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1

Discussion on angular asymmetry in the solutions of SLIP model

Kowalczyk Piotr, Płociniczak Łukasz, Wróblewska Zofia

Mathematica Applicanda

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2023

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Vol. 51, no. 2

223--237

EN

We consider a spring-mass model of human running which is built upon an inverted elastic pendulum. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). In our previous approach, we assumed that periodic solutions in the support phase are symmetrical with respect to the touch-down and take-off angles for the large spring constant (or small angle of attack). Based on proposed solutions, we introduce analytical approximations of an asymmetrical boundary value problem, which brings our model closer to real running. By appropriately concatenating asymptotic solutions for the two gait phases, we are able to reduce the dynamics to a one-dimensional apex to apex return map and then to investigate the existence and stability of periodic solutions. Unlike in the symmetrical version, we could not find sufficient conditions for this map to have a unique stable fixed point. Extending the model with the possibility of taking off with the angle other than during landing, the aforementioned asymmetry, is necessary in the context of real run considerations. Thanks to this, our work could be enriched by experimental results. In this paper, we will present the possible reasons for the instability of asymmetric solutions in conjunction with conclusions from the observation of real runs.

PL

W pracy rozważamy model biegu, w którym człowiek sprowadzony jest do punktu masy na nieważkiej sprężynie, a momencie kontaktu z podłożem staje się odwróconym sprężystym wahadłem. Sam model składa się z dwóch zestawów równań różniczkowych - jedno opisuje ruch środka masy biegacza podczas kontaktu stopy z podłożem (faza podparcia), a drugi fazę lotu. W naszym poprzednim podejściu zakładaliśmy, że rozwiązania okresowe w fazie podparcia są symetryczne względem kątów lądowania i odbicia dla dużej wartości sztywności nogi (lub małego kąta ataku). Na podstawie proponowanych rozwiązań wprowadzamy analityczne przybliżenia asymetrycznego problemu brzegowego, co zbliża nasz model do rzeczywistego biegu. Odpowiednio łącząc asymptotyczne rozwiązania dla obu faz biegu, jesteśmy w stanie zredukować dynamikę do jednego wymiaru i utworzyć odwzorowanie powrotu od wierzchołka do kolejnego wierzchołka praboli lotu, a następnie badać istnienie i stabilność rozwiązań okresowych. W odróżnieniu od wersji symetrycznej, nie mogliśmy znaleźć wystarczających warunków, aby to odwzorowanie miało jednoznacznie określony stabilny punkt stały. Rozszerzenie modelu o możliwość odbicia pod innym kątem, niż podczas lądowania (asymetria), jest konieczne w kontekście rozważań nad rzeczywistym biegiem. Dzięki temu nasza praca mogła zostać wzbogacona o wyniki eksperymentalne. W tym artykule przedstawimy możliwe przyczyny niestabilności asymetrycznych rozwiązań w połączeniu z wnioskami z obserwacji rzeczywistych biegów.

2

Semi-analytical scheme with its stability analysis for solving the fractional-order predator-prey equations by using Laplace-VIM

Adel Mohamed, Sweilam Nasser H., Khader Mohamed M.

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 4

5-17

EN

The article’s goal is to implement a semi-analytical technique named, the Laplace variational iteration method (LVIM), which is the combination of VIM and Laplace transform method. Although both the Laplace transform method and VIM cannot be applied to some nonlinear fractional differential equations (FDEs) individually, this combination will give a fast-convergent solution to the problem under study. The proposed scheme is used to numerically solve a biodynamic system called the Lotka-Volterra system, i.e. Predator-Prey Equations (PPEs). The system of FDEs can be used to represent this scenario, as well as the Caputo-Fabrizio fractional derivative will be used throughout the study. By assessing the residual error function, we can confirm that the given procedure is effective and accurate. The outcomes demonstrate that the technique used is an effective tool for simulating such models.

3

Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative

Mansouri Ikram, Bekkouche Mohammed Moumen, Ahmed Abdelaziz Azeb

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 1

46--56

EN

Within this work, we discuss the existence of solutions for a coupled system of linear fractional differential equations involving Caputo-Fabrizio fractional orders. We prove the existence and uniqueness of the solution by using the Picard-Lindel ̈of method and fixed point theory. Also, to compute an approximate solution of problem, we utilize the Adomian decomposition method (ADM), as this method provides the solution in the form of a series such that the infinite series converge to the exact solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

4

A method of lower and upper solutions for control problems and application to a model of bone marrow transplantation

Parajdi Lorand Gabriel, Precup Radu, Haplea Ioan Ştefan

International Journal of Applied Mathematics and Computer Science

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2023

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Vol. 33, no. 3

409--418

EN

A lower and upper solution method is introduced for control problems related to abstract operator equations. The method is illustrated on a control problem for the Lotka-Volterra model with seasonal harvesting and applied to a control problem of cell evolution after bone marrow transplantation.

