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Applications of Opial inequalities on time scales on dynamic equations with damping terms
Mathematical and Computer Modelling 58 (2013) 1777–1790
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Applications of Opial inequalities on time scales on dynamic equations with damping terms S.H. Saker Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
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Article history: Received 21 November 2011 Received in revised form 7 April 2013 Accepted 30 April 2013 Keywords: Opial’s inequality Disfocality Disconjugacy Oscillation Dynamic equations Time scales
abstract In this paper, we will employ some dynamic inequalities of Opial’s type on time scales to prove several results related to the spacing between consecutive zeros of a solution of a second order dynamic equation with a damping term. We also obtain several results related to the spacing between a zero of the solution and/or a zero of its derivative. As a special case, we will establish some new results for differential and difference equations with damping terms. For illustration, we will derive some well-known results obtained for differential equations without damping terms. The results yield conditions for disfocality and disconjugacy for dynamic equations with damping terms on time scales. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The study of dynamic equations on time scales which goes back to its founder Stefan Hilger [1] becomes an area of mathematics and recently has received a lot of attention. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale T, which may be an arbitrary closed subset of the real numbers R. In this paper, we consider the second-order half-linear dynamic equation with a damping term
r (t ) x∆ (t )
γ ∆
γ + p(t ) x∆ (t ) + q(t ) (xσ (t ))γ = 0,
t ∈ [α, β]T ,
(1.1)
where T is an arbitrary time scale and σ (t ) is the forward jump operator on T which is defined by σ (t ) := inf{s ∈ T : s > t }. The main aim is to apply some dynamic inequalities of Opial’s type on time scales to prove several results related to the problems: (i) obtain lower bounds for the spacing β − α where x is a solution of (1.1) and satisfies x(α) = x∆ (β) = 0, or x∆ (α) = x(β) = 0, (ii) obtain lower bounds for the spacing of zeros of a solution of (1.1). Throughout the paper, we will assume that γ ≥ 1 is a quotient of odd positive integers, r, p and q are real rd-continuous functions defined on T with r (t ) > 0 and µ(t ) |p(t )| ≤ r (t )/c where c is a positive constant such that c ≥ 1. We also assume that sup T = ∞, and define the time scale interval [a, b]T by [a, b]T := [a, b] ∩ T. We say that a solution x of (1.1) has a generalized zero at t if x(t ) = 0, and has a generalized zero in (t, σ (t )) in case x(t )xσ (t ) < 0 and µ(t ) > 0. Eq. (1.1) is disconjugate on the interval [t0 , b]T , if there is no nontrivial solution of (1.1) with two (or more) generalized zeros in [t0 , b]T . The solution x(t ) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Eq. (1.1) is said to be oscillatory if all its solutions are oscillatory. For
E-mail address: [email protected]. URL: http://www.mans.edu.eg/pcvs/30040/. 0895-7177/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mcm.2013.04.006
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more details of oscillation theory of dynamic equations on time scales, we refer to the book [2]. We say that (1.1) is right disfocal (left disfocal) on [α, β]T if the solutions of (1.1) such that x∆ (α) = 0 (x∆ (β) = 0) have no generalized zeros in [α, β]T . For Eq. (1.1) the point β > α is called a right focal point of α if the solution of (1.1) with initial conditions x(α) ̸= 0, x∆ (α) = 0 satisfies x(β) = 0. Left focal point is defined similarly. We note that, Eq. (1.1) in its general form covers several different types of differential and difference equations depending on the choice of the time scale T. For example, when T = R, we have σ (t ) = t, µ(t ) = 0, x∆ (t ) = x′ (t ) and (1.1) becomes the second-order differential equation
(r (t )(x′ (t ))γ )′ + p(t )(x (t ))γ + q(t )xγ (t ) = 0. ′
(1.2)
∆
When T = Z, we have σ (t ) = t + 1, µ(t ) = 1, x (t ) = ∆x(t ) = x(t + 1) − x(t ) and (1.1) becomes the second-order difference equation
∆(r (t ) (∆x(t ))γ ) + p(t ) (∆x(t ))γ + q(t )xγ (t + 1) = 0.
(1.3)
When T = hZ, h > 0, we have σ (t ) = t + h, µ(t ) = h, x∆ (t ) = ∆h x(t ) =
x(t + h) − x(t ) h
,
and (1.1) becomes the second-order difference equation
∆h (r (t )(∆h x(t ))γ ) + p(t ) (∆h x(t ))γ + q(t )xγ (t + h) = 0.
(1.4)
When T = q = {t : t = q , k ∈ N, q > 1}, we have σ (t ) = q t, µ(t ) = (q − 1)t , k
N
x∆ (t ) = ∆q x(t ) =
x(qt ) − x(t )
(q − 1)t
,
and (1.1) becomes the second-order q-difference equation
γ ∆q (r (t )(∆q x(t ))γ ) + p(t ) ∆q x(t ) + q(t )xγ (qt ) = 0. √ √ When T = N20 = {t 2 : t ∈ N0 }, we have σ (t ) = ( t + 1)2 and µ(t ) = 1 + 2 t, √ x(( t + 1)2 ) − x(t ) ∆ N x( t ) = , √ 1+2 t
(1.5)
and (1.1) becomes the second-order difference equation
√ ∆N (r (t )(∆N x(t ))γ ) + p(t ) (∆N x(t ))γ + q(t )xγ (( t + 1)2 ) = 0.
