Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 105, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
FOR SECOND-ORDER NONLINEAR INTEGRO-DYNAMIC
EQUATIONS ON TIME SCALES
RAVI P. AGARWAL, SAID R. GRACE, DONAL O’REGAN, AĞACIK ZAFER
Abstract. In this article, we study the asymptotic behavior of non-oscillatory
solutions of second-order integro-dynamic equations as well as the oscillatory
behavior of forced second order integro-dynamic equations on time scales. The
results are new for the continuous and discrete cases. Examples are provided
to illustrate the relevance of the results.
1. Introduction
We are concerned with the asymptotic behavior of non-oscillatory solutions of
the second-order integro-dynamic equation on time scales of the form
Z t
(r(t)(x∆ (t))α )∆ +
a(t, s)F (s, x(s))∆s = 0
(1.1)
0
and the oscillatory behavior of the second-order forced integro-dynamic equation
Z t
(r(t)(x∆ (t)))∆ +
a(t, s)F (s, x(s))∆s = e(t).
(1.2)
0
We take T ⊆ R+ = [0, ∞) to be an arbitrary time-scale with 0 ∈ T and sup T =.
By t ≥ s we mean as usual t ∈ [s, ∞) ∩ T.
We shall assume throughout that:
(i) e, r : T → R and a : T × T → R are rd-continuous and r(t) > 0, and
a(t, s) ≥ 0 for t > s, α is the ratio of positive odd integers and
Z t0
sup
a(t, s)∆s := k < ∞, t0 ≥ 0;
(1.3)
t≥t0
0
(ii) F : T × R → R is continuous and assume that there exist continuous
functions f1 , f2 : T × R → R such that F (t, x) = f1 (t, x) − f2 (t, x) for t ≥ 0;
(iii) there exist constants β and γ being the ratios of positive odd integers and
functions pi ∈ Crd (T, (0, ∞)), i = 1, 2, such that
xf1 (t, x) ≥ p1 (t)xβ+1
for x 6= 0 and t ≥ 0,
γ+1
for x 6= 0 and t ≥ 0.
xf2 (t, x) ≤ p2 (t)x
2000 Mathematics Subject Classification. 34N05, 45D05, 34C10.
Key words and phrases. Integro-dynamic equation; oscillation; time scales.
c
2014
Texas State University - San Marcos.
Submitted September 11, 2013. Published April 15, 2014.
1
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R. P. AGARWAL, S. R. GRACE, D. O’REGAN, A. ZAFER
EJDE-2014/105
We consider only those solutions of equation (1.1) (resp, (1.2)) which are nontrivial and differentiable for t ≥ 0. The term solution henceforth applies to such
solutions of equation (1.1). A solution x is said to be oscillatory if for every t0 > 0
we have inf t≥t0 x(t) < 0 < supt≥t0 x(t) and it is said to be non-oscillatory otherwise.
Dynamic equations on time-scales is a fairly new topic. For general basic ideas
and background, we refer the reader to the seminal book [2].
Although the oscillation and nonoscillation theory of differential equations and
difference equations is well developed, the problem for integro-differential equations
of Volterra type was discussed only in a few papers in the literature, see [3, 7, 10,
8, 9, 11] and their references. We refer the reader to [4, 5] for some initial papers
on the oscillation and nonoscillation of integro-dynamic and integral equations on
time scales.
To the best of our knowledge, there are no results on the asymptotic behavior of
non-oscillatory solutions of (1.1) and the oscillatory behavior of (1.2). Therefore,
the main goal of this article is to establish some new criteria for the asymptotic
behavior of non-oscillatory solutions of equation (1.1) and the oscillatory behavior
of equation (1.2).
2. Asymptotic behavior of the non-oscillatory solutions of (1.1)
In this section we study the asymptotic behavior of all non-oscillatory solutions
of equation (1.1) with all possible types of nonlinearities. We will employ the
following two lemmas, the second of which is actually a consequence of the first.