5

Positive solutions for fractional differential equation at resonance under integral boundary conditions

Wang Youyu, Huang Yue, Li Xianfei

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

238--253

EN

By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.

6

Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings

Mehmood Nayyar, Khan Israr Ali, Nawaz Muhammad Ayyaz, Ahmad Niaz

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

452--469

EN

In this article, fixed point results for self-mappings in the setting of two metrics satisfying F -lipschitzian conditions of rational-type are proved, where F is considered as a semi-Wardowski function with constant τ∈R instead of τ>0 . Two metrics have been considered, one as an incomplete while the other is orbitally complete. The mapping is taken to be orbitally continuous from one metric to another. Some examples are provided to validate our results. For applications, we present existence results for the solutions of a new type of ABC-fractional boundary value problem.

7

Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces

Kim Kyung Soo

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

81--89

EN

The aim of this paper is to study existence and uniqueness of coupled fixed point for a family of self-mappings satisfying a new coupled implicit relation in a Hilbert space. We also prove well-posedness of a coupled fixed point problem.

8

Computation of solution of integral equations via fixed point results

Alqudah Manar A., Garodia Chanchal, Uddin Izhar, Nieto Juan J.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

772--785

EN

The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.

9

Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

Karapınar Erdal, Romaguera Salvador, Tirado Pedro

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

939--951

EN

Involving w-distances we prove a fixed point theorem of Caristi-type in the realm of (non-necessarily T1) quasi-metric spaces. With the help of this result, a characterization of quasi-metric completeness is obtained. Our approach allows us to retrieve several key examples occurring in various fields of mathematics and computer science and that are modeled as non- T1 quasi-metric spaces. As an application, we deduce a characterization of complete G -metric spaces in terms of a weak version of Caristi’s theorem that involves a G-metric version of w-distances.

10

An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems

Okeke Chibueze C., Ugwunnadi Godwin C., Jolaoso Lateef O.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

506--527

EN

The purpose of this article is to study and analyse a new extragradient-type algorithm with an inertial extrapolation step for solving split fixed-point problems for demicontractive mapping, equilibrium problem, and pseudomonotone variational inequality problem in real Hilbert spaces. One of the advantages of the proposed algorithm is that a strong convergence result is achieved without a prior estimate of the Lipschitz constant of the cost operator, which is very difficult to find. In addition, the stepsize is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the cost operator. Some numerical experiments are reported to show the performance and behaviour of the sequence generated by our algorithm. The obtained results in this article extend and improve many related recent results in this direction in the literature.

11

A study of a meromorphic perturbation of the sine family

Domínguez Patricia, Vázquez Josué

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

8--27

EN

We study the dynamics of a meromorphic perturbation of the family λsinz by adding a pole at zero and a parameter μ , that is, fλ,μ(z)=λsinz+μ/z , where λ,μ∈C⧹{0} . We study some geometrical properties of fλ,μ and prove that the imaginary axis is invariant under fn and belongs to the Julia set when ∣λ∣≥1 . We give a set of parameters (λ,μ) , such that the Fatou set of fλ,μ has two super-attracting domains. If λ=1 and μ∈(0,2) , the Fatou set of f1,μ has two attracting domains. Also, we give parameters λ,μ such that ±π/2 are fixed points of fλ,μ and the Fatou set of fλ,μ contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of fλ,μ , where the Fatou set contains two types of domains, for λ,μ given.

12

New soft separation axioms and fixed soft points with respect to total belong and total non-belong relations

Al-shami Tareq M., Tercan Adnan, Mhemdi Abdelwaheb

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

196--211

EN

In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with respect to ordinary points, namely tt-soft pre Ti (i = 0, 1, 2, 3, 4) and tt-soft pre-regular spaces. The motivations to use these relations are, first, cancel the constant shape of soft pre-open and pre-closed subsets of soft pre-regular spaces, and second, generalization of existing comparable properties on classical topology. With the help of examples, we show the relationships between them as well as with soft pre Ti (i = 0, 1, 2, 3, 4) and soft pre-regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a tt-soft pre-regular space and demonstrate that it guarantees the equivalence of tt-soft pre Ti (i = 0, 1, 2). Furthermore, we investigate the behaviors of these soft separation axioms with the concepts of productand sum of soft spaces. Finally, we introduce a concept of pre-fixed soft point and study its main properties.

