STABILITY OF SOLUTIONS OF A NONSTANDARD ORDINARY DIFFEINTIAL SYSTEM BY LYAPUNOV ’ S SECOND METHODa

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. Earlier we established the existence of a (unique) solution of the nonstandard initial value problem y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper, we studied the stability of solutions of a nonstandard first order ordinary differential system.

X = [0,oo), denotes the n-dimensional real space equipped with the box norm given by I =.I=1, where z = column(zl, Z2,...,zn) {5 Rn, s = {z R"lzl <_ c}, where c is a positive constant, D is a connected subset of R n with the non empty interior, A denotes the interior of a subset A of C(J, Rn) is the set of all continuous vector valued functions defined on a subinterval J of I with values in Rn, and y" denote the first and second order derivatives of y, respectively, if they exist.
V 1 x V 2 is the Cartesian product of two non-empty sets V 1 and V2, taken in that order, f(t, y, z) = column(fl(t, y, z), f2(t, y, z),..., fn(t, y, z)) is a vector valued not necessarily linear function defined for (t,V, z) {5 1 x D x S with values in Rn, denotes jth(j = 1, 2,..., n) component of a function g defined on I x D x S with values in B-1 is the inverse matrix of the matrix B, if it exists.
(3) to Definition : By a solution of equation ( 3) we mean a function z C(J, Rn), where jr is some subinterval of I containing to, such that (ii') (Yo + f z(s)ds, z(t)) E D x S for all t E J to and t (iii') z(t) = f(t,y o + /z(s)ds, z(t)) holds for all t J.
Delinition 3: By a solution of system (1) we mean a continuously differentiable function y C(J, Rn), where J is some subinterval of I, such that conditions (ii) and (iii) of Definition 1 are true for t J.
Definition 4: Let b(t) be a solution of system (1) existing on the interval I.We say that (t) is a stable solution of system (1) if for every e > 0 and t o I there exists a 5(to, e ) > 0 such that whenever 19o-(to) < , where Yo E Do, then the solution y(t, to, Yo) exists for all t >_ t o and y(t, to, Yo)-ff(t) < e for t > t o.
We note that g(t, 0, 0) = 0, and that the stability of (t) is equivalent to the stability of the zero solution (w(t) =_ 0) of system (4).For this reason, from now on we take D = {y e Rnlyl <_ b}, where b > 0 is a constant, little later we assume that f(t, 0,0) = 0 and we study the stability of the zero solution of (1).

