DIFFERENCE EQUATIONS
RAVI P. AGARWAL, SAID R. GRACE, AND DONAL O’REGAN Received 28 August 2004
We establish some new criteria for the oscillation of third-order difference equations of the form∆((1/a2(n))(∆(1/a1(n))(∆x(n))α1)α2) +δq(n)f(x[g(n)])=0, where∆is the for- ward difference operator defined by∆x(n)=x(n+ 1)−x(n).
1. Introduction
In this paper, we are concerned with the oscillatory behavior of the third-order difference equation
L3x(n) +δq(n)fxg(n)=0, (1.1;δ) whereδ= ±1,n∈N= {0, 1, 2,. . .},
L0x(n)=x(n), L1x(n)= 1 a1(n)
∆L0x(n)α1, L2x(n)= 1
a2(n)
∆L1x(n)α2, L3x(n)=∆L2x(n).
(1.2)
In what follows, we will assume that
(i){ai(n)},i=1, 2, and{q(n)}are positive sequences and ∞
ai(n)1/αi= ∞, i=1, 2; (1.3) (ii){g(n)}is a nondecreasing sequence, and limn→∞g(n)= ∞;
(iii) f ∈Ꮿ(R,R),x f(x)>0, andf(x)≥0 forx=0;
(iv)αi,i=1, 2, are quotients of positive odd integers.
The domainᏰ(L3) ofL3 is defined to be the set of all sequences{x(n)},n≥n0≥0 such that{Ljx(n)}, 0≤j≤3 exist forn≥n0.
A nontrivial solution{x(n)}of (1.1;δ) is called nonoscillatory if it is either eventually positive or eventually negative and it is oscillatory otherwise. An equation (1.1;δ) is called oscillatory if all its nontrivial solutions are oscillatory.
Copyright©2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 345–367 DOI:10.1155/ADE.2005.345
The oscillatory behavior of second-order half-linear difference equations of the form
∆ 1 a1(n)
∆x(n)α1+δq(n)fxg(n)=0, (1.4;δ)
whereδ,a1,q,g, f, andα1are as in (1.1;δ) and/or related equations has been the sub- ject of intensive study in the last decade. For typical results regarding (1.4;δ), we refer the reader to the monographs [1,2,4, 8, 12], the papers [3,6,11,15], and the ref- erences cited therein. However, compared to second-order difference equations of type (1.4;δ), the study of higher-order equations, and in particular third-order equations of type (1.1;δ) has received considerably less attention (see [9,10,14]). In fact, not much has been established for equations with deviating arguments. The purpose of this paper is to present a systematic study for the behavioral properties of solutions of (1.1;δ), and therefore, establish criteria for the oscillation of (1.1;δ).
2. Properties of solutions of equation (1.1;1) We will say that{x(n)}is of typeB0if
x(n)>0, L1x(n)<0, L2x(n)>0, L3x(n)≤0 eventually, (2.1) it is of typeB2if
x(n)>0, L1x(n)>0, L2x(n)>0, L3x(n)≤0 eventually. (2.2) Clearly, any positive solution of (1.1;1) is either of typeB0orB2.In what follows, we will present some criteria for the nonexistence of solutions of typeB0for (1.1;1).
Theorem2.1. Let conditions (i)–(iv) hold,g(n)< nforn≥n0≥0, and
−f(−xy)≥ f(xy)≥ f(x)f(y) forxy >0. (2.3) Moreover, assume that there exists a nondecreasing sequence{ξ(n)}such thatg(n)< ξ(n)
< nforn≥n0.If all bounded solutions of the second-order half-linear difference equation
∆ 1 a2(n)
∆y(n)α2
−q(n)f
ξ(n)
k=g(n)
a1/α1 1(k)
fy1/α1ξ(n)=0 (2.4)
are oscillatory, then (1.1;1) has no solution of typeB0.
Proof. Let{x(n)}be a solution of (1.1;1) of typeB0.There existsn0∈Nso large that (2.1) holds for alln≥n0.Fort≥s≥n0, we have
x(s)=x(t+ 1)− t j=s
a1/α1 1(j) 1
a1/α1 1(j)∆x(j)≥
t
j=s
a1/α1 1(j)
−L1/α1 1x(t). (2.5)
Replacingsandtbyg(n) andξ(n) respectively in (2.5), we have xg(n)≥
ξ(n)
j=g(n)
a1/α1 1(j)
−L1/α1 1xξ(n) (2.6) forn≥n1∈Nfor somen1≥n0.Now using (2.3) and (2.6) in (1.1;1) and lettingy(n)=
−L1x(n)>0 forn≥n1, we easily find
∆ 1 a2(n)
∆y(n)α2−q(n)f
ξ(n)
j=g(n)
a1/α1 1(j)
fy1/α1ξ(n)≥0 forn≥n1. (2.7)
A special case of [16, Lemma 2.4] guarantees that (2.4) has a positive solution, a contra-
diction. This completes the proof.