Lemma 2.1 (Young inequality [6]). Let X and Y be nonnegative real numbers,
1
= 1. Then
n > 1 and n1 + m
XY ≤
1
1 n
X + Y m.
n
m
Equality holds if and only if X = Y .
Lemma 2.2 ([1]). If X and Y are nonnegative real numbers, then
X λ + (λ − 1)Y λ − λXY λ−1 ≥ 0
λ
λ
X − (1 − λ)Y − λXY
λ−1
≤0
for λ > 1,
(2.1)
for λ < 1,
(2.2)
where the equality holds if and only if X = Y .
We define
Z t
s 1/α
∆s,
R(t, t0 ) =
t0 r(s)
Note that due to monotonicity
t > t0 ≥ 0.
lim R(t, t0 ) 6= 0.
t→∞
(2.3)
Our first result is the following.
Theorem 2.3. Let conditions (i)–(iii) hold with γ = 1 and β > 1 and suppose
Z t
Z vZ u
1/α
β
1
1
1
lim
a(u, s)p11−β (s)p2β−1 (s)∆s∆u
∆v < ∞ (2.4)
t→∞ R(t, t0 ) t
r(v) t0 t0
0
for some t0 ≥ 0. If x is a non-oscillatory solution of (1.1), then
x(t) = O(R(t, t0 )),
as t → ∞.
(2.5)
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OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
3
Proof. Let x be a non-oscillatory solution of equation (1.1). Hence x is either
eventually positive or eventually negative. First assume x is eventually positive,
say x(t) > 0 for t ≥ t1 for some t1 ≥ t0 . Using conditions (ii) and (iii) with β > 1
and γ = 1 in equation (1.1), for t ≥ t1 , we obtain
Z t1
Z t
∆
r(t)(x∆ (t))α ≤ −
a(t, s)F (s, x(s))∆s +
a(t, s)[p2 (s)x(s) − p1 (s)xβ ]∆s.
0
t1
(2.6)
−1/β
1/β
If we apply (2.1) with λ = β, X = p1 x, and Y = ( β1 p2 p1
β
1
β
1
) β−1 we have
p2 (t)x(t) − p1 (t)xβ (t) ≤ (β − 1)β 1−β p11−β (t)p2β−1 (t),
t ≥ t1 .
Substituting (2.7) into (2.6) gives
∆
r(t)(x∆ (t))α
Z t
Z t1
β
1
β
1−β
≤−
a(t, s)p11−β (s)p2β−1 (s)∆s
a(t, s)F (s, x(s))∆s + (β − 1)β
(2.7)
(2.8)
t1
0
for all t ≥ t1 ≥ 0. Let
m := max{|F (t, x(t))| : t ∈ [0, t1 ] ∩ T}.
By assumption (i), we have
Z t1
Z
−
a(t, s)F (s, x(s))∆s ≤
0
t1
a(t, s)|F (s, x(s))|∆s ≤ mk := b.
(2.9)
0
Hence from (2.8) and (2.9), we obtain
Z t
β
1
α ∆
β
∆
1−β
a(t, s)p11−β (s)p2β−1 (s)∆s .
r(t) x (t)
≤ b + (β − 1)β
t1
Integrating this inequality from t1 to t leads to
α
x∆ (t)
α β
Z Z
β
1
r(t1 ) x∆ (t1 ) t − t1
(β − 1)β 1−β t u
≤
+b
+
a(u, s)p11−β (s)p2β−1 (s)∆s∆u
r(t)
r(t)
r(t)
t1 t1
or
β
α
c0 t
(β − 1)β 1−β
x∆ (t) ≤
+
r(t)
r(t)
Z tZ
t1
u
1
β
a(t, s)p11−β (s)p2β−1 (s)∆s∆u
t1
where
r(t1 )|(x∆ (t1 ))α |
+ b.
t1
By employing the well-known inequality
(a1 + b1 )λ ≤ σλ aλ1 + bλ1
for a1 ≥ 0, b1 ≥ 0, and λ > 0,
c0 =
(2.10)
where σλ = 1 if λ < 1 and σλ = 2λ−1 if λ ≥ 1 we see that there exists positive
constants c1 and c2 depending on α such that
1 Z tZ u
1/α
β
1
t 1/α
x∆ (t) ≤ c1
+ c2
a(t, s)p11−β (s)p2β−1 (s)∆s∆u
.
