Báo cáo hóa học: "OSCILLATION AND NONOSCILLATION THEOREMS FOR A CLASS OF EVEN-ORDER QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS" pdf

FOR A CLASS OF EVEN-ORDER QUASILINEARFUNCTIONAL DIFFERENTIAL EQUATIONS JELENA MANOJLOVI ´C AND TOMOYUKI TANIGAWA Received 13 November 2005; Accepted 30 January 2006 We are concerned with

FOR A CLASS OF EVEN-ORDER QUASILINEAR

FUNCTIONAL DIFFERENTIAL EQUATIONS

JELENA MANOJLOVI ´C AND TOMOYUKI TANIGAWA

Received 13 November 2005; Accepted 30 January 2006

We are concerned with the oscillatory and nonoscillatory behavior of solutions of order quasilinear functional differential equations of the type (| y(n)(t)| αsgny(n)(t))(n)+

even-q(t)| y(g(t))| βsgny(g(t)) =0, whereα and β are positive constants, g(t) and q(t) are

pos-itive continuous functions on [0,∞), andg(t) is a continuously differentiable functionsuch thatg (t) > 0, lim t →∞ g(t) = ∞ We first give criteria for the existence of nonoscilla-tory solutions with specific asymptotic behavior, and then derive conditions (sufficient aswell as necessary and sufficient) for all solutions to be oscillatory by comparing the aboveequation with the related differential equation without deviating argument

Copyright © 2006 J Manojlovi´c and T Tanigawa This is an open access article uted under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properlycited

By a solution of (A) we mean a function y : [T y, ∞)→ Rwhich isn times

continu-ously differentiable together with| y(n) | αsgny(n)and satisfies (A) at all sufficiently large

t Those solutions which vanish in a neighborhood of infinity will be excluded from our

consideration A solution is said to be oscillatory if it has a sequence of zeros clusteringaround∞, and nonoscillatory otherwise

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 42120, Pages 1 22

DOI 10.1155/JIA/2006/42120

The objective of this paper is to study the oscillatory and nonoscillatory behavior ofsolutions of (A) InSection 2we begin with the classification of nonoscillatory solutions

of (A) according to their asymptotic behavior ast → ∞ It suffices to restrict our eration to eventually positive solutions of (A), since if y(t) is a solution of (A), then so

consid-is−y(t) Let P denote the totally of eventually positive solutions of (A) It will be shownthat it is natural to divideP into the following two classes:

InSection 5we derive criteria for all solutions of (A) to be oscillatory Our derivationsdepend heavily on oscillation theory of even-order nonlinear differential equations

y(n)(t)α

sgny(n)(t) (n)

+q(t)y(t)β

recently developed by Tanigawa in [7] Comparison theorems which will be established

inSection 4enable us to deduce oscillation of an equation of the form (A) from that of asimilar equation with a different functional argument

We note that oscillation properties of second-order functional differential equationsinvolving nonlinear Sturm-Liouville-type differential operators have been investigated byKusano and Lalli [2], Kusano and Wang [4], and Wang [9] Moreover, in a recent paper

by Tanigawa [6] oscillation criteria for fourth-order functional differential equations

have been presented

2 Classification and integral representations of positive solutions

Our purpose here is to make a detailed analysis of the structure of the setP of all possible

positive solutions of (A)

Classification of positive solutions Let y(t) be an eventually positive solution of (A)

on [t0,∞),t0≥0 Then, we have the following lemma which was proved by Tanigawaand Fentao in [8] and which is a natural generalization of the well-known Kiguradzelemma [1]

It will be convenient to make use of the symbolsL i, =1, 2, , 2n −1, to denote the

“quasiderivatives” generating the differential operator L2n y =(|y(n) | αsgny(n))(n):

Lemma 2.1 If y(t) is a positive solution of ( A ) on [t0,∞ ), then there exist an odd integer

k ∈ {1, 3, , 2n −1} and a t1> t0such that

L i y(t) > 0, t ≥ t1, for i =0, 1, , k −1,(−1)i − k L i y(t) > 0, t ≥ t1, for i = k, k + 1, , 2n −1. (2.2)

We denote byP kthe subset ofP consisting of all positive solutions y(t) of (A) satisfying(2.2) The above lemma shows thatP has the decomposition

Observing that by L’Hospital’s rule, we have, for everyj ∈ {1, 2, , 2n −1}, that

PIIk

Integral representations for positive solutions We will establish the existence of eventually

positive solutions for each of the above classesP(I) and P(II) For this purpose a crucial

role will be played by integral representations forP(I j) andP(II k) types of solutions of

(2.9)

If j ∈ {0, 1, , n −1}, then first integrating (A) 2n − j(= n + (n − j)) times from t to

