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Journal articles on the topic 'Fixed-point equation'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Xu, Ding, Jinglei Xu, and Gongnan Xie. "Revisiting Blasius Flow by Fixed Point Method." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/953151.

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The well-known Blasius flow is governed by a third-order nonlinear ordinary differential equation with two-point boundary value. Specially, one of the boundary conditions is asymptotically assigned on the first derivative at infinity, which is the main challenge on handling this problem. Through introducing two transformations not only for independent variable bur also for function, the difficulty originated from the semi-infinite interval and asymptotic boundary condition is overcome. The deduced nonlinear differential equation is subsequently investigated with the fixed point method, so the original complex nonlinear equation is replaced by a series of integrable linear equations. Meanwhile, in order to improve the convergence and stability of iteration procedure, a sequence of relaxation factors is introduced in the framework of fixed point method and determined by the steepest descent seeking algorithm in a convenient manner.

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2

Brzdęk, Janusz, Liviu Cădariu, and Krzysztof Ciepliński. "Fixed Point Theory and the Ulam Stability." Journal of Function Spaces 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/829419.

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The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach's fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

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3

Sihombing, S. C., and L. Lia. "Fixed point theorem on volterra integral equation." Journal of Physics: Conference Series 1375 (November 2019): 012064. http://dx.doi.org/10.1088/1742-6596/1375/1/012064.

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4

Hammad, Hasanen A., Hassen Aydi, and Manuel De la Sen. "Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions." Complexity 2021 (February 22, 2021): 1–13. http://dx.doi.org/10.1155/2021/5730853.

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This paper involves extended b − metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.

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5

Garakoti, Neeraj, Joshi Chandra, and Rohit Kumar. "Fixed point for F⊥-weak contraction." Mathematica Moravica 25, no. 1 (2021): 113–22. http://dx.doi.org/10.5937/matmor2101113g.

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6

Amattouch, Mohamed Ridouan, and Hassan Belhadj. "A modified fixed point method for biochemical transport." Boletim da Sociedade Paranaense de Matemática 40 (February 2, 2022): 1–5. http://dx.doi.org/10.5269/bspm.46947.

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This work is devoted to a modified fixed point method applied to the bio-chemical transport equation. To have a good accuracy for the solution we treat, we apply an implicit scheme to this equation and use a modified fixed point technique to linearize the problem of transport equation with a generalized nonlinear reaction and diffusion equation. Next, we apply this methods in particular to the the dynamical system of a bio-chemical process. Eventually, we accelerate these algorithms by the optimized domain decomposition methods.Several test-cases of analytical problems illustrate this approach and show the efficiency of the proposednew method.

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7

Lowenthal, Franklin, Arnold Langsen, and Clark T. Benson. "Merton's Partial Differential Equation and Fixed Point Theory." American Mathematical Monthly 105, no. 5 (May 1998): 412. http://dx.doi.org/10.2307/3109802.

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8

Lowenthal, Franklin, Arnold Langsen, and Clark T. Benson. "Merton's Partial Differential Equation and Fixed Point Theory." American Mathematical Monthly 105, no. 5 (May 1998): 412–20. http://dx.doi.org/10.1080/00029890.1998.12004903.

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9

Babiarz, Artur, Jerzy Klamka, and Michał Niezabitowski. "Schauder’s fixed-point theorem in approximate controllability problems." International Journal of Applied Mathematics and Computer Science 26, no. 2 (June 1, 2016): 263–75. http://dx.doi.org/10.1515/amcs-2016-0018.

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AbstractThe main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

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10

Park, Choonkil, and Ji-Hye Kim. "The Stability of a Quadratic Functional Equation with the Fixed Point Alternative." Abstract and Applied Analysis 2009 (2009): 1–11. http://dx.doi.org/10.1155/2009/907167.

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Lee, An and Park introduced the quadratic functional equationf(2x+y)+f(2x−y)=8f(x)+2f(y)and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in Banach spaces.

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11

Došlá, Zuzana, Mauro Marini, and Serena Matucci. "A fixed-point approach for decaying solutions of difference equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2191 (January 4, 2021): 20190374. http://dx.doi.org/10.1098/rsta.2019.0374.

