BOUNDEDNESS AND ASYMPTOTIC STABILITY IN THE LARGE OF SOLUTIONS OF AN ORDINARY DIFFERENTIAL SYSTEM

Differential equations of the form y’= f(t,y,y’), where f is not necessarily linear in its arguments, represent certain physical phenomena and solutions have been known for quite some time. The well known Clairut’s and Chrystal’s equations fall into this category. Earlier existence of solutions of first order initial value problems and stability of solutions of first order ordinary differential system of the above type were established. In this paper we study boundedness and asymptotic stability in the large of solutions of an ordinary differential system of the above type under certain natural hypotheses on f.


INTRODUCTION
Differential equations of the form y'= f(t,y,y') where f is not necessarily linear in its arguments represent certain physical phenomena and are known for quite some time.The well known Clairut's and Chrystal's equations fall into this category [1].A few authors, notably E.L. Ince [2], H.T. Davis [1] et. al. have given some methods of finding solutions of equations of the above type.Apart from these, to the authors knowledge, there does not seem to exist any systematic study of these equations.
In our earlier papers [4,5,6], we studied the initial value problems and stability (in the sense of Lyapunov) of solutions of equations of the above type.In the present paper we study 1Received: February, 1991.Revised: December, 1991.the boundedness and asymptotic stability in the large of solutions of this new class of problems.
There is yet another type of stability called "Practical Stability" associated with the systems of the form y'= g(t,y) and a recent book by Professor V. Lakshmikantham et.al. [3] gives a very good account of practical stability.But since practical stability is neither weaker nor stronger than Lyapunov stability, in the present paper we confine ourselves to Lyapunov stability and in a subsequent paper we shall study the practical stability of y' = f(t, y, y').
Before proceeding to the main theorems, we present a few preliminary results under certain natural assumptions.Let I = [0,oo) and let R n denote the n-dimensional real space equipped with the box norm given by zl = E ]a:il.LetG=IxRnxRn.i=1   Consider the initial value problem (IVP) where f is an n-vector and (to, Yo) (5 I x Rn.
The following local ezistence and uniqueness result is an immediate consequence of Result 2 [6].If f satisfies conditions (I)-(III), then IVP (1), (2) has a unique solution y(t, to, Yo) existing on the interval [t o r, t o + r] f'l I, where r = min (1-k2 b k l "d, a Here, y(t, to, Yo) denotes the (continuous) dependence of the solution y(t) on (to, Yo).
Below, we present a continuation result.
Result 2 (Continuation of the solution of IVP (1), ( 2)): Suppose that f(t,y,z) satisfies conditions (I)-(III).Also, suppose that the solution (t, to,o), for as long as it exists, is strictly bounded by for some > 0. Then (t, t0,0) is continuable up to any t.
Then by Result 1, the solution y(t, to, Yo) exists on [t o r,t o + r], where 1--k 2/ ) r = min I i c, to" Now, if possible, let t o + r _< 7 < be such that the solution 9(t, o, o) can be continued only upto 7. Then we have I )-9ol < 2, and consider the set Dl={(t,y,z) qGI It I<a, ly y(7)l <2 lyo-y(7)l Izl <c} where a 1 > 0 is such that ' +al _< a. Clearly D C D. Then, by Result 1, the solution y(t, to, Yo) can be continued up to + r 1, where ) r = min kl , c ,al This is certainly a contradiction and hence the proof is complete.
Whenever the solution y(t, to, Yo) is continuable up to any t, > 0, we say that y(t, t o, Yo) exists for all future times and write y(t, o, Yo) exists for t E t 0.
Remark 1: In addition to assumption 1, if f has continuous first order partial derivatives with respect to (t,y,z) G and that k,k 2 in condition (III) denote the upper of and of bounds for jj (j = 1,2,...,n), respectively, then it can be easily verified that y(t, to, Yo) is continuously differentiable with respect to t and that Ofi 1( Of where E is the (n x n)identity matrix, and (), xojJ are the Jacobian matrices.We call a real valued function V(t,y,z) defined on G a Lyapunov function if V(t,y,z) is continuously differentiable with respect to (t,y,z) E G.
Result 3: Suppose that f(t,y,z) satisfies the conditions of Remark 1. Also, suppose that there exists a Lyapunov function V(t,y,z) defined on G satisfying the conditions and v'(t,v,z)<_o for all (t,y,z) .Gs uch that z = f(t,y,z).Then all solutions of system (1) are continuable up to any t.
The proof follows along the lines of the proof of Theorem 3.4 [7] and hence is omitted.
In the rest of the work, we assume that the conditions of Result 3 are true.Hence all solutions of (1) are continuable up to any t.(B1) equi-bounded if, for each a>O, toI (B2) (83) (84) there exists a positive constant = (to, a) such that Yo <_ a implies Y(t, to, Y0) < , > to; uniformly bounded if the/3 in (81) is independent of to; ultimately bounded if there exist a B > 0 and a T > 0 such that for every solution y(t, t0,Y0) of (1), lY(t, t0,Yo) <B for all t>_to+T, where B is independent of the particular solution while T may depend upon each solution; equi-ultimately bounded if there exists a B > 0 and if, for each a > 0, o I, there exists a T = T(to, a ) > 0 such that Y01 < a implies y(t, to, Yo) < B, t> to+T; (Bs) uniform-ultimately bounded if the T in (B4) is independent of 0.
