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2. Some Common Fixed Point Results of Tower Mappings in (Pseudo)modular Metric Spaces.
- Author
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Francis, Daniel, Okeke, Godwin Amechi, and Khan, Safeer Hussain
- Subjects
- *
METRIC spaces , *FIXED point theory - Abstract
In this paper, we prove the existence and uniqueness of common fixed point of tower type contractive mappings in complete metric (pseudo)modular spaces involving the theoretic relation. However, the newly introduced contraction in this paper further characterize and includes in their full strength several existing results in metrical fixed point theory. Some nontrivial supportive examples were given to justify our result. Our results generalize, improve, and unify some existing results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Parameters optimization of three-element dynamic vibration absorber with inerter and grounded stiffness.
- Author
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Baduidana, Marcial and Kenfack-Jiotsa, Aurelien
- Subjects
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VIBRATION absorbers , *FIXED point theory , *STEADY-state responses , *EQUATIONS of motion - Abstract
Improving the control performance of dynamic vibration absorbers has recently been effective by introducing a grounded negative stiffness device. However, the negative stiffness structure is unstable and difficult to achieve in engineering practice , and its major drawback is that it amplif ies the vibration response of the primary system at low frequency region. Meanwhile, some mechanical devices can be combined to make the DVA work even better with a grounded positive stiffness. For this purpose, this paper combines for the first time the control effect of the inerter device and grounded positive stiffness into a three-element DVA model in order to better improve vibration reduction of an undamped primary system under excitation. First, the dynamic equation of motion of the system is written according to Newton 's second law. Then, the steady-state displacement response of the primary system under harmonic excitation is calculated. In order to minimize the resonant response of the primary system around its natural frequency, the extended fixed point theory is applied. Thus, the optimized parameters such as the tuning frequency ratio, the stiffness ratio , and the approximate damping ratio are determined as a function of mass ratio and inerter – mass ratio. From the results analysis, it was found that the inerter – mass ratio has a better working range to guarantee the stability of the coupled system. Then , study on the effect of inerter – mass ratio on the primary system response is carried out. It can be seen that increasing the inerter – mass ratio in the optimal working range can reduce the response of the primary system beyond its uncontrolled static response. However, it is necessary to avoid the situation where the inerter – mass ratio is very large because it can lead to unrealistic optimal parameters. Finally, comparison with other DVA models is show n under harmonic and random excitation of the primary system. It is found that the proposed DVA model in this paper has high control performance and can be used in many engineering practice s. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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4. Coupled Fixed Point Theory in Subordinate Semimetric Spaces.
- Author
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Alharbi, Areej, Noorwali, Maha, and Alsulami, Hamed H.
- Subjects
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FIXED point theory , *MONOTONE operators - Abstract
The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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5. On Prešić-Type Mappings: Survey.
- Author
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Achtoun, Youssef, Gardasević-Filipović, Milanka, Mitrović, Slobodanka, and Radenović, Stojan
- Subjects
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FIXED point theory , *FUNCTIONAL analysis , *RESEARCH personnel - Abstract
This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known as Prešić–Menger and Prešić–Ćirić–Menger contractions, thereby enriching the literature on Prešić-type mappings. The paper endeavors to furnish young researchers with a comprehensive resource in functional and nonlinear analysis. The relevance of Prešić's method, which generalizes Banach's theorem from 1922, remains significant in metric fixed point theory, as evidenced by recent publications. The overview article addresses the growing importance of Prešić's approach, coupled with new ideas, reflecting the ongoing advancements in the field. Additionally, the paper establishes the existence and uniqueness of fixed points in Menger spaces, contributing to the filling of gaps in the existing literature on Prešić's works while providing valuable insights into this specialized domain. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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6. φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers.
- Author
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Joshi, Vishal, Singh, Naval, Singh, Deepak, and Sahoo, Prasanna K.
- Subjects
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FIXED point theory , *SET-valued maps , *MATHEMATICAL mappings , *METRIC spaces , *GENERALIZED spaces - Abstract
The purpose of this paper is twofold. Firstly, certain common fixed point theorems are established via φ-contractive multivalued mappings involving point-dependent control functions as coefficients in the framework of complex valued metric spaces. Our results improve and extend several results in the existing literature. Moreover, this section is equipped by some illustrative examples in support of our results. Secondly, we point out some slip-ups in the examples of some recent papers based on multivalued contractive mappings in complex valued metric spaces. Our observations are also authenticated with the aid of some appropriate examples. Some rectifications to correct the erratic examples are also suggested. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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7. Fractional Dynamics of Cassava Mosaic Disease Model with Recovery Rate Using New Proposed Numerical Scheme.
