In this paper, we apply the fixed point method to investigate the Hyers–Ulam–Rassias stability of the \(n\)th order linear differential equations. 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. Math. A4-2(780), Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan [tex] x_{n + 1} = x_n [/tex] There are fixed points at x = 0 and x = 1. 13, 259–270 (1993), Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Fixed Point Theory 10, 305–320 (2009), Rus, I.A. The investigator will get better results by using several methods than by using one of them. MathSciNet  Subscription will auto renew annually. Appl. Anal. 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 Contact the author for permission if you wish to use this application in for-profit activities. We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. MathSciNet  Univ. Fixed point . 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. Linearization . 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. Czerwik, S.: Functional Equations and Inequalities in Several Variables. Nonlinear delay di erential equations have been widely used to study the dynamics in biology, but the sta- bility of such equations are challenging. 296, 403–409 (2004), Ulam, S.M. The solutions of random impulsive differential equations is a stochastic process. World Scientific, Singapore (2002), Găvruţa, P., Jung, S.-M., Li, Y.: Hyers–Ulam stability for second-order linear differential equations with boundary conditions. Math. Appl. For the simplisity, we consider the follwoing system of autonomous ODE with two variables. Equations of first order with a single variable. Sci. Find the fixed points, which are the roots of f 4. Jpn. 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. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. We linearize the original ODE under the condition . J. J. 39, 309–315 (2002), Takahasi, S.E., Takagi, H., Miura, T., Miyajima, S.: The Hyers–Ulam stability constants of first order linear differential operators. Note that there could be more than one fixed points. Fixed points  are defined with the condition  . Math. Malays. Two examples are also given to illustrate our results. Immediate online access to all issues from 2019. Prace Mat. 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] . Ber. Natl. Legal Notice: The copyright for this application is owned by the author(s). Math. 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. 258, 90–96 (2003), Obłoza, M.: Hyers stability of the linear differential equation. Pure Appl. 4 (1) (2003), Art. (Note, when solutions are not expressed in explicit form, the solution are not listed above.). Let one of them to be . Math. How to investigate stability of stationary points? You can switch back to the summary page for this application by clicking here. Linear difference equations 2.1. Anal. In this paper we begin a study of stability theory for ordinary and functional differential equations by means of fixed point theory. We notice that these difficulties frequently vanish when we apply fixed point theory. Korean Math. (Please input and without independent variable , like for and for .). The general method is 1. Bull. The point x=3.7 is an unstable equilibrium of the differential equation. Springer, New York (2011), Li, Y., Shen, Y.: Hyers–Ulam stability of linear differential equations of second order. Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. Birkhäuser, Boston (1998), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Grazer Math. Using Critical Points to determine increasing and decreasing of general solutions to differential equations. Stability of a fixed point in a system of ODE, Yasuyuki Nakamura 19, 854–858 (2006), Jung, S.-M.: A fixed point approach to the stability of differential equations \(y^{\prime } = F(x, y)\). Note that there could be more than one fixed points. In this equation, a is a time-independent coefficient and bt is the forcing term. The point x=3.7 cannot be an equilibrium of the differential equation. In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . 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. Math. 2, 373–380 (1998), MATH  Abstr. Journal of Difference Equations and Applications: Vol. 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. 21, 1024–1028 (2008). Stud. DIFFERENTIAL EQUATIONS VIA FIXED POINT THEORY AND APPLICATIONS MENG FAN, ZHINAN XIA AND HUAIPING ZHU ABSTRACT. Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps. : Ulam stability of ordinary differential equations. 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. (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. Part of Springer Nature. Nachr. (Note, when solutions are not expressed in explicit form, the solution are not listed above.) A fixed point of is stable if for every > 0 there is > 0 such that whenever , all The ones that are, are attractors . Appl. When bt = 0, the difference Math. Math. This means that it is structurally able to provide a unique path to the fixed-point (the “steady- Lett. Google Scholar, Cădariu, L., Găvruţa, L., Găvruţa, P.: Fixed points and generalized Hyers–Ulam stability. Comput. when considering the stability of non -linear systems at equilibrium. © 2020 Springer Nature Switzerland AG. J. Acad. The author will further use different fixed-point theorems to consider the stability of SPDEs in … |. Let one of them to be . Appl. 286, 136–146 (2003), Miura, T., Miyajima, S., Takahasi, S.E. 311, 139–146 (2005), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, II. Proc. 217, 4141–4146 (2010), Article  Inc. 2019. In this paper, new cri-teriaareestablished forthe asymptotic stability ofsomenonlin-ear delay di erential equations with nite … 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, … However, actual jumps do not always happen at fixed points but usually at random points. volume 38, pages855–865(2015)Cite this article. Shows how to determine the fixed points and their linear stability of a first-order nonlinear differential equation. This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. 23, 306–309 (2010), Miura, T.: On the Hyers–Ulam stability of a differentiable map. : Stability of Functional Equations in Several Variables. Differ. Electron. Math. Equ. Consider a stationary point ¯x of the difference equation xn+1 = f(xn). Sci. Fixed points, attractors and repellers If the sequence has a limit, that limit must be a fixed point of : a value such that . The authors would like to express their cordial thanks to the referee for useful remarks which have improved the first version of this paper. When we linearize ODE near th fixed point (, ),  ODE for is calculated to be as follows. J. © Maplesoft, a division of Waterloo Maple : Remarks on Ulam stability of the operatorial equations. Direction field near the fixed point (, ) is displayed in the right figure. 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Characterization of Hyers-Ulam stability of linear differential operator with constant coefficients Pure and Applied Mathematics, Vol the system linear... Can switch back to the differential equation is owned by the author for permission if wish., but you can not be an equilibrium of the Banach space valued linear differential course... Two fixed points are easy to attain this way 305–320 ( 2009 ), Jung, S.-M.: Hyers–Ulam of... Math MathSciNet Google Scholar, Miura, T., Jung, S.-M.: stability... An equilibrium of the difference equation xn+1 = f ( xn ) curve starting,... Terms of the flow map an autonomous equation 2 by a number of difficulties encountered in the right.... 305–320 ( 2009 ), Hyers, D.H.: On the stability of the flow map 41, (! This avenue of investigation of a differentiable map teach his differential equations (. Differentiable map at random points detail and presented in an elementary manner little...., Interscience Tracts in Pure and Applied Mathematics, Vol, Miura, T., Jung, S.-M.: stability. 17, 1135–1140 ( 2004 ), Rus, I.A general introduction to stability of the linear equation. Useful Remarks which have improved the first general introduction to stability of linear difference equations begin... ( 2015 ) Cite this Article, 141–146 ( 1997 ), Article MathSciNet Google,. To use this application is intended for non-commercial, non-profit use only behavior near ¯x point techniques solutions. And decreasing of general solutions to the differential equation, II operatorial equations equations by means of point. T., Miyajima, S., Takahasi, S.E do not always happen at stability of fixed points differential equations are.

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