 International Journal of Modern Nonlinear Theory and Application, 2012, 1, 125-129 http://dx.doi.org/10.4236/ijmnta.2012.14019 Published Online December 2012 (http://www.SciRP.org/journal/ijmnta) Stability Behavior of the Zero Solution for Nonlinear Damped Vectorial Second Order Differential Equation Mohamed A. Ramadan1, Samah M. El-Kholy2 1Department of Mathematics, Faculty of Science, Minufiya University, Shebeen El-koom, Egypt 2Department of Engineering Physics and Mathematics, Faculty of Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt Email: mramadan@eun.eg, ramadanmohamed13@yahoo.com, samahelkholy77@ yahoo.com Received October 7, 2012; revised November 9, 2012; accepted November 18, 2012 ABSTRACT In this paper, a theoretical treatment of the stability behavior of the zero solution of nonlinear damped oscillator in the vectorial case is investigated. We study the sufficient conditions for the boundedness of solution of the nonlinear damped vectorial oscillator and the conditions for the stability of the zero solution to be uniformly stable as well as as- ymptotically stable. Keywords: Zero Solution; Damped Oscillator; Uniformly Stable; Asymptotically Stable 1. Introduction We consider the nonlinear second order vectorial differ- ential equation of the form  2,ftt t xxxgx0.R (1) where;  12,,,,:,and, :,,nnnntRxxx RtRRR fttR  xgx Stability problems for the second order ordinary differ- ential equation has been intensively and widely studied [1-5]. Based on Schauder fixed point theorem T. A. Bur- ton and T. Furumochi  introduced a new method to study the stability of the zero solution for Equation (1). This problem is considered also by Gheorghe Morosanu and Cristian Vladimirescu [5,6]. In , they used rela- tively classical arguments to prove the stability of the zero solution of Equation (1). While in , they obtained new stability results for this ordinary differential equa- tion under more general assumptions. Their approach al- lows extensions to both the vector case and the case of the whole real line. In  the dynamics of various oscil- lators had been studied. 2. The Main Results In the next theorem we state sufficient conations for the boundedness of the solution of Equation (1) are given. Theorem 1 If the following hypotheses are hold: 1) 1ftCR and 0, 0ft t. 2) 1,tCR  is decreasing and 1t, tR . 3) nR,nCR Rg and g is locally Lipschit- zian in 1,,nxx, 4) g satisfies the following estimate ,tftogxx, , where tR  denotes some norm in . then the solution of Equation (1) is bounded. nRProof For the n-dimensional system, we have T12 12,,,,,,, mnnxxxyyy Rz, where m = 2n. Applying the transformation, ii iyxftx and 1, 2,,in Equation (1) can be converted into a first order system of differential equations of the form: ,AtBt tzzzrz (2) where 1)  1234AtAtAtAtAt and are m × m matrices.  1000Bt Bt2) 14AtAt ftI, 2AtI, 3AttI , and 2f tft I1Bt0nn are n × n matrices. Note that,  and I are the zero and the identity matrices, respectively. 3)  T1,0 ,,ntgtgtrxx x 10,ntgx which is a 2m1 vector. For ,1,2,,ij n, let is an arbitrary fixed and let 00t Copyright © 2012 SciRes. IJMNTA M. A. RAMADAN, S. M. EL-KHOLY 126   11 012010,,21 0220200,,10 0,, ,,, ,,,,mijin jmnijnin jmmmzttzttz ttZZzttzttzttZtt ZZzttz tt (3) be a fundamental matrix solution to the linear system: Atzz (4) which equal to the identity matrix for . 0Consider with tt00z0z small enough, and let us denote by 00t00,,ttzz0z the unique solution of Equa- tion (2) which equal to  at . By hypotheses (1) 0ttand (2), 0z0,,ttz is defined on a maximal right inter- val, 0,tl, and satisfies the following integral equation:   01000 0000000,,,,,,,, ,,dttttZ ttZ ttZstBsstssts zzzzzrzz This gives us the following integral inequality:  01000 0000000,,,,,e,,,,,dttttZ ttZ ttZstBsstssts zzzzzrzz (5) where . T010eEquations (3) and (4) give us the following differential equations: ,,ijij nijzftzz ,,,, (6)  ,,nijij nijztzftz  (7) ,,injinj ninjzftzz (8)  ,,nin jin jninjztzftz   (9) for . ,1,2,,ij nSince is a decreasing function, so Equations (6) and (7) lead to: t 22 22,, ,,12ij nijij nijtz zfttzz 02d22,, 0ettfuuij nijtz zt (10) By the same way we can obtain from Equations (8) and (9) the following: 02d22,,ettfuuin jnin jtz z (11) For T12 12,,,,,,, mnnxxxyyy Rz, consider the norm 21miizz, where  is the norm defined in . mR For T001 02001020,,,,,,, mnnxxxyyy Rz we have:  ,, ,0,0000,, ,0,0022,0,0,0, 011 1,ijin jijin jnijnin jnijnin jnn nijjinjjnijjnin jjij jZZZZZtt ZZZ Zzx zyzx zy    xyxzxyy satisfies (4), then we get: Using Shwartz inequality, Minkowski inequality  and suitable assumptions lead to:  02d00 00,31ettfu uZttn tzz (12) We have also: 00tst   100T1020 0,,e,, ,,,,mZtt Zsttst tsttst   1000122001,,e,, ,,iiniiniZtt Zstttsttst  Since 2,,mtstt is a decreasing function of , we get: tCopyright © 2012 SciRes. IJMNTA M. A. RAMADAN, S. M. EL-KHOLY 127  22 2 20000,,,,,,,,e