Distribution of Zeros of Solutions to Functional ... - Semantic Scholar
Available online at www.sciencedirect.com
MATHEMATICAL AND
sc,s.c~ ~D,n.:cv. ELSEVIER
COMPUTER MODELLING
Mathematical and Computer Modelling 42 (2005) 193-205 www.elsevier,com/locate/mcm
D i s t r i b u t i o n of Zeros of S o l u t i o n s to F u n c t i o n a l E q u a t i o n s A. DOMOSHNITSKY Department of Mathematics and Computer Science The College of Judea and Samaria Ariel 44837, Israel adom~yosh, ac. il
M. DRAKHLIN The Research Institute The College of Judea and Sarnaria Ariel 44837, Israel
I. P. S T A V R O U L A K I S Department of Mathematics University of Ioannina 45110 Ioannina, Greece (Received and accepted February 2004) A b s t r a c t - - I n this paper, distribution of zeros of solutions to functional equations is studied. It will be demonstrated that oscillation properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Distances between zeros of solutions are estimated. On this basis, zones of solutions positivity to the Dirichlet boundary value problem for delay PDEs are estimated. @ 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - F u n c t i o n a l equations, Oscillation, Nonoscillation, Spectral radius.
1. I N T R O D U C T I O N I n this p a p e r , we i n v e s t i g a t e t h e o s c i l l a t o r y b e h a v i o r of solutions to t h e following f u n c t i o n a l equation, y(t) = (Ty) (t) + f (t), t E [0,+oc), (1.1) w h e r e T : L[0,+~) ~ L °~ [0,+~) is a linear p o s i t i v e o p e r a t o r (L[0~+~) is t h e space of m e a s u r a b l e e s s e n t i a l l y b o u n d e d functions y : [0, + ~ ) ~ [0, + o c ) ) , f e L ~ , + ~ ) is a given function. T h e following i n t e g r o - f u n c t i o n a l e q u a t i o n ,
y(t)----~p~(t)y(g~(t))+ 4=1
fh2(t) Jhl(t)
k ( t , s ) y ( s ) ds,
t e [0,
(1.2)
The research of the first two authors was supported by the KAMEA and Giladi Programs of the Ministry of Absorption of the State of Israel. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.02.043
Typeset by ~4flAN-TEX
194
A. DOMOSHNITSKY et al.
y ([) = ~(~),
~ < 0.
(1.3)
is a particular case of equation (1.1). Let us assume that pi : [0, +oo) --~ [0, +oo) (i = 1 , . . . , m) and k(., .) : [0, +oo) x (-co, +oo) ~ [0, +oo) are measurable essentially bounded nonnegative functions, gi (i = 1 , . . . , m) are measurable functions, hi and h2 are continuous functions, such that the differences t - hi(t) (j = 1, 2) and t - g~(t) (i = 1 , . . . , m) are bounded on [0, +oc), and among these functions there are both delays (g~(t) ~ t - z, i = 1 , . . . , n, hi(t) < t - z) and advances (gi(t) > t, i = n + 1 , . . . , r n , h2(t) >_ t). The following functional equation, y (t) = E P i
(t) y (gi (t)),
t e [0, + o c ) ,
(1.4)
~ < 0,
(1.5)
i=1
y(~) = p (~),
is an important particular case of equation (1.2),(1.3). The theory of difference equations was intensively developed during the last two decades. Note in this connection the known monographs by Agarwal [1] and Lakshmikantham and Trigante [2]. Let us stress now that the following difference equation, Yn ~- ~
PkYn-k,
n E [1,+00),
(1.6)
n < O,
(1.7)
~:--m
y (n) = ~Pn,
can be actually considered as a particular case of equation (1.4), (1.5). If we set in equation (1.4) g~(t) = t - i, where i is an integer number, and the coefficients pi(t) and the initial function ~(t) are constants on each interval [k, k + 1), we obtain that corresponding solutions y(t) are constants (y(t) = yk) on each interval [k, k + 1). In this case, we actually have a difference equation (1.6),(1.7). Obviously, each assertion obtained for functional equation (1.4) is also true for difference equation (1.6). One of the main aims of our approach is to derive conclusions about the behavior of solutions to functional equation (1.4) using corresponding properties of difference equation (1.6). Our approach will be a corresponding analogue for functional equations of the method of upper and lower solutions for difference equations. This method can be combined with the monotone iterative technique. For difference equations we can note in this connection monographs [1,2] and papers [3-9]. Note that questions of the action of the operator, m