- Author
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Abdullah, Tariq Q. S., Huang, Gang, Al-Sadi, Wadhah, Aboelmagd, Yasser, and Mobarak, Wael
- Subjects
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FIXED point theory , *MOSAIC diseases , *AGRICULTURAL pests , *PLANT viruses , *FOOD security , *CASSAVA - Abstract
Food security is a basic human right that guarantees humans an adequate amount of nutritious food. However, plant viruses and agricultural pests cause real damage to food sources, leading to negative impacts on meeting the human right of obtaining a sufficient amount of food. Understanding infectious disease dynamics can help us to design appropriate control and prevention strategies. Although cassava is among the most produced and consumed crops and greatly contributes to food security, cassava mosaic disease causes a decrease in photosynthesis and reduces cassava yield, resulting in a lack of crops. This paper developed a fractional model for cassava mosaic disease (CMD) dynamics based on the Caputo–Fabrizio (CF) fractional derivative to decrease cassava plant infection. We used fixed-point theory to study the existence of a unique solution in the form of the CMD model. A stability analysis of the model was conducted by using fixed-point theory and the Picard technique. A new numerical scheme was proposed for solving the nonlinear system of a fractional model in the sense of the CF-derivative and applied to obtain numerical simulations for a fractional model of the dynamics of CMD. The obtained results are described using figures that show the dynamics and behaviors of the compartments of CMD, and it is concluded that decreasing the population of whitefly vectors can prevent cassava plants from becoming infected better than increasing the recovery rate of the infected cassava plants. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. On the solutions of a nonlinear system of q-difference equations.
- Author
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Turan, Nihan, Başarır, Metin, and Şahin, Aynur
- Subjects
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NONLINEAR equations , *BOUNDARY value problems , *INITIAL value problems , *DIFFERENCE equations , *EQUATIONS - Abstract
In this paper, we examine the existence and uniqueness of solutions for a system of the first-order q-difference equations with multi-point and q-integral boundary conditions using various fixed point (fp) theorems. Also, we give two examples to support our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations.
- Author
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Filali, Doaa, Dilshad, Mohammad, and Akram, Mohammad
- Subjects
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FRACTIONAL differential equations , *BOUNDARY value problems , *METRIC spaces , *FIXED point theory , *CONTRACTIONS (Topology) - Abstract
After the initiation of Jachymski's contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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10. Three Existence Results in the Fixed Point Theory.
- Author
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Zaslavski, Alexander J.
- Subjects
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FIXED point theory , *METRIC spaces , *SET-valued maps , *GENERALIZATION - Abstract
In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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11. A Novel Fixed-Point Iteration Approach for Solving Troesch's Problem.
- Author
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Filali, Doaa, Ali, Faeem, Akram, Mohammad, and Dilshad, Mohammad
- Subjects
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GREEN'S functions , *BOUNDARY value problems , *NONLINEAR differential equations , *FIXED point theory , *BANACH spaces - Abstract
This paper introduces a novel F fixed-point iteration method that leverages Green's function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green's function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method's efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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12. Positive Radial Symmetric Solutions of Nonlinear Biharmonic Equations in an Annulus.
- Author
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Li, Yongxiang and Yang, Shengbin
- Subjects
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FIXED point theory , *NONLINEAR equations , *CONTINUOUS functions , *CONES , *BIHARMONIC equations - Abstract
This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → R + is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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13. Exploring solutions to specific class of fractional differential equations of order 3<uˆ≤4.
- Author
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Aljurbua, Saleh Fahad
- Subjects
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CAPUTO fractional derivatives , *FUNCTION spaces , *FRACTIONAL differential equations , *FIXED point theory , *DIFFERENTIAL equations - Abstract
This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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14. Zeros of a Functional Associated with a Family of Search Functionals. Corollaries for Coincidence and Fixed Points of Mappings of Metric Spaces.
- Author
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Kurbanov, A. É. and Fomenko, T. N.
- Subjects
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METRIC spaces , *FIXED point theory , *NONEXPANSIVE mappings , *FUNCTIONALS , *BANACH spaces , *COINCIDENCE - Abstract
The study of the zero existence problem for a nonnegative set-valued functional on a metric space is continued. The zero existence problem for a functional related by a certain -continuity condition to a parametric family of -search functionals on an open subset of a metric space is examined. A theorem containing several sufficient conditions for this functional to have zeros is proved. As corollaries of this result, theorems on the existence of coincidence and fixed points are also proved for set-valued mappings related by the -continuity condition to families of set-valued mappings with the property that the existence of coincidence and fixed points in an open subset of a metric space is preserved under parameter variation. For uniformly convex metric spaces, analogs of M. Edelstein's 1972 asymptotic center theorem and M. Frigon's 1996 fixed point theorem for nonexpansive mappings of Banach spaces are obtained and compared with the main results of the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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15. A sub-super solution method to continuous weak solutions for a semilinear elliptic boundary value problems on bounded and unbounded domains.