(1.6)
Some special cases of Eq. (1.2) have been studied by some authors, we refer to the papers by Hartman and Wintner [3], Fink and Mary [4], Ha [5], Pachpatte [6], Lee et al. [7], Saker [8] and Tiryaki, Ünal and Çakmak [9]. In these papers the authors established some inequalities of Lyapunov’s type. These inequalities give implicit lower bounds on the distance between zeros of a nontrivial solution and also lower bounds for the spacing between zeros of a solution and/ or its derivative. On time scales, the best known existence results of Lyapunov’s type for a special case of (1.1) (when γ = 1 and r (t ) = 1 and p(t ) = 0) are due to Bohner et al. [10]. They extended the Lyapunov inequality obtained for differential equations in [11] to the dynamic equation x∆∆ (t ) + q(t )xσ (t ) = 0,
(1.7)
where q(t ) is a positive rd-continuous function defined on T. The results in [10] have been proved by employing the quadratic b functional equation z(x) := a (x∆ (t ))2 − q(t ) (xσ )2 ∆t = 0. In [12] the author proved some new Opial type inequalities on time scales and applied these inequalities to establish new Lyapunov type inequalities for the equation
∆
r ( t ) x∆ ( t )
+ q(t )xσ (t ) = 0,
t ∈ [α, β]T ,
(1.8)
on an arbitrary time scale T, where r, q are rd-continuous functions satisfying β
α
1/r (t )∆t < ∞,
and α
β
|q(t )| dt < ∞.
(1.9)
In [13] the author considered the half-linear dynamic equation
∆
r (t )ϕ(x∆ )
+ p(t )ϕ(xσ (t )) = 0,
t ∈ [a, b]T , γ −1
(1.10)
on an arbitrary time scale T, where ϕ(u) = |u| u, γ > 0 is a positive constant, r and p are real rd-continuous positive functions and established some Lyapunov type inequalities by employing Hölder’s inequality.
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
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In this paper, we will employ a technique depends on the applications of some dynamic inequalities of Opial’s type on time scales established by the author in [14,15] to prove several results related to the problems (i)–(ii) above. The technique that we will apply in this paper is not only different from the techniques employed in the above mentioned papers, when T = R, but also give new results for Eqs. (1.2)–(1.6) with damping terms. To illustrate the main results, as special cases, we will derive some results obtained for differential equations without damping terms. The results yield conditions for disfocality and disconjugacy for dynamic equations with damping terms. Of particular interest in this paper is when q is oscillatory. To the best of the author’s knowledge nothing is known regarding the distribution of zeros of the dynamic equation (1.1) with a damping term on time scales, so the paper initiates the study. 2. Main results During the past decade a number of dynamic inequalities have been established by some authors which are motivated by some applications. For contributions, we refer the reader to [16–20,14,21–26] and the references cited therein. In this section, to prove the main results we will need some dynamic inequalities of Opial’s type that has been established in [14,15]. These inequalities are presented below in Theorems 2.1–2.6, but before we present these inequalities, for completeness, we recall the following concepts related to the notion of time scales. A function f : T → R is said to be right-dense continuous (rd-continuous) provided that f is continuous at right-dense points and at left-dense points in T, the left hand limits exist and are finite. The set of all such rd-continuous functions is denoted by Crd (T). The graininess function µ for a time scale T is defined by µ(t ) := σ (t ) − t, and for any function f : T → R the notation f σ (t ) denotes f (σ (t )). We will make use of the following product and quotient rules for the derivative of the product fg and the quotient f /g (where gg σ ̸= 0, here g σ = g ◦ σ ) of two differentiable functions f and g ∆
∆
σ ∆
∆
∆ σ
∆
(fg ) = f g + f g = fg + f g ,
f
and
=
g
f ∆ g − fg ∆ gg σ
.
(2.1)
The integration by parts formula is given by b
∆
f (t )g (t )∆t = [f (t )g (t )
]ba
−
a
b
f ( t ) ∆t = a
b a
f (t )∆t =
(2.2)
b a
f (t )dt and the integration formula on a discrete time scale is defined by
f (t )µ(t ).