( s y ) (t) =
(t) y
(t)),
t e [0,
(1.8)
i=1
where
y
= 0,
< 0,
(1.9)
in the space L~,+~) and estimates of its spectral radius were considered by Drakhlin {see, for part of his results were reflected in the known monograph by Azbelev, Maksimov and Rakhmatullina [11]. For the action of the operator S : L ~[0,+o~) -~ L ~[0,+o~) we assume
example, [101) and
that for each set M from the equality m e s M = O, it follows the equality mesg~-l(M) = 0 for i = 1 , . . . , m. The necessary condition, which is practicMly very close to the sufficient condition, is the following. There are no intervals [9, #], such that g~(t) = const for t C [~,,#]. DEFINITION 1.1. A function y C L[0~+oo), satisfying equation (1.1) for almost every t E [0, +oo) is called a solution of (1.1). Considering solutions in the space L[0~+oc) we should define oscillation for noncontinuous functions.
Distribution of Zeros
195
DEFINITION 1.2. A point t, is a zero o f a function y E L~,+~) if either p r o d u c t lira ess sup y ( s ) . lim ess inf y(s)
(1.10)
lim ess inf y ( s ) . lim ess sup y ( s ) t--.~sc[t,~] t-,v+ ~[.,q
(1.11)
t-*v-
s~[t,v]
t~v+
sE[v,t]
or
is nonpositive.
Obviously, if y is a continuous function, then a point v is its zero if and only if y(v) = 0. Zeros determined by this definition include "traditional" zeros as well as points of sign change. Note, for example, that the function y(t) = sin l/t, t E [-1, 1] has a zero at the point t = 0 according to Definition 1.2.
DEFINITION 1.3.
W e say that a function y E L °° [0,+~) nonoscillates if there exists v, such that y(t) does not have zeros for t E (v, +oc). I f there exists a sequence of zeros o f a function y tending to infinity, we say that a function y E L[0~+oc) oscillates.
DEFINITION 1.4. W e say that a function y E L °°Iv,,] is essentially positive if y(t) > 0 for almost all t c [v, #] and y does n o t have zeros there. Note that the function ]sint I is positive for t E [0, +oo) but not essentially positive because of zeros at the points ~rn (n = 0, 1,2,... ). There are many results about oscillation of all solutions of differential equations (see, for example, the known monograph by Gyori and Ladas [12]), functional equations (see, for example, [13-20]) and difference equations (see, for example, [21-26]). Detailed results about distribution of zeros of solutions (for example, about estimates of distance between zeros) have rarely been seen in the literature. Various estimates of the distance between adjacent zeros of oscillating solutions for delay differential equations of second-order were obtained by Azbelev [27], Domshlak [28,29], Eliason [30], Myshkis [31], Norkin [32] and in the works [33-35]. Distribution of zeros for first-order delay equation was considered in the recent paper by Zhang and Zhow [36] and for difference equations in the paper by Domshlak [37]. One of our main aims in this paper is the study of zeros' distribution of a corresponding functional equation for (1.1). Oscillation of solutions to boundary value problems for partial differential equations was considered in the publications [35,38-43]. We will also consider this problem as well as estimates of zones of solutions positivity for partial differential equations. Let L~,~], where 0 ", such that f , , + ~ ( t ) > 0 for t E [-,vl]. Now, y(t) = {(I - T,,+oo)-lf,,+oo}(t) = {(I + Tv,+oo + T~,+oo + T3.+oo + " " ) f , , + ~ } ( t ) > f , , + ~ ( t ) > 0 for t E [-,-11. This contradiction completes the proof. THEOREM 3.2. Let the following inequality be fulfilled, sup
m fh2(t) E p ~ (t) + k (t,s) ao,oo (s) ds < 1.