Theorem2.2. Let conditions (i)–(iv) and (2.3) hold, and assume that there exists a nonde- creasing sequence{ξ(n)}such thatg(n)< ξ(n)< nforn≥n0.Then, (1.1;1) has no solution of typeB0if either one of the following conditions holds:
(S1)
fu1/(α1α2)
u ≥1 foru=0, (2.8)
lim sup
n→∞
n−1 k=ξ(n)
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
f
ξ(n)
i=ξ(k)
a1/α2 2(i)
1/α1
>1, (2.9) (S2)
u
fu1/(α1α2)−→0 asu−→0, (2.10)
lim sup
n→∞
n−1 k=ξ(n)
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
f
ξ(n)
i=ξ(k)
a1/α2 2(i)
1/α1
>0. (2.11) Proof. Let {x(n)} be a solution of (1.1;1) of type B0. Proceeding as in the proof of Theorem 2.1to obtain the inequality (2.7), it is easy to check thaty(n)>0 and∆y(n)<0 forn≥n1.Letn2> n1be such that infn≥n2ξ(n)> n1.Now
y(σ)=y(τ+ 1)− τ j=σ
a1/α2 2(j) 1
a2(j)
∆y(j)α2 1/α2
≥
τ
j=σ
a1/α2 2(j)
1 a2(τ)
−∆y(τ)α2 1/α2
forτ≥σ≥n2.
(2.12)
Replacingσandτbyξ(k) andξ(n) respectively in (2.12), we have
yξ(k)≥
ξ(n)
j=ξ(k)
a1/α2 2(j)
1
a2
ξ(n)
−∆yξ(n)α2 1/α2
forn≥k≥n2. (2.13)
Summing (2.7) fromξ(n) to (n−1) and lettingY(n)=(−∆y(n))α2/a2(n) forn≥n2, we get
Yξ(n)≥Y(n) +
n−1 k=ξ(n)
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
×f
ξ(n)
i=ξ(k)
a1/α2 2(i)
Y1/α2ξ(n)
1/α1
forn≥n2.
(2.14)
Using condition (2.3) in (2.14), we have Yξ(n)≥fY1/(α1α2)ξ(n)
×
n−1 k=ξ(n)
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
f
ξ(n)
i=ξ(k)
a1/α2 2(i)
1/α1
, n≥n2. (2.15) Using (2.8) in (2.15) we have
1≥
n−1 k=ξ(n)
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
f
ξ(n)
i=ξ(k)
a1/α2 2(i)
1/α1
. (2.16)
Taking lim sup of both sides of the above inequality asn→ ∞, we obtain a contradiction to condition (2.9).
Next, using (2.10) in (2.15) and taking lim sup of the resulting inequality, we obtain a contradiction to condition (2.11). This completes the proof.
Theorem2.3. Let the hypotheses ofTheorem 2.2hold. Then, (1.1;1) has no solutions of type B0if one of the following conditions holds:
(O1)
f1/α2u1/α1
u ≥1 foru=0, (2.17)
lim sup
n→∞
n−1 k=ξ(n)
a1/α2 2(k)
n−1
j=k
q(j)f
ξ(j)
i=g(j)
a1/α1 1(i)
1/α2
>1, (2.18)
(O2)
u
f1/α2u1/α1−→0 asu−→0, (2.19) lim sup
n→∞
n−1 k=ξ(n)
a1/α2 2(k)
n−1
j=k
q(j)f
ξ(j)
i=g(j)
a1/α1 1(i)
1/α2
>0. (2.20)
Proof. Let{x(n)}be a solution of (1.1;1) of typeB0.As in the proof ofTheorem 2.1, we obtain the inequality (2.7) forn≥n1.Also, we see thaty(n)>0 and∆y(n)<0 forn≥n1. Next, we letn2≥n1 be as in the proof ofTheorem 2.2, and summing inequality (2.7) froms≥n2to (n−1), we have
1 a2(s)
−∆y(s)α2≥ 1 a2(n)
−∆y(n)α2+
n−1 k=s
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
fy1/α1ξ(k), (2.21) which implies
−∆y(s)≥a1/α2 2(s)
n−1
k=s
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
fy1/α1ξ(k)
1/α2
. (2.22)
Now,
y(v)=y(n) +
n−1 s=v
−∆y(s)≥
n−1 s=v
−∆y(s) forn−1≥s≥n2. (2.23)
Substituting (2.23) in (2.22) and settingv=ξ(n), we have
yξ(n)≥
n−1 s=ξ(n)
a1/α2 2(s)
n−1
k=s
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
fy1/α1ξ(k)
1/α2
≥f1/α2y1/α1ξ(n)
n−1 s=ξ(n)
a1/α2 2(s)
n−1
k=s
q(k)f
ξ(k)
j=g(k)
a1/α1 1(j)
1/α2
.
