed by 371
Abstract
Differential and difference equations play an important role in many branches of mathematics [...] Full article
Open AccessArticle
On the Asymptotic Behavior of a Class of Second-Order Non-Linear Neutral Differential Equations with Multiple Delays
Axioms 2020, 9(4), 134; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040134 - 18 Nov 2020
Cited by 3 | Viewed by 434
Abstract
In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear [...] Read more.
In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results. Full article
(This article belongs to the Special Issue Axioms on Advanced Differential Equations for Mathematical Modeling)
Open AccessArticle
Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces
Axioms 2020, 9(4), 133; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040133 - 18 Nov 2020
Viewed by 413
Abstract
In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in [...] Read more.
In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed. Full article
(This article belongs to the Special Issue Topological Algebra)
Open AccessArticle
Fixed Points of g-Interpolative Ćirić–Reich–Rus-Type Contractions in b-Metric Spaces
Axioms 2020, 9(4), 132; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040132 - 16 Nov 2020
Viewed by 395
Abstract
We use interpolation to obtain a common fixed point result for a new type of Ćirić–Reich–Rus-type contraction mappings in metric space. We also introduce a new concept of g-interpolative Ćirić–Reich–Rus-type contractions in b-metric spaces, and we prove some fixed point results [...] Read more.
We use interpolation to obtain a common fixed point result for a new type of Ćirić–Reich–Rus-type contraction mappings in metric space. We also introduce a new concept of g-interpolative Ćirić–Reich–Rus-type contractions in b-metric spaces, and we prove some fixed point results for such mappings. Our results extend and improve some results on the fixed point theory in the literature. We also give some examples to illustrate the given results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
Open AccessArticle
Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefficients and Rapidly Changing Kernels
Axioms 2020, 9(4), 131; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040131 - 13 Nov 2020
Viewed by 355
Abstract
In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the [...] Read more.
In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the influence of the integral term on the asymptotic behavior of the solution of the original problem. Full article
Open AccessArticle
Deep Reinforcement Learning Agent for S&P 500 Stock Selection
Axioms 2020, 9(4), 130; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040130 - 10 Nov 2020
Viewed by 796
Abstract
This study investigated the performance of a trading agent based on a convolutional neural network model in portfolio management. The results showed that with real-world data the agent could produce relevant trading results, while the agent’s behavior corresponded to that of a high-risk [...] Read more.
This study investigated the performance of a trading agent based on a convolutional neural network model in portfolio management. The results showed that with real-world data the agent could produce relevant trading results, while the agent’s behavior corresponded to that of a high-risk taker. The data used were wide in comparison with earlier reported research and was based on the full set of the S&P 500 stock data for twenty-one years supplemented with selected financial ratios. The results presented are new in terms of the size of the data set used and with regards to the model used. The results provide direction and offer insight into how deep learning methods may be used in constructing automatic trading systems. Full article
(This article belongs to the Special Issue Soft Computing in Economics, Finance and Management)
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Open AccessArticle
On Some Coupled Fixed Points of Generalized T-Contraction Mappings in a bv(s)-Metric Space and Its Application
Axioms 2020, 9(4), 129; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040129 - 09 Nov 2020
Viewed by 375
Abstract
Common coupled fixed point theorems for generalized T-contractions are proved for a pair of mappings S:X×XX and g:XX in a bv(s)-metric space, which generalize, extend, and improve some [...] Read more.
Common coupled fixed point theorems for generalized T-contractions are proved for a pair of mappings S:X×XX and g:XX in a bv(s)-metric space, which generalize, extend, and improve some recent results on coupled fixed points. As an application, we prove an existence and uniqueness theorem for the solution of a system of nonlinear integral equations under some weaker conditions and given a convergence criteria for the unique solution, which has been properly verified by using suitable example. Full article
(This article belongs to the Special Issue Theory and Application of Fixed Point)
Open AccessArticle
Dissipative Dynamics of Non-Interacting Fermion Systems and Conductivity
Axioms 2020, 9(4), 128; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040128 - 03 Nov 2020
Viewed by 385
Abstract
In this paper, Non-Equilibrium Steady State that is induced by electric field and the conductivity of non-interacting fermion systems under the dissipative dynamics is discussed. The dissipation is taken into account within a framework of the quantum dynamical semigroup introduced by Davies (1977). [...] Read more.
In this paper, Non-Equilibrium Steady State that is induced by electric field and the conductivity of non-interacting fermion systems under the dissipative dynamics is discussed. The dissipation is taken into account within a framework of the quantum dynamical semigroup introduced by Davies (1977). We obtain a formula of the conductivity for the stationary state, which is applicable to arbitrary potentials. Our formula gives a justification of an adiabatic factor that is often introduced in practical calculation while using the Kubo formula. In addition, the conductivity of crystals (i.e., periodic potentials) is also discussed. Full article
(This article belongs to the Special Issue Quantum Information, Foundations and Measurement)
Open AccessArticle
Inertial Iterative Self-Adaptive Step Size Extragradient-Like Method for Solving Equilibrium Problems in Real Hilbert Space with Applications
Axioms 2020, 9(4), 127; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040127 - 31 Oct 2020
Viewed by 417
Abstract
A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this [...] Read more.