t C [0,-t-oo) i = 1
Jh~(t)
(3.3)
Distribution of Zeros
199
Then,
(1) the spectral radius of the operator T : L~,+~) -~ L[o,+~)~ determined by formula (1.15) (with u = 0 and # = oc) is less than one, (2) if in addition the initial function ~ are positive (negative) functions and the inequality,
~p~ i=l
(t) +
k (t, s) ~o,~ (s) d~ > 0,
t c [0, + ~ ) ,
(3.4)
l(t)
is satisfied, then each solution y E L~,+~) of equation (1.2) is positive (negative) for
t e [o, +~). The proof follows from Corollary 3.3. Let us consider the following equation, b(t) y(t) = a(t) y ( t + m) + c(t) y ( t - k), (~) = ~ (~),
t E [0, #), < 0 or ~ > #,
(3.5) (3.6)
where a, b, c are continuous positive functions, k and m are natural numbers. A corresponding operator T : L[0~ ) ~ L[0~,,) is of the following form, a(t)
(Ty) (t) : - ~ y y(~)=0,
c(t)
(t + ,~) + ~(t) y (t - k) , ~#.
(3.~) (3.8)
THEOREM 3.3. Let there exist a real positive number fl, such that
b (t) ~ > a (t) ~m+k + c (t),
t e [0, ~),
(3.9)
then the spectral radius of the operator T is less than one and a solution of equation (3.5), (3.6) is positive for each positive initial function ~.
In order to prove Theorem 3.3 we choose v = ~3t in Condition (3) of Theorem 3.1. Inequality (3.9) implies that the spectral radius r of the operator T defined by equality (3.7) is less than one. Now, Lemma 3.1 implies nonoscillation of solutions with the positive initial function ~. Let us write now the following particular cases of equation (3.5), b(t) y (t) : a (t) y (t + 1) + c(t) y ( t -
1),
t e [0,,),
(3.1o)
t ~ [0,/_t),
(3.11)
and boy (t) = aoy (t + 1) + coy (t - 1),
in which a0, b0 and co are positive constants. Condition (3.9) for equation (3.11) can be written as follows, no/32 - bo/3 + Co < 0.
(3.12)
COROLLARY 3.4. Let the following inequality be fulfilled, bo2 > 4aoco,
(3.13)
then the spectral radius of the operator T, where m = k = 1, is less than one and a solution of equation (3.11),(3.6) is positive for each positive initial function ~.
The proof follows from Theorem 3.3 and the fact that inequality (3.12) has a real solution if and only if condition (3.13) is fulfilled. REMARK
3.1. It is known (see, for example, [14]) that the inequality, b~ < 4aoco,
implies oscillation of all solutions of equation (3.11),(3.6). improved.
(3.14) Thus~ inequMity (3.13) cannot be
A . DOMOSHNITSKY et al.
200
THEOREM 3.4. Let inequalities
(3.13)and a(t) < ao, b(t) >_ bo, c(t) #.
(3.15) (3.16)
is less than one. By Theorem 3.1 there exists an essentially positive function v E L ~[0,~), such that v(t) > (Tov)(t) for t E [0,#). The inequalities a(t) < ao, b(t) >_ bo, c(t) _ (Tv)(t), where the operator T is determined by the formula,
(Ty) (t) = - - y
(t + 1) + b--~y (t - 1),
(3.17)
and (3.16). For this function v the inequality v(t) > (Tov)(t) for t E [0, p) is satisfied. Theorem 3.1 completes the proof of Theorem 3.4. REMARK 3.2. Sufficient conditions of solutions' nonnegativity for difference equations on a finite interval were obtained in Theorems 2.1-2.4 of the recent paper [3] in which a corresponding hypothesis on the size of this interval (see Condition (iv)) was assumed.
4. E I G E N F U N C T I O N S
OF T H E O P E R A T O R Tv,~
Let us consider the equation,
boy (t) = aoy (t + 1) + coy (t - 1),
t e [0, + c o ) ,
(4.1)
where ao, bo, co are positive constants,
y (~) = 0,
~ < 0,
(4.2)
and construct eigenfunctions of the operator, CO (TO, x) (t) = ao bo x ( t + l ) + - - bo x (t - 1),
x ( ( ) = 0,
~ < ~,
~ > p.