r(t)
r(t) t1 t1
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R. P. AGARWAL, S. R. GRACE, D. O’REGAN, A. ZAFER
EJDE-2014/105
Integrating this inequality from t1 to t ≥ t1 , we obtain
|x(t)| ≤ |x(t1 )| + c1 R(t, t1 )
Z t
Z vZ u
1/α
β
1
1
a(u, s)p11−β (s)p2β−1 (s)∆s∆u
+ c2
∆v
t1 r(v) t1 t1
≤ |x(t1 )| + c1 R(t, t0 )
Z vZ u
Z t
1/α
β
1
1
a(u, s)p11−β (s)p2β−1 (s)∆s∆u
∆v.
+ c2
t0 r(v) t0 t0
(2.11)
Dividing both sides of (2.11) by R(t, t0 ) and using (2.3) and (2.4), we see that (2.5)
holds. The proof is similar if x is eventually negative.
Next, we present the following simple result.
Theorem 2.4. Let conditions (i) and (ii) hold with f2 = 0 and xf1 (t, x) > 0 for
x 6= 0 and t ≥ 0. If x is a non-oscillatory solution of equation (1.1), then (2.5)
holds.
Proof. Let x(t) be a non-oscillatory solution of equation (1.1) with f2 = 0. First
assume x is eventually positive, say x(t) > 0 for t ≥ t1 for some t1 ≥ t0 . From (1.1)
we find that
Z t
Z t1
∆
α ∆
(r(t)(x (t)) ) = −
a(t, s)f1 (s, x(s))∆s ≤
a(t, s)f1 (s, x(s))∆s.
0
0
Using (1.3) (see (2.9)) in the above inequality, we obtain (r(t)(x∆ (t))α )∆ ≤ b. The
rest of the proof is similar to that of Theorem 2.3 and hence is omitted.
Theorem 2.5. Let conditions (i)–(iii) hold with β = 1 and γ < 1 and suppose
Z t
Z vZ u
1/α
γ
1
1
1
a(u, s)p1γ−1 (s)p21−γ (s)∆s∆u
∆v < ∞ (2.12)
lim
t→∞ R(t, t0 ) t
r(v) t0 t0
0
for some t0 ≥ 0. If x is a non-oscillatory solution of equation (1.1), then (2.5)
holds.
Proof. Let x be a non-oscillatory solution of (1.1). First assume x is eventually
positive, say x(t) > 0 for t ≥ t1 for some t1 ≥ t0 . From conditions (ii) and (iii)
with β = 1 and γ < 1 in equation (1.1) we have
Z t1
Z t
∆
α ∆
r(t)(x (t))
≤−
a(t, s)F (s, x(s))∆s +
a(t, s)[p2 (s)xγ (s) − p1 (s)x]∆s
0
t1
(2.13)
for all t≥ t1 . Hence,
∆
r(t)(x (t))
α ∆
Z
t
≤b+
a(t, s)[p2 (s)xγ (s) − p1 (s)x]∆s,
t1
−1
1/γ
1
where b is as in (2.9). Applying (2.2) with λ = γ, X = p2 x and Y = ( γ1 p1 p2γ ) γ−1 ,
we obtain
γ
γ
1
p2 (t)xγ (t) − p1 (t)x(t) ≤ (1 − γ)γ 1−γ p1γ−1 (t)p21−γ (t),
t ≥ t1 .
Using (2.14) in (2.13) we have
Z t
γ
1
α ∆
γ
r(t) x∆ (t)
≤ b + (1 − γ)γ 1−γ
a(t, s)p1γ−1 (s)p21−γ (s)∆s
t1
(2.14)
t ≥ t1 .
EJDE-2014/105
OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
The rest of the proof is similar to that of Theorem 2.3 and hence is omitted.