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A boundary value problem associated with the difference equation with advanced argument * Δ ( a n Φ ( Δ x n ) ) + b n Φ ( x n + p ) = 0 , n ≥ 1 is presented, where Φ ( u ) = | u | α sgn u , α > 0, p is a positive integer and the sequences a , b , are positive. We deal with a particular type of decaying solution of (*), that is the so-called intermediate solution (see below for the definition). In particular, we prove the existence of this type of solution for (*) by reducing it to a suitable boundary value problem associated with a difference equation without deviating argument. Our approach is based on a fixed-point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future research complete the paper. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

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12

Wangwe, Lucas, and Santosh Kumar. "A Common Fixed Point Theorem for Generalised F -Kannan Mapping in Metric Space with Applications." Abstract and Applied Analysis 2021 (April 16, 2021): 1–12. http://dx.doi.org/10.1155/2021/6619877.

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This paper is aimed at proving a common fixed point theorem for F -Kannan mappings in metric spaces with an application to integral equations. The main result of the paper will extend and generalise the recent existing fixed point results in the literature. We also provided illustrative examples and some applications to integral equation, nonlinear fractional differential equation, and ordinary differential equation for damped forced oscillations to support the results.

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13

Sana, Gul, Muhammad Aslam Noor, Mahmood Ul Hassan, and Zakia Hammouch. "Some Recent Modifications of Fixed Point Iterative Schemes for Computing Zeros of Nonlinear Equations." Complexity 2022 (March 31, 2022): 1–17. http://dx.doi.org/10.1155/2022/4331899.

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In computational mathematics, it is a matter of deep concern to recognize which of the given iteration schemes converges quickly with lesser error to the desired solution. Fixed point iterative schemes are constructed to be used for solving equations emerging in many fields of science and engineering. These schemes reformulate a nonlinear equation f s = 0 into a fixed point equation of the form s = g s ; such application determines the solution of the original equation via the support of fixed point iterative method and is subject to existence and uniqueness. In this manuscript, we introduce a new modified family of fixed point iterative schemes for solving nonlinear equations which generalize further recursive methods as particular cases. We also prove the convergence of our suggested schemes. We also consider some of the mathematical models which are categorically nonlinear in essence to authenticate the performance and effectiveness of these schemes which can be seen as an expansion and rationalization of some existing techniques.

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14

Gao, Dongjie. "Existence and Uniqueness of the Positive Definite Solution for the Matrix EquationX=Q+A∗(X^−C)−1A." Abstract and Applied Analysis 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/216035.

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We consider the nonlinear matrix equationX=Q+A∗(X^−C)−1A, whereQis positive definite,Cis positive semidefinite, andX^is the block diagonal matrix defined byX^=diag(X,X,…,X). We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

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15

Sumati Kumari, Panda, Obaid Alqahtani, and Erdal Karapınar. "Some Fixed-Point Theorems in b-Dislocated Metric Space and Applications." Symmetry 10, no. 12 (December 2, 2018): 691. http://dx.doi.org/10.3390/sym10120691.

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In this article, we prove some fixed-point theorems in b-dislocated metric space. Thereafter, we propose a simple and efficient solution for a non-linear integral equation and non-linear fractional differential equations of Caputo type by using the technique of fixed point.

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16

.Malathi, B., and S. Chelliah. "Existence of solutions for fixed point theorem." Journal of University of Shanghai for Science and Technology 23, no. 10 (October 8, 2021): 247–66. http://dx.doi.org/10.51201/jusst/21/10741.

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The development of a mathematical model based on diffusion has received a great dealof attention in recent years, many scientist and mathematician have tried to apply basicknowledge about the differential equation and the boundary condition to explain anapproximate the diffusion and reaction model. The subject of fractional calculus attracted much attentions and is rapidly growing area of research because of itsnumerous applications in engineering and scientific disciplines such as signal processing, nonlinear control theory, viscoelasticity, optimization theory [1], controlled thermonuclear fusion, chemistry, nonlinear biological systems, mechanics,electric networks, fluid dynamics, diffusion, oscillation, relaxation, turbulence, stochastic dynamical system, plasmaphysics, polymer physics, chemical physics, astrophysics, and economics. Therefore, it deserves an independent theoryparallel to the theory of ordinary differential equations (DEs).In the development of non-linear analysis, fixed point theory plays an important role. Also, it has been widely used in different branches of engineering and sciences. Banach fixed point theory is a essential part of mathematical analysis because of its applications in various area such as variational and linear inequalities, improvement and approximation theory. The fixed-point theorem in diffusion equations plays a significant role to construct methods to solve the problems in sciences and mathematics. Although Banach fixed point theory is a vast field of study and is capable of solving diffusion equations. The main motive of the research is solving the diffusion equations by Banach fixed point theorems and Adomian decomposition method. To analysis the drawbacks of the other fixed-point theorems and different solving methods, the related works are reviewed in this paper.