A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this paper, we introduce a modified iterative method to solve equilibrium problems in real Hilbert space. This method can be seen as a modification of the paper titled “A new two-step proximal algorithm of solving the problem of equilibrium programming” by Lyashko et al. (Optimization and its applications in control and data sciences, Springer book pp. 315–325, 2016). A weak convergence result has been proven by considering the mild conditions on the cost bifunction. We have given the application of our results to solve variational inequality problems. A detailed numerical study on the Nash–Cournot electricity equilibrium model and other test problems is considered to verify the convergence result and its performance. Full article
(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)
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Open AccessArticle
Nonlinear Approximations to Critical and Relaxation Processes
Axioms 2020, 9(4), 126; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040126 - 28 Oct 2020
Viewed by 363
Abstract
We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels [...] Read more.
We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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Open AccessArticle
Shifting Operators in Geometric Quantization
Axioms 2020, 9(4), 125; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040125 - 28 Oct 2020
Viewed by 356
Abstract
The original Bohr-Sommerfeld theory of quantization did not give operators of transitions between quantum quantum states. This paper derives these operators, using the first principles of geometric quantization. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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Open AccessArticle
Distributed-Order Non-Local Optimal Control
Axioms 2020, 9(4), 124; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040124 - 25 Oct 2020
Viewed by 575
Abstract
Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. [...] Read more.
Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions. Full article
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Open AccessArticle
Approximate Solutions of the Model Describing Fluid Flow Using Generalized ρ-Laplace Transform Method and Heat Balance Integral Method
Axioms 2020, 9(4), 123; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040123 - 24 Oct 2020
Cited by 5 | Viewed by 538
Abstract
This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that [...] Read more.
This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations. In this work, both the ρ-Laplace homotopy transform method (ρ-LHTM) and the heat balance integral method (HBIM) are successfully combined to solve the fractional incompressible second-grade fluid differential equations. Numerical simulations and their physical interpretations of the mentioned incompressible second-grade fluid model are ensured to illustrate the main findings. It is also proposed that one can recognize the differences in physical analysis of diffusions such as ballistic diffusion, super diffusion, and subdiffusion cases by considering the impact of the orders ρ and φ. Full article
(This article belongs to the collection Mathematical Analysis and Applications)
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Open AccessArticle
A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel
Axioms 2020, 9(4), 122; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040122 - 22 Oct 2020
Viewed by 538
Abstract
The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The [...] Read more.
The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator. Full article
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Open AccessArticle
Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters
Axioms 2020, 9(4), 121; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040121 - 20 Oct 2020
Viewed by 696
Abstract
The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is [...] Read more.
The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem. Full article
Open AccessArticle
Local Spectral Theory for R and S Satisfying RnSRn = Rj
Axioms 2020, 9(4), 120; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040120 - 19 Oct 2020
Viewed by 388
Abstract
In this paper, we analyze local spectral properties of operators R,S and RS which satisfy the operator equations RnSRn=Rj and SnRSn=Sj for same integers j [...] Read more.
In this paper, we analyze local spectral properties of operators R,S and RS which satisfy the operator equations RnSRn=Rj and SnRSn=Sj for same integers jn0. We also continue to study the relationship between the local spectral properties of an operator R and the local spectral properties of S. Thus, we investigate the transmission of some local spectral properties from R to S and we illustrate our results with an example. The theory is exemplified in some cases. Full article
Open AccessArticle
On the Bernstein Affine Fractal Interpolation Curved Lines and Surfaces
Axioms 2020, 9(4), 119; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040119 - 18 Oct 2020
Viewed by 430
Abstract
In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein [...] Read more.
In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein affine fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein affine fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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Modified Viscosity Subgradient Extragradient-Like Algorithms for Solving Monotone Variational Inequalities Problems
Axioms 2020, 9(4), 118; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040118 - 15 Oct 2020
Cited by 1 | Viewed by 568
Abstract
Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities [...] Read more.
Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities problems in real Hilbert space. The result of the strong convergence of the method is well established without the information of the operator’s Lipschitz constant. There are proper mathematical studies relating our newly designed method to the currently state of the art on several practical test problems. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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Open AccessArticle
Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions
Axioms 2020, 9(4), 117; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040117 - 15 Oct 2020
Viewed by 432
Abstract
In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions [...] Read more.
In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well. Full article
(This article belongs to the Special Issue Special Functions and Their Applications)
Open AccessArticle
Finite Series of Distributional Solutions for Certain Linear Differential Equations
Axioms 2020, 9(4), 116; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040116 - 13 Oct 2020
Viewed by 389
Abstract
In this paper, we present the distributional solutions of the modified spherical Bessel differential equations t2y(t)+2ty(t)[t2+ν(ν+1)]y [...] Read more.
In this paper, we present the distributional solutions of the modified spherical Bessel differential equations t2y(t)+2ty(t)[t2+ν(ν+1)]y(t)=0 and the linear differential equations of the forms t2y(t)+3ty(t)(t2+ν21)y(t)=0, where νN{0} and tR. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of ν. The results of several examples are also presented. Full article
Open AccessArticle
Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems
Axioms 2020, 9(4), 115; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9040115 - 13 Oct 2020
Cited by 1 | Viewed by 575
Abstract
A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, [...] Read more.
A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
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