5
Theorem 2.6. Let conditions (i)–(iii) hold with β > 1 and γ < 1 and assume that
there exists a positive rd-continuous function ξ : T → T such that
Z t
Z vZ u
1
1
a(u, s)
lim
t→∞ R(t, t0 ) t
r(v) t0 t0
0
(2.15)
1/α
1
1
β
γ
1−β
1−γ
β−1
γ−1
× c1 ξ
(s)p1 (s) + c2 ξ
(s)p2 (s) ∆s ∆u
∆v < ∞
γ
β
for some t0 ≥ 0, where c1 = (β − 1)β 1−β and c2 = (1 − γ)γ 1−γ . If x is a nonoscillatory solution of equation (1.1), then (2.5) holds.
Proof. Let x be a non-oscillatory solution of equation (1.1). First assume x is
eventually positive, say x(t) > 0 for t ≥ t1 for some t1 ≥ t0 . Using (ii) and (iii) in
equation (1.1) we obtain
Z t1
Z t
∆
r(t)(x∆ (t))α ≤ −
a(t, s)F (s, x(s))∆s +
a(t, s)[ξ(s)x(s) − p1 (s)xβ (s)] ∆s
Z
0
t
+
t1
a(t, s)[p2 (s)xγ (s) − ξ(s)x(s)] ∆s.
t1
As in the proof of Theorems 2.3 and 2.5, one can easily show that
∆
r(t)(x∆ (t))α
Z t1
≤−
a(t, s)F (s, x(s))∆s
0
Z t
i
h
1
1
γ
β
γ
β
+
a(t, s) (β − 1)β 1−β ξ β−1 (s)p11−β (s) + (1 − γ)γ 1−γ ξ 1−γ (s)p21−γ (s) ∆s.
t1
The rest of the proof is similar to that of Theorem 2.3 and hence is omitted.
Theorem 2.7. Let conditions (i)–(iii) hold with β > 1 and γ < 1 and suppose that
there exists a positive rd-continuous function ξ : T → T such that
Z t
Z vZ u
1/α
1
β
1
1
∆v < ∞
lim
a(u, s)ξ β−1 (s)p11−β (s) ∆s ∆u
t→∞ R(t, t0 ) t
r(v) t0 t0
0
and
1
t→∞ R(t, t0 )
lim
Z t
Z vZ u
1/α
1
γ
1
a(u, s)ξ γ−1 (s)p21−γ (s) ∆s ∆u
∆v < ∞
t0 r(v) t0 t0
for some t0 ≥ 0. If x is a non-oscillatory solution of equation (1.1), then (2.5)
holds.
For the cases when both f1 and f2 are superlinear (β > γ > 1) or else sublinear
(1 > β > γ > 0), we have the following result.
Theorem 2.8. Let conditions (i)–(iii) hold with β > γ and assume
Z t
Z vZ u
1/α
γ
β
1
1
lim
a(u, s)p1γ−β (s)p2β−γ (s) ∆s ∆u
∆v < ∞ (2.16)
t→∞ R(t, t0 ) t
r(v) t0 t0
0
for some t0 ≥ 0. If x is a non-oscillatory solution of equation (1.1), then (2.5)
holds.
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R. P. AGARWAL, S. R. GRACE, D. O’REGAN, A. ZAFER
EJDE-2014/105
Proof. Let x be a non-oscillatory solution of (1.1). First assume x is eventually
positive, say x(t) > 0 for t ≥ t1 for some t1 ≥ t0 . Using conditions (ii) and (iii) in
equation (1.1) we have
∆
r(t)(x∆ (t))α
Z t1
Z t
(2.17)
≤−
a(t, s)F (s, x(s))∆s +
a(t, s)[p2 (s)xγ (s) − p1 (s)xβ (s)] ∆s.