- Author
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Ghanmi, Abdeljabbar, Alzumi, Hadeel Z., and Zeddini, Noureddine
- Subjects
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BOUNDARY value problems , *GREEN'S functions , *FIXED point theory , *NONLINEAR theories , *SEMILINEAR elliptic equations - Abstract
In this paper, we prove the existence of solutions for an elliptic system. More precisely, we combine the potential theory with the sub-super solution method and use the properties of the well-known Kato class to justify our existence results. The novelty of our study is that we consider either the bounded or the exterior domain; Also, the nonlinearities may be singular near the boundary. Some examples are presented to validate our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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16. A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator.
- Author
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Zhang, Xinguang, Chen, Peng, Li, Lishuang, and Wu, Yonghong
- Subjects
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FIXED point theory , *NONLINEAR operators - Abstract
In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. Common Attractors for Generalized F -Iterated Function Systems in G -Metric Spaces.
- Author
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Nazir, Talat and Silvestrov, Sergei
- Subjects
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FIXED point theory , *METRIC spaces - Abstract
In this paper, we study the generalized F-iterated function system in G-metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F-Hutchinson operators are also established. We prove that the triplet of F-Hutchinson operators defined for a finite number of general contractive mappings on a complete G-metric space is itself a generalized F-contraction mapping on a space of compact sets. We also present several examples in 2-D and 3-D for our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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18. Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem.
- Author
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Arıbaş, Özge, Gölgeleyen, İsmet, and Yıldız, Mustafa
- Subjects
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BURGERS' equation , *NONLINEAR equations , *GRONWALL inequalities , *CAPUTO fractional derivatives , *EIGENFUNCTION expansions , *INVERSE problems , *ELLIPTIC operators - Abstract
In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young's convolution and generalized Grönwall's inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. Fixed Point Theory in Bicomplex Metric Spaces: A New Framework with Applications.
- Author
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Alamri, Badriah
- Subjects
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METRIC spaces , *CONTRACTIONS (Topology) , *FIXED point theory , *INTEGRAL equations , *VOLTERRA equations - Abstract
This paper investigates the existence of common fixed points for mappings satisfying generalized rational type contractive conditions in the framework of bicomplex valued metric spaces. Our findings extend well-established results in the existing literature. As an application of our leading result, we explore the existence and uniqueness of solutions of the Volttera integral equation of the second kind. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. CHROMATIC FIXED POINT THEORY AND THE BALMER SPECTRUM FOR EXTRASPECIAL 2-GROUPS.
- Author
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KUHN, NICHOLAS J. and LLOYD, CHRISTOPHER J. R.
- Subjects
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FIXED point theory , *HOMOTOPY theory , *SEARCH theory , *K-theory - Abstract
In the early 1940s, P. A. Smith showed that if a finite p-group G acts on a finite dimensional complex X that is mod p acyclic, then its space of fixed points, XG, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a G-space X is a equivariant homotopy retract of the p-localization of a based finite G-C.W. complex, given H < G and n, what is the smallest r such that if XH is acyclic in the (n+r)th Morava K-theory, then XG must be acyclic in the nth Morava K-theory? Barthel et. al. then answered this when G is abelian, by finding general lower and upper bounds for these "blue shift" numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equiv- alent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod p homology, and thus applies to all finite dimensional G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for K(n)*-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava K-theories of all real Grassmanians. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. SEARCH OF MINIMAL METRIC STRUCTURE IN THE CONTEXT OF FIXED POINT THEOREM AND CORRESPONDING OPERATOR EQUATION PROBLEMS.
- Author
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SAVALIYA, JAYESH, GOPAL, DHANANJAY, SRIVASTAVA, SHAILESH KUMAR, and RAKOČEVIČ, VLADIMIR
- Subjects
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OPERATOR equations , *CONTRACTIONS (Topology) , *FIXED point theory , *NONLINEAR integral equations , *BOUNDARY value problems , *FRACTIONAL differential equations , *METRIC spaces - Abstract
The paper contains a brief summary of the generalization of metrical structure regarding the fixed point theorem and corresponding operator equation problems. We observed that many researcher either tried to weaken the metrical structure, the contraction condition, or both. The idea behind this paper is to look for a minimal metrical structure to establish fixed point theorems. In this connection, we present new variants of the known fixed point theorem under non-triangular metric space (namely F-contraction, (A; S)-contraction, (A; S)-contraction). We also apply the obtain result in solving various types of operator equation problems. e.g., high-order fractional differential equation with non-local boundary conditions and non-linear integral equation problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Iterated function system of generalized cyclic F-contractive mappings.