∞and then integrating j times from t0tot, we have

Proof (the “only if ” part) Suppose that (A) has a positive solution y(t) of class P(I j).Notice that sincey(t) satisfies asymptotic relations (2.7)(i) and (iii), there exist positiveconstantsc j,C jsuch that

These together with (3.3), show that the conditions (3.1) and (3.2) are satisfied

(The “if ” part.) We will distinguish two cases for j ∈ {0, 1, , n −1} and for j ∈ {n,

n + 1, , 2n −1}

Case 1 Let j ∈ {n, n + 1, , 2n −1}and suppose that (3.2) is satisfied Letc > 0 be an

arbitrarity fixed constant and chooset0> 0 such that



···



n+ j − n α

β

c1− β/α,(3.5)where

A =2− β/α if 2n − j −1 is even, A =2−1 if 2n − j −1 is odd. (3.6)Define the constantsk1andk2by

(j − n)! 1/α

1 + (j − n)/α

···n + ( j − n)/α, i =1, 2, , (3.7)where

LetY denote the set

t ≥ t0,

Ᏺjy(t) =0, t ∗ ≤ t ≤ t0,

(3.11)and forj = n,

It can be verified thatᏲjmapsY continuously into a relatively compact subset of Y

First, we can show thatᏲj(Y ) ⊂ Y by using the expression

subinterval of [t ∗,∞) Then, by virtue of the Lebesgue convergence theorem it followsthat the sequence{Ᏺjy m(t)}converges toᏲjy0(t) on compact subintervals of [t ∗,∞),which implies the continuity of the mappingᏲj Finally, since the setsᏲj(Y ) andᏲ

j(Y )

= {(Ᏺjy) :y ∈ Y }are locally bounded on [t ∗,∞), the Arzel´a theorem implies thatᏲj(Y )

is relatively compact inC[t ∗,∞) Thus, all the hypotheses of the Schauder-Tychonoff fixedpoint theorem are satisfied, and so there exists a y ∈ Y such that y =Ᏺj y In view of

(3.11) and (3.12) the fixed elementy = y(t) is a solution of the integral equation which is

a special case of (2.9) withζ(t) =0,ξ j( t) =(c/( j − n)!)(t − t0)j − nas well as it is a specialcase as of (2.10) withζ(t) =0,ω n = c By di fferentiation of these integral equations 2n

times, we see thaty(t) is a solution of the differential equation (A) on [t ∗,∞) satisfying

L j y(∞)= c, that is, y ∈ P(I j).

Case 2 Let j ∈ {0, 1, , n −1}and suppose that (3.1) is satisfied Letc > 0 be any given

constant and chooset0> 0 so that

B =2− β/α if 2n − j −1 is even, B =2−1 if 2n − j −1 is odd. (3.15)Define the constantsk1andk2as follows:

k1= c j!, k2=2c

and define the setY by (3.10) with thesek1,k2 We define the mappingᏲj:Y → C[t ∗,∞)

in the following manner: forj ∈ {1, 2, , n −1},

relatively compact inC[t ∗,∞) Consequently, there exists a fixed elementy ∈ Y such that

y =Ᏺj y, which is the integral equation (2.13) withω0= c for j =0 as well as it is theintegral equation (2.12) withζ ∗ j(t) =(c/ j!)(t − t0)jforj ∈ {1, 2, , n −1} It is clear thatthe fixed element y = y(t) is a solution of (A) belonging toP(I j) This completes the

Unlike the solutions of classP(I) it seems to be very difficult (or impossible) to acterize the existence of solutions of classP(II), and we will be content to give sufficientconditions under which (A) possesses such solutions

char-Theorem 3.2 (i) Let k be an odd integer less than n Equation ( A ) has a solution of class P(II k) if

Proof (i) Let k be an odd integer less than n The desired solution y(t) will be obtained

as a solution of the integral equation

Ᏻk is continuous in the topology of C[t ∗,∞) and that Ᏻk(Y ) is relatively compact in C[t ∗,∞), there exists a fixed element y ofᏳk in Y Repeated differentiation of (3.26)shows that

wherec > 0 is an arbitrary fixed constant Define the mappingᏳn:Y → C[t ∗,∞), withthe setY defined by (3.25), in the following way:

mappingᏳn, which clearly satisfiescϕ n −1(t) ≤ y(t) ≤2cϕ n( t) for t ≥ t ∗ Likewise we canshow thatL n −1 y(∞)= ∞andL n y(∞)=0, which implies thaty(t) ∈ P(II k).