0
t1
By applying Lemma 2.1 with
n=
β
,
γ
X = xγ (s),
Y =
γp2 (s)
,
βp1 (s)
m=
m
β−γ
we obtain
β
γ p2 (s) γ γ
p1 (s)[xγ (s)
− (x (s))β/γ ]
γ
β p1 (s) β
1
β
= p1 (s)[XY − X n ]
γ
n
β
1 m
≤ p1 (s)
Y
γ
m
β
γ
β −γ γ
[ p2 (s)] β−γ (p1 (s)) γ−β .
=
γ
β
The rest of the proof is similar to that of Theorem 2.3 and hence is omitted.
p2 (s)xγ (s) − p1 (s)xβ (s) =
Remark 2.9. If in addition to the hypotheses of Theorems 2.3–2.8,
lim R(t, t0 ) < ∞,
t→∞
then every non-oscillatory solution of (1.1) is bounded.
Remark 2.10. The results given above hold for equations of the form
Z t
∆
α ∆
(r(t)(x (t)) ) +
a(t, s)F (s, x(s))∆s = e(t)
(2.18)
0
if the additional condition
Z t
Z v
1/α
1
1
lim
|e(s)| ∆s
∆v < ∞
t→∞ R(t, t0 ) t
r(v) t0
0
is satisfied.
3. Oscillation results for (1.2)
This section we study of the oscillatory properties of (1.2). For this end hypotheses (i) and (ii) are replaced by the assumptions:
(I) e, r : T → R and a : T × T → R are rd-continuous, r(t) > 0 and a(t, s) ≥ 0
for t > s and there exist rd-continuous functions k, m : T → R+ such that
a(t, s) ≤ k(t)m(s),
t≥s
(3.1)
with
Z
k1 := sup k(t) < ∞,
t≥0
t≥0
t
m(s)∆s < ∞.
k2 := sup
0
In this case condition (1.3) is satisfied with k = k1 k2 .
EJDE-2014/105
OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
7
(II) F : T × R → R is continuous and assume that there exists rd-continuous
function, q : T → (0, ∞) and a real number β with 0 < β ≤ 1 such that
xF (t, x) ≤ q(t)xβ+1 ,
for x 6= 0 and t ≥ 0.
(3.2)
In what follows
g± (t, p) = e(t) ∓ k1 (1 − β)β
β/(1−β)
Z
t
pβ/(β−1) (s)q(s)1/(1−β) m1/(1−β) (s)∆s, (3.3)
0
where 0 < β < 1, p ∈ Crd (T, (0, ∞)).
We first give sufficient conditions under which non-oscillatory solutions x of
equation (1.2) satisfy
x(t) = O(t), as t → ∞.
(3.4)
Theorem 3.1. Let 0 < β < 1, conditions (I) and (II) hold, assume the function
t/r(t) is bounded, and for some t0 ≥ 0,
Z ∞
s
∆s < ∞.
(3.5)
r(s)
t0
Let p ∈ Crd (T, (0, ∞)) such that
Z
∞
sp(s) ∆s < ∞.
(3.6)
t0
If
Z
Z u
1 t 1
lim sup
g− (s, p)∆s∆u < ∞,
t→∞ t t0 r(u) t0
Z
Z u
1 t 1
lim inf
g+ (s, p) ∆s ∆u > −∞,
t→∞ t t r(u) t
0
0
then every non-oscillatory solution x(t) of (1.2) satisfies
lim sup
t→∞
(3.7)
|x(t)|
< ∞.
t
Proof. Let x be a non-oscillatory solution of (1.1). First assume x is eventually
positive, say x(t) > 0 for t ≥ t1 for some t1 ≥ t0 .
Using condition (3.2) in (1.2) we have
Z t1
Z t
∆
β
∆
r(t)(x (t)) ≤ e(t) −
a(t, s)F (s, x(s))∆s +
a(t, s)q(s)x (s)∆s, (3.8)
0
t1
for t ≥ t1 . Let
c := max |F (t, x(t)| < ∞.
0≤t≤t1
By assumption (3.1), we obtain
Z t1
Z
−
a(t, s)F (s, x(s))∆s ≤ c
0
t1
a(t, s)∆s ≤ ck 1 k2 =: b,
t ≥ t1 .
