On Stability of Dynamic Equations on Time Scales Via Dichotomic Maps

Transcription

1 Available at Appl. Appl. Math. ISSN: Vol. 7, Issue (December ), pp Applications and Applied Mathematics: An International Journal (AAM) On Stability of Dynamic Equations on Time Scales Via Dichotomic Maps Veysel Fuat Hatipoğlu and Zeynep Fidan Koçak Department of Mathematics Mugla University Mugla, Turkey * Corresponding author Deniz Uçar* Department of Mathematics Usak University Usak, Turkey deniz.ucar@usak.edu.tr Received: May 8, ; Accepted: October 6, Abstract Dichotomic maps are used to check the stability of ordinary differential equations and difference equations. In this paper, this method is extended to dynamic equations on time scales; the stability and asymptotic stability to the trivial solution of the first order system of dynamic equations are examined using dichotomic and strictly dichotomic maps. This method, in dynamic equations, also involves Lyapunov s direct method. Keywords: Dichotomic Maps; Stability; Time Scales MSC No.: 34D, 34N5. Introduction and Preliminaries The extension of Lyapunov s direct method to discrete equations has been given by Carvalho and Ferreira (988). Improvement of this method on ordinary differential equations have been made by Carvalho (99), Carvalho and Cooke (99). Many researches on this subect were done after those papers [Bená and Dos Reis (998), Marconato (5)]. Lyapunov stability 5

2 AAM: Intern. J., Vol. 7, Issue (December ) 5 theory on time scales was studied by Kaymakcalan (99). Hoffacker and Tisdell (5) also studied Lyapunov s direct method for the following dynamical equation. In this study, we examine the stability of the trivial solution x to the first order system of dynamic equations (): x f t, x, t t, n xd, () using dichotomic map, where, for all t, t t, so that x is a solution to equation (). tt, and D is a compact set. We also assume f t, D, Bohner () has defined the time scale as a nonempty closed subset of the real numbers, and has given the definition of - derivative as follows. Definition. n Fix t and let x: x t to be the vector (if it exists) with the property that given any, there is a neighbourhood U of t such that. Define xi t xi s xi t t s t s, for all s U and each i,..., n said that x is delta differentiable at t.. The function x t is the delta derivative of x at t, and it is Definition. A function :, r, increasing on, r with is of class if it is well defined, continuous, and strictly. Guseinov (3) stated the following theorem. Theorem. (Mean Value Theorem). Let f be a continuous function on [a,b], which is -differentiable on ab,. Then, there exist, ab, such that f f b f a b a f.

3 5 Veysel Fuat Hatipoğlu et al. Definition 3. The system () is stable if for any and t, there exists, t such that x implies xt; t, x for all t t. The system () is asymptotically stable; if it is stable and there exists a such that if x then lim xt; t, x. t Hoffacker and Tisdell (5) has stated Theorem. Theorem. If there exists a continuously differentiable positive definite function V in a neighborhood of zero with V x t negative semidefinite, then the trivial solution xt to equation () is stable.. Dichotomic Map on Time Scales If T, T is a given constant and is a neighborhood of the origin having xt, y defined for t t,, t : t, t,, t t and y, then T y : V x T, y V x T, y, T y : V x T, y V x T, y, y: V x., y, y: V x., y, and let T T T Definition 4.. n We say a - differentiable map V : is dichotomic with respect to (), if there are a n constant T and a neighborhood of the origin in T. Definition 5. such that We say that a given map V is strictly dichotomic with respect to (), if it is dichotomic with * T and respect to () and moreover, it satisfies the supplementary condition that T.

4 AAM: Intern. J., Vol. 7, Issue (December ) 53 We see that Lyapunov functions are automatically dichotomic with respect to the given equation. Note that there may exist points y which do not belong to. N Now define, for a given y, and c max V x t, y : T t T,,, t min t: T t T, c V x t, y. where is a class function for all,,, Now, we state the following lemma. Lemma. If V is positive definite and We have that Lemma. c for some, then x t, y for t t. V x t, y and, since V x implies x, the result follows. If V is dichotomic with respect to () and then c c. If t T x t, y is a solution of () defined for t T c V x t y c, then by the definition of c, we have, t T, and note that V xt, y V xt, y for T t T Theorem there is t t, t with,,, gives V xt, y x t, y x t, y as t t xt, y. Since V is dichotomic V xt, y V x t, y c V xt, y V x t, y c.,. Now if. By Mean Value V x t y V x t y V x t y t t, and this, which immediately means that, so that

5 54 Veysel Fuat Hatipoğlu et al. Lemma 3. If V is strictly dichotomic with respect to () and xt, y is a solution of () defined for t T and t T, then c c. Since xt, y implies xt, y, Theorem 3.,, c V x t y V x t y c. If V is strictly dichotomic with respect to () and xt, y is a solution of () defined for t T and c, then c c. Under the above hypothesis, we know from Lemma that c c c. Furthermore, from, then c c so that c c t T, we have either c c or c c. In the first case we again have c c c. In the case that c c, one notes that T and c, and sees that if one had Lemma 3, we know that if t T t T, then one could not have V xt y,. In fact, for t T, one could not have V xt, y contradict with t T Also, one could not have V xt y contradict with t T. Then we conclude that t T c c c. Theorem 4.. In the case, since that would immediately,, since that would when c c, so that If V is a positive definite map that is dichotomic with respect to equation (), then the trivial solution to system () is stable. Take, and take any r with sup, :, Set, and note that. If y R and V y Considering the continuity of V and the map t, y xt, y, we can take d, r R with y R inf V y: r y R x t y y r t T R., then y r. such that