(2.24)
The rest of the proof is similar to that ofTheorem 2.2and hence is omitted.
Theorem2.4. Let conditions (i)–(iv), (2.3) hold,g(n)=n−τ,whereτis a positive integer and assume that there exist two positive integers such thatτ > τ >τ.˜ If the first-order delay equation
∆y(n) +q(n)f
n−τ
j=n−τ
a1/α1 1(j)
f
n−˜τ
i=n−τ
a1/α2 2(i)
1/α1
fy1/(α1α2)[n−τ]˜ =0 (2.25) is oscillatory, then (1.1;1) has no solution of typeB0.
Proof. Let{x(n)}be a solution of (1.1;1) of typeB0.As in the proof ofTheorem 2.1, we obtain (2.6) forn≥n1, which takes the form
x[n−τ]≥
n−τ
j=n−τ
a1/α1 1(j)
−L1/α1 1x[n−τ] forn≥n1. (2.26)
Similarly, we find
−L1x[n−τ]≥
n−τ˜
i=n−τ
a1/α2 2(i)
L1/α2 2x[n−τ]˜ forn≥n2≥n1. (2.27)
Combining (2.26) with (2.27) we have
x[n−τ]≥
n−τ
j=n−τ
a1/α1 1(j)
n−τ˜
i=n−τ
a1/α2 2(i)
1/α1
L1/(α2 1α2)x[n−τ]˜ forn≥n3≥n2. (2.28) Using (2.3) and (2.28) in (1.1;1) and settingZ(n)=L2x(n), we have
∆Z(n) +q(n)f
n−τ
j=n−τ
a1/α1 1(j)
f
n−τ˜
i=n−τ
a1/α2 2(i)
1/α1
×fZ1/(α1α2)[n−τ]˜ ≤0 forn≥n3.
(2.29)
By a known result in [2,12], we see that (2.25) has a positive solution which is a contra-
diction. This completes the proof.
As an application ofTheorem 2.4, we have the following result.
Corollary2.5. Let conditions (i)–(iv), (2.3) hold,g(n)=n−τ,τis a positive integer and let there exist two positive integersτ,τ˜such thatτ > τ >τ.˜ Then, (1.1;1) has no solution of typeB0if either one of the following conditions holds:
(I1)in addition to (2.8), lim inf
n→∞
n−1 k=n−τ
q(k)f
k−τ
j=k−τ
a1/α1 1(j)
f
k−τ˜
i=k−τ
a1/α2 2(i)
1/α1
>
τ˜ 1 + ˜τ
τ+1˜
, (2.30) (I2)
±0
du
fu1/(α1α2)<∞, (2.31) ∞
k=n0
q(k)f
k−τ
j=k−τ
a1/α1 1(j)
f
k−τ˜
i=k−τ
a1/α2 2(i)
1/α1
= ∞. (2.32)
Next, we will present some criteria for the nonexistence of solutions of typeB2 of (1.1;1).
Theorem2.6. Let conditions (i)–(iv) and (2.3) hold. If ∞
q(j)f
g(j)−1
i=n0
a1/α1 1(i)
= ∞, (2.33)
then (1.1;1) has no solution of typeB2.
Proof. Let{x(n)}be a solution of (1.1;1). There exists an integern0∈N so large that (2.2) holds forn≥n0.From (2.2), there exist a constantc >0 and an integern1≥n0such that
1 a1(n)
∆L0x(n)α1=L1x(n)≥c, (2.34) or
∆x(n)≥
ca1(n)1/α1 forn≥n1. (2.35) Summing (2.35) fromn1tog(n)−1(≥n1) we obtain
xg(n)≥c1/α1
g(n)−1 j=n1
a1/α1 1(j). (2.36)
Using (2.3) and (2.36) in (1.1;1) we have
−L3x(n)=q(n)fx[g(n)]
≥q(n)fc1/α1f
g(n)− 1 j=n1
a1/α1 1(j)
forn≥n2≥n1. (2.37)
Summing (2.37) fromn2ton−1(> n2) we obtain
∞> L2x(n2)≥ −L2x(n) +L2x(n2)
≥fc1/α1
n−1 k=n2
q(k)f
g(k)−1
j=n1
a1/α1 1(j)
−→ ∞ asn−→ ∞, (2.38)
a contradiction. This completes the proof.