(iii) Letk be an odd integer greater than n and less than 2n In this case, we let c > 0

and chooset0≥0 large enough so thatt ∗ =min{t0, inft ≥ t0g(t)} ≥1 and



···



n + k − n α

4 Comparison theorems

In order to establish criteria (preferably sharp) for all solutions of (A) to be oscillatory, weare essentially based on the following oscillation result of Tanigawa [7] for the even-ordernonlinear differential equation (B

Theorem 4.1 (i) Let α > β All solutions of ( B ) are oscillatory if and only if

(ii)h, k, and l are continuously differentiable functions on [0,∞) such thath (t) > 0,

k (t) > 0, l (t) > 0, lim t →∞ h(t) =limt→∞ k(t) =limt→∞ l(t) = ∞;

(iii)F and G are continuous functions on [0,∞)× Rsuch thatuF(t, u) ≥0,uG(t, u) ≥

0 andF(t, u), G(t, u) are nondecreasing in u for any fixed t ≥0

Theorem 4.2 Suppose that

h(t) ≥ k(t), t ≥0,

F(t, x) sgn x ≥ G(t, x) sgn x, (t, x) ∈[0,∞)× R. (4.7)

If all the solutions of ( 4.5 ) are oscillatory, then so are all the solutions of ( 4.4 ).

Theorem 4.3 Suppose that l(t) ≥ h(t) for t ≥ 0 If all the solutions of ( 4.6 ) are oscillatory, then so are all the solutions of ( 4.4 ).

These theorems can be regarded as generalizations of the main comparison principlesdeveloped in the papers [3,5] to differential equations involving higher-order nonlin-ear differential operators To prove these theorems we need the following lemma whichcompares the differential equation (4.4) with the differential inequality

providedt1> 0 is su fficiently large Put t ∗ =min{t1, inft ≥ t1h(t)} Let us now consider theset

Define the mappingᏴkby

Define the mappingᏴkby

Ᏼku(t) = z

t1

+

clearly satisfies the integral equations (4.13), (4.16), and (4.19) on [t ∗,∞), respectively,that is,

a positive solution of (4.4) This completes the proof ofLemma 4.4 

Proof of Theorem 4.2 It is sufficient to prove that if (4.4) has an eventually positive tion, then so does (4.5)

solu-Letu(t) be an eventually positive solution of (4.4) Note thatu(t) is monotone

in-creasing for all sufficiently large t In view of (4.7), we see that there existst0> 0 such that u(h(t)) ≥ u(k(t)), t ≥ t0, and

This together yields

Proof of Theorem 4.3 The statement of the theorem is equivalent to the statement that if

there exists an eventually positive solution of (4.4) then the same is true of (4.6)

Letu(t) be an eventually positive solution of (4.4) The following inequalities are sible for some oddk ∈ {1, 3, , 2n −1}:



h −1

l(ρ),u

Ifk = n, then u(t) satisfies the inequality

If 1≤ k < n, then u(t) satisfies the inequality

of the corresponding integral inequalities (4.12), (4.15), and (4.18) Proceeding here in

a similar way, on the basis thatu(t) satisfies (4.28), (4.29), and (4.30), respectively, weconclude that there exists a positive solution for each of the following equations:

for 1≤ k < n It can be checked by di fferentiation that w(t) is a positive solution of

the differential equation (4.6) in each of the three cases This completes the proof of

5 Oscillation criteria

The aim of this section is to establish criteria (preferably sharp) for all solutions of (A)

to be oscillatory Oscillation theorems will be established first in the sublinear case of (A)forα > β as well as in the superlinear case for α < β We first give the sufficient conditionfor all of solutions of sublinear equation (A) to be oscillatory

Theorem 5.1 Let α > β Suppose that there exists a continuously differentiable function

h : [0, ∞)→(0,∞ ) such that h (t) > 0, lim t →∞ h(t) = ∞, and

then all solutions of ( A ) are oscillatory.

Proof Let us consider the equations

It will be shown below that there is a class of sublinear equations of the type (A) forwhich the oscillation situation can be completely characterized

Theorem 5.2 Let α > β and suppose that

Proof That the oscillation of (A) implies (5.7) is an immediate consequence ofTheorem3.1.

Assume now that (5.7) is satisfied The condition (5.6) means that there exists a stantc > 1 such that

has only oscillatory solutions Comparison of (A) with (5.11) viaTheorem 5.2then leads

Oscillation criteria for (A) in the superlinear case are given in the following theorems

Theorem 5.3 Let α < β Suppose that there exists a continuously di fferentiable function

h : [0, ∞)→(0,∞ ) such that h (t) > 0, lim t →∞ h(t) = ∞, and ( 5.1 ) is satisfied If

then all solutions of ( A ) are oscillatory.

The proof ofTheorem 5.3is similar to the proof ofTheorem 5.1, so it will be omitted

Theorem 5.4 Let α < β and suppose that

lim inf

t →∞

g(t)

Then, all solutions of ( A ) are oscillatory if and only if either ( 4.2 ) or ( 4.3 ) holds.

