A Continuation Result for a Bidimensional System of Differential Equations


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1 Revista INTEGRACIÓN Universidad Industrial de Santander Escuela de Matemáticas Vol. 13, No 2, p , juliodiciembre de 1995 A Continuation Result for a Bidimensional System of Differential Equations In this note we present 'sorne sufficient conditions for the global existence of all solutions of the bidirnensional system x' = Q (y)  IJ (y) f (x), y' = a(t) 9 (3:), which contains the classical Lieuard's equation. x' = o(y) {3(y)f(x), y' = a(t)g(x), where the dots indicate differentiation with respect to t and o, {3,f and 9 are continuous realvalued functions and a is a positive continuously differentiable function on [O,+oJ, and define g(x) = fox g(s)ds, A(y) = foy o(r)dr. If in (1) o(y) = y, {3(y) == 1 and a(t) == 1 this system reduces to wellknown generalized Lienard's equation:
2 Let C(lR) and C I(lR) denote the family of continuous functions and continuous increasing functions on lr, and let: CS(lR) = {h E C(lR) : xh(x) > O for x # O}, CC(lR) = {h E: (CI(R)nCS(lR)}, CP(lR) = {hec(lr):h(x»ofora11x}. Some attempts [1,2,4,5,1l,12} have been made to find.~ufficient condi~ions on and 9 for solutions of (2) and its nonautonomous form: to be continued in the future urider the condition 9 E C S(I R). In [5} and [9} we gave some conditions so that a11solutions of (1) are continuable to the future, considering that,9 E C S (IR). In this paper we obtain sufficient conditions for the global existence of solutions of (1) without make use of above condition. The problem of continuability of solutions is of particular importance in the qualitative theory. So, in various earlier papera (see, for example [61O,13}) we studied various qualitative properties of solutions of (1), taking in account the results of [5} and [9}. On the other hand, the goal of this work is to illustrate how theresults obtained in [3} for the equation (2), can be generalized to the system (1). In this paper, we assume the uniqueness of the solutions of (1) and z(t; ta, zo) denote the unique solution of (1) with z(to; to, zo) = zo where z() = (x(), y()). Throughout a) a E CC(lR), b) {3 E C P (lr), e),9 E C(lR), this paper we also assume the following conditions hold: d) a E C P ([O, +00 1) n C 1([O, +00 )) where C 1denote the family of functions with first continuous derivate.
3 3. Bf(x}g(x} ~  [G(x} + 'Y] lor &11 x. 4. A (±oo) = ±oo. a' (t) 5. a(t} > 1 lor al1 t, Proof. that: for sorne T ~ to. Choose To < T sufficiently close to T so that sorne solution i (t) = (x (t ), y (t}) of (1) exist on [To, T] by an application oí local existence theorern. Define L = {z~ = Az(To} + (1  A)i(To} : O ~ A ~ 1} and let: Since zo = i(to}, by continuous dependence we see that O < A. ~ 1. We clairn that z(t; To, z~.) does not exist up to t = T. If it did, then by continuous dependence, there would exist a neighborhood of z~. such that a11 solutions passing through that neighborhood exist at t = T, contradicting the definition of A. This establishes the clairn. Thus we have: for sorne To < T. ~ T. We sha11 proof that the set 6. = {z(t.; TO,Z~.) : O ~ A < A.} is unbounded. By continuous dependence and (4), there exist sequence {tn} and {An} such that tn  T. Y An  A. as n and
