Counterexamples to theorems for zeros of solutions of second order linear differential equation

Bradley, J. S.

Varner, J. T., III

English

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33 John So Bradley and John T, Varner, I11 
1 
1 , Introduction. Recently b i g h t o n  [l] obtained some in te res t ing  
conditions f o r  a solution t o  a d i f f e ren t i a l  system y" + py = 0 , 
y(a) = 0 t o  have a zero i n  en in te rva l  (a,b] 
t h i s  paper is t o  present counterexamples to Theorems 2 and h of 111 and 
t o  show how the hypotheses of those theorems may be strengthened so t h a t  
the conclusions a r e  valid,  
of Theorem 3 of [l] since the  proof there relies on Theorem 2. 
same time, we present modest generalizations of those and other  theorems 
of [l]; i n  par t icular ,  t he  requirement that the coef f ic ien t  function p 
be of c l a s s  C' and conva,  or  concave i n  one theorem, i s  replaced by the 
conditions that p be continuous and s a t i s f y  a cer ta in  in t eg ra l  inequal- 
ity. 
The primary purpose of 
We also present a d i f f e ren t  proof of a pa r t  
A t  the 
2. A generalization and extention of Sturm's Comparison Theorem, - - 
The following generalization of Sturm's Comparison Theorem was proved 
i n  [1J f o r  the case r ( x )  p, 1 . 
~ 
'This research was p a r t i a l l y  supported by NASA Grant NGR 43-001-029, 
Portions of this paper formed the bas i s  of the second author's Masters 
Thesis a t  the University of Tennessee. 
' 2Ftresent address: Department of Mathematics, Wofford College, 
Spartanburg, S, C, 
https://ntrs.nasa.gov/search.jsp?R=19670004873 2020-03-16T17:11:09+00:00Z
2 
THEOREN 2.1 Suppose t h a t  p , q , r , continuous on [a, by , 
r ( x )  0 on [a, b] , and tha t  z i s  a nonnull solution of the  
- -
- --  -- 
system 
then a nonnull solut ion y -- of the  system --
must have a zero on the in te rva l  (a, b] e 
---L--- 
PROOF, F i r s t  note that t h e  limit of z/y exists a t  a 
and that (yzf - ziyt)/y is not the zero function on (a, b] To 
verify this l a s t  statement suppose t h a t  
f o r  a l l  x i n  (a, b] This implies t h a t  ([yzt - zy t ] /yy  (x) = 0 
f o r  a l l  x i n  (a, b] and hence 
c y z t  - zyt]/y) (x)  = 0 
X 
o = ( x ( y ~ f  - zp) /y2  = [ (z/y)t = 0 fo r  a l l  x i n  (a,  b l  
which implies t h a t  z(x) = 0 for a l l  x i n  (a, b] But this 
4 -  -r.-+r.r- +A +ha kvmn+hae.la 
&LO b v r r u r a r y  v u  uaa\r 
Suppose t h a t  y(x)  0 for  a l l  x i n  (a, b] Then 
Integrat ing the  l e f t  member of (2.1) by pa r t s  y ie lds  the  inequal i ty  
3 
which implies t h a t  lb r Dyz'  - zyt )/d 5 0 which is impossible. 
a 
The nex t  theorem and the  lemma used t o  prove it a re  apparently 
su f f i c i en t ly  well-known t h a t  references t o  them i n  t he  l i t e r a t u r e  
a r e  scarce3 w e  include them here f o r  completeness. 
LEMMA 2.1. L e t  p be continuous on (a, b] , p1 continuous on - - - - 
[a, b] and p,(x) 2 p(x) on (a, b] , let u be a solut ion t o  
yfl + = 0 on (a, b] , Urn u ( t )  = 0 = u(b) , and suppose t h a t  
there  exists a deleted r i g h t  neighborhood N of a such t h a t  
u ' ( t )  is pos i t ive  f o r  a l l  t i n  N . If v i s  a solution t o  the 
- - - -  - 
-t + a  - 
--- - - 
- - - -  -- - --
system y1I + ply = 0 , v(a)  = 0 then there  exists a c i n  '---- - 
(a, b] such t h a t  v(c)  = 0 . 