t.J
Let M be the closed subset of C(J, Rn) defined by M = {z e C(J, n")l I I II -< }- Denote by F the map on M defined by t (Fz)(t) = f(t,y 0 + /z(s)ds, z(t)),z .Ma nd t e J. to Using conditions (I)-(III), it is easy to see that F is a contraction on M and by the contraction mapping theorem [6], F has a unique fLxed point in M. Hence by Result 1, IVP (1), ( 2) has a unique solution existing on the interval J and the proof is complete.
The following corollary is an immediate consequence of Result 2: Corollar 1.Let 0 < e < b be a constant and let conditions (I)-(III) be satisfied.Then for every Yo with uol <_,, 1VP (1),(2) has a unique solution ezisting on the interval J = [t o a, t o + ] f'l I, where a is any real number, such that 0 < < rain((1-k2)lkl, (b--e)/c).(7)   In other words, every IVP (1),(2) with yol _< e has a unique solution existing on the common interval J.
The next result follows from condition (IIl) and the Gronwall's inequality [3], and can be easily established.
Re.salt 3: Let t o E I, and let Yo,' o DO.Suppose that conditions (I)-(III) are true.If the solution y(t, t o, Yo) and (t, to,' o) exist on a common interval J then k 1 y(t, to, Z/o)-' (t,t Now, suppose that f satisfies the following condition: f has all continuous partial derivatives upto order p with respect to (t, y,z) I x D S, where p > 1 is an integer, and and 0 _< k 2 < 1 are constants.
Of 17..I < 2 (J = 1,2,...,n), where Then, it is easy to verify that condition (IV) implies conditions (I) and (III), and hence we have the following result: Re.sail : Suppose that conditions (II) and (IV) are satisfied.Then IVP (1),(2) has a unique solution y existing on the interval J = [to-a,to-t-a]I, where a is given by ( 6) with k = k (i = 1,2).Moreover, y is p / 1 times continuously differentiable on J and -Ofi, I Of -Of i)f, "=(z-Cj)-(7+(j where I is n x n identity matrix Ofl Oz x Oz. Of.
Oz 1 Of, Oz n and \j] is defined similarly.Relation (9) follows from the quotient y'(t + h)-yt(t) f(t -t-h, y(t + h), y"(t + h))-f(t, y(t), yt(t)) by applying the mean value theorem for vector valued functions on the right hand side function, by making use of condition (IV) and, finally, by allowing h tend to zero.
2. STABILrI THEOREMS Throughout this section, we assume that the function f satisfies conditions (II) and (IV) and that f(t, O, O) = 0 for all t E I.
Consider the system y' = f t, y, y').
(10) Definition 5: We say that y(t) =_ 0 is a stable solution of (10) if for every e > 0 and any t o e I there exists a 6(e, to) > 0 such that whenever [Y0[ < 6 then the solution y(t, to, Yo) exists for all t >_ t o and [y(t, to, Yo) < e for all t > t 0. Definition 6: We say that y(t) =_ 0 is a uniformly stable solution of (10) if in Definition 5 is independent of t o Definition 7: We say that y(t) = 0 is an asymptotically stable solution of (10) if y(t) =_ 0 is stable (i) and (ii)   there exists a 6o(to)> 0 such that whenever yol < then the solution y(t, t o, Yo) tends to zero as (i) and (ii) Definition 8: We say that y(t) --0 is an uniformly asymptotically stable solution of (10) if y(t) --0 is uniformly stable there exists a 6 0 > 0 and for every e > 0 there exists a T(e)> 0 such that whenever Note 1: We note that uniform [asymptotic] stability always implies [asymptotic] stability of the zero solution of (10).Below, we shall see that the converse is also true if f is either periodic in t or autonomous.
Yo < 60 then y(t, to, Yo) < e for all t >_ t o + T(e).
Theorem 1: Let f(t,y,z) be periodic in t of period w.
the zero solution of ( 10) is The theorem can be proved easily, using Result 3, along the same lines as of Theorems 7.3 and 7.4.[8], and hence the proof will be omitted here.
Delinilion 9: We call a real valued function V(t,y,z), defined on I x D x S, a Lyapunov function, if: (i) V(t,y,z) is continuous with respect to (t,y,z) E I xDxS and (it) V(t, y, z) is continuously differentiable with respect to (t, y, z) E I x D o x S .
Definition 11: A real valued function a(r) defined on the interval [0,b] is called positive definite if at0) = 0 and a(r) > 0 for r E (0, b].Now, we shall present few theorems for [uniform] stability and [uniform] asymptotic stability of the zero solution of system (10).
Theorem (Stability o] the zero sol=ion ): Suppose that there erists a Lyapnnov function V(t, y, z) defined on I x D x S satisfying the following conditions: (i) V(t, O, O)= 0 for all t E I, (it) there exists a continuous positive definite function a(r) defined on [0,b] satisfying inequality a ( Y ) <_ v(t,y,) fo lt (t,y, ) z D x S h that z = .f(t,y,z)and (iii) V*(t,y,z) <_O for all (t,y,z) e lxDxS such that z = f(t,y,z).
Then the zero solution of system (10) is stable.
We note that for every (t,y) I x D, in view of conditions (II) and (IV), there exists a unique z S such that z = f(t,y,z).and it can be shown as in the above that (t) < e for an t e [t o, to + 2a].
Proceeding in this way, indefinitely, we get a sequence of solutions {yn(t)} and if we define ,(t), t e [to, to +,1, u(t), t e [to + ,, to + 2a], yn(t), t e [t o + (-" 1)c, 0 + then clearly y(t) is a unique solution of system (10) with y(to)= Yo" Moreover, y(t) exists on the entire interval [to, + oo) and ly(t) < c for all t >_ t o.This shows that the zero solution of ( 10) is stable and the proof is complete.
Theorem $ (Uniform stability a the Nero solution): Let condition (ii) in Theorem 2 be replaced by the condition: (ii') a(lYl ) _< W(t,y,z) < b(lYl) for art (t,y,z) I O S such that z = y(t,y,z), were a(r) and b(r) are continuous, positive definite functions ad o [0,1.
Then the zero solution of system (10) is uniformly stable.
Proof: Let e be such that 0 < e < b and (k 1/(1-k 2))e < c.Choose a ()> 0 such that b(5) < a(e) and < e.Now, let o E I and Yol < di.By Corollary 1, the solution y(t, to, Yo) exists initially on the interval [to, t o + a], where a is given by (7).Then the zero solution of ( 10) is uniformly asymptotically stable.
Proof: By Theorem 3, the zero solution of ( 10) is uniformly stable.Choose r/> 0 such that and (T 1/(1 -T 2))r/< c.Then there exists a 5 o > 0 such that if t o I and Yol < 6o, then to, Yo) < r/for all t _> t o.Also for a given e > 0 with < r/, there exists a 6(e) > 0 such that if I and Yo[ < , then y(t, to, Yo) < e for all t _> t o.

Now, let
Claim: For every solution y(t, to, Yo) of (I0) such that t o E I and lol < ao, there exists at least one point t I E [to, t o + T] such that y(tl, to, Yo) < 6.
For, if possible, let there exist a solution y(t, to, Yo) of (10), for some t o I and some Yo with Yol < 6o, such that y(t, to, Yo)[ >-6 for all t e [t o, t o + T].Then, there exists a ' > t o + T such that y(t, to, Yo) > 5/2 for all t [to,' ).Now, from conditions (i) and (ii) and note 2, we get that M 1 <_ a(ly(t, to, yo) l) <_ v(t,y(t, to, Yo),y'(t, to, yo)) <-V(to, Yo, Yt(to)) M3(t to) which implies that t t o < (M2 M1)/M 3 = T for all t e [to,7 ), which is certainly a contradiction.Hence our claim is true.
Thus, we have that if t o E I and yol < 60, then y(h, to, Yo) < 6 for some [to, t o + T] and consequently y(t, to, yo) < for all t >_ t o + T. The proof is complete.
The following corollary follows immediately from Theorem 5.
Then the zero solution of ( 10) is uniformly asymptotically stable.
3. EXAMPLES 1.In the study of free oscillations of positively damped systems, we encounter the following differential equation [1]: ,, + o +,(,)3(,, where w is a nonzero constant and 0 < e < < 1. Equation ( 15) is equivalent to the system y' = f(t,y,y') where and Choose the constants b > 0 and e > 0 such that 3e < 11c 4 and b < c(1-ec4)1(1 -b w2).( 7) Let D = {y E R21 yl < b} and S = {z R21 z _< }. w easily verify that fCt, y, z) is defined on I x D x S and satisfies the following conditions: A) y(t, 0, 0) =0, B) f(t, y, z) e S for (t, y, z) e I x D x S, C) f(t,y,z) is continuously differentiable with respect to (t,y,z) .IxDxS,a nd Of w 2