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Initial functions defining dominant positive solutions of a linear differential equation with delay

Josef Diblik* and Maria Kudelcikova

"Correspondence: josef.diblik@fhv.uniza.sk Department of Mathematics, University of Zilina, Zilina, Slovak Republic

Abstract

Linear differential equation y(t) = -c(t)y(t - r),

where c(t) is a positive continuous function and delay r is a positive constant, is considered for t ^ro. It is proved that, under certain assumptions on the function c(t) and delay r, a class of positive linear initial functions defines dominant positive solutions with positive limit for t ^ ro. MSC: 34K15; 34K25

Keywords: linear differential equation with delay; initial function; retract method; dominant solution; asymptotic behavior of solution

ringer

1 Introduction

This article is devoted to the problem of the asymptotic behavior of solutions of delayed equations of the type

y(t) = -c(t)y(t - r) (1)

with a positive continuous function c(t) on the set [to - r, ro), to e R, 0 < r = const in the non-oscillatory case. The following results on the asymptotic behavior of solutions, needed in the following analysis, are taken from [1] (see [2] as well).

Theorem 1 (Theorem 18 in [1]) Let there exist a positive solution y of (1) on [t0 - r, ro). Then there are two positive solutions yi andy2 of (1) on [t0 - r, ro) satisfying

lim y-f =0. (2)

t^ro y1(t)

Moreover, every solution y of (1) on [t0 - r, ro) is represented by the formula

y(t)=Ky 1(t) +O(y2(t)), (3)

where t e [t0 - r, ro) and a coefficient K e R depends on y.

© 2012 Diblik and Kudelcikova; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

In [2] it is shown that in representation (3) an arbitrary couple y and y2 of two positive solutions of (1) satisfying (2) can be used, i.e., the following theorem holds.

Theorem 2 Assume thaty1 andy2 are two positive solutions of (1) on [to - r, to) satisfying (2). Then every solution y of (1) on [to - r, to) is represented by formula (3), where t e [to - r, to) and a coefficient K e R depends on y.

This is the reason for introducing the following definition.

Definition 1 [2] Let y1 and y2 be fixed positive solutions of (1) on [to - r, to) with property (2). Then (y1, y2) is called a pair of dominant and subdominant solutions on [to - r, to).

We note that in the literature one can find numerous criteria of positivity of solutions not only to (1), but more complicated, as well as lots of properties of such solutions and explanation of their importance (see, e.g., books [3-9], papers [1, 2,1o-2o], and the references therein). They are formulated as implicit criteria (simultaneously both sufficient and necessary) or as explicit sufficient criteria. In the paper we employ the following explicit criterion (assumptions are slightly modified to restrict the criterion to the considered case).

Theorem 3 [17] If

for t e [to, to), then (1) has a non-oscillatory solution on [to - r, to).

In this paper we prove that every positive linear initial function given on the initial interval [to - r, to] and satisfying certain restrictions, defines a positive solution y = y(t) of (1) on [to - r, to). Moreover, we show that this positive solution is a dominant solution and its limit y(TO) is positive.

The paper is organized as follows. The main result (Theorem 5 below) in Section 3 is proved by the sensitive and flexible retract method. It is shortly described in Section 2. Its applicability is performed via Theorem 4, where an important role is played by a system of initial functions (see Definition 3). Proper choice of such a system of initial functions together with the application of Theorem 4 form the mainstay of the proof ofTheorem 5.

2 Preliminaries - Wazewski's retract principle

Let C([a, b], Rn), where a, b e R, a < b,be the Banach space of the continuous mappings from the interval [a, b] into Rn equipped with the supremum norm

where || • || is the maximum norm in Rn. In the case a = -r < o and b = o, we shall denote this space as Cn, that is,

||f||c = sup ||f(6)||, f e C([a,b],Rn),

6e[a,b]

Cn := C([-r,o],Rn).

If a e R, A > o, and y e C([a - r, a + A],Rn), then, for each t e [a, a + A], we define yt e Cn by yt(6) = y(t + 6), 6 e [-r,o].

In this section we present Wazewski's principle for a system of retarded functional differential equations

y(t)=F (t, yt), (4)

where F: fi* ^ Rn is a continuous quasi-bounded map which satisfies a local Lipschitz condition with respect to the second argument and fi* is an open subset in R x Cnr.

