Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps. The authors would like to express their cordial thanks to the referee for useful remarks which have improved the first version of this paper. Electron. Soon-Mo Jung. Subscription will auto renew annually. MathSciNet  J. The system x' = -y, y' = -ay - x(x - .15)(x-2) results from an approximation of the Hodgkin-Huxley equations for nerve impulses. 2, 373–380 (1998), MATH  The intersection near is an unstable fixed point. Transform it into a first order equation [math]x' = f(x)[/math] if it's not already 3. 4, http://jipam.vu.edu.au, Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. How to investigate stability of stationary points? The results can be generalized to larger systems. A4-2(780), Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan Equations of first order with a single variable. I found the Jacobian to be: [0, -1; -3x^2 + 4.3x - 0.3, -a] However, this gives me an eigenvalue of 0, and I'm not sure how to do stability here. Immediate online access to all issues from 2019. Appl. Lett. J. Inequal. If the components of the state vector x are (x1;x2;:::;xn)and the compo-nents of the rate vector f are (f1; f2;:::; fn), then the Jacobian is J = 2 6 6 6 6 6 4 ∂f1 ∂x1 ∂f1 ∂x2::: ∂f1 ∂xn We linearize the original ODE under the condition . 55, 17–24 (2002), MathSciNet  Math. 2. 346, 43–52 (2004), MATH  Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea, Department of Mathematics, College of Sciences, Yasouj University, 75914-74831, Yasouj, Iran, You can also search for this author in Part of Springer Nature. The stability of the trajectories of this system under perturbations of its initial conditions can also be addressed using the stability theory. Malays. In this paper, we apply the fixed point method to investigate the Hyers–Ulam–Rassias stability of the \(n\)th order linear differential equations. Prace Mat. A fixed point of is stable if for every > 0 there is > 0 such that whenever , all This application is intended for non-commercial, non-profit use only. Appl. Soc. J. Korean Math. Make sure you've got an autonomous equation 2. Fixed point . 13, 259–270 (1993), Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . Comput. However, actual jumps do not always happen at fixed points but usually at random points. We discuss the stability of solutions to a kind of scalar Liénard type equations with multiple variable delays by means of the fixed point technique under an exponentially weighted metric. Anal. Pure Appl. (Please input and without independent variable , like for and for .). Suitable for advanced undergraduates and graduate students, it contains an extensive collection of new and classical examples, all worked in detail and presented in an elementary manner. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton (2003) [3] , Zhang (2005) [14] , Raffoul (2004) [13] , and Jin and Luo (2008) [12] . The point x=3.7 is a semi-stable equilibrium of the differential equation. Therefore: a 2 × 2 system of differential equations can be studied as a mathematical object, and we may arrive at the conclusion that it possesses the saddle-path stability property. : Hyers–Ulam–Rassias stability of the Banach space valued linear differential equations \(y^{\prime } = \lambda y\). MathSciNet  Malays. As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. Appl. Bull. Inc. 2019. We are interested in the local behavior near ¯x. In this paper, new cri-teriaareestablished forthe asymptotic stability ofsomenonlin-ear delay di erential equations with nite … This means that it is structurally able to provide a unique path to the fixed-point (the “steady- Let us start with equations in one variable, (1) xt +axt−1 = bt This is a first-order difference equation because only one lag of x appears. Czerwik, S.: Functional Equations and Inequalities in Several Variables. A dynamical system can be represented by a differential equation. Stability of a fixed point can be determined by eigen values of matrix  . Graduate School of Information Science, Nagoya University Solution curve starting (, ) can also diplayed with animation. Proc. 21, 1024–1028 (2008). Math. 17, 1135–1140 (2004), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, III. Jpn. Sci. World Scientific, Singapore (2002), Găvruţa, P., Jung, S.-M., Li, Y.: Hyers–Ulam stability for second-order linear differential equations with boundary conditions. 23, 306–309 (2010), Miura, T.: On the Hyers–Ulam stability of a differentiable map. when considering the stability of non -linear systems at equilibrium. volume 38, pages855–865(2015)Cite this article. The point x=3.7 cannot be an equilibrium of the differential equation. The author will further use different fixed-point theorems to consider the stability of SPDEs in … DIFFERENTIAL EQUATIONS VIA FIXED POINT THEORY AND APPLICATIONS MENG FAN, ZHINAN XIA AND HUAIPING ZHU ABSTRACT. differential equation: x˙ = f(x )+ ∂f ∂x x (x x )+::: = ∂f ∂x x (x x )+::: (2) The partial derivative in the above equation is to be interpreted as the Jacobian matrix. Abstract: Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. J. Inequal. The general method is 1. Google Scholar, Hyers, D.H., Isac, G., Rassias, T.M. Korean Math. When we linearize ODE near th fixed point (, ),  ODE for is calculated to be as follows. Sci. https://doi.org/10.1007/s40840-014-0053-5. Stud. 38, 855–865 (2015). Math. Learn more about Institutional subscriptions, Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. 