0
Hence from (3.8) we have
r(t)(x∆ (t))
∆
Z
t
≤ e(t) + b + k1
β
[m(s)q(s)x (s) − p(s)x(s)]∆s
t1
Z
(3.9)
t
p(s)x(s)∆s,
+ k1
t1
t ≥ t1 .
8
R. P. AGARWAL, S. R. GRACE, D. O’REGAN, A. ZAFER
EJDE-2014/105
Applying (2.2) of Lemma 2.2 with
λ = β,
X = (qm)1/β x,
Y =
1
1
p(mq)−1/β β−1
β
we have
β
m(s)q(s)x (s) − p(s)x(s) ≤ (1−β)β β/(1−β) pβ/(β−1) (s)m1/(1−β) (s)q 1/(1−β) (s).
Thus, we obtain
r(t)(x∆ (t))
∆
Z
t
≤ g+ (t, p) + b+k 1
p(s)x(s)∆s for t ≥ t1 .
(3.10)
t1
Integrating (3.10) from t1 to t we have
Z t
Z tZ
∆
∆
r(t)x (t) ≤ r(t1 )x (t1 ) +
g+ (s, p)∆s + b(t − t1 ) + k1
t1
t1
u
p(s)x(s)∆s ∆u,
t1
(3.11)
for t ≥ t1 . Employing [10, Lemma 3] to interchange the order of integration, we
obtain
Z t
Z t
∆
∆
r(t)x (t) ≤ r(t1 )x (t1 ) +
g+ (s, p)∆s + b(t − t1 )+k 1 t
p(s)x(s)∆s, t ≥ t1
t1
t1
and so,
x∆ (t) ≤
r(t1 )x∆ (t1 )
1
+
r(t)
r(t)
Z
t
g+ (s)∆s+
t1
b(t − t1 ) k1 t
+
r(t)
r(t)
Z
t
p(s)x(s)∆s,
t ≥ t1 .
t1
Integrating this inequality from t1 to t and using (3.5) and the fact that the function
t/r(t) is bounded for t ≥ t1 , say by k3 we see that
Z t
Z u
Z t
1
1
∆s +
g+ (s)∆s∆u
x(t) ≤ x(t1 ) + r(t1 )x∆ (t1 )
t1 r(u) t1
t1 r(s)
Z t
Z tZ u
s
+b
∆s+k 1 k3
p(s)x(s)∆s∆u, t ≥ t1 .
t1 t1
t1 r(s)
Once again, using [10, Lemma 3] we have
Z t
Z u
Z t
1
1
x(t) ≤ x(t1 ) + r(t1 )x∆ (t1 )
∆s +
g+ (s) ∆s ∆u
t1 r(u) t1
t1 r(s)
Z t
Z t
s
+b
∆s+k 1 k3 t
p(s)x(s) ∆s, t ≥ t1
t1 r(s)
t1
(3.12)
and so,
Z t
x(s) x(t)
≤ c1 +c2
sp(s)
∆s, t ≥ t1 ;
(3.13)
t
s
t1
note (3.5) and (3.7), c2 = k1 k3 and c1 is an upper bound for
Z t
Z t
Z u
Z t
1h
1
1
s
x(t1 ) + r(t1 )x∆ (t1 )
∆s +
g+ (s) ∆s ∆u+b
∆s]
t
r(s)
r(u)
r(s)
t1
t1
t1
t1
for t ≥ t1 . Applying Gronwall’s inequality [2, Corollary 6.7] to inequality (3.13)
and then using condition (3.6) we have
lim sup
t→∞
x(t)
< ∞.
t
(3.14)
EJDE-2014/105
OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
9
If x(t) is eventually negative, we can set y = −x to see that y satisfies equation
(1.2) with e(t) replaced by −e(t) and F (t, x) replaced by −F (t, −y). It follows in
a similar manner that
−x(t)
lim sup
< ∞.
(3.15)
t
t→∞
The proof is complete.
Next, by employing Theorem 3.1 we present the following oscillation result for
equation (1.2).