13

L ∞-error estimates of a finite element method for Hamilton-Jacobi-Bellman equations with nonlinear source terms with mixed boundary condition

Miloudi Madjda, Saadi Samira, Haiour Mohamed

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

452--461

EN

In this paper, we introduce a new method to analyze the convergence of the standard finite element method for Hamilton-Jacobi-Bellman equation with noncoercive operators with nonlinear source terms with the mixed boundary conditions. The method consists of combining Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart and then between the continuous solution and the approximate solution.

14

Convergence theorems for generalized hemicontractive mapping in p-uniformly convex metric space

Ugwunnadi Godwin C., Izuchukwu Chinedu, Mewomo Oluwatosin T.

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

221--229

EN

In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in p-uniformly convex metric spaces, and prove both Delta-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete p-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.

15

Convergence theorem for a finite family of asymptotically demicontractive multi-valued mappings in CAT(0) spaces

Ugwunnadi Godwin C., Mewomo Oluwatosin T., Izuchukwu Chinedu

Journal of Applied Analysis

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2020

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Vol. 26, nr 1

117--130

EN

In this paper, we introduce the class of asymptotically demicontractive multivalued mappings and establish a strong convergence theorem of the modified Mann iteration to a common fixed point of a finite family of asymptotically demicontractive multivalued mappings in a complete CAT(0) space. We also give a numerical example of our iterative method to show its applicability.

16

Existence results for nonlinear coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type

Nabil Tamer

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

236--248

EN

The combined systems of integral equations have become of great importance in various fields of sciences such as electromagnetic and nuclear physics. New classes of the merged type of Urysohn Volterra-Chandrasekhar quadratic integral equations are proposed in this paper. This proposed system involves fractional Urysohn Volterra kernels and also Chandrasekhar kernels. The solvability of a coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type is studied. To realize the existence of a solution of those mixed systems, we use the Perov’s fixed point combined with the Leray-Schauder fixed point approach in generalized Banach algebra spaces.

17

An iterative algorithm for the system of split mixed equilibrium problem

Karahan Ibrahim, Jolaoso Lateef Olakunle

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

309--324

EN

In this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

18

Vibration control of a rotating shaft by passive mass-spring-disc dynamic vibration absorber

Nguyen Duy Chinh

Archive of Mechanical Engineering

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2020

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LXVII, nr 3

279--297

EN

Shaft is a machine element which is used to transmit rotary motion or torque. During transmission of motion, however, the machine shaft doesn’t always rotate with a constant angular velocity. Because of unstable current or due to sudden acceleration and deceleration, the machine shaft will rotate at a variable angular velocity. It is this rotary motion that generates the moment of inertial force, causing the machine shaft to have torsional deformation. However, due to the elasticity of the material, the shaft produces torsional vibration. Therefore, the main objective of this paper is to determine the optimal parameters of dynamic vibration absorber to eliminate torsional vibration of the rotating shaft that varies with time. The new results in this paper are summarized as follows: Firstly, the author determines the optimal parameters by using the minimum quadratic torque method. Secondly, the maximization of equivalent viscous resistance method is used for determining the optimal parameters. Thirdly, the author gives the optimal parameters of dynamic vibration absorber based on the fixed-point method. In this paper, the optimum parameters are found in an explicit analytical solutions, helping the scientists to easily find the optimal parameters for eliminating torsional vibration of the rotating shaft.

19

On some fixed point theorems for expansive mappings in dislocated cone metric spaces with Banach algebras

Auwalu Abba, Hinçal Evren, Mishra Lakshmi Narayan

Journal of Mathematics and Applications

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2019

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Vol. 42

21--33

EN

In this paper, we introduced the notion of generalized expansive mappings in dislocated cone metric spaces with Banach algebras. Furthermore, we prove some ﬁxed point theorems for generalized expansive mappings in dislocated cone metric spaces with Banach algebras without the assumption of normality of cones. Moreover, we give an example to elucidate our result. Our results are signiﬁcant extension and generalizations of many recent results in the literature.

20

On the convergence of sigmoidal fuzzy grey cognitive maps

Harmati István Á., Kóczy László T.

International Journal of Applied Mathematics and Computer Science

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2019

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Vol. 29, no. 3

453--466

EN

Fuzzy cognitive maps (FCMs) are recurrent neural networks applied for modelling complex systems using weighted causal relations. In FCM-based decision-making, the inference about the modelled system is provided by the behaviour of an iteration. Fuzzy grey cognitive maps (FGCMs) are extensions of fuzzy cognitive maps, applying uncertain weights between the concepts. This uncertainty is expressed by the so-called grey numbers. Similarly as in FCMs, the inference is determined by an iteration process which may converge to an equilibrium point, but limit cycles or chaotic behaviour may also turn up. In this paper, based on the grey connections between the concepts and the parameters of the sigmoid threshold function, we give sufficient conditions for the existence and uniqueness of fixed points of sigmoid FGCMs.