- Author
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NAZIR, TALAT, ABBAS, MUJAHID, and LODHI, HIRA HALEEM
- Subjects
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FIXED point theory , *METRIC spaces , *COLLECTIONS - Abstract
The aim of this paper is to study the sufficient conditions for the existence of attractor of a generalized cyclic iterated function system composed of a complete metric space and a finite collection of generalized cyclic Γ-contraction mappings. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. A note on the paper 'Fixed point theorems for cyclic weak contractions in compact metric spaces'.
- Author
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Kadelburg, Zoran, Radenović, Stojan, and Vujaković, Jelena
- Subjects
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FIXED point theory , *COMPACT spaces (Topology) , *MATHEMATICAL mappings , *METRIC spaces , *MATHEMATICAL analysis - Abstract
We show that the result on cyclic weak contractions of Harjani et al. (J. Nonlinear Sci. Appl. 6:279-284, ) holds without the assumption of compactness of the underlying space, and also without the assumption of continuity of the given mapping. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
24. Existence and uniqueness of continuous solutions for iterative functional differential equations in Banach algebras.
- Author
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BEN AMARA, Khaled
- Subjects
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BANACH algebras , *FUNCTIONAL differential equations , *DIFFERENTIAL equations - Abstract
This paper is devoted to studying the existence and uniqueness of continuous solutions of the following iterative functional differential equation ... By using of Boyd-Wong's fixed point theorem and under suitable conditions, we establish the existence and uniqueness of a continuous solution. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Common Fixed Point Theorems on S-Metric Spaces for Integral Type Contractions Involving Rational Terms and Application to Fractional Integral Equation.
- Author
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Saluja, G. S., Nashine, Hemant Kumar, Jain, Reena, Ibrahim, Rabha W., and Nabwey, Hossam A.
- Subjects
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FRACTIONAL integrals , *INTEGRAL calculus , *CONTRACTIONS (Topology) , *INTEGRAL equations , *FIXED point theory , *FRACTIONAL calculus , *INTEGRALS - Abstract
It has been shown that the findings of d -metric spaces may be deduced from S -metric spaces by considering d ϖ , ϰ = Λ ϖ , ϖ , ϰ . In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete S -metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper's findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. S-Pata-type contraction: a new approach to fixed-point theory with an application.
- Author
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Chand, Deep, Rohen, Yumnam, Saleem, Naeem, Aphane, Maggie, and Razzaque, Asima
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FIXED point theory , *CONTRACTIONS (Topology) , *ORDINARY differential equations , *MATHEMATICAL mappings - Abstract
In this paper, we introduce new types of contraction mappings named S-Pata-type contraction mapping and Generalized S-Pata-type contraction mapping in the framework of S-metric space. Then, we prove some new fixed-point results for S-Pata-type contraction mappings and Generalized S-Pata-type contraction mappings. To support our results, we provide examples to illustrate our findings and also apply these results to the ordinary differential equation to strengthen our conclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Common Fixed Point Theorems for Contractive Mappings of Integral Type in -Metric Spaces.
- Author
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Guan, Hongyan and Gou, Jinze
- Subjects
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FIXED point theory , *INTEGRALS - Abstract
This paper is the first to introduce a fixed point problem of integral type in a -metric space. We study sufficient conditions for the existence and uniqueness of a common fixed point of contractive mappings of integral type. We also give two examples to support our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Study on a Nonlocal Fractional Coupled System Involving (k , ψ)-Hilfer Derivatives and (k , ψ)-Riemann–Liouville Integral Operators.
- Author
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Samadi, Ayub, Ntouyas, Sotiris K., and Tariboon, Jessada
- Subjects
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FRACTIONAL differential equations , *FIXED point theory , *FRACTIONAL integrals , *INTEGRAL operators - Abstract
This paper deals with a nonlocal fractional coupled system of (k , ψ) -Hilfer fractional differential equations, which involve, in boundary conditions, (k , ψ) -Hilfer fractional derivatives and (k , ψ) -Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel'skiĭ's fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. On Common Fixed Point for Contractive Mappings in p-Pompeiu-Hausdorff Metric Spaces.
- Author
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Ekayanti, Arta, Marjono, Marjono, Muslikh, Mohamad, and Fitri, Sa’adatul
- Subjects
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METRIC spaces , *FIXED point theory , *SET-valued maps - Abstract
In this paper we establish the existence of a common fixed point from a pair of setvalued mappings. By utilizing the concept of convergence of set-valued mappings’ sequences, both ordinary and pointwise convergence, we establish a common fixed point theorem. This our newly result is a generalization of common fixed point theorem of set-valued mappings on partial metric spaces. Further, we establish newly common fixed point theorem under ϕ-contraction on partial metric spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Exploring multivalued probabilistic ψ-contractions with orbits in b-Menger spaces.