Theorem 3.2. Let 0 < β < 1, conditions (I), (II), (3.5), (3.6), and (3.7) hold,
assume the function t/r(t) is bounded , and there is a function p ∈ Crd (T, (0, ∞))
such that (3.6) holds. If for every 0 < M < 1,
Z u
Z t
i
h
1
g− (s, p)∆s∆u = ∞,
lim sup M t +
t→∞
t0 r(u) t0
(3.16)
Z t
Z u
h
i
1
lim inf M t +
g+ (s, p)∆s∆u = −∞,
t→∞
t0 r(u) t0
then (1.2) is oscillatory.
Proof. Let x be a non-oscillatory solution of equation (1.2), say x(t) > 0 for t ≥ t1
for some t1 ≥ t0 . The proof when x(t) is eventually negative is similar. Proceeding
as in the proof of Theorem 3.1 we arrive at (3.12). Therefore,
Z t
Z u
Z ∞
1
1
∆
∆s +
g+ (s, p)∆s∆u
x(t) ≤ x(t1 ) + r(t1 )x (t1 )
t1 r(u) t1
t1 r(s)
Z ∞
Z ∞
s
x(s) +b
∆s+k1 k3 t
sp(s)
∆s, t ≥ t1 .
s
t1 r(s)
t1
Clearly, the conclusion of Theorem 3.1 holds. This together with (3.5) imply that
Z u
Z t
1
g+ (s, p)∆s∆u,
(3.17)
x(t) ≤ M1 + M t +
t1 r(u) t1
where M1 and M are positive real numbers. Note that we make M < 1 possible by
increasing the size of t1 . Finally, taking liminf in (3.17) as t→ ∞ and using (3.16)
result in a contradiction with the fact that x(t) is eventually positive.
Corollary 3.3. Let 0 < β < 1 and condition (I), (II), (3.5), and (3.6) hold, assume
the function t/r(t) is bounded, and for some t0 ≥ 0 suppose
Z
Z u
Z
Z u
1 t 1
1 t 1
lim sup
e(s) ∆s∆u < ∞, lim inf
e(s)∆s∆u > −∞
t→∞ t t r(u) t
t→∞ t t0 r(u) t0
0
0
(3.18)
and
Z
Z u
1 t 1
lim
pβ/(β−1) (s)q(s)1/(1−β) m1/(1−β) (s)∆s∆u < ∞.
(3.19)
t→∞ t t r(u) t
0
0
If for every 0 < M < 1,
Z u
i
1
e(s)∆s∆u = ∞,
t→∞
t0 r(u) t0
Z t
Z u
h
i
1
lim inf M t +
e(s)∆s∆u = −∞,
t→∞
t0 r(u) t0
Z
h
lim sup M t +
t
(3.20)
10
R. P. AGARWAL, S. R. GRACE, D. O’REGAN, A. ZAFER
EJDE-2014/105
then (1.2) is oscillatory.
Similar reasoning to that in the sublinear case guarantees the following theorems
for the integro-dynamic equation (1.2) when β = 1.
Theorem 3.4. Let β = 1, conditions (I), (II), (3.5) and (3.18) hold, assume the
function t/r(t) is bounded, and for some t0 ≥ 0 suppose
Z t
sm(s)q(s)∆s < ∞.
(3.21)
lim sup
t→∞
t0
Then every non-oscillatory solution of equation (1.2) satisfies
lim sup
t→∞
|x(t)|
< ∞.
t
Theorem 3.5. Let β = 1, conditions (I), (II), (3.5), (3.18), (3.20), and (3.21)
hold, assume the function t/r(t) is bounded. Then (1.2) is oscillatory.
Remark 3.6. We note that the results of Section 3 can be obtained by using
the hypothesis (i) with the additional assumption that the function a(t, s) is nonincreasing with respect to the first variable. In this case, k1 m(t) which appeared in
the proofs and m(t) which appeared in the statements of the theorems should be
replaced by a(t, t). The details are left to the reader.
4. Examples
As we already mentioned the results of the present paper are new for the cases
when T = R (the continuous case) or when T = Z (the discrete case).
