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1

Singular quasilinear convective systems involving variable exponents

Moussaoui Abdelkrim, Nabab Dany, Vélin Jean

Opuscula Mathematica

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2024

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Vol. 44, no. 1

105--134

EN

The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.

2

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.

3

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.

4

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.

5

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.

6

An other proof of the Reich fixed point theorem

Miarka Marek

Fasciculi Mathematici

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2023

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nr 66

77--79

EN

We give a simple and nonconstructive proof of the Reich fixed point theorem which generalizes both Banach and Kannan fixed point theorems.

7

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.

8

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.

9

A generalization of Matthews partial metric space and fixed point theorems

Antal A., Gairola U. C., Khantwal D., Matkowski J., Negi S.

Fasciculi Mathematici

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2021

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nr 65

5--15

EN

In the present paper, we introduce the notion of a generalized partial metric space which is an extension of the partial metric space due to S. G. Matthews (Partial metric topology, Papers on general topology and applications, Ann. New York Acad. Sci.,728 (1994), 183-197). We investigate some basic properties of the generalized partial metric spaces and establish some new fixed point theorems for linear and non-linear contraction on such spaces.

10

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.

11

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.

12

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.

13

Implicit functions under fixed point consideration in Probabilistic Menger Spaces

Murthy P. P., Prasad K. N. V. V. V., Majumdar J.

Fasciculi Mathematici

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2020

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nr 63

45--61

EN

In the present paper, we have introduced a pair of weakly-biased maps in the Probabilistic Menger Spaces under the implicit relation. Our results proved herein is the partial extension and mild improvement of the results due to Imdad, Tanveer and Hasan [10]. We have discussed an example in support of our main theorem.

14

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.

15

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.

16

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.

17

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.

18

The equivalence of convergence results between modified Ishikawa and modified Mann iterations

Mogbademu A. A., Kim J. K.

Fasciculi Mathematici

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2019

|

nr 62

93--102

EN

In [11], the author discussed a new class of nearly weak uniformly L-Lipschitzian mappings and prove some strong convergence results of the modified Ishikawa iteration with errors in real Banach spaces. And the author has given the open problem as follows: Are there any difference on convergence between the Mann iteration and Ishikawa iteration? Can we prove the equivalence on convergence between these two iterations? In this paper, we given an affirmative answer to the open problem.

19

Solving a nonlinear Volterra-Fredholm integro-differential equation with weakly singular kernels

Touati Sami, Lemita Samir, Ghiat Mourad, Aissaoui Mohamed-Zine

Fasciculi Mathematici

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2019

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nr 62

155--168

EN

The aim of the current work is to investigate the numerical study of an integro-differential nonlinear Volterra-Fredholm equation with a weakly singular kernels. Our approximation technique is based on the product integration method in conjunction with an iterative scheme. The existence and uniqueness of the solution have been proved. We conclude the paper with numerical examples to illustrate the effectiveness of our method.

20

A general fixed point theorem for weakly subsequentially continuous mappings

Popa Valeriu

Fasciculi Mathematici

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2019

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nr 62

103--119

EN

In this paper a general fixed point theorem for two pairs of subsequentially mappings compatible of type E is proved, which generalize the results by [2]-[4], [6] and other results. As applications, new results for mappings satisfying contractive conditions of integral type, φ-contractive conditions and weak contractive conditions are obtained.