Proof We need only to prove the “if ” part of the theorem, since the “only if ” part follows

immediately fromTheorem 3.1

In view of (5.14) there exists a positive constantc < 1 such that

u(n)(t)α

sgnu(n)(t) (n)

+q(t)u(ct)β

sgnu(ct) =0, (5.19)and to conclude that (5.19) has the same oscillatory behavior as (5.16) Since (5.15) holds,applying another comparison principle,Theorem 5.1, we conclude that all the solutions

of (A) are necessarily oscillatory This completes the proof 

From the proofs of Theorems5.2and5.4we see that in caseα > β or α < β, the

oscil-lation of the functional differential equation

Example 6.1 Consider the equation

whereα, β, γ are fixed positive constants and λ is a varying parameter.

It is easy to check that, written for (6.1),

(3.1) is equivalent toλ > n + α(n − j) + βγ j, (6.2)(3.2) is equivalent toλ > 2n − j +



n + j − n α



so that fromTheorem 3.1we see that (6.1) has a positive solution belonging to the class

P(I j) if and only if

λ > n + α(n − j) + βγ j, j ∈ {0, 1, , n −1},

λ > 2n − j +



n + j − n α

α > βγ The conclusions which follow fromTheorem 3.2are

(i) (6.1) has solutions ofP(II k) (1 ≤ k ≤ n) if

α > βγ, n + α(n − k) + βγk < λ ≤ n + α(n − k) + βγk + α − βγ; (6.7)(ii) (6.1) has solutions ofP(II k) (n < k ≤2n −1) if

α > βγ, 2n − k + βγ



n + k − n α



< λ ≤2n − k + βγ



n + k − n α

+ 1− βγ

α . (6.8)

We now want oscillation criteria for (6.1)

Suppose thatα > β If γ ≤1, then fromTheorem 5.2we conclude that all solutions of(6.1) are oscillatory if and only if

Suppose thatα < β If γ > 1, then fromTheorem 5.4we conclude that all solutions of(6.1) are oscillatory if and only if

The research of the first author was done during her six-month stay as a Visiting Scholar

at the Department of Applied Mathematics of the Fukuoka University in Japan, supported

by the Matsumae International Foundation She wishes to express her sincere gratitudefor warm hospitality of the host scientist, Professor Naoki Yamada The second author’sresearch was supported in part by Grant-in-Aid for Young Scientist (B) (no 16740084)

by the Ministry of Education, Culture, Sports, Science, and Technology, Japan

References

[1] I T Kiguradze, On the oscillatory character of solutions of the equation d m u/dt m+a(t) | u | nsignu =

0, Matematicheskii Sbornik New Series 65 (107) (1964), 172–187 (Russian).

[2] T Kusano and B S Lalli, On oscillation of half-linear functional-di fferential equations with

devi-ating arguments, Hiroshima Mathematical Journal 24 (1994), no 3, 549–563.

[3] T Kusano and M Naito, Comparison theorems for functional-di fferential equations with deviating

arguments, Journal of the Mathematical Society of Japan 33 (1981), no 3, 509–532.

[4] T Kusano and J Wang, Oscillation properties of half-linear functional-di fferential equations of the

second order, Hiroshima Mathematical Journal 25 (1995), no 2, 371–385.

[5] W E Mahfoud, Comparison theorems for delay di fferential equations, Pacific Journal of

Mathe-matics 83 (1979), no 1, 187–197.

[6] T Tanigawa, Oscillation and nonoscillation theorems for a class of fourth order quasilinear tional differential equations, Hiroshima Mathematical Journal 33 (2003), no 3, 297–316.

func-[7] , Oscillation criteria for a class of higher order nonlinear di fferential equations, Memoirs

on Differential Equations and Mathematical Physics 37 (2006), 137–152.

[8] T Tanigawa and W Fentao, On the existence of positive solutions for a class of even order ear differential equations, Advances in Mathematical Sciences and Applications 14 (2004), no 1,

E-mail address:jelenam@pmf.ni.ac.yu

Tomoyuki Tanigawa: Department of Mathematics, Faculty of Science Education,

Joetsu University of Education, Niigata 943-8512, Japan

E-mail address:tanigawa@juen.ac.jp

Mathe-matics 83 (1979), no 1, 187–197.

[6] T Tanigawa, Oscillation and nonoscillation theorems for a class of fourth order quasilinear tional differential equations,...

[3] T Kusano and M Naito, Comparison theorems for functional- di fferential equations with deviating

arguments, Journal of the Mathematical Society of Japan... class= "text_page_counter">Trang 14

Define the mappingᏴkby