PROW. Suppose v i s  not zero on (a, b] , and note t h a t  
Assume v(x) > 0 f o r  a l l  x i n  (a, b] and t h a t  u(x) > 0 f o r  
a l l  x i n  (a, b) j hence, ut(b) 4 0 and v t ( a )  5 0 . Integra%- 
irig (2.2) frsm t +io b nne has 
v(b) ut(b) - v ( t )  u ' ( t )  + u ( t )  v ' ( t )  = 
. 
~ 4 
For t suff ic ient ly  near a the l e f t  member of (2.3) 5s negative, 
which is impossible s ince the right member is non-negative. Hence 
v w i l l  vanish i n  (a, b] 
THEOREM 2.2, 
and l e t  u be a aolution t o  yn + p(x) y 0 with 
Let p be continuous and decreasing on (xl, x4] 
111 - - 111 
-- -.I) ..LI -
lh u ( t )  * 0 - u(%) . Suppsae there exists a deleted r i g h t  --- 
t-l 
neighborhood N of xl such t h a t  u ' ( t )  %e positive f o r  a l l  
t i n  N v is a solution t o  yfl + p(x)y = 0 wfth 
3s x4' where xl < x then 
__m_ - 3 '  -consecutive zeros a t  x 
- -- - --
e--- --
-- 
X 2 ' - 5 2 X 4 ' X 3 '  
u(x + 5 - x3) and g by g(x) = p ( t )  = p(x + xl - x3) ... > p(x) . 
For x i n  (x3 x4] , a is a aolution t o  zw + g(x)e = 0 With 
lira z(x) l i m  u ( t )  * o and v is a solut ion t o  
T P ~  + p(x) v = o with v(x,) = o = v(x ) 4 
of ~ ( x )  after x 
Lemma 2 , l  that  
t -b  Xl * j X 3  
Since the first sero 
x1 is 
3 - xl + x3 2 x4 , 
x * 3 - xl + x3 9 f t  follows f r o m  
3o Distribution of 4;emso Throughout t h i s  section y = kx + m -- 
--P )(;LAL .I quat ion of 'rise -joP,ning +kr - 4 n + a  
W & b U  yw*&.YY (8. p(a)) =?Id 
(b, p(b)) , where p is the coeffiefent function in t he  d i f f e r e n t i a l  
equation y" 6 pg = 0 , 
of TheorPcnsa 1 of [l] fs sccompl%ahed using baa%cally the 8ame 
techn%qaes a s  used i n  [I], but o w  proof does not rely on the 
concept of prfncipal  solut ion as does the proof of h igh ton ' s  t h e o r e a  
The proof of the following generalisation 
5 
THEOREM 3.1. If p is posit ive and continuous on [a , b] , and y 
i s  a nonnull solution t o  the svstem 
- - - - -
Y" + PY = 0 , y(a)  = 0 = y(b)  
for  which -- 
(b + In - p(x) ) y2 (x)  dx 2 0 , 
then -
where a i s  the first positive zero of the Bessells function J - -I- --- 
(a i s  appraximately 2.91, - 
PROOF. The conclusion is obvious i f  p(a)  = p(b) hence we assume 
that  p(a) # p(b) A nonnull solution t o  the d i f f e r e n t i a l  system 
z" + (kx + m)z = 0 , z(a) - 0 m u s t  vanish again on (a , b] by 
Theorem 2.1, Following Leighton, we make the change of variable 
t = lor + m from which it follows t h a t  z is  a solution t o  
z " 
is 
t o  
+ (kx + m)z = 0 i f ,  and only i f  y defined by y ( t )  = z(x) 
a solnution t o  yl ' ( t )  + t/k2 y ( t )  = 0 Hence, any solution z 
the equation + (kx + m)z = 0 may be wri t ten as 
Suppose a 2 0 , l e t  2; be of the form (301) and suppose 
z(a)  = 0 . Suppose the next larger  zero of z(x) occurs a t  bl . 