4 for some p E 6.. On the other hand by local existence at (T.,p), the solution :(t; T., p) exists on T.  f ~ t ~ T. for some f > O. Therefore by continuous dependence: contradicting (5). Hence 6. is unbounded. Thus we can choose a sequence {z(t.;tq, ZAn)} of solutions such that: A(y) V (t, x, y) = a (t) + G (x) +'Y. I a'(t) A(y) V(l)(t, x, y) =  () ()  {3(y)f (x )g(x), a t a t V(1)(t,x,y) ~ V(t,x,y). I3y the continuity of V and the compactness of L,there exists WQ > O such that:
5 ror all n, whiclr ilja contradiction to (9). Hence, there exists a constant k > O euch that: Iv(T. To,zAn)1 ~ k for 8011 n. From this and by the continuity of Q we have that: Integrating the first equation of (1) on to ~ t < T and taking in account (10) we obtain: xo + a( M)(t  to)  [t (3(y(r))! (x (r)) dr lto ~ x (t) ~ XQ + a(m)(t  to)  [t (3(y(r))! (x (r)) dr. lto Supposc that x(t) as t  T, then exists é > Osuch tliat z(t} > O on [T 6, T). Making t  T in the right hand side of the abov~ inequality wc havc: 1: 3 (y (T))! (x (T)) dr ht l Tf ~. Tf ~ 00 = {3(Y(T))!(x(r))dr+ {3(y(T))!(X(T))dr [T (3(y(r))!(x(r))dr > o, ltf contradicting (11). Thus x(t) f as t  T. By similar way we obtain that x(t) f as t  T. This is a contradiction with (3). Hence, 8011 solutions of (1) exist in the future. The fo11owingexample does not satisfy the conditions of the Lemma 1 oí [9J, but do ours in the above theorem: X + 2c,!(x) = x(x  d)(x + d) and g(x) =. x, { x  2c, x <c Ixl ~ c c < x
6 Remark 1. le Q (y) = y, {3(y) == 1 and a(t) == 1, the conditions oe Theorem reduce to the obtained in [3, Th5.1]., Remark 2. Notice the advantages oepresent prooe on the [3,T h.5.1], taking a unique Liapunov's Eunction (7). [1] T. A. BURTON, A Continuation result 101' differential equations, Proc. Amer. Math. Soc., 67 (1977), pp [2] J. R. GRAEF, On the generalized Liéna1'd equation with negative damping,.j. Differential Equations, 12 (1972), pp [3J T. HARA, T. YONEYAMA ANO.J.SUGIE, Continuation r'esults 101' differential e~lj,tions by two Liapunov lunctions, A~m. Mat. pura appl. (IV), XXXIII (1983), pp [.(]  Necessary and Sufficíent Conditions lor' the Continuability 01 solutions 01 the system x' = y  F(x), y' = g(; ;), Appl. Anal., 19 (1985), pp [5] J. E. N ÁPOLES. On the continuability 01 solutions 01 bidimensional system", submited for publication. [6] On the global stability 01 non autonomous systems, submited for publication. [7] J. E. N ÁPOLES ANO.J.A. REPILAOO, On the continuation and non oscillation 01 solutions 01 bidimensional systems x' = a(y)  l3(y) f(; ;), y' = a(t)g(: ;), Rey. Ciencias Matemáticas, XV, Nos 23 (1994),Uniy. of Hayana, to appear. [8] On the boundedness and the asymptotic stability in the whole 01 solutions 01 a bidimensional system 01 differ'ential equations" Rey. Ciencias Matemáticas,Uniy. of Hayana, to appear. [9] Continuability, Oscíllability and Boundedness 01 bidimensional system ;1;' = a(y)  l3(y) f(x), y' == a,(t)g(x), Rev. Ciencias Matemáticas, XV, Nos 2 3 (1994), Uniy. of Hayana, to appear. [10] Sufficíent conditions 101' the oscillability 01 solutions 01 system x' = a(y)  l3(y) f(x), y' = a(t)g(x), Rey. Ciencias Matemáticas, XV, Nos 23 (1994), Uniy. of Hayana, to appear. [11] J. E. NÁPOLES ANO A. 1. RUIZ, On the behaviout' olthe solutions olthe differ' ential equation x" +g(x)x' + a(t)f(; ;) = O, (1), Rey. Ciencias Matemáticas, VI, No.1 (1985), Univ. of Habana, 1985, pp (spanish). [12]  On the behaviout' 01 the solutions of the differ'ential equation ; ;" + g(x ); ;' + a(t) f(x) = O, (I1), Rey. Ciencias Matemáticas, VII, No.3 (1986), Uniy. of Hayana, pp (spanish). [13] A. 1 RUIZ ANO J. E. N ÁPOLES, Conver'gence in nonlinear' system with 10r'CÍ'nq ter'm, submited for publication.
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