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17

Bodaghi, Abasalt. "Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach." Mathematica Slovaca 71, no. 1 (January 29, 2021): 117–28. http://dx.doi.org/10.1515/ms-2017-0456.

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Abstract In this article, by using a new form of multi-quadratic mapping, we define multi-m-Jensen-quadratic mappings and then unify the system of functional equations defining a multi-m-Jensen-quadratic mapping to a single equation. Using a fixed point theorem, we study the generalized Hyers-Ulam stability of multi-quadratic and multi-m-Jensen-quadratic functional equations. As a consequence, we show that every multi-m-Jensen-quadratic functional equation (under some conditions) can be hyperstable.

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18

Ramzanpour, Elahe, and Abasalt Bodaghi. "Approximate multi-Jensen-cubic mappings and a fixed point theorem." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 19, no. 1 (December 1, 2020): 141–54. http://dx.doi.org/10.2478/aupcsm-2020-0011.

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AbstractIn this paper, we introduce multi-Jensen-cubic mappings and unify the system of functional equations defining the multi-Jensen-cubic mapping to a single equation. Applying a fixed point theorem, we establish the generalized Hyers-Ulam stability of multi-Jensen-cubic mappings. As a known outcome, we show that every approximate multi-Jensen-cubic mapping can be multi-Jensen-cubic.

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19

Sangare, Daouda. "Fixed-Point and STILS Method to Solve a Coupled System of Transport Equations." Journal of Applied Mathematics 2022 (August 22, 2022): 1–6. http://dx.doi.org/10.1155/2022/2705591.

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In this paper, a coupled system of two transport equations is studied. The techniques are a fixed-point and Space-Time Integrated Least Square (STILS) method. The nonstationary advective transport equation is transformed to a “stationary” one by integrating space and time. Using a variational formulation and an adequate Poincare inequality, we prove the existence and the uniqueness of the solution. The transport equation with a nonlinear feedback is solved using a fixed-point method.

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20

Xu, Ding, Xian Wang, and Gongnan Xie. "Spectral Fixed Point Method for Nonlinear Oscillation Equation with Periodic Solution." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/538716.

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Based on the fixed point concept in functional analysis, an improvement on the traditional spectral method is proposed for nonlinear oscillation equations with periodic solution. The key idea of this new approach (namely, the spectral fixed point method, SFPM) is to construct a contractive map to replace the nonlinear oscillation equation into a series of linear oscillation equations. Usually the series of linear oscillation equations can be solved relatively easily. Different from other existing numerical methods, such as the well-known Runge-Kutta method, SFPM can directly obtain the Fourier series solution of the nonlinear oscillation without resorting to the Fast Fourier Transform (FFT) algorithm. In the meanwhile, the steepest descent seeking algorithm is proposed in the framework of SFPM to improve the computational efficiency. Finally, some typical cases are investigated by SFPM and the comparison with the Runge-Kutta method shows that the present method is of high accuracy and efficiency.

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21

Samadi, A., and M. B. Ghaemi. "An extension of Darbo fixed point theorem and its applications to coupled fixed point and integral equations." Filomat 28, no. 4 (2014): 879–86. http://dx.doi.org/10.2298/fil1404879s.

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In this paper, an extension of Darbo fixed point theorem is introduced. By applying our extension, we obtain a coupled fixed point theorem and a solution for an integral equation. The proofs of our results are based on the technique of measure of noncompactness.

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22

Nyein, Ei Ei, and Aung Khaing Zaw. "A fixed point method to solve differential equation and Fredholm integral equation." Journal of Nonlinear Sciences and Applications 13, no. 04 (February 28, 2020): 205–11. http://dx.doi.org/10.22436/jnsa.013.04.05.

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23

Uddin, Fahim, Umar Ishtiaq, Naeem Saleem, Khaleel Ahmad, and Fahd Jarad. "Fixed point theorems for controlled neutrosophic metric-like spaces." AIMS Mathematics 7, no. 12 (2022): 20711–39. http://dx.doi.org/10.3934/math.20221135.