5, pp. Stability of Unbounded Differential Equations in Menger k-Normed Spaces: A Fixed Point Technique Masoumeh Madadi 1, Reza Saadati 2 and Manuel De la Sen 3,* 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran; mahnazmadadi91@yahoo.com 2 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, … S.-M. Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 4 (1) (2003), Art. Math. (Note, when solutions are not expressed in explicit form, the solution are not listed above.). (2012), Article ID 712743, p 10. doi:10.1155/2012/712743, Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. In this equation, a is a time-independent coefficient and bt is the forcing term. Fixed points  are defined with the condition  . 2006 edition. Math. Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. Google Scholar, Cădariu, L., Găvruţa, L., Găvruţa, P.: Fixed points and generalized Hyers–Ulam stability. Appl. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Acad. In this paper, we apply the fixed point method to investigate the Hyers–Ulam–Rassias stability of the ... Cimpean, D.S., Popa, D.: On the stability of the linear differential equation of higher order with constant coefficients. Soc. Math. Comput. Autonomous Equations / Stability of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stability, Long- term behavior of solutions, direction fields, Population dynamics and logistic equations Autonomous Equation: A differential equation where the independent variable does not explicitly appear in its expression. J. : A characterization of Hyers-Ulam stability of first order linear differential operators. Prace Mat. However, the Ackermann numbers are an example of a recurrence relation that do not map to a difference equation, much less points on the solution to a differential equation. 286, 136–146 (2003), Miura, T., Miyajima, S., Takahasi, S.E. Lett. Fixed points are defined with the condition . 39, 309–315 (2002), Takahasi, S.E., Takagi, H., Miura, T., Miyajima, S.: The Hyers–Ulam stability constants of first order linear differential operators. Stability of a fixed point in a system of ODE, Yasuyuki Nakamura Jung, SM., Rezaei, H. A Fixed Point Approach to the Stability of Linear Differential Equations. But not all fixed points are easy to attain this way. Contact the author for permission if you wish to use this application in for-profit activities. An attractive fixed point of a function f is a fixed point x0 of f such that for any value of x in the domain that is close enough to x0, the iterated function sequence Journal of Difference Equations and Applications: Vol. PubMed Google Scholar. Math. Bull. In this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory. © Maplesoft, a division of Waterloo Maple 449-457. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. Appl. Appl. Note that there could be more than one fixed points. Birkhäuser, Boston (1998), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Ber. 1. Stability, in mathematics, condition in which a slight disturbance in a system does not produce too disrupting an effect on that system.In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x. 296, 403–409 (2004), Ulam, S.M. [tex] x_{n + 1} = x_n [/tex] There are fixed points at x = 0 and x = 1. Equ. Shows how to determine the fixed points and their linear stability of a first-order nonlinear differential equation. Thus one can solve many recurrence relations by rephrasing them as difference equations, and then solving the difference equation, analogously to how one solves ordinary differential equations. This is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. The object of the present paper is to examine the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of a nonlinear Volterra integro-differential equation by using the fixed point method. J. Springer, New York (2011), Li, Y., Shen, Y.: Hyers–Ulam stability of linear differential equations of second order. : Hyers–Ulam stability of linear differential operator with constant coefficients. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. In this paper we begin a study of stability theory for ordinary and functional differential equations by means of fixed point theory. Note that there could be more than one fixed points. Hi I am unsure about stability of fixed points here is an example. The point x=3.7 is an unstable equilibrium of the differential equation. Linear difference equations 2.1. So I found the fixed points of (0,0) (0.15,0) and (2,0). http://www.phys.cs.is.nagoya-u.ac.jp/~nakamura/, Let us consider the following system of ODE. 258, 90–96 (2003), Obłoza, M.: Hyers stability of the linear differential equation. Fixed Point. © 2020 Springer Nature Switzerland AG. : Remarks on Ulam stability of the operatorial equations. For that reason, we will pursue this avenue of investigation of a little while. Google Scholar, Cimpean, D.S., Popa, D.: On the stability of the linear differential equation of higher order with constant coefficients. You can switch back to the summary page for this application by clicking here. Fixed Point Theory 10, 305–320 (2009), Rus, I.A. In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . MATH  Math. This is a preview of subscription content, log in to check access. Google Scholar, Miura, T., Jung, S.