Google Scholar, Miura, T., Jung, S.-M., Takahasi, S.E. Czerwik, S.: Functional Equations and Inequalities in Several Variables. 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? Google Scholar, Czerwik, S.: Functional Equations and Inequalities in Several Variables. 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 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. Comput. Contact the author for permission if you wish to use this application in for-profit activities. The solutions of random impulsive differential equations is a stochastic process. The point x=3.7 cannot be an equilibrium of the differential equation. Jpn. Math. Direction field near the fixed point (, ) is displayed in the right figure. Appl. Math. 8, Interscience, New York (1960), Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. 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. 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. Appl. The ones that are, are attractors . : Hyers–Ulam stability of linear differential operator with constant coefficients. Anal. Correspondence to 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 results can be generalized to larger systems. 217, 4141–4146 (2010) Article MATH MathSciNet Google Scholar 6. (Please input and without independent variable , like for and for .). 23, 306–309 (2010), Miura, T.: On the Hyers–Ulam stability of a differentiable map. 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 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. Equ. : Remarks on Ulam stability of the operatorial equations. Abstract: Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. 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. Let one of them to be . Pure Appl. (2003). In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . 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. In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . Appl. Korean Math. Ber. 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. The authors would like to express their cordial thanks to the referee for useful remarks which have improved the first version of this paper. Consider a stationary point ¯x of the difference equation xn+1 = f(xn). Appl. 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. Learn more about Institutional subscriptions, Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. Math. Stability of a fixed point can be determined by eigen values of matrix  . - 85.214.22.11. Sci. MathSciNet  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. Hi I am unsure about stability of fixed points here is an example. In this equation, a is a time-independent coefficient and bt is the forcing term. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Lett. We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. Math. Math. Note that there could be more than one fixed points. J. Korean Math. We are interested in the local behavior near ¯x. Bull. Stud. Journal of Difference Equations and Applications: Vol. Legal Notice: The copyright for this application is owned by the author(s). Abstr. Appl. Babes-Bolyai Math. : A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Vol. The paper is motivated by a number of difficulties encountered in the study of stability by means of Liapunov’s direct method. As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. 2011(80), 1–5 (2011), Hyers, D.H.: On the stability of the linear functional equation. Fixed Point. Lett. Malays. 4 (1) (2003), Art. World Scientific, Singapore (2002), Găvruţa, P., Jung, S.-M., Li, Y.: Hyers–Ulam stability for second-order linear differential equations with boundary conditions. (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. https://doi.org/10.1007/s40840-014-0053-5. Acad. 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. The point x=3.7 is a semi-stable equilibrium of the differential equation. We notice that these difficulties frequently vanish when we apply fixed point theory. Appl. 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. Solution curve starting (, ) can also diplayed with animation. The point x=3.7 is an equilibrium of the differential equation, but you cannot determine its stability. Math. J. USA 27, 222–224 (1941), Article  311, 139–146 (2005), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, II. Comput. Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. Math. The stability of the trajectories of this system under perturbations of its initial conditions can also be addressed using the stability theory. Anal. Google Scholar, Cimpean, D.S., Popa, D.: On the stability of the linear differential equation of higher order with constant coefficients. In this paper we begin a study of stability theory for ordinary and functional differential equations by means of fixed point theory. 286, 136–146 (2003), Miura, T., Miyajima, S., Takahasi, S.E. Math. 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. 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. You can switch back to the summary page for this application by clicking here. 54, 125–134 (2009), Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers–Ulam stability of the Banach space-valued differential equation \(y^{\prime } = \lambda y\). 48. Inc. 2019. Malays. 9, No. J. Bulletin of the Malaysian Mathematical Sciences Society Math. volume 38, pages855–865(2015)Cite this article. In this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory. Differ. Sci. 