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<abstract> <p>In this paper, we establish the concept of controlled neutrosophic metric-like spaces as a generalization of neutrosophic metric spaces and provide several non-trivial examples to show the spuriousness of the new concept in the existing literature. Furthermore, we prove several fixed point results for contraction mappings and provide the examples with their graphs to show the validity of the results. At the end of the manuscript, we establish an application to integral equations, in which we use the main result to find the solution of the integral equation.</p> </abstract>

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24

Hannabou, Mohamed, Khalid Hilal, and Ahmed Kajouni. "Existence and Uniqueness of Mild Solutions to Impulsive Nonlocal Cauchy Problems." Journal of Mathematics 2020 (November 12, 2020): 1–9. http://dx.doi.org/10.1155/2020/5729128.

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In this paper, a class of nonlocal impulsive differential equation with conformable fractional derivative is studied. By utilizing the theory of operators semigroup and fractional derivative, a new concept on a solution for our problem is introduced. We used some fixed point theorems such as Banach contraction mapping principle, Schauder’s fixed point theorem, Schaefer’s fixed point theorem, and Krasnoselskii’s fixed point theorem, and we derive many existence and uniqueness results concerning the solution for impulsive nonlocal Cauchy problems. Some concrete applications to partial differential equations are considered. Some concrete applications to partial differential equations are considered.

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25

Filobello-Nino, Uriel, Hector Vazquez-Leal, Jesús Huerta-Chua, Jaime Martínez-Castillo, Agustín L. Herrera-May, Mario Alberto Sandoval-Hernandez, and Victor Manuel Jimenez-Fernandez. "The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations." Mathematics 10, no. 20 (October 14, 2022): 3797. http://dx.doi.org/10.3390/math10203797.

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This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.

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26

Alsmeyer, Gerold, and Matthias Meiners. "A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation." Journal of Applied Probability 44, no. 3 (September 2007): 586–606. http://dx.doi.org/10.1239/jap/1189717531.

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After suitable normalization the asymptotic root value W of a minimax game tree of order b ≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval of R appears to be a particular solution of the stochastic maximin fixed-point equation W ξ max1≤i≤bmin1≤j≤bWi,j, where Wi,j are independent copies of W and denotes equality in law. Moreover, ξ= g'(α) > 1, where g(x) := (1 − (1 − x)b)b and α denotes the unique fixed point of g in (0, 1). This equation, which takes the form F(t) = g(F(t/ξ)) in terms of the distribution function F of W, is studied in the present paper for a reasonably extended class of functions g so as to encompass more general stochastic maximin equations as well. A complete description of the set of solutions F is provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic fixed-point equations for weighted minima and maxima is also discussed.

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27

Gao, Dongjie. "A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations." Journal of Mathematics 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/167049.

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By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equationX=Q+A⁎f(X)A, wherefis a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equationX=kQ+A⁎(X^-C)qAand prove that the equation has a unique positive definite solution whenQ^≥Candk>1and0<q<1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.

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Jia, Zhifu, Xinsheng Liu, and Cunlin Li. "Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump." Symmetry 12, no. 5 (May 6, 2020): 765. http://dx.doi.org/10.3390/sym12050765.

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No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model.

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Haider, Syed Sabyel, and Mujeeb Ur Rehman. "Construction of fixed point operators for nonlinear difference equations of non integer order with impulses." Fractional Calculus and Applied Analysis 23, no. 3 (June 25, 2020): 886–907. http://dx.doi.org/10.1515/fca-2020-0045.

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AbstractIn this article, we establish a technique for transforming arbitrary real order delta difference equations with impulses to corresponding summation equations. The technique is applied to non-integer order delta difference equation with some boundary conditions. Furthermore, the summation formulation for impulsive fractional difference equation is utilized to construct fixed point operator which in turn are used to discuss existence of solutions.

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Boulares, Hamid, Abbes Benchaabane, Nuttapol Pakkaranang, Ramsha Shafqat, and Bancha Panyanak. "Qualitative Properties of Positive Solutions of a Kind for Fractional Pantograph Problems using Technique Fixed Point Theory." Fractal and Fractional 6, no. 10 (October 14, 2022): 593. http://dx.doi.org/10.3390/fractalfract6100593.