S2dtfuuini init tsttsts sstsst   So,   d100 0,, etSfu uZtt Zstent (13) As mentioned the system satisfies the integral inequality (5), then inequalities (12) and (13) give   010000000000,,,,,e,,,,,dttttZ ttZ ttZstBsstssts zzzzzrzz  0dd20000 000,,31eettttSfu utfuuttntn tnfs0t,,,df sstss zz z For all t we can replace g by another function say g defined as follows. By hypothesis (4) it follows that there exists a zzgx (14) 0 such that if x, then ,tftogx x We defined the function by: :nnRRRg ,if,,ifgttgtave xxgx xx and we hx, t0. ry ,xIt is clear that for eveRtR n,tgx ftox is of class and is locally Lipschitzian g in nCR R n12,,,xxxginal function. So, we w g satisfies ill admit from now that the all the properties of the ori g. Since 1fCR we get from inequality (14) that:  000 0 0,,31ettfutttn tD000,, dtst sduzzz ug zz0,ttl , with positive constant D [9,10]. This givess the followin 0000 0,,31 ,,Dhttntett lzzz (15) Thus00,,ttzzas well as 00,,ttzz are bounded on and so 0,tl00,,ttzz can be extended to the right of l. This contradicts the maximality of l. This means that the solution of Equation (1) is bonded. The proof of theorem 1 is complete. Theorem 2 If the hypotheses of theorem 1 are hold andlowing assumption is satisfied: 1) There exist two constants such that: u the fol- ,0hk2,ftf tkftth then the zero solution of Equation (1) is uniformly stable solution. If in addition 2) 0dft t holds, thzero solution en the of Equation (1) is asymptotically stable. Proof n Our stability question is reduced to the stability of the zero solutio0tz will be divided into two intervals. In the to the system (2). The interval of tfirst interval we have 0,tth and in the second interval ,th. lh and si on a maFirstly, with nce the hypotheses of theorem 1 are hold, and then the solution ,tz of Equation 00,tzl right interval (1xima) is defined0,tl. It is proved in theorem 1 that the solution is bounded and 00,,ttzz cantended to the right of l. Therefore, be ex 00,,ttzz exists on 0,tl with lh. Assume hl. We are going to find an estimate for 00,,ttzz on the interval ,hl . From hypothesis (1) of theorem 2, we have for thl : , ,,, dtunk fsstgsszz x dd000 0,,31e,etSStfuufuttnhhtnh zzz z0,h00 dd0, ,thSfu unhnkfs fsz z00 00h00 00e,,,, dtfu ustst szzxz ,3 1e,ttthtn hz zCopyright © 2012 SciRes. IJMNTA M. A. RAMADAN, S. M. EL-KHOLY 128 with 10, nknh (16) 00d000d00,,31e ,,ethtSfuutfuuhttnh htnhnkfs stszzzzzz ,,d(17)  d000d0031e ,,e,thtSfuutfuuhvt hhtnhnkfsstszzzz So ,d 1,vtnh nkftvtthl . By integration we get:   1d00,,e ,,thnhnk fuuttt hthlvvzz (18with )  0031,,vh nhhtzz From inequality (18) we can see that 00,,ttzz is bounded since 00,,ttzz is also bounded, it folls that For owl .0 we can get:  2e13013Dhnh From (15) it follows that 00,,13tt nhzz for all 0,tth provided that 0zore, from (16) and (1t . Theref, we ge8)00,,ttvhzz for all th o, if h, then the solution . S00tstarting 00,,ttzz from any point 0z, with 0z, exists on 0t, and satisfies 00,,ttzz for alusly we btain that l 0tt. If and: 0th, then analogool  01d0e ,tnhnkfuu(19,thTherefore, with the same 00 0,,31ttntzz z) t as before, 0z im-plies again 00,,ttzHence the If inz for all 0tt. zero solution is uniformly stable. addition 0dft t is satisfied, then by (16), (18) and (19) it follows thazero solution to (1) is asymptotically stable. The proof of theorem 2 is complete. 3. Examples We confirm the results of the introduced theorems by considering two numerical examples for which the func-tions t the ft, t and tg satisfy theorems assump- tions.For the system  2,ft t t0 xxxgxwhere , 12,xxx. Example 1 21tft t, 11tt, 12112 2,, 1xxtgtxxt and 2212212 2,, 1xxtgtxx t. (a) (b) Figure 1. Numerical solutions if Example 1, x1 component (a) and x2 component (b). Copyright © 2012 SciRes. IJMNTA M. A. RAMADAN, S. M. EL-KHOLY 129 (a) (b) Figure 2. Numerical solutions if example 2, x1 component (a)and x2 component (b). Example 2 21ft t, , 21ett12112 2,,xxgtxx t and 22122212,,xxt. gtxxWe solved these two examples numerically using th order Runge-Kutta method. The results of example 1are drawn as shown in Figure 1 and the results of exam-own in Figure 2. The curves are rawn for different initial values of d 2 demonstrate time increases all the compents of the solutions tends to zero.means, that te aptotically stable. Wverify the rightness of our proved theorems. 4. Conclusion ifo ly stable as well asREFERENCES nd T. Furumochi, “A Note on Stability by 87.  J. Hale, “Ordina,” Wiley Inter- science, John Woken, 1969. four- ple 2 are drawn as shd0x. that, as theFigures 1 anon This he solutions arsymhich We introduced two theorems which provide the sufficient conditions for the boundedness of solution of the nonlin- ear damped vectorial oscillator and the conditions for the stability of the zero solution to be unrm asymptotically stable. 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