The principle below was for the first time introduced by Wazewski [21] for ordinary differential equations and later extended to retarded functional differential equations by Rybakowski [22].

We recall that the functional F is quasi-bounded if F is bounded on every set of the form [ti, t2] x Cl C fi", where ti < t2, Cn,L := C([-r,o],L) and L is a closed bounded subset of Rn (see [5, p.3o5]).

In accordance with [23], a function y(t) is said to be a solution of the system (4) on [a -r, a + A) if there are a e R and A > o such thaty e C([a - r, a + A), Rn), (t,yt) e fi" and y(t) satisfies the system (4) for t e [a, a + A). For given a e R, y e C, we say y(a, y) is a solution of the system (4) through (a, y) e fi* if there is an A > o such thaty(a, y) is a solution of the system (4) on [a - r, a + A) and ya (a, y) = y. In view of the above conditions, each element (a, y) e fi* determines a unique solution y(a, y) of the system (4) through (a, y) e fi* on its maximal interval of existence Ia,y = [a, a), a < a <to which depends continuously on initial data [23]. A solution y(a, y) of the system (4) is said to be positive ifyi(a,y) > o on[a -r,a] UIa,y for each i = 1,2,...,n.

As usual, if a set w C R x Rn, then int w and dw denote the interior and the boundary of w, respectively.

Definition 2 [22] Let the continuously differentiable functions li(t,y), i = 1,2,...,p and mj(t,y), j = 1,2,...,q,p2 + q2 > o be defined on some open set wo C R x Rn. The set

w" = {(t,y) e wo: li(t,y) < o,mj(t,y) < o, i = 1, ...,p,j = 1,...,q} (5)

is called a regular polyfacial set with respect to the system (4), provided it is nonempty and the conditions (a) to (y ) below hold:

(a) For (t, n) e R x Cnr such that (t + 9, n (9)) e a>' for 9 e [-r,0), we have (t, n) e a". (fi) For all i = 1,2, ...,p, all (t,y) e dw for which k(t,y) = 0, and all n e Cnr for which n(0) = y and (t + 9, n(9)) e w , 9 e [-r,0), it follows that Dh(t,y) > 0, where

mn ^ ^ 9li(t,y),u ^ 9li(t,y) Dli(t, y) => —-fk (t, n ) + .

dyk d t

(y) For all j = 1,2,...,q, all (t,y) e Bw" for which mj(t,y) = 0, and all n e Cr for which n (0) = y and (t + 9, n(9)) e w , 9 e [-r,0), it follows that Dmj(t,y) < 0, where

n u \ ^dy\u \ dy) Dmj (t, y) = ^ —j-fk (t, n )+ j

k=1 dyk dt

The elements (t, n) e R x Cn in the sequel are assumed to be such that (t, n) e

Definition 3 A system of initial functions pAw> with respect to the nonempty sets A and w , where A c w' c R x Rn, is defined as a continuous mapping p: A ^ Cnr such that (i) and (ii) below hold:

(i) If z = (t,y) e A n int W, then (t + Q,p(z)(&)) e W for Q e [-r,0].

(ii) If z = (t,y) e A n dw , then (t + Q,p(z)(Q)) e W for Q e [-r,0) and (t,p(z)(0)) = z.

Definition 4 [24] If A c B are subsets of a topological space and n: B ^ A is a continuous mapping from B onto A such that n (p)= p for every p e A, then n is said to be a retraction of B onto A. When a retraction of B onto A exists, A is called a retract of B.

The following lemma describes the main result of the paper [22].

Lemma 1 Let w c w0 be a regularpolyfacial set with respect to the system (4), and let W be defined as follows:

W = {(t,y) e dw":mj(t,y) <0,j = 1,2,..., q}.

Let Z c W U w be a given set such that Z n W is a retract of W but not a retract of Z. Then, for each fixed system of initial functions pz,w", there is a point z0 = (a0, y0) e Z n w such that for the corresponding solution y(a0, p(z0))(t) of (4) we have

(t,y(oo,p(z0))(t)) e w

for each t e D^pte).