-M., Takahasi, S.E. 14, 141–146 (1997), Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003), Rus, I.A. Differ. The ones that are, are attractors . 2013R1A1A2005557). 217, 4141–4146 (2010), Article  Lett. |. Using Critical Points to determine increasing and decreasing of general solutions to differential equations. Find the fixed points, which are the roots of f 4. Nonlinear delay di erential equations have been widely used to study the dynamics in biology, but the sta- bility of such equations are challenging. The paper is motivated by a number of difficulties encountered in the study of stability by means of Liapunov’s direct method. Appl. Appl. By this work, we improve some related results from one delay to multiple variable delays. For the simplisity, we consider the follwoing system of autonomous ODE with two variables. 9, No. Let one of them to be . Math. In this paper we just make a first attempt to use the fixed-point theory to deal with the stability of stochastic delay partial differential equations. Rocznik Nauk.-Dydakt. 54, 125–134 (2009), Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers–Ulam stability of the Banach space-valued differential equation \(y^{\prime } = \lambda y\). When bt = 0, the difference Direction field near the fixed point (, ) is displayed in the right figure. 19, 854–858 (2006), Jung, S.-M.: A fixed point approach to the stability of differential equations \(y^{\prime } = F(x, y)\). The solutions of random impulsive differential equations is a stochastic process. In this case there are two fixed points that are 1-periodic solutions to the differential equation. Legal Notice: The copyright for this application is owned by the author(s). Math. ); jrwang@gzu.edu.cn (J.W.) : A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Vol. 311, 139–146 (2005), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, II. Anal. Bulletin of the Malaysian Mathematical Sciences Society Rocznik Nauk.-Dydakt. Sci. We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. Appl. Babes-Bolyai Math. 217, 4141–4146 (2010) Article MATH MathSciNet Google Scholar 6. The fixed-point theory used in stability seems in its very early stages. Let one of them to be . Abstr. Math. Nachr. Grazer Math. Anal. MathSciNet  Appl. Linearization . Consider a stationary point ¯x of the difference equation xn+1 = f(xn). Anal. 2011(80), 1–5 (2011), Hyers, D.H.: On the stability of the linear functional equation. - 85.214.22.11. Univ. Google Scholar, Czerwik, S.: Functional Equations and Inequalities in Several Variables. Tax calculation will be finalised during checkout. Math. Correspondence to : Ulam stability of ordinary differential equations. Find the fixed points and classify their stability. For this purpose, we consider the deviation of the elements of the sequence to the stationary point ¯x: zn:= xn −x¯ zn has the following property: zn+1 = xn+1 −x¯ = f(xn)− ¯x = f(¯x+zn)− ¯x. : Stability of Functional Equations in Several Variables. The investigator will get better results by using several methods than by using one of them. J. Fixed points, attractors and repellers If the sequence has a limit, that limit must be a fixed point of : a value such that . 48. It is different from deterministic impulsive differential equations and also it is different from stochastic differential equations. A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations Kui Liu 1,2, Michal Feckanˇ 3,4,* and JinRong Wang 1,5 1 Department of Mathematics, Guizhou University, Guiyang 550025, China; liuk180916@163.com (K.L. nakamura@nagoya-u.jp https://doi.org/10.1007/s40840-014-0053-5, DOI: https://doi.org/10.1007/s40840-014-0053-5, Over 10 million scientific documents at your fingertips, Not logged in Soc. Equilibrium Points and Fixed Points Main concepts: Equilibrium points, fixed points for RK methods, convergence of fixed points for one-step methods Equilibrium points represent the simplest solutions to differential equations. Math. This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. Sci. 41, 995–1005 (2004), Miura, T., Miyajima, S., Takahasi, S.E. (2003). It has the general form of y′ = f (y). In general when talking about difference equations and whether a fixed point is stable or unstable, does this refer to points in a neighbourhood of those points? Soc. USA 27, 222–224 (1941), Article  Included are most of the standard topics in 1st and 2nd order differential equations, Laplace transforms, systems of differential eqauations, series solutions as well as a brief introduction to boundary value problems, Fourier series and partial differntial equations. 33(2), 47–56 (2010), Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. Math. Natl. In terms of the solution operator, they are the fixed points of the flow map. The stability of a fixed point can be deduced from the slope of the Poincaré map at the intersection point or by computing the Floquet exponents, which is done in this Demonstration. 8, Interscience, New York (1960), Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. Two examples are also given to illustrate our results. Lett. Bull. Appl. (Note, when solutions are not expressed in explicit form, the solution are not listed above.) We notice that these difficulties frequently vanish when we apply fixed point theory. The point x=3.7 is an equilibrium of the differential equation, but you cannot determine its stability.