JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 103260 10.1155/2013/103260 103260 Research Article On the Global Stability Properties and Boundedness Results of Solutions of Third-Order Nonlinear Differential Equations Ateş Muzaffer Pomares Hector Department of Electrical and Electronics Engineering Yüzüncü Yıl University 65080 Van Turkey yyu.edu.tr 2013 14 11 2013 2013 07 06 2013 12 09 2013 28 09 2013 2013 Copyright © 2013 Muzaffer Ateş. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We studied the global stability and boundedness results of third-order nonlinear differential equations of the form x+ψ(x,x˙,x¨)x¨+f(x,x˙,x¨)=P(t,x,x˙,x¨). Particular cases of this equation have been studied by many authors over years. However, this particular form is a generalization of the earlier ones. A Lyapunov function was used for the proofs of the two main theorems: one with P0 and the other with P0. The results in this paper generalize those of other authors who have studied particular cases of the differential equations. Finally, a concrete example is given to check our results.

1. Introduction

We will consider here the equations of the form (1)x+ψ(x,x˙,x¨)x¨+f(x,x˙,x¨)=P(t,x,x˙,x¨). Now, (1) has an equivalent system (2)x˙=y,y˙=z,z˙=-ψ(x,y,z)z-f(x,y,z)+p(t,x,y,z), where ψC(R×R×R,R), fC(R×R×R,R), and PC([0,)×R×R×R,R). We also assume that the real functions ψ, f, and P depend only on the arguments displayed explicitly. The dots denote differentiation with respect to t.

Global stabilities of some special cases of (1) have been studied by a number of authors.

In 1953, Šimanov  investigated the global stability of the zero solution of the equation(3a)x+ψ(x,x˙)x¨+bx˙+cx=0, where b and c are constants.

Later, Ezeilo  and Ogurtsov  discussed the global stability of the zero solution of the equation of the form (3b)x+ψ(x,x˙)x¨+ϕ(x˙)+g(x)=0. Then, Goldwyn and Narendra  studied on the same subject for the following differential equation: (3c)x+h(x˙)x¨+μ(x˙)x˙+k(x)x=0. Recently, Qian  and Omeike  have discussed the global stability, and in a recent paper, Tunç  has investigated the boundedness of solutions of the following differential equations: (3d)x+ψ(x,x˙)x¨+f(x,x˙)=p(t,x,x˙,x¨). Plus, Tunç  and Omeike [9, 10] have studies on the global stability of solutions of the differential equation of the form (3e)x+ψ(x,x˙,x¨)x¨+f(x,x˙)=0. Moreover, Tunç and Omeike have studies on the asymptotic behavior of the following differential equations: (3f)x+ψ(x,x˙,x¨)x¨+f(x,x˙)=P(t,x,x˙,x¨),(3g)x+ψ(x,x˙)x¨+f(x,x˙)=P(t),respectively.

Motivation of this study has been based on recent studies of Qian , Tunç [7, 8, 11], and Omeike [6, 9, 10]. Equation (1) is a quite general third-order nonlinear differential equation. Equations (3a), (3b), (3c), (3d), (3e), (3f), and (3g) are some special cases of (1), and our study is reducible to the studies in , but the inversions are not possible. Thus, the studies which have been done in  are some special cases of our study. Hence, our results extend and include those results obtained in .

2. Preliminaries

Before introducing our main results, we state some basic theorems and a Lyapunov function which will be required in future. Consider the autonomous system (4)x˙=f(x), where f:ΩRn is continuous, with Ω being an open set in Rn containing the origin. Let f(0)=0 and f(x)0 for x0.

Theorem 1.

Suppose that there exists a scalar function V(x) such that

V(x) is positive definite on Rn, and V(x) as x.

V˙(x)0 on Rn.

Then, all solutions of (4) are bounded as t (i.e., (4) is Lagrange stable).

Proof.

See .

Theorem 2.

In addition to the assumptions of Theorem 1, if the origin is the only invariant subset of E, then the zero solution of (4) is globally asymptotically stable.

Proof.

E is the set where V˙=0, and assume that (0,0) is to be the only invariant subset of E; then (0,0) solution of (4) is globally asymptotically stable.

Theorem 3.

In Theorem 1, if (ii) is replaced by the condition that V˙(x) is negative definite at all points xRn, then the zero solution of (4) is globally asymptotically stable.

Proof.

See .