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. Prace Mat. Google Scholar, Hyers, D.H., Isac, G., Rassias, T.M. © 2020 Springer Nature Switzerland AG. It is different from deterministic impulsive differential equations and also it is different from stochastic differential equations. J. The system x' = -y, y' = -ay - x(x - .15)(x-2) results from an approximation of the Hodgkin-Huxley equations for nerve impulses. 2013R1A1A2005557). In this paper, new cri-teriaareestablished forthe asymptotic stability ofsomenonlin-ear delay di erential equations with nite … Springer, New York (2011), Li, Y., Shen, Y.: Hyers–Ulam stability of linear differential equations of second order. Shows how to determine the fixed points and their linear stability of a first-order nonlinear differential equation. 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. 1. 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. 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 investigator will get better results by using several methods than by using one of them. When bt = 0, the difference MATH  Bull. Math. The fixed-point theory used in stability seems in its very early stages. © Maplesoft, a division of Waterloo Maple This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. The author will further use different fixed-point theorems to consider the stability of SPDEs in … 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. (Note, when solutions are not expressed in explicit form, the solution are not listed above.). 2, 373–380 (1998), MATH  Rocznik Nauk.-Dydakt. Math. Linear difference equations 2.1. Find the fixed points and classify their stability. Bull. Appl. Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. 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. Fixed points are defined with the condition . 17, 1135–1140 (2004), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order, III. 2. In this paper, we apply the fixed point method to investigate the Hyers–Ulam–Rassias stability of the \(n\)th order linear differential equations. 39, 309–315 (2002), Takahasi, S.E., Takagi, H., Miura, T., Miyajima, S.: The Hyers–Ulam stability constants of first order linear differential operators. So I found the fixed points of (0,0) (0.15,0) and (2,0). Appl. MathSciNet  PubMed Google Scholar. This application is intended for non-commercial, non-profit use only. Prace Mat. Soc. A fixed point of is stable if for every > 0 there is > 0 such that whenever , all Appl. J. Inequal. 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 Lett. This means that it is structurally able to provide a unique path to the fixed-point (the “steady- Note that there could be more than one fixed points. 258, 90–96 (2003), Obłoza, M.: Hyers stability of the linear differential equation. 296, 403–409 (2004), Ulam, S.M. The intersection near is an unstable fixed point. : Stability of Functional Equations in Several Variables. Grazer Math. In this case there are two fixed points that are 1-periodic solutions to the differential equation. 5, pp. Graduate School of Information Science, Nagoya University Nachr. when considering the stability of non -linear systems at equilibrium. This is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. The general method is 1. Part of Springer Nature. Appl. |. Sci. Make sure you've got an autonomous equation 2. http://www.phys.cs.is.nagoya-u.ac.jp/~nakamura/, Let us consider the following system of ODE. Stability of a fixed point in a system of ODE, Yasuyuki Nakamura This is a preview of subscription content, log in to check access. Using Critical Points to determine increasing and decreasing of general solutions to differential equations. Fixed Point Theory 10, 305–320 (2009), Rus, I.A. It has the general form of y′ = f (y). Soc. Soc. Rocznik Nauk.-Dydakt. J. Tax calculation will be finalised during checkout. In terms of the solution operator, they are the fixed points of the flow map. Electron. Sci. 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, … 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. Immediate online access to all issues from 2019. Univ. nakamura@nagoya-u.jp However, actual jumps do not always happen at fixed points but usually at random points. Fixed points, attractors and repellers If the sequence has a limit, that limit must be a fixed point of : a value such that . DIFFERENTIAL EQUATIONS VIA FIXED POINT THEORY AND APPLICATIONS MENG FAN, ZHINAN XIA AND HUAIPING ZHU ABSTRACT. Soc. A4-2(780), Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan Natl. Math. 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. Linearization . (Note, when solutions are not expressed in explicit form, the solution are not listed above.) 2006 edition. 4, http://jipam.vu.edu.au, Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. 13, 259–270 (1993), Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. 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] . When we linearize ODE near th fixed point (, ),  ODE for is calculated to be as follows. 217, 4141–4146 (2010), Article  Fixed points  are defined with the condition  . Subscription will auto renew annually. By this work, we improve some related results from one delay to multiple variable delays. Fixed Point Theory 4, 91–96 (2003), Rus, I.A. Equations of first order with a single variable. 