283
Internat. J. Math. & Math. Sci.
VOL. 17 NO.2 (1994) 283-286
GHIZZETTI’S THEOREM FOR PIECEWISE CONTINUOUS SOLUTIONS
MANUEL PINTO
Departamento de MateYnhticas
Facultad de Ciencias, Universidad de Chile
Casilla 653, Santiago Chile
(Received March 30, 1992)
ABSTRACT. We obtain that certain second order differential equations have discontinuous
solutions which behaves asymptotically as straight lines.
KEY WORDS AND PHRASES. Fixed impulse times, discontinuous solutions.
1991 AMS SUBJECT CLASSIFICATION CODES. 34A37, 34G20.
INTRODUCTION.
Many applied problems in physics, biology, economics, etc., are submitted to noncontinuous
perturbations. Biological systems such as heart beats, blood flows, pulse frequency modulated
1.
systems and models for biological neural nets exhibit an impulsive behavior. This justify the
increasing interest on the differential equations with impulse action.
We wish to study the second order impulsive differential equation
y’(t) f(t,y(t),y’(t)),
(P1)
ti, > a > 0
Ay(t)
gl(t,y(t),y’(t)),
(P2)
Ay’(t)
g2(t,y(t),y’(t)),
(P3)
Y(to) YO ’a < tO < tl
where
(P4)
AY(ti) y(t t )- y(t i)
and
v
{ti}
CI
[a,c),
< + 1-cx) as
i-oc.
We assume the following basic hypotheses
(F) f: I x C2-,C a continuous function such that
f(t, yl,Y2) <_ Al(t) ly 11 +A2(t) lY2l-
(G)
()
gi:vx G’C(i
(s)
(1.1)
1,2) are two continuous functions such that
gi(t, Yl,Y2) <_ 7i(t)(lYl + Y21)i 1,2.
Sl(S + 2(s) LI(I), 7(tk) tk72(tk)+ 71(tk)E
(1.2)
1.
We will demonstrate that for every o > a > 0, any solution y of problem
of [t0,o and as t--.o, it satisfies
y(t) tl + 0(A(t)) + (t2 + 0(A(t)))t
and
(P)
is defined on all
(1.3)
284
M. PINTO
v’(*)
where 6i(i
(1.4)
2 + 0(A(t)),
1,2) are constants and
(1.5)
+
A(t)
tk G (t,c)
Furthermore, under the stronger condition
(I’)
S2Al(S),SA2(s) E Ll(I);tg72(tk),tkTl(tk) ’1"
the relation
holds, where
c
s(s)ds +
Al(t)
tk
In this way,
we extend
(1.6)
61 +62t + 0(Al(t)), t-oo
y(t)
E(t, oe) tkT(tk)"
(1.7)
the classical Ghizzetti’s Theorem
[2]
for ordinary differential
(see [5] to [9]) to problem (P).
Let I v I
and Cv+ (I) {u 6 C(lv)/U(t i1,2,.
u(ti) and u(t" exist
t--,t
and
as
and
denote
the
left
of
limit
right
u(t)
u(t)
u(tf-)
i.
Next, we establish without proof (see [1], [3]) a Gronwall’s inequality.
LEMMA 1. Let be 0 < u,, 6 Cv+ ([t0,oo)),7:v-.[0,oo) and c _> 0. The inequality
equations
to ,(s)u(s)ds + Et) 7(tk)u(tk)’
(t0,
u(t) < c +
implies
u(t) < cezp
to
As usual,
> o
(t0’t
(t).
where
-r(t)=
(t0,t)
}.
(to, t)
k
MAIN RESULTS.
We will prove the following results:
THEOREM 1. Under conditions (F), (G), and (I), for every o _> a > 0 any solution v of
problem (P) is defined on all of [t0,oo and it satisfies formulae (1.3) and (1.4).
THEOREM 2. Under conditions (F), (G), and (I), there exist nonoscillatory solutions v of
problem (P).
THEOREM 3. Under conditions (F), (G), and (I’), the conclusion of Theorem is true and
formula (1.6) holds. Moreover, there exists solutions v of problem (P) satisfying (1.6) with i # 0
2.
(i
1,2).
PROOF’ OF" THEOREM 1. Let
y=
(2.1)
A(t)+ B(t)t
under the condition
At(t) + Bt(t)t
Then
0
for #
(2.2)
i.
(2.3)
v’(t) B(t)
and hence
B’(t) (t), for #
where
(t)= (t,v(t),v’(t)). Solving equations (2.1)-(2.4) we get for #
A(t) A(to)- f s(s)ds
B(t) B(tO) + f ioTf(s)ds.
o
(2.4)
i,
i.
(2.5)
GHIZZETTI’S THEOREM FOR PIECEWISE CONTINUOUS SOLUTIONS
285
For the impulse effect, from
AA
Av
we
AA + tAB
A(Bt)
get
AA + tAB
1
.:,
AB
where
i(t) i(t,y(t),y’(t)).