Remark 1 When Lemma 1 is applied, a lot of technical details should be fulfilled. In order to simplify necessary verifications, it is useful, without loss of generality, to vary the first coordinate t in the definition of the set w in (5) within a half-open interval open at the right. Then the set w is not open, but tracing the proof of Lemma 1, it is easy to see that for such sets it remains valid. Such possibility is used below. Similar remark and explanation can be applied to sets of the type which serve as domains of definitions of functionals on the right-hand sides of equations considered.

Continuously differentiable functions li(t,y), i = 1,2, ...,p and m;(t,y), j = 1,2,..., q,p2 + q2 > 0 mentioned in Definition 2 are often used in the form:

li(t,y) = {yi - Pi(t)){yi - Si(t)), i = 1,2, ...,p, mj(t,y) =yj - Pj(t)) y - Sj(t)), j = p + 1,p + 2,..., n, mn+1(t, y) = -t + t0 - r,

where p, S are continuous vector functions

P = (P1, P2,..., Pn), S = (S1, S2,..., Sn): [t0 - r, ro) ^ Rn,

with p(t) « S(t) for t e [t0 - r, ro) (the symbol « here and below means pi(t) < Si(t) for all i = 1,2,...,n), continuously differentiable on [t0,ro). Hence, the shape of the regular polyfacial set w* from Definition 2 can be simplified to

w" := {(t,y): t e [t0 - r, ro), p(t) « y « S(t)}.

In the sequel we employ the result from [11, Theorem 1].

Theorem 4 Let there beap e{0,..., n} such that:

(i) If t > t0, $ e Cnr and (t + Q, $(Q)) e w" for any Q e [-r,0), then

for any i = 1,2, ...,p. (If p = 0, this condition is omitted.) (ii) If t > t0, $ e Cnr and (t + Q, $(Q)) e w" for any Q e [-r,0), then

(pi)'(t)<Fi(t,$) when $i(0) = pi(t), (Si)'(t) > Fi(t, $) when $i(0) = Si(t)

for any i = p + 1,p + 2,..., n. (Ifp = n, this condition is omitted.) Then, for each fixed system of initial functions pz,w", where the set Z is defined as

Z = {(t0,y),y e [p(t0), S(t0)]},

there is a point z0 = (a0, y0) e Z n w such that for the corresponding solution y(a0,p(z0))(t) of (4) we have

(t,y(a0,p(z0))(t)) e w

for each t e Da0p(z0), i.e., then there exists an uncountable set Y of solutions of (4) on [t0 -r, ro) such that each y e Y satisfies

p(t) « y(t) « S(t), t e [t0 - r, ro).

The original Theorem 1 is in [11] proved using the retract technique combined with Razumikhin-type ideas known in the theory of stability of retarded functional differential equations.

3 Main result

In this section we consider scalar differential equation (1), where r >0 and c: [t0 - r, ro) ^ R+ = (0, ro) is a continuous function satisfying

(SO'(t) < Fi(t, $) when $i(0) = Si(t), (pi)'(t)>Fi(t,$) when $i(0) = pi(t)

(6) (7)

for t e [t0 - r, to) and

c(s)ds < to. (9)

The first condition (8), in accordance with Theorem 3, guarantees the existence of a positive solution y = y(t) on the interval [t0 - r, to). Then Theorems 1 and 2 are valid and equation (1) has two different positive solutions (dominant and subdominant) y = y1(t) and y = y2(t) on the interval [t0 - r, to). Condition (9), as will be seen from the explanation below, implies that the dominant solution has a positive limit for t ^to. We set

C := exp ^-e ^ c(t)d?j > 0, where the constant C is well defined due to (9), and

y(t):=exp^-ej^" c(s)ds^ - C > 0, t e [t0 - r, to). Obviously, p(to) = 0. Denote

m = min {|y'(t)|} = min f ec(t) exp ( -e / c(s)ds) [.

[t0-r,t0]11 [t0-r,t0][ \ Jt0-r /)

Due to positivity of c(t) on [to - r, to], we have m >0. Let e Cl be a linear initial function defined on the interval [t0 - r, t0] as

<^(t0+ 9):=K + fi9, 9 e [-r,0], (10)

where K, f e R and |f| < m. The following theorem gives sufficient conditions for the property

y(t0, VK,f)(t) >0, t e [t0 - r, to)

together with

lim y(to, yK,^)(t) = K (yK^),

t—>+TO

where K*(yK,x) is a positive constant depending on the choice of the initial linear function

yK,ix..