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The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation has a positive solution by utilising the Schauder fixed point theorem (SFPT) and the upper and lower solutions method. Another method for proving the existence of a singular positive solution is the Banach fixed point theorem (BFPT). Finally, we provide an example that illustrates and explains our conclusions.

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Grayna, J., V. Kavitha, and Soumya George. "A Result on ULAM Stability of Impulsive Fractional Volterra Differential Equation Using a Fixed Point Approach." Journal of Advanced Research in Dynamical and Control Systems 11, no. 10-SPECIAL ISSUE (October 31, 2019): 1388–97. http://dx.doi.org/10.5373/jardcs/v11sp10/20192984.

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Jung, Soon-Mo. "A Fixed Point Approach to the Stability of an Integral Equation Related to the Wave Equation." Abstract and Applied Analysis 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/612576.

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Sawangsup, Kanokwan, and Wutiphol Sintunavarat. "Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions." Open Mathematics 15, no. 1 (February 26, 2017): 111–25. http://dx.doi.org/10.1515/math-2017-0012.

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Abstract The aim of this work is to introduce the notion of weak altering distance functions and prove new fixed point theorems in metric spaces endowed with a transitive binary relation by using weak altering distance functions. We give some examples which support our main results where previous results in literature are not applicable. Then the main results of the paper are applied to the multidimensional fixed point results. As an application, we apply our main results to study a nonlinear matrix equation. Finally, as numerical experiments, we approximate the definite solution of a nonlinear matrix equation using MATLAB.

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Shammakh, Wafa, A. George Maria Selvam, Vignesh Dhakshinamoorthy, and Jehad Alzabut. "Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins." Symmetry 14, no. 9 (September 8, 2022): 1877. http://dx.doi.org/10.3390/sym14091877.

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The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations.

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OGROWSKY, A., and B. SCHMALFUSS. "DISCRETIZATION OF STATIONARY SOLUTIONS OF SPDE'S BY EXTERNAL APPROXIMATION IN SPACE AND TIME." International Journal of Bifurcation and Chaos 20, no. 09 (September 2010): 2835–50. http://dx.doi.org/10.1142/s0218127410027398.

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We consider a stochastic partial differential equation with additive noise satisfying a strong dissipativity condition for the nonlinear term such that this equation has a random fixed point. The goal of this article is to approximate this fixed point by space and space-time discretizations of a stochastic differential equation or more precisely, a conjugate random partial differential equation. For these discretizations external schemes are used. We show the convergence of the random fixed points of the space and space-time discretizations to the random fixed point of the original partial differential equation.

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Liu, Yang, and Zhang Weiguo. "Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order." Abstract and Applied Analysis 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/540846.

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We investigate the existence of multiple positive solutions for a class of boundary value problems of nonlinear differential equation with Caputo’s fractional order derivative. The existence results are obtained by means of the Avery-Peterson fixed point theorem. It should be point out that this is the first time that this fixed point theorem is used to deal with the boundary value problem of differential equations with fractional order derivative.

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Haque, Inzamamul, Javid Ali, and M. Mursaleen. "Solvability of Implicit Fractional Order Integral Equation in ℓ p 1 ≤ p < ∞ Space via Generalized Darbo’s Fixed Point Theorem." Journal of Function Spaces 2022 (May 20, 2022): 1–8. http://dx.doi.org/10.1155/2022/1674243.

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We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in ℓ p 1 ≤ p < ∞ space. The fundamental tool in the presentation of our proofs is the measure of noncompactness mnc approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.

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Ahmad, Sahibzada Waseem, Muhammad Sarwar, Gul Rahmat, and Fahd Jarad. "Existence of Unique Solution of Urysohn and Fredholm Integral Equations in Complex Double Controlled Metric Type Spaces." Mathematical Problems in Engineering 2022 (May 29, 2022): 1–11. http://dx.doi.org/10.1155/2022/4791454.

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Complex Urysohn integral equations and complex Fredholm integral equation of the second kind have intensified the attention of appreciable researchers to their solution due to their comprehensive applications. This study is devoted to the existence and uniqueness of solution to these integral equations in the setting of complete complex double controlled metric spaces via fixed point theory. For this motive, a fixed point result together with a numerical example for the convergence behavior of operator to the fixed point is analyzed in the context of the quoted metric spaces.