We note that the uniform (-ultimately) boundedness of solutions of system (1) implies the equi (-ultimately) boundedness of solutions of (1).Below, we shall show that the converse is also true if f is either periodic in or autonomous.
Theorem 1: Let f(t,y,z) be such that f(t+w,y,z)=f(t,y,z) for all (t,y,z) (5 G, where w > 0 is a constant.If the solutions of (1) are equi (-ultimately) bounded, then they are uniform (-ultimately) bounded.
The proof follows, using result 3 [6], along the lines of proof of Theorems 9.2 and 9.3 [7] and hence is omitted.
Under the hypotheses of Result 3, Proof: Let o (5 I and a > 0 be given.For Vo with yol-< a, consider the solution y(t, to, Yo).Using condition (II), we choose a constant c > 0 such that on the set D={(t,y,z)GlO<_t<to, lYl <a, Izl _<c}, we have If(t,y,z) < c.Let and define a map F: MM by M = (z(sRnl Izl F(z) = f(t0, Yo, z).
Clearly, F maps M into itself and, by (III), is a contraction on M. Hence F has a unique fixed point z in M. Consequently, '(t0, to, o)   and y'(t0, to, y0) < .
Clearly T > 0 and is independent of 0. It can be proved as in part (B), that there exists a 1 E [t o, o + T] such that (tx, to, _< Consequently, Y(t, to, Yo)[ -< B1 for all t>t o+T.complete.
For 0 </ < a, T can be assigned any positive value and the proof is The following corollary follows immediately from Theorem 3 (C).
Corollary 1: If we replace in Theorem 3(C), the condition --c(lyl) by <_ for all (t,y,z) G satisfying z = f(t,y,z), where c is a positive constant, then solutions of (1)   are uniform-ultimately bounded.
3. ASYMPTOTIC STABILITY IN THE LARGE OF SOLUTIONS OF SYSTEM (1) In addition to the assumptions made earlier, in this section we also assume that f(t,O,O) = O, t I. Thus y =_ 0 is a solution of system (1).It is quite easy to verify that the study of stability of solutions of y' = f(t,y,y') with f(t,O,O) 0 is equivalent to the study of stability of the zero solution of an equivalent system and thus f(t,O,O) = O,t I is not a severe restriction on f (see [6]).Also, for the definitions of stability and uniform stability refer to [6].
Definition 4: The solution y(t) = 0 of system (1) is ($1) asymptotically stable in the large, if it is stable and every solution of (1) tends to zero as equi-asymptotically stable in the large, if it is stable, and for each a > 0, e > 0 t o E I, there exists a T = T(to, e,a > 0 such that Yol < a implies Y(t, o, Yo) < e, >_ o -I-T; uniform-asymptotically stable in the large, if it is uniformly stable, and for each c > 0, e > 0, there exists a T = T(e,a) > 0 such that o E I and yol _< c, implies ly(t, to, e) <e, t>t o+T, and the solutions of (1) are uniformly bounded; exponential-asymptotically stable in the large, if there exists a c > 0 and for each a > 0, there exists a constant k = k(a) > 0 such that Y01 < a implies y(t, o, yo) _< :e (= =o) yo l, t >_ o.
We note that the uniform-asymptotic stability in the large implies the asymptotic stability in the large.The next theorem shows that the converse is also true if f is either periodic in t or autonomous.
Theorem 4: Let f(t,y,z) be such that f(t+w,y,z)=f(t,y,z), for all (t,y,z) G, where w is a positive constant.If the zero solution of ( 1) is asymptotically stable in the large, then it is uniform.asymptoticallystable in the large.
The proof of this theorem follows, using Result 3 [6] and Theorem 1, along the lines of the proof of Theorem 7.4 [7] and hence is omitted.
Suppose that v'(t,y,z)<_ -(lyl), for all (t,y,z) G such that z = f(t,y,z).Then the zero solution of ( 1) is asymptotically stable in the large.
(Equi.asymptotically stable in the large of the zero solution): If condition (iii) of part (A) is replaced by v'(t, u, ) <_ v(t, u, ), where e is a positive constant, then the zero solution of (1) is equi.asymptoticallystable in the large.
This completes the proof.
The following corollary is an immediate consequence of Theorem 5 (C).
Coronary 2: /f the condition V'(t,y,z) < -c( u i Theorem 5 (C) is replaced by V*(t,y,z) <_ -cV(t,y,z), where c is a positive constant, then the zero solution of (1) is uniform asymptotically stable in the large.
Finally, we end this section by presenting a theorem on the exponential-asymptotical stability of the zero solution.
Theorem 6: Suppose that V(t,y,z) is a ,yapunov function defined on G and satisfies the following conditions: (i) For each a > O, there ezists a constant k = k(a) > 0 such that and (ii) v'(t,u,z) <_ -v(t,u,z), LetT=T(e,o)=(M-k)/k 2.