It is well known that the stability is a very important problem in the theory and applications of differential equations. So far, the most effective method to study the stability of nonlinear differential equations is still Lyapunov’s second method. The major advantage of this method is that stability in large can be obtained without any prior knowledge of solutions. Today, this method is widely recognized as an excellent tool not only in the study of differential equations but also in the theory of control systems, dynamical systems, systems with time lag, power system analysis, time-varying nonlinear feedback systems, and so on. Its chief characteristic is the construction of a scalar function, namely, the Lyapunov function. Unfortunately, it is sometimes very difficult to find a proper Lyapunov function for a given system. Therefore, in this work, we construct a suitable Lyapunov function which is an excellent tool in the proof of the main theorems. Here, this function, V=V(t)=V(x,y,z), is defined by (5)V(x,y,z)=0xf(u,0,0)du+1a0yf(x,v,0)dv+0yψ(x,v,0)vdv+12az2+yz. Rewrite the function V(x,y,z) as follows: (6)V(x,y,z)=12a(ay+z)2+12ab[f(x,0,0)+by]2+0y[ψ(x,v,0)-a]vdv+1a0y[fv(x,θ1v,0)-b]vdv+0x[1-1abfu(u,0,0)]f(u,0,0)du, where (7)fv(x,θ1v,0)=f(x,v,0)-f(x,0,0)v,(v0,0θ11).

3. Main Result

In the case P0, we have the following.

Theorem 4.

Let δ0, a, b, and c be positive constants such that δ0 is sufficiently small, and ab>c, and assume that the following conditions are satisfied:

f(x,0,0)/xδ0  (x0),

fx(x,0,0)c,

ψ(x,y,z)a for all x, y, and z,

fy(x,y,0)b for all x, y, and z,

yψz(x,y,z)0, fz(x,y,z)0 for all x, y and z,

a[f(x,y,z)-f(x,0,0)-0yψx(x,v,0)vdv]yy0yfx(x,v,0)dv.

Then, the zero solution of (1) is globally asymptotically stable.

Proof.

From conditions (i)–(iv) of Theorem 4, we obtain (8)V(x,y,z)=12a(ay+z)2+12ab[f(x,0,0)+by]2+12δ1x2, where δ1=(1/ab)(ab-c)δ0>0. It follows that there exists a constant D0>0 small enough that (9)V(x,y,z)D0(x2+y2+z2). Hence, V is a positive definite function (see Global Asymptotic Stability on page 223, Theorem 5.2.12 that of Rao ).

Now, we show that the derivative of V with respect to t along the solution path of system (2) is negative semidefinite.

Let (10)V˙=-U, where (11)U=[ψ(x,y,z)-ψ(x,y,0)zy0+1af(x,y,z)-f(x,y,0)z]z2+[1aψ(x,y,z)-1]z2+y[0yfx(x,v,0)dvf(x,y,z)-f(x,0,0)0000-0yψx(x,v,0)vdv0000-1a0yfx(x,v,0)dv]. First, from condition (v) of Theorem 4, we obtain (12)[ψ(x,y,z)-ψ(x,y,0)z]yz2=yψz(x,y,θ2z)z20,dddddddddddddddddddddddddddddddd0θ21,[f(x,y,z)-f(x,y,0)z]z2=fz(x,y,θ3z)z20,dddddddddddddddddddddddddddddd0θ31. Next, from conditions (iii), (v), and (vi) of Theorem 4 we obtain that U0.

Hence (13)V˙(2)(x,y,z)0. In addition, we can conclude that V(x,y,z) as x2+y2+z2 (see Global Asymptotic Stability on page 223, Theorem 5.2.12 of Rao ).

The whole discussions (conditions of Theorems 1, 2, 3, and 4) show that the zero solution of system (2) is globally asymptotically stable (also see Theorem 1.5 that of Reissing et al. ). Then, the rest of the proof may now follow as in . Thus, the proof of Theorem 4 is completed.

In the case P(t,x,y,z)0, we have the following.

Theorem 5.

Suppose that the following conditions are satisfied:

all the conditions of Theorem 4 hold;

|P(t,x,y,z)|q(t), where qL1(0,), L1(0,) is a space of integrable Lebesgue functions.

Then, there is a finite positive constant K such that every solution (x(t),y(t),z(t)) of system (2) satisfies (14)|x(t)|2K,|y(t)|2K,|z(t)|2K.

Proof.