Theorem 5 Let inequalities (8), (9) be valid, a constant C > 3/4 and yKe C be defined by (1o). Then the solution y(to, yK,x)(t), where yKis defined by (1o), K, ¡x e R and\x\ < m, is positive including the value y(to, yK,x)(TO), i.e.

y(t0, VKf)(t) >0, t e [t0 - r, to)

lim y(to, VK,v)(t)=K'((pK,v) > 0.

Proof We will employ Theorem 4 withp = n = 1, i.e., the case (i) only. Set

F(t,$) :=-c(t)$(-r), p(t):=pi(t), 5(i):=p2(i),

where functions pj: [t0 - r, to) ^ R are defined as

P2(t) := C + p(t), Pi(t) := C - p(t).

We have

lim pi(t) = C, i = 1,2

and since C > 3/4 > 1/2 (i.e., C > p(t0 - r)): p2(t) > p1(t) > 0 on [t0 - r, to). Now, we define

a> := {(t,y): t e [to - r, to), pi(t) <y < (2(t)}

Z := {(to,y),y e [pi(to),P2(to^}.

We verify inequality (6). For t > to, $ e Cl, and (t + 0, $(0)) e 0 e [-r,0), with $(o) 5(t) = p2(t) = C + p(t), i.e., for

pi(t + 0)<$(0)<P2(t + 0), 0 e [-r,0), $(0)= p2(t) = C + p(t),

we have

F (t, $)-8<(t) = -c№(-r)-p<2 (t)

= -c(t)$(-r) + ec(t) expl -e / c(s)ds

V ito-r

> -c(t)p2(t - r)+ ec(t) expI -e / c(s)ds

V ito-r

= -c(t)[C + p(t - r)] +ec(t) exp^-ej^" c(s)ds^

= -c(t) expl -e / c(s)ds I + ec(t) expl -e / c(s)ds V Jto-r J \ J to

¡to-r

= c(t) exp ( -e / c(s)ds

e-expl e

; J^ c(s)ds^

> [we use (8)]

> c(t) exp( -e / c(s)ds

e - exp l e •

Therefore, F(t, $) > S'(t) and (6) holds. Inequality (7) holds as well because for t > t0, $ e C- and (t + 6, $(6)) e a>', 6 e [-r,0), with $(0) = p(t)= y-(t) = C - y(t), i.e., for

y-(t + 6)<$(6)<y2(t + 6), 6 e [-r,0), $(0)= y-(t) = C - y(t)

we have

F (t, $)-p '(t) = -c(t)$(-r)-y- (t)

= -c(t)$(-r) +

-ec(t) expl -e / c(s)ds

V ./t0-r

< -c(t)y1(t - r)- ec(t) expl -e / c(s)ds

V ./t0-r

= -c(t)[C - y(t - r)] -ec(t) exp^-e £ c(s)ds^ = -c(t)

2C - exp( -e / c(s)ds

- ec(t) exp ( -e / c(s)ds

= c(t)

= c(t)

< [we use (8)]

< c(t)

-2C-eexp( -e / c(s)dsl + expl -e / c(s)ds

Jt0-r / \ Jt0-r

-2C + exp( -e / c(s)ds 11 -e + expl e / c(s)ds

Jt0-r ' \ \ Jt-r

-2C + exp( -e J^ c(s)ds^ + exp^e • -

= -2c(t)C <0.