(4.3) (4.4)
Let us find a solution of equation (4.1). Set y(t) = 1 for t C [0, 1) and find from equation (4.1) that y(t) = bo/ao for t E [1, 2). Then, from equation (4.1), we can find y(t) = (b2o- aoco)/a2o for t C [2, 3) and so on. We obtain the following solution y of equation (4.1) which is also an eigenfunction of the operator T [o,~]" O
1,
tE [0, 1),
bo
t ~ [1, 2),
a0 '
v (t)
=
b~ - aoco -----a~-[--'
t C [2, 3),
~ ( b o ~ - 2aoco),
t c [3, 4),
{bo (bo2 - 3aoco) +a2c 2} ~1, ao
t e [4,5),
(4.5)
Distribution of Zeros
201
l,et us consider b(t) y(t) = a(t) y(t + 1) + c ( t ) y ( t - 1),
t E [0,+oo),
(4.6)
where a, b, c are positive functions and introduce the corresponding operator T~, • L °°[~,.] ~ L ~[~,.] by equalities a (t)
.
(Tu,y) (t) : ~(t) x (t + 1) + b--~x (t - 1)
(4.7)
and (4.4). THEOREM 4.1. Let the function y determined by equality (4.5) be negative on the interval (#, p + 1) and at least one of the following two conditions either, bo>b(t),
ao0,
a0b(t),
ao aoC0, on intervals such that t~ - u > 5. REMARK 4.1. All solutions of equation (4.1) oscillate if b0~ < 4aoco [14], and our method allows to estimate a zone of their positivity. COROLLARY 4.1. Let the following conditions be fulfilled, a(t) > ao > O, for t E [u,+oc), lim b ( t ) = O ,
lira
lira c ( t ) = O ,
t--,+oo
t--*+c~
b 2 (t) c(t)
- O,
(4.12)
then ai1 solutions of equation (4.6) oscillate. EXAMPLE 4.1.
Consider the equation, b0 Co -~-y (t) = y (t + 1) + ~-~y (t - 1),
t E [1, +oo),
(4.13)
Observe that the conditions of Corollary 4.1 are satisfied. Therefore, all solutions oscillate. Let us consider the following integral equation, t+l
y (t) =
k (t, s) y (s) ds,
t E [0, +oo),
(4.14)
dt--i
where y
=
< o.
(4.15)
202
A. DOMOSHNITSKY
et al.
THEOREM 4.3. Let the following inequaBties be fuiIilled
~
O+l
k (t,s) ds ~> 1,
t,O ~
(4.16)
[0, T o o ) ,
then there are no positive (negative) solutions on the interval [c~,,8] if/3 - c~ > 2. In order to prove this assertion let us set v = 1 in Lemma 4.1. We obtain that r.~ > 1 if # - ~, > 1. Theorem 2.1 completes the proof. 5. A P P L I C A T I O N S TO OSCILLATION OF PARTIAL DIFFERENTIAL EQUATIONS In this part oscillation properties of solutions of the following equation,
A(t) ~ " (0 (t), ~) _ the(t) k (t, ~) ~ " (~, x) d~ J h i (t)
(5.1)
m
+Ep~(t)u(gi(t),x)=O,
t E [0, +oc),
x E [0, w],
i=I
with the periodic u~'
u(t,O)=u(t,w),
(t,0)
= u x'
(t,~)
t c [0, + ~ ) ,
,
(5.2)
Neumann, U x' ( t , 0 )
~ U x' ( t , ~ )
0,
t e [0, + ~ ) ,
(5.3)
t E [0, + o c ) ,
(5.4)
and Dirichlet,
~ (t,0)= ~(t,~)=0, boundary conditions will be studied.