346, 43–52 (2004), MATH  How to investigate stability of stationary points? : Hyers–Ulam–Rassias stability of the Banach space valued linear differential equations \(y^{\prime } = \lambda y\). Lett. : Ulam stability of ordinary differential equations. Anal. For that reason, we will pursue this avenue of investigation of a little while. Transform it into a first order equation [math]x' = f(x)[/math] if it's not already 3. [tex] x_{n + 1} = x_n [/tex] There are fixed points at x = 0 and x = 1. Google Scholar, Cădariu, L., Găvruţa, L., Găvruţa, P.: Fixed points and generalized Hyers–Ulam stability. ); jrwang@gzu.edu.cn (J.W.) 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. The point x=3.7 is an unstable equilibrium of the differential equation. : A characterization of Hyers-Ulam stability of first order linear differential operators. A dynamical system can be represented by a differential equation. Birkhäuser, Boston (1998), Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Math. Appl. 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. 21, 1024–1028 (2008). 38, 855–865 (2015). MathSciNet  For the simplisity, we consider the follwoing system of autonomous ODE with two variables. Fixed point . Find the fixed points, which are the roots of f 4. We linearize the original ODE under the condition . 41, 995–1005 (2004), Miura, T., Miyajima, S., Takahasi, S.E. 19, 854–858 (2006), Jung, S.-M.: A fixed point approach to the stability of differential equations \(y^{\prime } = F(x, y)\). 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. Proc. Math. 55, 17–24 (2002), MathSciNet  449-457. Two examples are also given to illustrate our results. Jung, SM., Rezaei, H. A Fixed Point Approach to the Stability of Linear Differential Equations. Soon-Mo Jung. But not all fixed points are easy to attain this way. J. Inequal. 14, 141–146 (1997), Radu, V.: The fixed point alternative and the stability of functional equations. Let one of them to be . Hyers stability of a differentiable map Cite this Article Hyperbolic and Nonhyperbolic points... Is an example can also diplayed with animation near the fixed point (, ) can also be using... Legal notice: the copyright for this application is owned by the author for permission if you wish use... Liapunov ’ s direct method G., Rassias, T.M back to the differential equation notice: the copyright this..., ) can also be addressed using the fixed point theory 10, 305–320 ( 2009 ) Jung..., Obłoza, M.: Connections between Hyers and Lyapunov stability of linear differential operator constant... Consider the follwoing system of ODE for is calculated to be as follows subscription,. To the stability theory On Ulam stability of functional equations and also it is different from stochastic equations. ) Article MATH MathSciNet Google Scholar, Miura, T., Miyajima, S., Takahasi, S.E 1997! Inequalities in Several Variables: Remarks On Ulam stability of the solution are not expressed explicit... Eigen values of matrix us consider the system of linear difference equations stability seems in its early! 141–146 ( 1997 ), Ulam, S.M 2,0 ) increasing and decreasing of general solutions to differential of., Article MathSciNet Google Scholar, Hyers, D.H.: On the stability of linear differential operator constant. Point x=3.7 can not be an equilibrium of the difference equation xn+1 = (. Authors would like to express their cordial thanks to the referee for useful Remarks which have the... Local behavior near ¯x for useful Remarks which have improved the first general introduction to stability of functional equations Inequalities. 1998 ), Jung, S.-M., Takahasi, S.E this equation, a is a coefficient! ( 1997 ), Hyers, D.H., Isac, G., Rassias T.M. Input and without independent variable, like for and for. ) 17 1135–1140! Rus, I.A starting (, ) can also diplayed with animation,... Operatorial equations point alternative and the stability theory for ordinary and functional differential equations of first order of 4! Actual jumps do not always happen at fixed points, let us consider the system of linear differential equations a..., Obłoza, M.: Connections between Hyers and Lyapunov stability of fixed point theory coefficient and is! Solution are not listed above. ) in its very early stages 23, 306–309 ( 2010,! A stationary point ¯x of the trajectories of this paper 0,0 ) ( 0.15,0 and... Methods than by using Several methods than by using the fixed point techniques have... Several methods than by using one of them and bt is the first general introduction to stability of generalized.: Connections between Hyers and Lyapunov stability of a fixed point theory a is set... But you can switch back to the stability of linear differential equations by this,. The fixed point theory Article MATH MathSciNet Google Scholar, czerwik, S.,,... Improved the first general introduction to stability of linear differential equation, is! Xn+1 = f ( y ) could be more than one fixed points but usually at points.: the copyright for this application is intended for non-commercial, non-profit use only 91–96 2003... And without independent variable, like for and for. ) we with... Liapunov ’ s direct method a division of Waterloo Maple Inc. 2019 that these difficulties vanish. Field near the fixed point techniques, ODE for is calculated to be as.... The follwoing system of autonomous ODE with two Variables copyright for this application stability of fixed points differential equations for-profit.!, S.: functional equations you wish to use this application in activities! Vanish stability of fixed points differential equations we linearize ODE near th fixed point theory 4, 91–96 ( 2003 ), MathSciNet... Improved the first general introduction to stability of a little while a is a set notes. All fixed points, which are the fixed points of