6 AAM: Intern. J., Vol. 7, Issue (December ) 55 for t T and y d x t, y r. Now take, one has V xt, y, xt, y R. For such t, y one has, and note that if xt, y r for T t T, then xt, y Tt T, and since c y, we have c y. Thus V xt, y Tt T, and we find that xt, y r for T t T we see that xt, y r for all t. argument on the index Theorem 5. R for for. From this inductive If V is a positive definite map that is strictly dichotomic with respect to equation (), then the trivial solution to system () is asymptotically stable. Let V be a strictly dichotomic and take y having y d d, r. By Theorem 4 we already know that xt, y is stable and by Lemma 3 we have c c. Thus we only need to show that c as. Suppose there exists a positive constant c c such that c c as. Since c d and the only point y with y d, where and T vanish simultaneously, is y, the standing hypothesis guarantee the existence of a positive constant hc such that either hc or T hc for all y, c y d. As a consequence we must have c c h c,,,, with which leads us to conclude that c, a contradiction. So, c and the theorem is proved. It is clear from the Theorem 4 and Theorem 5 that we can obtain stability results for dynamic equations in the spirit of Lyapunov s direct method even when V is not negative semi-definite. Example. Consider the dynamic equation x p x. () Rewrite equation () as, x py

7 56 Veysel Fuat Hatipoğlu et al. y px with x x and y y. Take V x, y x y Calculating V by using product rule we get V t, x, yx y x t x t x t x t y t y t y t y t x t x t t x t y t y t t y t pxy t p y pxy t p x pxy p t x y. pyx tpy pxy t px Since t, V t, x, y. Then V is positive definite. is not negative semi definite, so V is not a Lyapunov function for this equation. The solution of the equation () is cos, sin, x t x t t y t t p p sin, cos, y t x t t y t t. p p If we choose T a on time scale with period a (i.e. let T, is periodic with period T provided t implies t T V x a, y a V x, y. then a time scale ) then we obtain, So we have for B, for instance, xy, : pxy a B. So that a, and from Theorem 4, we can say that given equilibrium is stable.

8 AAM: Intern. J., Vol. 7, Issue (December ) Conclusion Stability and instability for dynamic equations on time scales has been studied using Lyapunov functions. In addition to previous studies, using dichotomic and strictly dichotomic maps, we obtain stability results for dynamic equations on time scales in the spirit of Lyapunov s direct method even when V is not negative semi-definite. Investigating the stability of dynamic equations by using dichotomic maps is more useful for the choice of Lyapunov function V. Acknowledgement The authors would like to thank the referees for their encouraging attitude and valuable suggestions in the review process of the work. REFERENCES Bená, M.A. and Dos Reis, J.G. (998). Some results on stability of retarded functional differential equations using dichotomic map techniques, Positivity,, N. 3, pp Bohner, M. and Peterson, A. (). Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston. Bohner, M. and Martynyuk, A.A. (7). Elements of Lyapunov stability theory for dynamic equations on time scale, International Applied Mechanics, Vol. 43, No. 9, pp Carvalho, L.A.V. (99). On the Stability of Discrete Equations and Ordinary Differential Equations, In Delay Differential Equations and Dynamical Systems Lecture Notes in Mathematics, (S. Busenberg and M. Martelli, editors), Vol. 475, pp , Springer- Verlag, New York. Carvalho, L.A.V. and Ferreira, R. R. (988). On a new extension of Lyapunov's direct method to discrete equations, Quart. Appl. Math., 66, pp Carvalho, L.A.V. and Cooke, K.L. (99). On dichotomic maps for a class of differentialdifference equations, Proc. Royal Soc. Edinburgh, 7A, pp Carvalho, L.A.V. and Marconato, S.A.S. (997). On dichotomic maps for differential equations with piecewise continuous argument, Communications in Applied Analysis,, N., pp. 3-. Guseinov, G. Sh. (3). Integration on time scales, Journal of Mathematical Analysis and Applications, 85, pp Hoffacker, J. and Tisdell, C. C. (5). Stability and instability for dynamic equations on time scales, Computers and Mathematics with Applications, 49, pp Kaymakcalan, B. (99). Lyapunov stability theory for dynamic systems on time scales, J. Appl. Math. Stochastic. Anal., 5(5), pp LaSalle, J.P. (976). Stability of Dynamical Systems, SIAM, Philadelphia. Marconato, S. A. S. (5). The relationship between differential equations with piecewise constant argument and the associated discrete equations via dichotomic maps, Dynamics of Continuous Discrete and Impulsive Systems,, pp