6 
Let 5 and a2 be given by 
Since 5 and % have the same sign and s ince the seros of J1 
3 
and J are symmetric i n  the l i n e  x = 0 the aastnnption 
that a S O  and , a? are both posi t ive doela not s ac r i f i ce  
generali ty,  It follara that (1 and % are consecutive zeros 
of a l i n e a r  combination of J1 and J2. since kx + m # 0 on 
1 
-3 
- 
-- 7 3 
[a, b] Recall that J1 and J 1 are independent solutions t o  -- 3 3 
(3.3) 
and that the transformation y = u ( x ) / m  t ransform (3*3) i n t o  
For x > 0 the general solution of (Job) may be writ ten as 
1 
s ince  u(x) * 2 J ~ ( X )  i a  a solut ion of (3.4) satisfging 
~ ( 0 )  - 0 = u(a) s 5 and 9 are consecutive zeros of a solut ion 
7 
a 2 a2 - % From (3.2) one find8 that 
Since b - a < b - a it follows from (3.5) that - 1 
5 and cL2 and since the asauslption that 5 5 OC2 or) aince a + 
are positive @lies k > 0 
After using 5 as given by (3 ,2 )  and observing that 
ka + m - p(a) a straight forward calculat ion yields 
Ths followfng theorun gives a s u f f i o b n t  condition which is 
related t o  the necessary conditions given i n  Theorem 3.1, 
a solut ion t o  the syatem + (kx + m)z; = 0 %(a) = 0 f o r  which -- -I ... 
I 
' .  . .  
8 . 
then a solution to the ayatem y" + pg o , y(a) = o will --- --- 
vanish again on (a, b] e - - 
PROOFo The condition8 (3e6) 
which may be written as 
2 
2 
(kb + ni)2 
Thw, since the distance 
equation of' order $ is 
system z" + (k + m)z = 
and (3*7)  imply that 
a 
(ka + m)2 - >7k 
between consecutive zeros of solutions of Bessells 
less  than 
0 z(a) - 0 given by the solution to the 
z(a) - o 
9 
COROLLILRII, If p is positive, concave and of c lass  C' on the - ---- --
hold then a solut ion t o  the syatem Jm + py - 0 , y(a)  = 0 w i l l  -9 -- -- -
v m i q  on 
0 
- the interval 
If' the condition (3.7) is removed then the above corol lary is 
the stme as Theorem 2 i n  [l]. However, the following example 
shows that Lefghton*s Theorem 2 %a f a l se ,  
t h a t  the condition (307) automatically holds when p(a> p(b) , 
( I t  should be remarked 
so that i n  t h i s  case Theorem 2 i n  I11 is t rue) ,  Let 
p(2) - -2(x - 1) on the fnterval  [a ; $1 , Then p is 
concave, pos i t ive  and of class Ct and condition (3*6)  holds, 
If a solut ion t o  the system 7" + py - o y($) - o has a 
zero on ($ , tj' 8 then Sturmcs comparison t h e o r e m  implies that 
sin(+ - ij) haa a %em on @ t] ,, Therefore no solut ion of 
the aystsrm 7" + py - 0 , y ( $ )  - 0 can vanish on ($ , i] 
contrary t o  the conclusion of Theorem 2 of [l],, 
The next theorem appeared as Theorem 3 f n  [ll0 h igh ton ' s  
however, Theorem 2 proof' was based on Theorem 2 of the same paper; 
cannot be used as atated s ince tihe hypothesis require8 that 
p(a)  > 0 and p(b) s 0 One a l so  needs the condition (3,7) of 
Theorsm jo2 which is necsssrii=ji iii eem k is nega,ti~s. 
The theorem depends on the folluuing lama. 