So
,XA
(t)- O(t),
(t)
AB
Next,
we consider the impulsive integral system
A(t)
r
a(to)_ ] t,?(,)a" +
o
B(to) +
(2.5)-(2.6). It is equivalent
to the system
[gl(k,Y(tk),Y’(tk))- tkg2(tk, Y(tk),Y’(tk))]
(2.7)
(to,
t’j t(s)ds + t eZ(tO, t)g2(tk’Y(tk)’Yt(tk))
(2.8)
tO
From (2.1), (2.3) d the by,theses th v(s) s-
a(,) + n(,)l we get
{l(s,y(s),y’(s))l < Al(S) ly(s) + A2(s) ly’(s)
< SAl(S)
s
1a(s) + B(s) + A2(s)
B(s) < (SAl(S) + A2(s))v(s)
and
gl(tk, Y(tk),y’(tk)) < 71(tk){ A(t k) + tkB(tk) + B(tk)
< tkTl(tk)v(tk),
(2.9)
2(t,y(tk),y’(tk) <
Now, tom (9..7) d (9..9) w obta to >_ t0
Ia(t)
Ia(to) + [
-<
J
to
< to-- A(t0) +
l(s)ld,+ ’t0 < tk <
/o t(sxl(, + X(s))v(s)as +
tO < t < t[’l(tk + tk72(tk)]V(tk
(2.8) and (2.9) we get
< IB(t0)I
to
Adding (2.0) to (2.1) we get
Similarly, from
+/t(SAl(S)+A2(s))v(s)ds+(t’t)tkV2(tk)V(tk)"
(2.11)
(to, t)
o
SAl(S + A2(s), 7(8) 872(8) + 71(8). From Lemma i, we obtain
v(t) <_ v(to) lI t)(1 + 27(tk))" ezp(2 tA(8)ds)(t > to).
(to,
o
Since a e LI,- e tl(V we get that v is bounded and hence both t- 1A(t),B(t) are also bounded.
From (2.9) we get that l(s,y(8),I/(s)) Ll([a, oo)) and #l(tley(tl),y’(tk)),g2(tley(tl),y’(tk) e tl" Then,
from (2.8), B converges, i.e.,
where A(s)
f
and by
(2.8)
where
B(t)=2+O(A(t))
(t, oo)
On the other hand, the functions of
M. PINTO
286
(t-tk)
I’(s) lds and
EL1;i(tk) E.l(i=l,2 ).
are increasing and bounded since
Therefore the functions
(t-s)
(t-t k)
(s) ds and
Then they converge as
1,2)
i(tk)
(i
(t s)
(s)ds
(t0, t)
O
converge as t-.. So, by
1,2)
Ii(tt) l(i
(to, t)
o
(2.7) and (2.8), we get
A(t)
A(O)
+ B(t)=--+ B(to)+
It
o
+
Then
(t-(to, t)tl(tk)+ (to, ttk)?t2(tk
t-1A(t)+ B(t) converges at t--,oo and hence t-1A(t) converges also, i.e.,
A(t)
(1 + o(1))t.
PROOF OF THEOREMS 2 AND 3. From the proof of Theorem 1,
1A(t) + B(t) and B converge as t--,oo. Furthermore
’
ly(t)
lira
t--,oo
T
(s)ds +
so
that
2(tk) < Yll
Thus (2.8) implies that limt__,oo B(t) timt__,oo y(t)
Finally, Theorem 3 follows at once from
0
get that
(2.12)
y’(t)
lira
Let y’(t0) 1 # 0. Then there exists T large enough
oo
we
IB(t0)
and Theorem 2 follows from
(2.12).
+ (t, oo)2(tk ))
<
s(s)ds+ ’(t, oo)tk2(tk)O as
because of condition (I’) implies s[f(s)e L and tk2(tk)et 1. In this case, by (2.7) A itself
converges as t and in the same way as it was proved previously we can demonstrate that
there exist A so that A(oo) # 0, i.e., in formula (1.6), 1 can be taken nonzero in this case.
REFERENCES
BAINOV, D.; LAKSHMIKANTHAM, V. & SIMEONOV, P., Theory of Impulsive
Differential System, World Scientific, 1989.
2. GHIZZETTI, A., Un teorema sul comportamento asintotico degli integrali delle equazion:
differenziali lineari omegenee, Rend. Mat. e A,,t. S (5) (949), 28-42.
3. PINTO, M., Impulsive integral inequalities of Bihari-Type, Libertas Math XII (1992), 57-70.
4. PINTO, M., Second order impulsive differential equations, submitted.
5. KONG, Q., Asymptotic behavior of a class of nonlinear differential equations of n-th order,
Proc. Amer. Math. Soc. 103 (1988), 831-838.
6. KREITH, K., Extremal solutions for a class of nonlinear differential equations, Proc. Amer.
Math. Soc. 79 (1980), 415-421.
7. TONG, J., The asymptotic behavior of a class of nonlinear differential equations of second
order, Proc. Amer. Math. Soc. 84 (1982), 235-236.
8. PINTO, M., Continuation, nonoscillation and asymptotic formulae of solutions of second
order nonlinear differential equations, Nonlinear Analysis T.M.A. 16 (1991), 981-995.
9. PINTO, M., Integral inequalities of Bihari-Type and applications, Funck. Ekvac. 33 (1990),
1.
387-403.
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