- Author
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Achtoun, Youssef, Radenović, Stojan, Tahiri, Ismail, and Sefian, Mohammed Lamarti
- Subjects
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METRIC spaces , *ORBITS (Astronomy) , *ORBIT method , *FIXED point theory , *CONTRACTIONS (Topology) , *COINCIDENCE - Abstract
Introduction/purpose: The paper presents a novel approach to certain well-established fixed point theorems for multivalued probabilistic contractions in b-Menger spaces, leveraging the boundedness of the orbits. The aim was to generalize and enhance the results previously derived by Fang and Hadžić. Methods: The boundedness of orbits in b-Menger spaces is used to establish their approach for multivalued probabilistic contractions. Results: The findings of the study not only generalized the existing fixed point theorems but also enhanced them significantly. The effectiveness of the approach in extending the results originally proposed by Fang and Hadžić was showcased. Moreover, the applicability of the coincidence fixed point theorem in fuzzy b-metric spaces was demonstrated. Conclusions: The study presented a novel perspective on fixed point theorems in multivalued probabilistic contractions within b-Menger spaces. By leveraging boundedness and introducing a coincidence fixed point theorem for fuzzy b-metric spaces, the work contributed to the advancement in this field. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Nonlocal Cahn-Hilliard type model for image inpainting.
- Author
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Jiang, Dandan, Azaiez, Mejdi, Miranville, Alain, and Xu, Chuanju
- Subjects
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FIXED point theory , *INPAINTING - Abstract
This paper proposes a Cahn-Hilliard type inpainting model equipped with a nonlocal diffusion operator. A rigorous analysis of the well-posedness of the stationary solution is established using Schauder's fixed point theory. We construct a time stepping scheme based on the convex splitting method with the nonlocal term treated implicitly and the fidelity term treated explicitly. We prove the consistency, stability and convergence of the semidiscrete-in-time scheme. To the best of our knowledge, this is the first study to present such an analysis for semidiscrete-in-time problems of this model, which provides valuable guidance for parameter selection. Numerical experiments validate the effectiveness of the proposed nonlocal model, which shows superior performance compared to both local and classical total variation models in preserving fine textures and recovering image edges. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Approximate Controllability and Ulam Stability for Second-Order Impulsive Integrodifferential Evolution Equations with State-Dependent Delay.
- Author
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Bensalem, Abdelhamid, Salim, Abdelkrim, Benchohra, Mouffak, and N'Guérékata, Gaston
- Subjects
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INTEGRO-differential equations , *EVOLUTION equations , *FIXED point theory , *RESOLVENTS (Mathematics) , *OPERATOR theory , *CARLEMAN theorem , *IMPULSIVE differential equations - Abstract
In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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33. A novel stability analysis of functional equation in neutrosophic normed spaces.
- Author
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Aloqaily, Ahmad, Agilan, P., Julietraja, K., Annadurai, S., and Mlaiki, Nabil
- Subjects
- *
NORMED rings , *FUNCTIONAL equations , *FUNCTIONAL analysis , *QUADRATIC equations , *NEUTROSOPHIC logic , *FIXED point theory - Abstract
The analysis of stability in functional equations (FEs) within neutrosophic normed spaces is a significant challenge due to the inherent uncertainties and complexities involved. This paper proposes a novel approach to address this challenge, offering a comprehensive framework for investigating stability properties in such contexts. Neutrosophic normed spaces are a generalization of traditional normed spaces that incorporate neutrosophic logic. By providing a systematic methodology for addressing stability concerns in neutrosophic normed spaces, our approach facilitates enhanced understanding and control of complex systems characterized by indeterminacy and uncertainty. The primary focus of this research is to propose a novel class of Euler-Lagrange additive FE and investigate its Ulam-Hyers stability in neutrosophic normed spaces. Direct and fixed point techniques are utilized to achieve the required results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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34. On an m-dimensional system of quantum inclusions by a new computational approach and heatmap.
- Author
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Ghaderi, Mehran and Rezapour, Shahram
- Subjects
- *
FIXED point theory , *DIFFERENTIAL equations , *BOUNDARY value problems , *RESEARCH personnel , *PHENOMENOLOGICAL theory (Physics) - Abstract
Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler's fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution's existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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35. Equivalent Condition of the Measure Shadowing Property on Metric Spaces.
