Rocznik Nauk.-Dydakt. Lett. when considering the stability of non -linear systems at equilibrium. 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. 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. In this equation, a is a time-independent coeﬃcient and bt is the forcing term. Nachr. It has the general form of y′ = f (y). Immediate online access to all issues from 2019. Malays. Bull. Consider a stationary point ¯x of the diﬀerence equation xn+1 = f(xn). We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. The general method is 1. Prace Mat. Learn more about Institutional subscriptions, Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. Soc. In this paper we begin a study of stability theory for ordinary and functional differential equations by means of fixed point theory. Anal. 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. You can switch back to the summary page for this application by clicking here. 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. Equ. Appl. 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. Appl. By this work, we improve some related results from one delay to multiple variable delays. J. This application is intended for non-commercial, non-profit use only. Sci. The intersection near is an unstable fixed point. Linearization . : Hyers–Ulam–Rassias stability of the Banach space valued linear differential equations $$y^{\prime } = \lambda y$$. The point x=3.7 cannot be an equilibrium of the differential equation. Jpn. Fixed point . Prace Mat. 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 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 J. Inequal. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Google Scholar, Cimpean, D.S., Popa, D.: On the stability of the linear differential equation of higher order with constant coefficients. (2003). 33(2), 47–56 (2010), Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. The stability of the trajectories of this system under perturbations of its initial conditions can also be addressed using the stability theory. Contact the author for permission if you wish to use this application in for-profit activities. In this paper, new cri-teriaareestablished forthe asymptotic stability ofsomenonlin-ear delay di erential equations with nite … : A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Vol. 13, 259–270 (1993), Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Math. Birkhäuser, Boston (1998), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. This is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. 39, 309–315 (2002), Takahasi, S.E., Takagi, H., Miura, T., Miyajima, S.: The Hyers–Ulam stability constants of first order linear differential operators. The system x' = -y, y' = -ay - x(x - .15)(x-2) results from an approximation of the Hodgkin-Huxley equations for nerve impulses. 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. Legal Notice: The copyright for this application is owned by the author(s). Hi I am unsure about stability of fixed points here is an example. Proc. Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps. Czerwik, S.: Functional Equations and Inequalities in Several Variables. MathSciNet  Part of Springer Nature. Soc. The point x=3.7 is a semi-stable equilibrium of the differential equation. World Scientific, Singapore (2002), Găvruţa, P., Jung, S.-M., Li, Y.: Hyers–Ulam stability for second-order linear differential equations with boundary conditions. 217, 4141–4146 (2010) Article MATH MathSciNet Google Scholar 6. Math. In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . 286, 136–146 (2003), Miura, T., Miyajima, S., Takahasi, S.E. MathSciNet  The investigator will get better results by using several methods than by using one of them. Lett. 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. Subscription will auto renew annually. (Please input and without independent variable , like for and for .). Abstr. $$x_{n + 1} = x_n$$ There are fixed points at x = 0 and x = 1. Stud. The point x=3.7 is an equilibrium of the differential equation, but you cannot determine its stability. Google Scholar, Czerwik, S.: Functional Equations and Inequalities in Several Variables. Anal. 4 (1) (2003), Art. We notice that these difficulties frequently vanish when we apply fixed point theory. 21, 1024–1028 (2008). 2013R1A1A2005557). Sci. Nonlinear delay di erential equations have been widely used to study the dynamics in biology, but the sta- bility of such equations are challenging. Appl. The solutions of random impulsive differential equations is a stochastic process. 8, Interscience, New York (1960), Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. So I found the fixed points of (0,0) (0.15,0) and (2,0). Korean Math. The authors would like to express their cordial thanks to the referee for useful remarks which have improved the first version of this paper. The paper is motivated by a number of difficulties encountered in the study of stability by means of Liapunov’s direct method. http://www.phys.cs.is.nagoya-u.ac.jp/~nakamura/, Let us consider the following system of ODE. Grazer Math. In this case there are two fixed points that are 1-periodic solutions to the differential equation. 5, pp. Math. 54, 125–134 (2009), Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers–Ulam stability of the Banach space-valued differential equation $$y^{\prime } = \lambda y$$. We are interested in the local behavior near ¯x. J. : Hyers–Ulam stability of linear differential operator with constant coefficients. MATH  Journal of Difference Equations and Applications: Vol. Equations of ﬁrst order with a single variable. 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 Babes-Bolyai Math. Appl. Sci. Fixed Point. 296, 403–409 (2004), Ulam, S.M. When we linearize ODE near th fixed point (, ),  ODE for is calculated to be as follows. This is a preview of subscription content, log in to check access. 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. : Ulam stability of ordinary differential equations. (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. 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. Fixed points are defined with the condition . 258, 90–96 (2003), Obłoza, M.: Hyers stability of the linear differential equation. We linearize the original ODE under the condition . Natl. Linear difference equations 2.1. Pure Appl. 311, 139–146 (2005), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, II. Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. MathSciNet  Rocznik Nauk.-Dydakt. (Note, when solutions are not expressed in explicit form, the solution are not listed above.). PubMed Google Scholar. This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. : Remarks on Ulam stability of the operatorial equations. Appl. 55, 17–24 (2002), MathSciNet  Math. Bull. Comput. Appl. Note that there could be more than one fixed points. (Note, when solutions are not expressed in explicit form, the solution are not listed above.) Find the fixed points, which are the roots of f 4. Math. Fixed Point Theory 10, 305–320 (2009), Rus, I.A. Stability of a fixed point in a system of ODE, Yasuyuki Nakamura Using Critical Points to determine increasing and decreasing of general solutions to differential equations. 9, No. 14, 141–146 (1997), Radu, V.: The fixed point alternative and the stability of functional equations. 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. In terms of the solution operator, they are the ﬁxed points of the ﬂow map. USA 27, 222–224 (1941), Article  48. As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. volume 38, pages855–865(2015)Cite this article. Math. Fixed points  are defined with the condition  . Soc. However, actual jumps do not always happen at fixed points but usually at random points. For that reason, we will pursue this avenue of investigation of a little while. 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, … 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. Soc. Stability of a fixed point can be determined by eigen values of matrix  . Find the fixed points and classify their stability. 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. Math. Comput. Sci. In this paper, we apply the fixed point method to investigate the Hyers–Ulam–Rassias stability of the $$n$$th order linear differential equations. Solution curve starting (, ) can also diplayed with animation. : Stability of Functional Equations in Several Variables. Appl. Make sure you've got an autonomous equation 2. 2006 edition. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. 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? Graduate School of Information Science, Nagoya University Soon-Mo Jung. Math. Two examples are also given to illustrate our results. 2, 373–380 (1998), MATH  Differ. In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . 2011(80), 1–5 (2011), Hyers, D.H.: On the stability of the linear functional equation. Lett. Appl. - 85.214.22.11. In this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory. The point x=3.7 is an unstable equilibrium of the differential equation. J. Inequal. 19, 854–858 (2006), Jung, S.-M.: A fixed point approach to the stability of differential equations $$y^{\prime } = F(x, y)$$. Lett. Math. The fixed-point theory used in stability seems in its very early stages. nakamura@nagoya-u.jp Math. 17, 1135–1140 (2004), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, III. How to investigate stability of stationary points? © Maplesoft, a division of Waterloo Maple 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. Springer, New York (2011), Li, Y., Shen, Y.: Hyers–Ulam stability of linear differential equations of second order. 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. This means that it is structurally able to provide a unique path to the fixed-point (the “steady- 346, 43–52 (2004), MATH  449-457. 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. J. 2. https://doi.org/10.1007/s40840-014-0053-5. J. Korean Math. When bt = 0, the diﬀerence Inc. 2019. Acad. Anal. Fixed points, attractors and repellers If the sequence has a limit, that limit must be a fixed point of : a value such that . Appl. 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] . : A characterization of Hyers-Ulam stability of first order linear differential operators. A fixed point of is stable if for every > 0 there is > 0 such that whenever , all 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. 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. Correspondence to A dynamical system can be represented by a 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. |. Univ. Abstract: Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. Google Scholar, Miura, T., Jung, S.-M., Takahasi, S.E. 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. The author will further use different fixed-point theorems to consider the stability of SPDEs in … Google Scholar, Cădariu, L., Găvruţa, L., Găvruţa, P.: Fixed points and generalized Hyers–Ulam stability. The ones that are, are attractors . For the simplisity, we consider the follwoing system of autonomous ODE with two variables. Google Scholar, Hyers, D.H., Isac, G., Rassias, T.M. Ber. 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. 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. Let one of them to be . 217, 4141–4146 (2010), Article  38, 855–865 (2015). Transform it into a first order equation $x' = f(x)$ if it's not already 3. Math. Shows how to determine the fixed points and their linear stability of a first-order nonlinear differential equation. 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. Jung, SM., Rezaei, H. A Fixed Point Approach to the Stability of Linear Differential Equations. ); jrwang@gzu.edu.cn (J.W.) Appl. It is different from deterministic impulsive differential equations and also it is different from stochastic differential equations. DIFFERENTIAL EQUATIONS VIA FIXED POINT THEORY AND APPLICATIONS MENG FAN, ZHINAN XIA AND HUAIPING ZHU ABSTRACT. 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 Anal. Let one of them to be . J. 41, 995–1005 (2004), Miura, T., Miyajima, S., Takahasi, S.E. Note that there could be more than one fixed points. Electron. A4-2(780), Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan Malays. 1. 23, 306–309 (2010), Miura, T.: On the Hyers–Ulam stability of a differentiable map. Math. Math. Math. But not all fixed points are easy to attain this way. The results can be generalized to larger systems. Tax calculation will be finalised during checkout. Appl. Bulletin of the Malaysian Mathematical Sciences Society Direction field near the fixed point (, ) is displayed in the right figure. 4, http://jipam.vu.edu.au, Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Bull. © 2020 Springer Nature Switzerland AG. Fixed Point Theory 4, 91–96 (2003), Rus, I.A.