Theorem2.7. Let conditions (i)–(iv) and (2.3) hold, andg(n)=n−τ,n≥n0≥0,where τis a positive integer. If the first-order delay equation
∆y(n) +q(n)f
n−τ−1 k=n0
a1(k)
k−1 j=n0
a1/α2 2(j)
1/α1
fy1/(α1α2)[n−τ]=0 (2.39)
is oscillatory, then (1.1;1) has no solution of typeB2.
Proof. Let{x(n)}be a solution of (1.1;1) of typeB2.There exists an integern0≥0 so large that (2.2) holds forn≥n0.Now,
L1x(n)=L1x(n0) +
n−1 j=n0
∆L1x(j)
=L1x(n0) +
n−1 j=n0
a1/α2 2(j)a−21/α2(j)∆L1x(j)
=L1x(n0) +
n−1 j=n0
a1/α2 2(j)L1/α2 2x(j)
≥L1/α2 2x(n)
n−1 j=n0
a1/α2 2(j) forn≥n1,
(2.40)
or
1 a1(n)
∆x(n)α1≥L1/α2 2x(n)
n−1 j=n0
a1/α2 2(j). (2.41) Thus,
∆x(n)≥
a1(n)
n−1 j=n0
a1/α2 2(j)
1/α1
L1/(α2 1α2)x(n) forn≥n0. (2.42) Summing (2.42) fromn0tog(n)−1> n0, we have
xg(n)≥
g(n)−1 k=n0
a1(k)
k−1 j=n0
a1/α2 2(j)
1/α1
L1/(α2 1α2)xg(n) forn≥n1≥n0. (2.43)
Using (2.3), (2.43),g(n)=n−τ, and lettingy(n)=L2x(n),n≥n1, we obtain
∆y(n) +q(n)f
k−τ−1 k=n0
a1(k)
k−1 j=n0
a1/α2 2(j)
1/α1
fy1/(α1α2)[n−τ]≤0. (2.44)
The rest of the proof is similar to that ofTheorem 2.4and hence is omitted.
Theorem2.8. Let conditions (i)–(iv) and (2.3) hold andg(n)> n+ 1forn≥n0∈N.If the half-linear difference equation
∆ 1 a2(n)
∆y(n)α2+q(n)f
g(n)−1
j=n
a1/α1 1(j)
fy1/α1(n)=0 (2.45)
is oscillatory, then (1.1;1) has no solution of typeB2.
Proof. Let{x(n)}be a solution of (1.1;1) of typeB2.Then there exists ann0∈Nsuffi- ciently large so that (2.2) holds forn≥n0.Now, form≥s≥n0we get
x(m)−x(s)=
m−1 j=s
a1/α1 1(j)L1/α1 1x(j), (2.46) or
x(m)≥
m−1
j=s
a1/α1 1(j)
L1/α1 1x(s). (2.47)
Replacingmandsin (2.47) byg(n) andn, respectively, we have xg(n)≥
g(n)−1
j=n
a1/α1 1(j)
L1/α1 1x(n) forg(n)≥n+ 1≥n1≥n0. (2.48)
Using (2.3) and (2.48) in (1.1;1) and lettingZ(n)=L1x(n) forn≥n1, we obtain
∆ 1 a2(n)
∆Z(n)α2+q(n)f
g(n)−1
j=n
a1/α1 1(j)
fZ1/α1(n)≤0 forn≥n1. (2.49)
By [16, Lemma 2.3], we see that (2.45) has a positive solution, a contradiction. This com-
pletes the proof.
Remark 2.9. We note that a corollary similar to Corollary 2.5 can be deduced from Theorem 2.7. Here, we omit the details.
Remark 2.10. We note that the conclusion of Theorems2.1–2.4can be replaced by “all bounded solutions of (1.1;1) are oscillatory.”
Next, we will combine our earlier results to obtain some sufficient conditions for the oscillation of (1.1;1).
Theorem2.11. Let conditions (i)–(iv) and (2.3) hold,g(n)< nforn≥n0∈N. Moreover, assume that there exists a nondecreasing sequence{ξ(n)}such thatg(n)< ξ(n)< nforn≥ n0.If either conditions (S1) or (S2) ofTheorem 2.2and condition (2.33) hold, the equation (1.1;1) is oscillatory.