- Author
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Miao, Jie and Yang, Yinong
- Subjects
- *
PHASE space , *COMPACT spaces (Topology) , *METRIC system , *DYNAMICAL systems , *FIXED point theory , *METRIC spaces - Abstract
The concept referred to as the measure shadowing property for a dynamical system on compact metric space has recently been introduced, acting as an extension of the classical shadowing property by using the property of the Borel measures on the phase space. In this paper, we extend the concept of the measure shadowing property of continuous flows from compact metric spaces to the general metric spaces and demonstrate the equivalence relation between the measure shadowing property and the shadowing property for flows on metric spaces via the shadowable points. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative.
- Author
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Zhang, Xinguang, Jiang, Yongsheng, Li, Lishuang, Wu, Yonghong, and Wiwatanapataphee, Benchawan
- Subjects
- *
BOUNDARY value problems , *FIXED point theory , *ALGORITHMS , *MACHINE learning , *ARTIFICIAL intelligence , *DIGITAL technology - Abstract
Let α ∈ (1 , 2 ] , β ∈ (0 , 1) with α − β > 1. This paper focused on the multiplicity of positive solutions for a singular tempered fractional boundary value problem { − 0 R D t α , λ u (t) = p (t) h (e λ t u (t) , 0 R D t β , λ u (t)) , t ∈ (0 , 1) , 0 R D t β , λ u (0) = 0 , 0 R D t β , λ u (1) = 0 , where h ∈ C ([ 0 , + ∞) × [ 0 , + ∞) , [ 0 , + ∞)) and p ∈ L 1 ([ 0 , 1 ] , (0 , + ∞)). By applying reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the above equation were established. The interesting points were that the nonlinearity contained the lower order tempered fractional derivative and that the weight function can have infinite many singular points in [ 0 , 1 ]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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37. On Certain Coupled Fixed Point Theorems Via C Star Class Functions in C*-Algebra Valued Fuzzy Soft Metric Spaces With Applications.
- Author
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Ushabhavani, C., Reddy, G. Upender, and Rao, B. Srinuvasa
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- *
METRIC spaces , *FIXED point theory , *HOMOTOPY theory - Abstract
The discussion of this paper is to aim to examine application of the notion of C*-algebra valued fuzzy soft metric to homotopy theory using common coupled fixed point results from C*-class functions. We also tried to provide an illustration of our major findings. The results attained expand upon and apply to many of the findings in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
38. Numerical analysis of COVID-19 model with Caputo fractional order derivative.
- Author
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Shahabifar, Reza, Molavi-Arabshahi, Mahboubeh, and Nikan, Omid
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CAPUTO fractional derivatives , *NUMERICAL analysis , *BASIC reproduction number , *FIXED point theory , *ORDINARY differential equations , *GLOBAL analysis (Mathematics) , *TRAPEZOIDS - Abstract
This paper focuses on the numerical solutions of a six-compartment fractional model with Caputo derivative. In this model, we obtain non-negative and bounded solutions, equilibrium points, and the basic reproduction number and analyze the stability of disease free equilibrium point. The existence and uniqueness of the solution are proven by employing the Picard–Lindelof approach and fixed point theory. The product–integral trapezoidal rule is employed to simulate the system of FODEs (fractional ordinary differential equations). The numerical results are presented in the form of graphs for each compartment. Finally, the sensitivity of the most important parameter (β) and its impact on COVID-19 dynamics and the basic reproduction number are reported. [ABSTRACT FROM AUTHOR]
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- 2024
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39. SOME NEW OBSERVATIONS ON w-DISTANCE AND F-CONTRACTIONS.
- Author
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Kadelburg, Zoran and Radenović, Stojan
- Subjects
- *
CONTRACTIONS (Topology) , *METRIC spaces , *FIXED point theory - Abstract
The aim of this paper is to present some new observations about w-distance (in the sense of O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica 44, 2 (1996), 381-391) and F-contractions (in the sense of D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012:94 (2012)). Both concepts have been examined separately a lot, but there have been few attempts to connect them. This article is a step in filling this gap. Besides, some comments and improvements of results in the existing literature are presented. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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40. On a generalization of a relatively nonexpansive mapping and best proximity pair.
- Author
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Chaira, Karim and Seddoug, Belkassem
- Subjects
- *
NONEXPANSIVE mappings , *FIXED point theory , *NORMED rings , *FUNCTIONAL equations , *COMMERCIAL space ventures , *GENERALIZATION - Abstract
Let A and B be two nonempty subsets of a normed space X, and let T : A ∪ B → A ∪ B be a cyclic (resp., noncyclic) mapping. The objective of this paper is to establish weak conditions on T that ensure its relative nonexpansiveness. The idea is to recover the results mentioned in two papers by Matkowski (Banach J. Math. Anal. 2:237–244, 2007; J. Fixed Point Theory Appl. 24:70, 2022), by replacing the nonexpansive mapping f : C → C with a cyclic (resp., noncyclic) relatively nonexpansive mapping to obtain the best proximity pair. Additionally, we provide an application to a functional equation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Dynamical Transmission and Mathematical Analysis of Ebola Virus Using a Constant Proportional Operator with a Power Law Kernel.