DEFINITION 5.1. We say that the solution u(t, x) of a P D E boundary value problem oscillates if for each to there exists a zero of the solution ( t l , z l ) , such that tl > to, Xl C (O, cd). The following partial differential-difference equation, u (p (t), x) + A (t) u ~ (t, x) + b (t, x) u (r (t), x) = 0,
t > 0,
x C [0, w],
(5.5)
where p and r are monotone increasing functions such that p(t) >_ t, r(t)
MATHEMATICAL AND
sc,s.c~ ~D,n.:cv. ELSEVIER
COMPUTER MODELLING
Mathematical and Computer Modelling 42 (2005) 193-205 www.elsevier,com/locate/mcm
D i s t r i b u t i o n of Zeros of S o l u t i o n s to F u n c t i o n a l E q u a t i o n s A. DOMOSHNITSKY Department of Mathematics and Computer Science The College of Judea and Samaria Ariel 44837, Israel adom~yosh, ac. il
M. DRAKHLIN The Research Institute The College of Judea and Sarnaria Ariel 44837, Israel
I. P. S T A V R O U L A K I S Department of Mathematics University of Ioannina 45110 Ioannina, Greece (Received and accepted February 2004) A b s t r a c t - - I n this paper, distribution of zeros of solutions to functional equations is studied. It will be demonstrated that oscillation properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Distances between zeros of solutions are estimated. On this basis, zones of solutions positivity to the Dirichlet boundary value problem for delay PDEs are estimated. @ 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - F u n c t i o n a l equations, Oscillation, Nonoscillation, Spectral radius.
1. I N T R O D U C T I O N I n this p a p e r , we i n v e s t i g a t e t h e o s c i l l a t o r y b e h a v i o r of solutions to t h e following f u n c t i o n a l equation, y(t) = (Ty) (t) + f (t), t E [0,+oc), (1.1) w h e r e T : L[0,+~) ~ L °~ [0,+~) is a linear p o s i t i v e o p e r a t o r (L[0~+~) is t h e space of m e a s u r a b l e e s s e n t i a l l y b o u n d e d functions y : [0, + ~ ) ~ [0, + o c ) ) , f e L ~ , + ~ ) is a given function. T h e following i n t e g r o - f u n c t i o n a l e q u a t i o n ,
y(t)----~p~(t)y(g~(t))+ 4=1
fh2(t) Jhl(t)
k ( t , s ) y ( s ) ds,
t e [0,
(1.2)
The research of the first two authors was supported by the KAMEA and Giladi Programs of the Ministry of Absorption of the State of Israel. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.02.043
Typeset by ~4flAN-TEX
194
A. DOMOSHNITSKY et al.
y ([) = ~(~),
~ < 0.
(1.3)
is a particular case of equation (1.1). Let us assume that pi : [0, +oo) --~ [0, +oo) (i = 1 , . . . , m) and k(., .) : [0, +oo) x (-co, +oo) ~ [0, +oo) are measurable essentially bounded nonnegative functions, gi (i = 1 , . . . , m) are measurable functions, hi and h2 are continuous functions, such that the differences t - hi(t) (j = 1, 2) and t - g~(t) (i = 1 , . . . , m) are bounded on [0, +oc), and among these functions there are both delays (g~(t) ~ t - z, i = 1 , . . . , n, hi(t) < t - z) and advances (gi(t) > t, i = n + 1 , . . . , r n , h2(t) >_ t). The following functional equation, y (t) = E P i
(t) y (gi (t)),
t e [0, + o c ) ,
(1.4)
~ < 0,
(1.5)
i=1
y(~) = p (~),
is an important particular case of equation (1.2),(1.3). The theory of difference equations was intensively developed during the last two decades. Note in this connection the known monographs by Agarwal [1] and Lakshmikantham and Trigante [2]. Let us stress now that the following difference equation, Yn ~- ~
PkYn-k,
n E [1,+00),
(1.6)
n < O,
(1.7)
~:--m
y (n) = ~Pn,
can be actually considered as a particular case of equation (1.4), (1.5). If we set in equation (1.4) g~(t) = t - i, where i is an integer number, and the coefficients pi(t) and the initial function ~(t) are constants on each interval [k, k + 1), we obtain that corresponding solutions y(t) are constants (y(t) = yk) on each interval [k, k + 1). In this case, we actually have a difference equation (1.6),(1.7). Obviously, each assertion obtained for functional equation (1.4) is also true for difference equation (1.6). One of the main aims of our approach is to derive conclusions about the behavior of solutions to functional equation (1.4) using corresponding properties of difference equation (1.6). Our approach will be a corresponding analogue for functional equations of the method of upper and lower solutions for difference equations. This method can be combined with the monotone iterative technique. For difference equations we can note in this connection monographs [1,2] and papers [3-9]. Note that questions of the action of the operator, m
( s y ) (t) =
(t) y
(t)),
t e [0,
(1.8)
i=1
where
y
= 0,
< 0,
(1.9)
in the space L~,+~) and estimates of its spectral radius were considered by Drakhlin {see, for part of his results were reflected in the known monograph by Azbelev, Maksimov and Rakhmatullina [11]. For the action of the operator S : L ~[0,+o~) -~ L ~[0,+o~) we assume
example, [101) and
that for each set M from the equality m e s M = O, it follows the equality mesg~-l(M) = 0 for i = 1 , . . . , m. The necessary condition, which is practicMly very close to the sufficient condition, is the following. There are no intervals [9, #], such that g~(t) = const for t C [~,,#]. DEFINITION 1.1. A function y C L[0~+oo), satisfying equation (1.1) for almost every t E [0, +oo) is called a solution of (1.1). Considering solutions in the space L[0~+oc) we should define oscillation for noncontinuous functions.