Proof. Let{x(n)}be a nonoscillatory solution of (1.1;1), say,x(n)>0 for n≥n0∈N. Then,{x(n)}is either of typeB0orB2.ByTheorem 2.2,{x(n)}is not of typeB0and by Theorem 2.6,{x(n)}is not of typeB2.This completes the proof.
Theorem2.12. Let conditions (i)–(iv), (2.3) hold,g(n)=n−τ,n≥n0∈N,whereτ is a positive integer. Moreover, assume that there exist two positive integersτandτ˜such that τ > τ >τ.˜ If both first-order delay equations (2.25) and (2.39) are oscillatory, then (1.1;1) is oscillatory.
Proof. The proof follows from Theorems2.4and2.7.
Next, we will apply Theorems2.11and2.12to a special case of (1.1;1), namely, the equation
∆ 1 a2(n)
∆ 1 a1(n)
∆x(n)α1 α2
+q(n)xαg(n)=0, (2.50)
whereαis the ratio of positive odd integers.
Corollary2.13. Let conditions (i)–(iv) hold,g(n)< nforn≥n0∈N, and assume that there exists a nondecreasing sequence{ξ(n)}such thatg(n)< ξ(n)< nforn≥n0.Equation (2.50) is oscillatory if either one of the following conditions holds:
(A1)α=α1α2,
∞ j=n0≥0
q(j)
g(j)−1
i=n0
a1/α1 1(i)
α
= ∞, (2.51)
lim sup
n→∞
n−1 j=ξ(n)
q(j)
ξ(j)
i=g(j)
a1/α1 1(i)
α
ξ(n)
i=ξ(j)
a1/α2 2(i)
α2
>1, (2.52)
(A2)α < α1α2and condition (2.51) hold, and
lim sup
n→∞
n−1 j=ξ(n)
q(j)
ξ(j)
i=g(j)
a1/α1 1(i)
α
ξ(n)
i=ξ(j)
a1/α2 2(i)
α2
>0. (2.53)
Corollary2.14. Let conditions (i)–(iv) hold,g(n)=n−τ,n≥n0∈N,whereτis a pos- itive integer, and assume that there exist two positive integersτ,τ˜such that τ > τ >τ.˜ If
the first-order delay equations
∆y(n) +q(n)
n−τ
j=n−τ
a1/α1 1(j)
α
n−˜τ
i=n−τ
a1/α2 2(i)
α2
Zα/(α1α2)[n−τ]˜ =0, (2.54)
∆Z(n) +q(n)
n−τ−1 j=n0
a1(j)
j−1
i=n0
a1/α2 2(i)
1/α1
α
Zα/(α1α2)[n−τ]=0 (2.55) are oscillatory, then (2.50) is oscillatory.
For the mixed difference equations of the form L3x(t) +q1(t)f1
xg1(n)+q2(n)f2
xg2(n)=0, (2.56) whereL3 is defined as in (1.1;1),{ai(n)},i=1, 2 are as in (i) satisfying (1.3),α1andα2
are as in (iv),{qi(n)},i=1, 2 are positive sequences,{gi(n)},i=1, 2 are nondecreasing sequences with limn→∞gi(n)= ∞,i=1, 2,fi∈Ꮿ(R,R),x fi(x)>0 and fi(x)≥0 forx=0 andi=1, 2. Also,f1, f2satisfy condition (2.3) by replacing f byf1and/orf2.
Now, we combine Theorems2.1and2.8and obtain the following interesting result.
Theorem2.15. Let the above hypotheses hold for (2.56),g1(n)< nandg2(n)> n+ 1for n≥n0∈Nand assume that there exists a nondecreasing sequence{ξ(n)}such thatg1(n)<
ξ(n)< nforn≥n0.If all bounded solutions of the equation
∆ 1 a2(n)
∆y(n)α2
−q1(n)f1
ξ(n)
k=g1(n)
a1/α1 1(k)
f1
y1/α1ξ(n)=0 (2.57)
are oscillatory and all solutions of the equation
∆ 1 a2(n)
∆Z(n)α2
+q2(n)f2
g(n)−1
j=n
a1/α1 1(j)
f2
Z1/α1(n)=0 (2.58)
are oscillatory, then (2.56) is oscillatory.
3. Properties of solutions of equation (1.1;-1) We will say that{x(n)}is of typeB1if
x(n)>0, L1x(n)>0, L2x(n)<0, L3x(n)≥0 eventually, (3.1) it is of typeB3if
x(n)>0, Lix(n)>0, i=1, 2, L3x(n)≥0 eventually. (3.2) Clearly, any positive solution of (1.1;-1) is either of typeB1orB3.In what follows, we will give some criteria for the nonexistence of solutions of typeB1for (1.1;-1).