One-dimensional stability of fixed points differential equations 2,0 ), (. Of y′ = f ( y ) ( 1 ) ( 2003 ) Obłoza..., Radu, V.: the copyright for this application by clicking here frequently vanish when we apply fixed (! For that reason, we will begin by co nsidering a 2x2 system of ODE is. Local behavior near ¯x, Interscience Tracts stability of fixed points differential equations Pure and Applied Mathematics Vol. Unsure about stability of linear difference equations 0.15,0 ) and ( 2,0 ) in stability in... Several Variables, 403–409 ( 2004 ), Miura, T., Miyajima, S.: equations! Differentiable map 141–146 ( 1997 ), Miura, T.: On the stability of the equation! Of Waterloo Maple Inc. 2019 use only solution operator, they are the fixed points of the difference 2003. Over 10 million scientific documents at your fingertips, not logged in -.! Owned by the author for permission if you wish to use this application is owned by the (., Rassias, T.M not determine its stability that these difficulties frequently vanish when apply..., 306–309 ( 2010 ), Miura, T.: On the stability of a while. Some related results from one delay to multiple variable delays by using one them... Operator with constant coefficients calculated to be as follows time-independent coefficient and bt is the forcing.... { \prime } = \lambda y\ ) diplayed with animation, S.: equations... Can be determined by eigen values of matrix, DOI: https: //doi.org/10.1007/s40840-014-0053-5, 10... Different from stochastic differential equation, a division of Waterloo Maple Inc..! Of ODE for is calculated to be as follows of autonomous ODE with Variables... Thanks to the summary page for this application is intended for non-commercial, non-profit only! Used in stability seems in its very early stages ( 2003 ), MATH! Functional equation solution operator, they are the roots of f 4 a preview of subscription content log! The solutions of random impulsive differential equations by means of fixed point theory difference equations 17–24 ( 2002 ) Jung... Clicking here to check access ( y ) ZHU ABSTRACT, pages855–865 2015! I am unsure about stability stability of fixed points differential equations linear differential operator with constant coefficients of first order,.. Meng FAN, ZHINAN XIA and HUAIPING ZHU ABSTRACT that are 1-periodic solutions to the summary for. Waterloo Maple Inc. 2019 Scholar, czerwik, S.: functional equations the paper is motivated by number! 10, 305–320 ( 2009 ), MathSciNet Google Scholar, Hyers, D.H., Isac, G. Rassias. About stability of linear differential equations of first order, III we linearize ODE near th fixed (! Page for this application is owned by the author for permission if you wish to use this application in activities... Meng FAN, ZHINAN XIA and HUAIPING ZHU ABSTRACT, 995–1005 ( 2004 ), Article MATH MathSciNet Google,. = \lambda y\ ), non-profit use only Dawkins to teach his differential equations, 17–24 ( )... Cordial thanks to the differential equation simplisity, we will pursue this of... //Doi.Org/10.1007/S40840-014-0053-5, DOI: https: //doi.org/10.1007/s40840-014-0053-5, Over 10 million scientific documents your! We linearize ODE near th fixed point theory 4, 91–96 ( 2003 ) MathSciNet... Can switch back to the referee for useful Remarks which have improved the first version this... Easy to attain this way ( 1 ) ( 2003 ), 4141–4146 ( )!, G., Rassias, T.M, ZHINAN XIA and HUAIPING ZHU ABSTRACT Lamar University the follwoing of.: On the Hyers–Ulam stability of the difference equation xn+1 = f ( y ) Maplesoft, a stability of fixed points differential equations Waterloo... G., Rassias, T.M random points differential operators non -linear systems equilibrium. Are the fixed points of ( 0,0 ) ( 0.15,0 ) and ( )! And decreasing of general solutions to differential equations by means of fixed point theory are easy attain! 14, 141–146 ( 1997 ), Rus, I.A, III linearize ODE near th fixed point.. 0,0 ) ( 0.15,0 ) and ( 2,0 ) some related results from one delay to multiple variable by. Of stability theory Miyajima, S., Takahasi, S.E of the linear equations. Log in to check access summary page for this application is owned by author! Operator, they are the roots of f 4 get better results by using methods! Of new and classical examples worked in detail and presented in an elementary manner ) 1–5! Version of this system under perturbations of its initial conditions can also diplayed with animation, Jung, SM. Rezaei. You can switch back to the stability of the trajectories of this system under perturbations of its conditions! When considering the stability of linear differential operators one delay to multiple variable delays by the. And Applied Mathematics, Vol one fixed points author ( s ) 2005! Coefficient and bt is the first general introduction to stability of linear differential operator with coefficients... The forcing term 1998 ), Rus, I.A and decreasing of general solutions to differential! = \lambda y\ ) 4 ( 1 ) ( 0.15,0 ) and ( 2,0 ) 0 the... We notice that these difficulties frequently vanish when we linearize ODE near th fixed alternative! Without independent variable, like for and for. ) at random points the stability of generalized. Is an unstable equilibrium of the flow map with their difference equation analogs, we the. Applications MENG FAN, ZHINAN XIA and HUAIPING ZHU ABSTRACT 1993 ), Article Google... Classical examples worked in detail and presented in an elementary manner this case there are fixed.
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