- Author
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Xu, Changjin and Farman, Muhammad
- Subjects
- *
EBOLA virus , *MATHEMATICAL analysis , *FIXED point theory , *FRACTIONAL differential equations , *POWER law (Mathematics) , *DECOMPOSITION method - Abstract
The Ebola virus continues to be the world's biggest cause of mortality, especially in developing countries, despite the availability of safe and effective immunization. In this paper, we construct a fractional-order Ebola virus model to check the dynamical transmission of the disease as it is impacted by immunization, learning, prompt identification, sanitation regulations, isolation, and mobility limitations with a constant proportional Caputo (CPC) operator. The existence and uniqueness of the proposed model's solutions are discussed using the results of fixed-point theory. The stability results for the fractional model are presented using Ulam–Hyers stability principles. This paper assesses the hybrid fractional operator by applying methods to invert proportional Caputo operators, calculate CPC eigenfunctions, and simulate fractional differential equations computationally. The Laplace–Adomian decomposition method is used to simulate a set of fractional differential equations. A sustainable and unique approach is applied to build numerical and analytic solutions to the model that closely satisfy the theoretical approach to the problem. The tools in this model appear to be fairly powerful, capable of generating the theoretical conditions predicted by the Ebola virus model. The analysis-based research given here will aid future analysis and the development of a control strategy to counteract the impact of the Ebola virus in a community. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
42. Critical remarks on “Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance”.
- Author
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Kadelburg, Zoran, Fabiano, Nicola, Savatović, Milica, and Radenović, Stojan
- Subjects
- *
DIFFERENTIAL equations , *FIXED point theory , *FUNCTIONAL analysis - Abstract
Introduction/purpose: In this paper, several critical remarks are presented concerning the paper of Iqbal & Rizwan: Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance from 2020. Methods: Conventional theoretical methods of functional analysis. Results: It is shown that their use of the non-decreasing “control” function F instead of a strictly increasing one in Wardowski-type results usually produces contradictions. Conclusion: It is shown that such results can be obtained in a more general class of metric-like spaces, where strict monotonicity is the only as sumption that has to be imposed on the function F. An example is presented showing that the obtained results are stronger than the classic ones. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Global Stability of Fractional Order HIV/AIDS Epidemic Model under Caputo Operator and Its Computational Modeling.
- Author
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Ahmad, Ashfaq, Ali, Rashid, Ahmad, Ijaz, Awwad, Fuad A., and Ismail, Emad A. A.
- Subjects
- *
HIV , *AIDS , *COMPUTATIONAL neuroscience , *FIXED point theory , *ORDINARY differential equations , *DIFFERENTIAL operators , *EPIDEMICS - Abstract
The human immunodeficiency virus (HIV) causes acquired immunodeficiency syndrome (AIDS), which is a chronic and sometimes fatal illness. HIV reduces an individual's capability against infection and illness by demolishing his or her immunity. This paper presents a new model that governs the dynamical behavior of HIV/AIDS by integrating new compartments, i.e., the treatment class T. The steady-state solutions of the model are investigated, and accordingly, the threshold quantity R 0 is calculated, which describes the global dynamics of the proposed model. It is proved that for R 0 less than one, the infection-free state of the model is globally asymptotically stable. However, as the threshold number increases by one, the endemic equilibrium becomes globally asymptotically stable, and in such case, the disease-free state is unstable. At the end of the paper, the analytic conclusions obtained from the analysis of the ordinary differential equation (ODE) model are supported through numerical simulations. The paper also addresses a comprehensive analysis of a fractional-order HIV model utilizing the Caputo fractional differential operator. The model's qualitative analysis is investigated, and computational modeling is used to examine the system's long-term behavior. The existence/uniqueness of the solution to the model is determined by applying some results from the fixed points of the theory. The stability results for the system are established by incorporating the Ulam–Hyers method. For numerical treatment and simulations, we apply Newton's polynomial and the Toufik–Atangana numerical method. Results demonstrate the effectiveness of the fractional-order approach in capturing the dynamics of the HIV/AIDS epidemic and provide valuable insights for designing effective control strategies. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
44. Preface to the Special Issue "Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications".
- Author
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Du, Wei-Shih, Kostić, Marko, Fedorov, Vladimir E., and Pinto, Manuel
- Subjects
- *
FIXED point theory , *INTEGRO-differential equations , *FRACTIONAL calculus - Abstract
This document is a preface to a special issue of the journal Mathematics, focusing on abstract fractional integro-differential equations and fixed point theory with applications. The preface explains that fractional calculus is used to model complex systems in various disciplines, and abstract fractional integro-differential equations arise from different scientific research areas. The special issue includes 12 research papers selected from 36 submissions, covering topics such as almost periodic solutions, stability analysis, and convex analysis. The authors express their appreciation to the contributors and reviewers, and hope that the papers will inspire future research in these areas. [Extracted from the article]
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- 2023
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45. Recent Advancements in KRH-Interpolative-Type Contractions.