Consider the function V(t)=V defined as above. Since P0, then the total derivative of V(t) can be revised as (15)V˙(t)V˙(2)+[y+1az]×P(t,x,y,z). Let D1=max(1,a-1). Then, we have (16)V˙(t)D1[|y|+|z|]q(t). Using the inequality (17)|y|1+y2, we obtain (18)V˙(t)D1[2+y2+z2]q(t). From (9), we have (19)V˙(t)D2q(t)+D3V(t)q(t), where (20)D2=2D1,D3=D1D0-1. Integrating (19) from 0 to t, we obtain (21)V(t)V(0)+D20tq(s)ds+D30tV(s)q(s)ds. Setting (22)D4=V(0)+D20tq(s)ds, then, we have (23)V(t)D4+D30tV(s)q(s)ds. Using Gronwall-Bellman inequality (see Rao ) yields (24)V(t)D4exp[D30tq(s)ds]. The proof of Theorem 5 is complete.

4. Example

Consider the equation (25)x+[(sinx)x˙+(x˙)2+ex˙x¨+3]x¨+(x˙)3+x˙+x1+x2+x˙ex˙x¨=11+t2+x2+(x˙)2+(x¨)2. Equation (25) is in the form of (1), where (26)ψ(x,y,z)=(sinx)y+y2+eyz+3,f(x,y,z)=y3+y+x1+x2+yeyz,P(t,x,y,z)=11+t2+x2+y2+z2.

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M111"><mml:mi>P</mml:mi><mml:mo>≡</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula>).

With a=2, b=1, c=1, from condition (vi) of Theorem 4, we have (27)2[(1-13cosx)y2+eyz+1]y21-x2(1+x2)2y2. Hence, condition (vi) of Theorem 4 is satisfied.

Then, it is easy to check that all the other conditions [(i)–(vi)] of Theorem 4 are satisfied. Hence, the trivial solution of (25) is globally asymptotically stable.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M116"><mml:mi>P</mml:mi><mml:mo>≠</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula>).

Assume that ψ, f, a, b, c are the same as in Case 1 of the example, and (28)P(t,x,y,z)=11+t2+x2+y2+z211+t2=q(t) and that (29)0q(s)ds=011+s2ds=π2<. Then, all the conditions of Theorem 5 are satisfied. Hence, the solutions of (25) are bounded.

Šimanov S. N. On stability of solution of a nonlinear equation of the third order Akademii Nauk SSSR 1953 17 369 372 MR0055523 Ezeilo J. O. C. On the stability of solutions of certain differential equations of the third order The Quarterly Journal of Mathematics 1960 11 64 69 MR0117394 10.1093/qmath/11.1.64 ZBL0090.06603 Ogurtsov A. I. On the stability of the solutions of some non-linear differential equations of third and fourth order Izvestiya Vysshikh Uchebnykh Zavedenii 1959 10 2000 2009 Goldwyn M. Narendra S. Stability of Certain Nonlinear Differential Equations Using the Second Method of Lyapunov 1963 Cambridge, Mass, USA Craft Laboratory, Harvard University Qian C. On global stability of third-order nonlinear differential equations Nonlinear Analysis: Theory, Methods and Applications 2000 42 4 651 661 2-s2.0-0034299685 10.1016/S0362-546X(99)00120-0 ZBL0969.34048 Omeike M. New results on the asymptotic behavior of a third-order nonlinear differential equation Differential Equations & Applications 2010 2 1 39 51 10.7153/dea-02-04 MR2654750 ZBL1198.34085 Tunç C. On the asymptotic behavior of solutions of certain third-order nonlinear differential equations Journal of Applied Mathematics and Stochastic Analysis 2005 1 29 35 10.1155/JAMSA.2005.29 MR2140325 ZBL1077.34052 Tunç C. Global stability of solutions of certain third-order nonlinear differential equations Panamerican Mathematical Journal 2004 14 4 31 35 MR2102487 ZBL1065.34047 Omeike M. O. Further results on global stability of third-order nonlinear differential equations Nonlinear Analysis: Theory, Methods & Applications 2007 67 12 3394 3400 10.1016/j.na.2006.10.021 MR2350895 ZBL1129.34323 Omeike M. O. Further results on global stability of solutions of certain third-order nonlinear differential equations Acta Universitatis Palackianae Olomucensis 2008 47 121 127 MR2482722 ZBL1177.34072 Tunç C. The boundedness of solutions to nonlinear third order differential equations Nonlinear Dynamics and Systems Theory 2010 10 1 97 102 MR2643196 Rao M. R. M. Ordinary Differential Equations 1980 New Delhi, India Affiliated East-West Press MR587850 Reissing R. Sansone G. Conti R. Nonlinear Differential Equations of Higher Order 1974 Leyden, The Netherlands Noordhoff International Publishing