Distribution of Zeros
195
DEFINITION 1.2. A point t, is a zero o f a function y E L~,+~) if either p r o d u c t lira ess sup y ( s ) . lim ess inf y(s)
(1.10)
lim ess inf y ( s ) . lim ess sup y ( s ) t--.~sc[t,~] t-,v+ ~[.,q
(1.11)
t-*v-
s~[t,v]
t~v+
sE[v,t]
or
is nonpositive.
Obviously, if y is a continuous function, then a point v is its zero if and only if y(v) = 0. Zeros determined by this definition include "traditional" zeros as well as points of sign change. Note, for example, that the function y(t) = sin l/t, t E [-1, 1] has a zero at the point t = 0 according to Definition 1.2.
DEFINITION 1.3.
W e say that a function y E L °° [0,+~) nonoscillates if there exists v, such that y(t) does not have zeros for t E (v, +oc). I f there exists a sequence of zeros o f a function y tending to infinity, we say that a function y E L[0~+oc) oscillates.
DEFINITION 1.4. W e say that a function y E L °°Iv,,] is essentially positive if y(t) > 0 for almost all t c [v, #] and y does n o t have zeros there. Note that the function ]sint I is positive for t E [0, +oo) but not essentially positive because of zeros at the points ~rn (n = 0, 1,2,... ). There are many results about oscillation of all solutions of differential equations (see, for example, the known monograph by Gyori and Ladas [12]), functional equations (see, for example, [13-20]) and difference equations (see, for example, [21-26]). Detailed results about distribution of zeros of solutions (for example, about estimates of distance between zeros) have rarely been seen in the literature. Various estimates of the distance between adjacent zeros of oscillating solutions for delay differential equations of second-order were obtained by Azbelev [27], Domshlak [28,29], Eliason [30], Myshkis [31], Norkin [32] and in the works [33-35]. Distribution of zeros for first-order delay equation was considered in the recent paper by Zhang and Zhow [36] and for difference equations in the paper by Domshlak [37]. One of our main aims in this paper is the study of zeros' distribution of a corresponding functional equation for (1.1). Oscillation of solutions to boundary value problems for partial differential equations was considered in the publications [35,38-43]. We will also consider this problem as well as estimates of zones of solutions positivity for partial differential equations. Let L~,~], where 0 ", such that f , , + ~ ( t ) > 0 for t E [-,vl]. Now, y(t) = {(I - T,,+oo)-lf,,+oo}(t) = {(I + Tv,+oo + T~,+oo + T3.+oo + " " ) f , , + ~ } ( t ) > f , , + ~ ( t ) > 0 for t E [-,-11. This contradiction completes the proof. THEOREM 3.2. Let the following inequality be fulfilled, sup
m fh2(t) E p ~ (t) + k (t,s) ao,oo (s) ds < 1.