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39

Murthy, Penumarthy Parvateesam, Chandra Prakash Dhuri, Santosh Kumar, Rajagopalan Ramaswamy, Muhannad Abdullah Saud Alaskar, and Stojan Radenovi’c. "Common Fixed Point for Meir–Keeler Type Contraction in Bipolar Metric Space." Fractal and Fractional 6, no. 11 (November 4, 2022): 649. http://dx.doi.org/10.3390/fractalfract6110649.

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In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using the extension of Meir–Keeler contraction in the setting of bipolar metric spaces have been proved. The derived results have been supplemented with non-trivial examples. Our results extend and generalise the results established in the past. We have provided an application to find an analytical solution to an Integral Equation to supplement the derived result.

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Watkins, Will. "Modified Wiener equations." International Journal of Mathematics and Mathematical Sciences 27, no. 6 (2001): 347–56. http://dx.doi.org/10.1155/s0161171201006561.

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This paper is concerned with a class of functional differential equations whose argument transforms are involutions. In contrast to the earlier works in this area, which have used only involutions with a fixed point, we also admit involutions without a fixed point. In the first case, an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. In our case, either two initial conditions or two boundary conditions are necessary for a solution; the equation is then reduced to a boundary value problem for a higher order ODE.

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Chacha, Chacha S. "Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I." Tanzania Journal of Science 47, no. 3 (August 14, 2021): 1033–40. http://dx.doi.org/10.4314/tjs.v47i3.14.

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The nonlinear matrix equation was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms. Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method

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42

Rosten, Oliver J. "A conformal fixed-point equation for the effective average action." International Journal of Modern Physics A 34, no. 05 (February 20, 2019): 1950027. http://dx.doi.org/10.1142/s0217751x19500271.

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A Legendre transform of the recently discovered conformal fixed-point equation is constructed, providing an unintegrated equation encoding full conformal invariance within the framework of the effective average action.

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43

Manzonetto, Giulio, Andrew Polonsky, Alexis Saurin, and Jakob Grue Simonsen. "The fixed point property and a technique to harness double fixed point combinators." Journal of Logic and Computation 29, no. 5 (July 10, 2019): 831–80. http://dx.doi.org/10.1093/logcom/exz013.

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Abstract The ${\lambda }$-calculus enjoys the property that each ${\lambda }$-term has at least one fixed point, which is due to the existence of a fixed point combinator. It is unknown whether it enjoys the ‘fixed point property’ stating that each ${\lambda }$-term has either one or infinitely many pairwise distinct fixed points. We show that the fixed point property holds when considering possibly open fixed points. The problem of counting fixed points in the closed setting remains open, but we provide sufficient conditions for a ${\lambda }$-term to have either one or infinitely many fixed points. In the main result of this paper we prove that in every sensible ${\lambda }$-theory there exists a ${\lambda }$-term that violates the fixed point property. We then study the open problem concerning the existence of a double fixed point combinator and propose a proof technique that could lead towards a negative solution. We consider interpretations of the ${\lambda } {\mathtt{Y}}$-calculus into the ${\lambda }$-calculus together with two reduction extension properties, whose validity would entail the non-existence of any double fixed point combinators. We conjecture that both properties hold when typed ${\lambda } {\mathtt{Y}}$-terms are interpreted by arbitrary fixed point combinators. We prove reduction extension property I for a large class of fixed point combinators. Finally, we prove that the ${\lambda }{\mathtt{Y}}$-theory generated by the equation characterizing double fixed point combinators is a conservative extension of the ${\lambda }$-calculus.

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Mureşan, Sorin, Loredana Florentina Iambor, and Omar Bazighifan. "New Applications of Perov’s Fixed Point Theorem." Mathematics 10, no. 23 (December 4, 2022): 4597. http://dx.doi.org/10.3390/math10234597.

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The goal of this paper is to consider a differential equation system written as an interesting equivalent form that has not been used before. Using Perov’s fixed point theorem in generalized metric spaces, the existence and uniqueness of the solution are obtained for the proposed system. The approximation of the solution is given, and as a novelty, the approximation of its derivative is also obtained using the same iteration steps.

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Alsmeyer, Gerold, and Matthias Meiners. "A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation." Journal of Applied Probability 44, no. 03 (September 2007): 586–606. http://dx.doi.org/10.1017/s0021900200003296.