Now, we will specify the system of initial functions mentioned in Theorem 4. For

z = (t0,y) e Z,

(y varies within the interval [yl(to),y2(t0)]), we define

p(z)(6) :=y* + ^6, 6 e [-r,0],|^|<m,

i.e., every initial function is a linear function described by formula (10). Since y'(t) < 0, t e [t0 - r, to), for the system of functions pz,ffl*, both assumptions (i), (ii) in Definition 3 are valid. Indeed, this property implies

y2(t) = y'(t) < 0 and y-(t) = -y'(t) > 0

if t e [to - r, to),

min {p2(t)^ = min {|p1 (t)^ = m

-m < pi(t) < m, t e [to - r, to], i = 1,2. Therefore, every segment

y(t)=y* + ¡xt, \x\<m, t e [to - r, to] satisfies inequalities

p2(t)<y(t)<p1(t) (2)

if y e intZ, t e [to - r, to]. Consequently, (i) in Definition 3 holds. If y e dZ, then inequalities (12) hold if t e [to - r, to) and (ii) is also valid. Theorem 4 is also valid for this system. Consequently, there exists a point

zo = (to,y*o) e Z n w

such that

(t,y(to,p(zo))(t)) e w, t e [to - r, to),

d(t) <y(to,p(zo))(t) < p2(t), t e [to - r, to). (13)

From inequalities (13) we conclude

lim y(to,p(zo))(t) = C

because of (11). This solution is positive, i.e.,

y(to,p(zo)) (t) > o, t e [to - r, to)

due to positivity of p1(t).

Since the statement of the theorem holds for initial functions with ¡ = o, we can also conclude that due to linearity of equation (1), every constant positive initial function defines a positive solution.

If the solution y(to,p(zo))(t) does not coincide with the solutiony(to, pK,x)(t), i.e., if yo = K, then due to linearity, the sum or the difference of y(to,p(zo))(t) and a suitable positive solution generated by a positive constant initial function gives the solution y(to, pK,x)(t).

It is only necessary to show that the solution y(t0, yK^)(t) will be again positive. The condition for positivity is

yi(to- r) >^2(io)-^i(io)

or, after some computations,

4C > 1 + 2exp^-ejf c(s)ds^.

The last inequality holds since

4C > 3>1 + 2exp^-ejf c(s)ds^.

We finish the proof with the conclusion that the existence of positive limit K"(yK^) is proved. □

Theorem 6 Let all assumptions of Theorem 5 be valid. Then the solution y(t°, VK.^) of equation (1) is a positive dominant solution.

Proof Every positive solution y = y(t) of equation (1) on [to - r,+cx)) is decreasing and therefore its limit limt^ro y(t) exists and is finite. The value of the limit can be either positive or zero. In the case of solution y(to, of equation (1), we have

lim y(to, VKtii)(t) = K (<pk,v) > 0.

By Theorem 1 there must exist another positive solution y = Y(t) of equation (1) on [t0 -r,+ro) such that either

lim y(t°, ^)(t) = o (14)

t^ro Y (t)

lim , Y (t> = 0. (15)

t^ro y(t°, vk ,ii)(t)

The first possibility (14) is impossible since in such a case there should exist a positive solution Y(t) of equation (1) on [t° - r,+ro) with the property

lim Y(t) = to,

which is obviously false. The possibility (15) remains. Then, by Definition l, a solution y(t0, of equation (1) is a dominant solution on [t0 - r,+cx>). □

Remark 2 It is well known [8, Theorem 3.3.1] that every continuous initial function y, defined on the interval [t0 - r, t0], such that y(t0) > 0, y(t0) > y(s), s e [t0 - r, t0), defines a positive solution on [t0 - r,+cx>) if the assumptions of Theorem 5 hold. But it is not known

if such a solution is dominant or subdominant or if its limit for t ^to is positive or equals zero. The statements of Theorems 5 and 6 give new results in this direction since, for a class of linear initial positive functions (not fully covered by known results), positivity of generated solutions (including positivity of their limits) is established together with dominant character of their asymptotical behavior. It is a problem for future investigation to find values of positive limits of solutions considered in the paper (e.g., by methods used in [25-27]) or to enlarge the presented method to more general classes of equations and initial functions.

The topic considered in this paper is also connected with problems on the existence of bounded solutions. We refer, e.g., to recent papers [28-30] and to the references therein.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors have made the same contribution. Allauthors read and approved the finalmanuscript.

Acknowledgements

This research was supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).

Received: 16 October 2012 Accepted: 27 November 2012 Published: 12 December 2012

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doi:10.1186/1687-1847-2012-213

Cite this article as: Diblik and Kudelakova: Initial functions defining dominant positive solutions of a linear differential equation with delay. Advances in Difference Equations 2012 2012:213.

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