t C [0,-t-oo) i = 1
Jh~(t)
(3.3)
Distribution of Zeros
199
Then,
(1) the spectral radius of the operator T : L~,+~) -~ L[o,+~)~ determined by formula (1.15) (with u = 0 and # = oc) is less than one, (2) if in addition the initial function ~ are positive (negative) functions and the inequality,
~p~ i=l
(t) +
k (t, s) ~o,~ (s) d~ > 0,
t c [0, + ~ ) ,
(3.4)
l(t)
is satisfied, then each solution y E L~,+~) of equation (1.2) is positive (negative) for
t e [o, +~). The proof follows from Corollary 3.3. Let us consider the following equation, b(t) y(t) = a(t) y ( t + m) + c(t) y ( t - k), (~) = ~ (~),
t E [0, #), < 0 or ~ > #,
(3.5) (3.6)
where a, b, c are continuous positive functions, k and m are natural numbers. A corresponding operator T : L[0~ ) ~ L[0~,,) is of the following form, a(t)
(Ty) (t) : - ~ y y(~)=0,
c(t)
(t + ,~) + ~(t) y (t - k) , ~#.
(3.~) (3.8)
THEOREM 3.3. Let there exist a real positive number fl, such that
b (t) ~ > a (t) ~m+k + c (t),
t e [0, ~),
(3.9)
then the spectral radius of the operator T is less than one and a solution of equation (3.5), (3.6) is positive for each positive initial function ~.
In order to prove Theorem 3.3 we choose v = ~3t in Condition (3) of Theorem 3.1. Inequality (3.9) implies that the spectral radius r of the operator T defined by equality (3.7) is less than one. Now, Lemma 3.1 implies nonoscillation of solutions with the positive initial function ~. Let us write now the following particular cases of equation (3.5), b(t) y (t) : a (t) y (t + 1) + c(t) y ( t -
1),
t e [0,,),
(3.1o)
t ~ [0,/_t),
(3.11)
and boy (t) = aoy (t + 1) + coy (t - 1),
in which a0, b0 and co are positive constants. Condition (3.9) for equation (3.11) can be written as follows, no/32 - bo/3 + Co < 0.
(3.12)
COROLLARY 3.4. Let the following inequality be fulfilled, bo2 > 4aoco,
(3.13)
then the spectral radius of the operator T, where m = k = 1, is less than one and a solution of equation (3.11),(3.6) is positive for each positive initial function ~.
The proof follows from Theorem 3.3 and the fact that inequality (3.12) has a real solution if and only if condition (3.13) is fulfilled. REMARK
3.1. It is known (see, for example, [14]) that the inequality, b~ < 4aoco,
implies oscillation of all solutions of equation (3.11),(3.6). improved.
(3.14) Thus~ inequMity (3.13) cannot be
A . DOMOSHNITSKY et al.
200
THEOREM 3.4. Let inequalities
(3.13)and a(t) < ao, b(t) >_ bo, c(t) #.
(3.15) (3.16)
is less than one. By Theorem 3.1 there exists an essentially positive function v E L ~[0,~), such that v(t) > (Tov)(t) for t E [0,#). The inequalities a(t) < ao, b(t) >_ bo, c(t) _ (Tv)(t), where the operator T is determined by the formula,
(Ty) (t) = - - y
(t + 1) + b--~y (t - 1),
(3.17)
and (3.16). For this function v the inequality v(t) > (Tov)(t) for t E [0, p) is satisfied. Theorem 3.1 completes the proof of Theorem 3.4. REMARK 3.2. Sufficient conditions of solutions' nonnegativity for difference equations on a finite interval were obtained in Theorems 2.1-2.4 of the recent paper [3] in which a corresponding hypothesis on the size of this interval (see Condition (iv)) was assumed.
4. E I G E N F U N C T I O N S
OF T H E O P E R A T O R Tv,~
Let us consider the equation,
boy (t) = aoy (t + 1) + coy (t - 1),
t e [0, + c o ) ,
(4.1)
where ao, bo, co are positive constants,
y (~) = 0,
~ < 0,
(4.2)
and construct eigenfunctions of the operator, CO (TO, x) (t) = ao bo x ( t + l ) + - - bo x (t - 1),
x ( ( ) = 0,
~ < ~,
~ > p.