- Author
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Abbas, Ansar, Ali, Amjad, Al Sulami, Hamed, and Hussain, Aftab
- Subjects
- *
FRACTIONAL differential equations , *METRIC spaces , *FIXED point theory , *CONTRACTIONS (Topology) - Abstract
The focus of this paper is to conduct a comprehensive analysis of the advancements made in the understanding of Interpolative contraction, building upon the ideas initially introduced by Karapinar in 2018. In this paper, we develop the notion of Interpolative contraction mappings to the case of non-linear Kannan Interpolative, Riech Rus Ćirić interpolative and Hardy–Roger Interpolative contraction mappings based on controlled function, and prove some fixed point results in the context of controlled metric space, thereby enhancing the current understanding of this particular analysis. Furthermore, we provide a concrete example that illustrates the underlying drive for the investigations presented in this context. An application of the proposed non-linear Interpolative-contractions to the Liouville–Caputo fractional derivatives and fractional differential equations is provided in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
46. Fixed point theorems and applications in p-vector spaces.
- Author
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Yuan, George Xianzhi
- Subjects
- *
FIXED point theory , *SET-valued maps , *VECTOR topology , *NONLINEAR analysis , *VECTOR spaces , *COINCIDENCE theory , *UNIFORM spaces - Abstract
The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex and p-vector spaces for p ∈ (0 , 1 ] . In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p-convex spaces and topological vector spaces for p ∈ (0 , 1 ] ; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. Fixed point theorems for generalized (α,ϕ)-Meir–Keeler type hybrid contractive mappings via simulation function in b-metric spaces.
- Author
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Abduletif Mamud, Mustefa and Koyas Tola, Kidane
- Subjects
- *
FIXED point theory , *FUNCTION spaces - Abstract
In this paper, we introduce the notion of generalized (α , ϕ) -Meir–Keeler hybrid contractive mappings of type I and II via simulation function and establish fixed point theorems for such mappings in the setting of complete b-metric spaces. Our results extend and generalize many related fixed point results in the existing literature. Finally, we provide an example in support of our main finding. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Positive solutions for a system of fractional $ q $-difference equations with generalized $ p $-Laplacian operators.
- Author
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Li, Hongyu, Wang, Liangyu, and Cui, Yujun
- Subjects
- *
DIFFERENCE equations , *LAPLACIAN operator , *FIXED point theory , *PARAMETER estimation , *MATHEMATICAL models - Abstract
In this paper, we consider the existence of positive solutions for a system of fractional q -difference equations with generalized p -Laplacian operators. By using Guo-Krasnosel'skii fixed point theorem, we obtain some existence results of positive solutions for this system with two parameters under some different combinations of superlinearity and sublinearity of the nonlinear terms. In the end, we give two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration.
- Author
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Srivastava, Rekha, Tassaddiq, Asifa, and Kasmani, Ruhaila Md
- Subjects
- *
FRACTAL analysis , *FIXED point theory , *COMPARATIVE studies , *FRACTALS - Abstract
Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Fractional derivative analysis of Asthma with the effect of environmental factors.
- Author
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JAN, Rashid, ZOBAER, M. S., YÜZBAŞI, Şuayip, Attaullah, JAWAD, Muhammad, and JAN, Asif
- Subjects
- *
FIXED point theory , *TOBACCO smoke , *HEALING , *AIR pollution , *SMOKING , *ASTHMA , *NICOTINE , *CIGARETTE smoke - Abstract
It is observed that the exposure to environmental factors such as indoors and outdoors air pollution, cigarette smoke, and allergens are highly related to asthma attacks. It is also reported that limited exposure to asthma riggers, cure due to medicine, the attacks of asthma can be minimized. In this paper, we formulate the dynamics of asthma with smoking and environmental factors classes in the fractional Caputo-Fabrizio (CF) framework to visualize its dynamical behaviour. We delineate the important properties of the CF derivative for the analysis of our model. The model is then analyzed for the basic properties and the uniqueness and existence of the hypothesized asthma system are investigated via the theory of fixed point. Furthermore, a novel numerical scheme is presented for the solution of our fractional system to illustrate the time series of asthma model. The dynamical behaviour of our asthma model is then highlighted numerically to show the impact of fractional-order ϧ on the system and to visualize the role of input factors on the dynamics of asthma disease. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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