3,1, If p is positive, of c la s s  C t  on [a, b] and if' 
.DI - --.- - I- 
. 
there exists a p i n t  c i n  (a, b) at which the tangent l i n e  to 
the curve paases through the point (b, 0) if ( 3 0 8 )  holds or 
through the point (a, 0)  if (3*9) holds, The number c is a 
--- 1111) --- -- 
-- - - --
0 - -- -(I 
solution of the 
-0 
equation a p'!.) (C - b) or of 
-I 
TFBOREM 3.3, If p 3.8 positive, convex and of class C1 on [a b] 
and if the conditions 
- - -'--- .I
--- 
conditione 
hold where c is the point guaranteed by b m  3 1 then a 
-@ - 0-- --.O 9 - r .  
nonnull solution of the dftf'erential system --- 
I 
11 
c 
PROOF, If conditions (3.10) hold, the tangent l i n e  assured by 
Lemma 3.1 has the  equation t ( x )  - pt (c)  (x - a) . Let e(x) be 
a solution of the  d i f f e ren t i a l  system s" + t (x )z  = 0 , 
Since p is convex, Ff z(x) has another zero on (a, b] , then 
by Stulnts  Comparison Theorem a solution of system (3.12) gluat 
%(a) - 0 , 
a lso  have a aero on (a, b] . By hypothesis, 
Multiplying (3.13) by ( p * ( ~ ) ) ~  , replacfng p'(c) (b - a) with 
t ( b )  and using the  f ac t  t ha t  t ( a )  = 0 we obtain 
Therefore by Theorem 3.2, where p(x) of t h a t  theorem is now 
t ( x )  (a, b] 
since a solution t o  the system z" + t (x)z  = 0, z(a) = 0 does, 
If' conditions ( j O 1 l )  hold, the slope of the tangent l i n e  
a solution t o  Syatsra (3,U) mu& have a zero on 
guaranteed by Lemma 3,,1 is negative, 
c lear ly  holds since t ( b )  - 0 i n  t h i a  ease, and we need only show 
t h a t  
Huwever, condition (3*6) 
(t(a)) ' + 3n p'(c) / 2 2 0 where t (x)  
is the equation of the tangent l i n e  guaranteed by Lemma 3,1, By 
hYpOtkred8 9 
c 
from which it follows t h a t  
or  
tb 
since p f ( c )  C 0 . Thusnconclusion again follows from Theorem 3.2. 
Finally,  we give an a l te rna te  for Theorem 4 of El]. 
THEOREM 3.ke Let  p be continuous, c f 0 - and d r e a l  -numbers. Suppose - -
tha t  Q: + d > 0 f o r  a l l  x on [a , b] and t h a t  z - -  is a solut ion - t o
the svstem 
- - -- I --
with the --
zft + (cx + d)z  = 0 , z(a) = 0 
property that 
2 (" [p(x) - (cx + d)]  z (IC)& - > 0 -- f o r  a l l  t - i n (a , b] . 
(3.14) 
and 
(3.15) 3nC /2 + (ca + d)  - 0 , 
- x 
w i l l  have a zero on (a , bl ----- 
I - *  
.. 
c 13 
PROCIF. The proof i s  similar t o  tha t  of Theorem 3.2. 
CORoLLAKJt. Let p be - -
p(a) f p(b), 
the curve y = p(x) 
a r c  y = p(x) (a 5 x 5 b) , If ca + d 2 0 , eb + d > - 0 'e- and if 
y 5 cx + d be an equation of the l i n e  tangent t o  -- --- - 
para l l e l  t o  the chord Joining the endpoints of the -- --- - -- 
-
bo i s  the smallest value la rger  than a f o r  which (3.U) and (3.15) - - -- -
hold then a nonnull solution of the system (3.16) w i l l  vanish on the -' -- -- - -- 
i n t e rva l  (a , bo] . 
If the requirement that condition (3.15) hold is removed from the  
Qpothesis ,  then the  above Corollary i s  the  same as Theorem 4 of [l], 
However, the conclusion is  not then t r u e  if  the function 
by p(x) = -(x - 3 )  on the i n t e rva l  [I, 23 
p is defined 
Bibliography 
1, h igh ton ,  Walter, "On the zeros of solutions of a second-order l i nea r  
d i f f e ren t i a l  equation,tt Journal - de Mathematiques, W, (1965), 
pp. 297-3100 
THE UMVERSITY 05' TENNESSEE 


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