(4.3) (4.4)
Let us find a solution of equation (4.1). Set y(t) = 1 for t C [0, 1) and find from equation (4.1) that y(t) = bo/ao for t E [1, 2). Then, from equation (4.1), we can find y(t) = (b2o- aoco)/a2o for t C [2, 3) and so on. We obtain the following solution y of equation (4.1) which is also an eigenfunction of the operator T [o,~]" O
1,
tE [0, 1),
bo
t ~ [1, 2),
a0 '
v (t)
=
b~ - aoco -----a~-[--'
t C [2, 3),
~ ( b o ~ - 2aoco),
t c [3, 4),
{bo (bo2 - 3aoco) +a2c 2} ~1, ao
t e [4,5),
(4.5)
Distribution of Zeros
201
l,et us consider b(t) y(t) = a(t) y(t + 1) + c ( t ) y ( t - 1),
t E [0,+oo),
(4.6)
where a, b, c are positive functions and introduce the corresponding operator T~, • L °°[~,.] ~ L ~[~,.] by equalities a (t)
.
(Tu,y) (t) : ~(t) x (t + 1) + b--~x (t - 1)
(4.7)
and (4.4). THEOREM 4.1. Let the function y determined by equality (4.5) be negative on the interval (#, p + 1) and at least one of the following two conditions either, bo>b(t),
ao0,
a0b(t),
ao aoC0, on intervals such that t~ - u > 5. REMARK 4.1. All solutions of equation (4.1) oscillate if b0~ < 4aoco [14], and our method allows to estimate a zone of their positivity. COROLLARY 4.1. Let the following conditions be fulfilled, a(t) > ao > O, for t E [u,+oc), lim b ( t ) = O ,
lira
lira c ( t ) = O ,
t--,+oo
t--*+c~
b 2 (t) c(t)
- O,
(4.12)
then ai1 solutions of equation (4.6) oscillate. EXAMPLE 4.1.
Consider the equation, b0 Co -~-y (t) = y (t + 1) + ~-~y (t - 1),
t E [1, +oo),
(4.13)
Observe that the conditions of Corollary 4.1 are satisfied. Therefore, all solutions oscillate. Let us consider the following integral equation, t+l
y (t) =
k (t, s) y (s) ds,
t E [0, +oo),
(4.14)
dt--i
where y
=
< o.
(4.15)
202
A. DOMOSHNITSKY
et al.
THEOREM 4.3. Let the following inequaBties be fuiIilled
~
O+l
k (t,s) ds ~> 1,
t,O ~
(4.16)
[0, T o o ) ,
then there are no positive (negative) solutions on the interval [c~,,8] if/3 - c~ > 2. In order to prove this assertion let us set v = 1 in Lemma 4.1. We obtain that r.~ > 1 if # - ~, > 1. Theorem 2.1 completes the proof. 5. A P P L I C A T I O N S TO OSCILLATION OF PARTIAL DIFFERENTIAL EQUATIONS In this part oscillation properties of solutions of the following equation,
A(t) ~ " (0 (t), ~) _ the(t) k (t, ~) ~ " (~, x) d~ J h i (t)
(5.1)
m
+Ep~(t)u(gi(t),x)=O,
t E [0, +oc),
x E [0, w],
i=I
with the periodic u~'
u(t,O)=u(t,w),
(t,0)
= u x'
(t,~)
t c [0, + ~ ) ,
,
(5.2)
Neumann, U x' ( t , 0 )
~ U x' ( t , ~ )
0,
t e [0, + ~ ) ,
(5.3)
t E [0, + o c ) ,
(5.4)
and Dirichlet,
~ (t,0)= ~(t,~)=0, boundary conditions will be studied.
DEFINITION 5.1. We say that the solution u(t, x) of a P D E boundary value problem oscillates if for each to there exists a zero of the solution ( t l , z l ) , such that tl > to, Xl C (O, cd). The following partial differential-difference equation, u (p (t), x) + A (t) u ~ (t, x) + b (t, x) u (r (t), x) = 0,
t > 0,
x C [0, w],
(5.5)
where p and r are monotone increasing functions such that p(t) >_ t, r(t)