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After suitable normalization the asymptotic root valueWof a minimax game tree of orderb≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval ofRappears to be a particular solution of the stochastic maximin fixed-point equationWξ max1≤i≤bmin1≤j≤bWi,j, whereWi,jare independent copies ofWanddenotes equality in law. Moreover, ξ=g'(α) > 1, whereg(x) := (1 − (1 −x)b)band α denotes the unique fixed point ofgin (0, 1). This equation, which takes the formF(t) =g(F(t/ξ)) in terms of the distribution functionFofW, is studied in the present paper for a reasonably extended class of functionsgso as to encompass more general stochastic maximin equations as well. A complete description of the set of solutionsFis provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic fixed-point equations for weighted minima and maxima is also discussed.

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Bousselsal, Mahmoud, and Sidi Hamidou Jah. "Integrable Solutions of a Nonlinear Integral Equation via Noncompactness Measure and Krasnoselskii's Fixed Point Theorem." International Journal of Analysis 2014 (March 16, 2014): 1–10. http://dx.doi.org/10.1155/2014/280709.

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We study the existence of solutions of a nonlinear Volterra integral equation in the space L1[0,+∞). With the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes on nonlinear integral equations. Our results extend and generalize some previous works. An example is given to support our results.

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Bodaghi, Abasalt, Idham Arif Alias, Lida Mousavi, and Sedigheh Hosseini. "Characterization and Stability of Multimixed Additive-Quartic Mappings: A Fixed Point Application." Journal of Function Spaces 2021 (November 11, 2021): 1–11. http://dx.doi.org/10.1155/2021/9943199.

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In this article, we introduce the multi-additive-quartic and the multimixed additive-quartic mappings. We also describe and characterize the structure of such mappings. In other words, we unify the system of functional equations defining a multi-additive-quartic or a multimixed additive-quartic mapping to a single equation. We also show that under what conditions, a multimixed additive-quartic mapping can be multiadditive, multiquartic, and multi-additive-quartic. Moreover, by using a fixed point technique, we prove the Hyers-Ulam stability of multimixed additive-quartic functional equations thus generalizing some known results.

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Dutta, Hemen, B. V. Senthil Kumar, and Khalifa Al-Shaqsi. "Approximation of Jensen type reciprocal mappings via fixed point technique." Miskolc Mathematical Notes 23, no. 2 (2022): 607. http://dx.doi.org/10.18514/mmn.2022.3631.

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In this paper, we solve a new Jensen type m-dimensional multiplicative inverse functional equation and then its various stability problems in the setting of non-negative real numbers and non-Archimedean spaces via fixed point method. The functional equation dealt in this study is linked with the famous relationship between arithmetic and harmonic mean of m values. The role of harmonic mean is very significant in many other fields such as traffic flow theory, industrial engineering, communication system, etc. By inversing the arithmetic mean of reciprocal values, we attain the harmonic mean. This property could be analyzed as an inverse problem via the functional equation dealt in this investigation.

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Nazam, Muhammad, Maha M. A. Lashin, Aftab Hussain, and Hamed H. Al Sulami. "Remarks on the generalized interpolative contractions and some fixed-point theorems with application." Open Mathematics 20, no. 1 (January 1, 2022): 845–62. http://dx.doi.org/10.1515/math-2022-0042.

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Abstract In this manuscript, some remarks on the papers [H. A. Hammad, P. Agarwal, S. Momani, and F. Alsharari, Solving a fractional-order differential equation using rational symmetric contraction mappings, Fractal Fract. 5 (2021), 159] and [A. Hussain, F. Jarad, and E. Karapinar, A study of symmetric contractions with an application to generalized fractional differential equations, Adv. Differ. Equ. 2021 (2021), 300] are given. In the light of remarks, we introduce a new property that makes it convenient to investigate the existence of fixed points of the interpolative contractions in the orthogonal metric spaces. We derive several new results based on known contractions from the main theorems. As an application, we resolve a Urysohn integral equation.

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50

EL-Fassi, Iz-iddine, Samir Kabbaj, and Abdellatif Chahbi. "A fixed point approach to the hyperstability of the general linear equation in β-Banach spaces." Analysis 38, no. 3 (August 1, 2018): 115–26. http://dx.doi.org/10.1515/anly-2017-0028.

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AbstractThe purpose of this paper is first to reformulate the fixed point theorem (see Theorem 1 of [J. Brzdȩk, J. Chudziak and Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 2011, 17, 6728–6732]) in β-Banach spaces. We also show that this theorem is a very efficient and convenient tool for proving the hyperstability results of the general linear equation in β-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it.

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