Example 4.1. Consider the integro-differential equations
0 Z t
1
t
(x0 (t))3 +
[sa x5 (s) − x3 (s)]ds = 0,
2 + s2
t
t
0
t>0
(4.1)
and
1
(x0 (t))1/3
t2
0
Z
+
0
t
t
[sb x5/7 (s) − sc x3/7 (s)]ds = 0,
t2 + s2
t > 0,
(4.2)
where a, b, and c are nonnegative real numbers satisfying 3a < 2 and 3b − 2 < 5c ≤
3b.
For (4.1), take α = 3, r(t) = 1/t, a(t, s) = t/(t2 + s2 ), p1 (t) = ta , p2 (t) = 1,
β = 5, γ = 3, R(t, 0) = (3/5)t5/3 . Since
Z t Z v Z u
1/3
1
u2
−5/3
−3a/2
t
v
s
dsdu
dv
2
2
0
0 u 0 u +s
Z t Z v
1/3
≤ c1 t−5/3
v
u−3a/2 du
dv
0
0
= c2 t−a/2 ,
where c1 and c2 are certain constants, condition (2.16) holds.
For (4.2), take α = 1/3, r(t) = 1/t2 , a(t, s) = t/(t2 + s2 ), p1 (t) = tb , p2 (t) = tc ,
β = 5/7, γ = 3/7, R(t, 0) = (1/10)t10 . Condition (2.16) holds, because
Z t Z v Z u
3
1
u2
−10
−3a/2+5c/2
v2
s
dsdu
dv
t
2
2
0
0 u 0 u +s
EJDE-2014/105
OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
≤ d1 t−10
Z t
v2
v
Z
0
11
3
u−3b/2+5c/2 du dv
0
= d2 t−9b/2+15c/2 ,
where d1 and d2 are certain constants.
As a result, we may conclude from Theorem 2.8 that every non-oscillatory solution of (4.1) and of (4.2) satisfies x = O(t5/3 ) and x = O(t10 ), respectively, as
t → ∞.
Example 4.2. Consider the integro-differential equation
Z t
xβ (s)
((1 + t)3 x0 )0 +
ds = t4 sin t,
2
4
0 (t + 1)(s +)
(4.3)
where β = 1/3 or β = 1.
We observe that r(t) = (1 + t)3 , k(t) = 1/(t2 + 1), m(s) = 1/(s4 + 1), q(t) = 1,
e(t) = t4 sin t. Letting p(t) = m(t), we see that the integral appearing in the
definition of g± (t, p) given by (3.3) becomes bounded. It is then not difficult to
show that all conditions of Theorem 3.2 for β = 1/3 are satisfied. On the other
hand, all conditions of Theorem 3.5 for β = 1 are also satisfied. Therefore, every
solution of equation (4.3) is oscillatory for β = 1/3 and β = 1.
References
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(2005i:34001).
[2] M. Bohner, A. Peterson; Dynamic Equations on Time Scales. An Introduction with Applications, Birkhusser, Boston, 2001. MR1843232 (2002c:34002).
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[7] J. J. Levin; Boundedness and oscillation of some Volterra and delay equations, J. Differential
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equations, Hiroshima Math.J. 20 (1990), 223–229. MR1063361 (91g:45002).
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Ravi P. Agarwal
Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363,
USA
E-mail address: agarwal@tamuk.edu
Said R. Grace
Department of Engineering Mathematics, Faculty of Engineering, Cairo University,
Orman, Giza 12221, Egypt
E-mail address: saidgrace@yahoo.com
12
R. P. AGARWAL, S. R. GRACE, D. O’REGAN, A. ZAFER
EJDE-2014/105
Donal O’Regan
School of Mathematics, Statistics and Applied mathematics, National University of
Ireland, Galway, Ireland
E-mail address: donal.oregan@nuigalway.ie
Ağacik Zafer
College of Engineering and Technology, American University of the Middle East,
Block 3, Egaila, Kuwait
E-mail address: agacik.zafer@gmail.com