JUAN E. NAPOLES VALDES. Abstract. In this work, we obtain some results on the existence of periodic solutions of a certain generalized system of Liénard type.
ON THE EXISTENCE OF PERIODIC SOLUTIONS OF SOME GENERALIZED LIÉNARD TYPE SYSTEM JUAN E. NÁPOLES VALDES Abstract. In this work, we obtain some results on the existence of periodic solutions of a certain generalized system of Liénard type. The presented remarks, show as some well-known results on the existence of such solutions, are particular cases of our results.
1. Introduction. We are interested in obtaining results on the existence of periodic solutions of a broad class of nonlinear system of the form:
x0
=
1 ( (y) h(x)
y0
=
h(x)g(x);
(y)F (x)) ;
(1.1) where h, arguments.
: R ! (0; 1); and c, F, g : R ! R are continuous functions of its
Existence of periodic solutions for various classes of second-order di¤erential Zx equations have been widely discussed in the literature. Let h 1, F (x) = f (s)ds, 0
1 and (u) = u then we have as particular cases of the system 1.1 the unforced Liénard equation:
(1.2)
x00 + f (x)x0 + g(x) = 0:
The equation 1.2 is equivalent to system:
2000 Mathematics Subject Classi…cation. Primary 34C25; Secondary 34D05. Key words and phrases. periodic solutions, generalized Liénard system, phase plane analysis. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1
2
JUAN E. NÁPOLES VALDES
x0
= y
y0
=
F (x); g(x):
The second order nonlinear di¤erential equation
(1.3)
x00 + f1 (x)x0 + f2 (x)x02 + g(x) = 0;
0 x 1 Z Zx @ A f2 (s)ds , F (x) = g(u)f1 (u)du, is equivalent to system 1.1 for h(x) = exp 1, (z) = z and y = h(x)x0 + F (x).
0
0
The equations 1.2 and ?? arising in many problems of applied sciences (theory of feedback electronic circuits, motion of rnass-spring system, cf. [1], [7] and [18]), and have been investigated extensively. Fortunately, several surveys of the literature have been made on the equation (2), in fact, the book of Sansone and Conti [22], contains an excellent summary of the results obtained up to 1960. Reissig, Sansone and Conti [20] updated the previous volume to 1962, and the papers by Burton [2] and Burton and Townsend [3,4] continued the e¤ort to the mid of 1960’s. The bibliography contained in [10], brings the literature up to date as of early 1970. Among the papers which were published in the last years we refer to the following ones [11], [13], [16], [17], [23] and [24]. For a description of the problems associated with the oscillatory nature of equation 1.2 the book by Minorsky [14] is recommended. Generalizations of equations 1.2 and ?? were considered in recent years, in [12] and [25] vector-valued functions are considered whereas in [5], [8] and the system 1.1 with 1 is analyzed. In [19] the system 1.1 is studied in an attempt to unify the methods know for particular cases in a general result on an important qualitative aspect, the boundedness of solutions. The purpose of this paper is to investigate the existence of periodic solutions of the nonlinear di¤erential system 1.1 under suitable assumptions. Our main results will be given in Theorems 3 and 8 and the examples and presented remarks, show that our theorems extend and improve previous results obtained by varied authors.
2. Existence of at least a periodic solutions. We impose some generic assumptions on the system 1.1 as follows:
(i) The functions F(x), g(x), and h(x) are continuous on R with F(0) = 0 and h(x) > 0 for all x.
PERIODIC SOLUTIONS
3
(ii) (z) is locally Lipschitz continuous and strictly increasing on R, with (0) = 0, and 1 is continuous on R such that (y) b > 0 for all y. (iii) xg(x)>0 if x 6= 0. Furthermore, we assume (see [21]) that …rst equation of system 1.1 de…nes implicitly, a function y = (x) such that : ( m; m) ! R and m > 0, (0) = 0, ( (x)) ( (x))F (x) 0, x 2 ( m; m), sgn (x) = sgnF (x) when F (x) 6= 0. For the convenience, let us denote following notations as in [21]: Y = f(0; y) : y > 0g, Y + = f(0; y) : y < 0g, X + = f(x; 0) : x > 0g, + = f(x; y) : y = (x); x > 0g; i.e. the right half of the characteristic curve. In this section, an existence criterion for periodic solutions of 1.1 is given. First we need some previous results. Let us consider the following condition on F: (iv) there exist x 2 < x 1 < 0 < x1 < x2 such that xF (x)>0 for all x 2 (x x 6= 0, and xF (x) 0 for all x 2 (x 2 ; x 1 ) ^ (x1 ; x2 ).
1 ; x1 ),
Lemma 1. Suppose that assumptions (i)- (iv) hold. Then the trivial solution of system 1.1 is locally asymptotically stable. Proof. Consider a function V0 on a neighborhood of the origin as (2.1)
where G(x) =
V0 (x; y) = A(y) + G(x);
Zx
2
h (s)g(s)ds and A(y) =
0
Zy
(s)ds.
0
From assumptions on h, g and is it clear that V0 is a positive de…nide and continuous and di¤erentiable function. Let C0 = maxfG(x 1 ); G(x1 )g and D = f(x; y) : V0 (x; y) 0 and yq < yq
0 and y 2.
y u (x) dx:
10
JUAN E. NÁPOLES VALDES
Proof. From above lemma, there exists x. in z,] such that A(x0) O. Let Yr = y(x), O— (x,y) intersects y+ at (O,y+) and O+(x7„y,) intersects y. at (O,y_). Hence A(z) = A(y) — A(y) O, since a is odd, A(y)=A(— y) for any y and thus ‘+ — (19) (i’) and (ji’) together with (19), it can be seen that the trajectory of system (1) passing through (xi„ y,) is symmetric about the origin, which implies that the trajectory is a closed orbit. Corollary 3. If assumptions of above theorem hold and if (1) there exist constants B,6,r > O such that cs’(y) (unknown char) (5, í3’(y) anci /3(y) (unknown char) B for any y, (2) [BF’(z)j2 + 4iF(z)[h2(.x)g(.x)j’ O and e(x) > O for O < x < z. — d “ — h2(x)g(x) — — dx y) - ¡3(y)F(x) — (20) — - (h2(x)g(x))’e + h2 (x)g(x)(y)F’(x) + h4(.x)g2(x)(a’(y) - ¡3’(y)F(x)) with i = 1, u (1) and (2) implies that the numerator of (19) has the same sign for all x E (O, Hence the sign of (,) depends on that of j. That j, (yU)lf O and (yj” (unknown char)O for O