(2.3)
t ∈(a,b)
For example, if T = Z, then
b a
f (t )∆t =
f (n) and if T = qN0 , where q > 1, then on the interval [a, b/q] we have
b−1 n=a
logq (b)−1
b
f ∆ (t )g σ (t )∆t .
a
Note that if T = R, then
b
f (t )∆t = (q − 1) a
qn f (qn ).
n=logq (a)
The books on the subject of time scales by Bohner and Peterson [27,28] summarize and organize much of time scale calculus and contain some results for dynamic equations on time scales. Now, we are ready to state the main inequalities that will be needed in the proofs of the main results. We begin with the inequalities proved in [14]. Theorem 2.1. Let T be a time scale with a, τ ∈ T and λ, δ be positive real numbers such that λ ≥ 1, and let A, B be nonnegative
τ
−1
rd-continuous functions on (a, τ )T such that a A λ+δ−1 (t )∆t < ∞. If x : [a, τ ] ∩ T → R+ is delta differentiable with x(a) = 0, (and x∆ does not change sign in (a, τ )T ), then we have τ
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ≤ K1 (a, τ , λ, δ)
τ
a
λ+δ
A(t ) x∆ (t )
∆t ,
(2.4)
a
where λ−1
K1 (a, τ , λ, δ) := 2
sup a≤t ≤τ
τ
×
λ
µ (t )
(B(t ))
λ+δ λ
B(t )
+2
A(t )
− λδ
(A(t ))
a
2λ−1
t
δ λ+δ
−1 A λ+δ−1
δ λ+δ
(t )∆t
(λ+δ−1)
λ λ+δ
∆t
.
(2.5)
a
Theorem 2.2. Let T be a time scale with τ , b ∈ T and λ, δ be positive real numbers such that λ ≥ 1, and let A, B be nonnegative
b
−1
rd-continuous functions on (t , b)T such that τ A λ+δ−1 (t )∆t < ∞. If x : [τ , b] ∩ T → R+ is delta differentiable with x(b) = 0,
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
(and x∆ does not change sign in (τ , b)T ), then we have b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ≤ K2 (t , b, λ, δ)
τ
b
λ+δ
A(t ) x∆ (t )
τ
∆t ,
(2.6)
where λ−1
K2 (τ , b, λ, δ) := 2
sup
τ ≤t ≤b
λ
µ (t )
b
(B(t ))
× τ
λ+δ λ
B(t )
2λ−1
+2
A(t )
(A(t ))
b
− λδ
δ λ+δ
−1 A λ+δ−1
δ λ+δ
(t )∆t
λ λ+δ
(λ+δ−1)
∆t
.
(2.7)
t
In the following, we assume that there exists τ ∈ (a, b) which is the unique solution of the equation K (λ, δ) = K1 (a, τ , λ, δ) = K2 (τ , b, λ, δ) < ∞,
(2.8)
where K1 (a, τ , λ, δ) and K2 (τ , b, λ, δ) are defined as in Theorems 2.1 and 2.2. Note that since b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t =
τ
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t
a
a
b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ,
+ τ
then the combination of Theorems 2.1 and 2.2 gives us the following result. Theorem 2.3. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λ ≥ 1, and let A, B be nonnegative −1
b
rd-continuous functions on (a, b)T such that a A λ+δ−1 (t )∆t < ∞. If x : [a, b] ∩ T → R+ is delta differentiable with x(a) = 0 = x(b), (and x∆ does not change sign in (a, b)T ), then we have b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ≤ K (λ, δ)
b
λ+δ
A(t ) x∆ (t )
∆t .
(2.9)
a
a
In the following, we present the inequalities that have been proved in [15]. Theorem 2.4. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λ + δ > 1, and let r, s be
τ
−1
nonnegative rd-continuous functions on (a, τ )T such that a r λ+δ−1 (t )∆t < ∞. If x : [a, τ ] ∩ T → R is delta differentiable with x(a) = 0 (and x∆ does not change sign in (a, τ )T ), then τ
δ s(t ) |x(t )|λ x∆ (t ) ∆t ≤ G1 (a, τ , λ, δ)
a
τ
λ+δ
r (t ) x∆ (t )
∆t ,
(2.10)
a
where G1 (a, τ , λ, δ) =
δ λ+δ
δ λ+δ
τ
(s(t ))
λ+δ λ
(r (t ))
− λδ
t
r
a
−1 λ+δ−1
(t )∆t
(λ+δ−1)
λ λ+δ
.
∆t
(2.11)
a
Theorem 2.5. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λ + δ > 1, and let r , s be
b
−1
nonnegative rd-continuous functions on (b, τ )T such that τ r λ+δ−1 (t )∆t < ∞. If x : [τ , b] ∩ T → R is delta differentiable with x(b) = 0, (and x∆ does not change sign in (τ , b)T ), then we have b
τ
δ s(t ) |x(t )| x∆ (t ) ∆t ≤ G2 (τ , b, λ, δ) λ
b
λ+δ
r (t ) x∆ (t )
τ
∆t ,
(2.12)
where G2 (τ , b, λ, δ) =
δ λ+δ
δ λ+δ
b
τ
(s(t ))
λ+δ λ
(r (t ))
− λδ
b
r
−1 λ+δ−1
(t )∆t
(λ+δ−1)
λ λ+δ
∆t
.
t
In the following, we assume that there exists τ ∈ (a, b) which is the unique solution of the equation G(λ, δ) = G1 (a, τ , λ, δ) = G2 (τ , b, λ, δ) < ∞,
(2.13)
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1781
where G1 (a, τ , λ, δ) and G2 (τ , b, λ, δ) are defined as in Theorems 2.4 and 2.5. Note that since b
δ
s(t ) |x(t )|λ x∆ (t ) ∆t =
a
τ
δ
s(t ) |x(t )|λ x∆ (t ) ∆t +
b
τ
a
δ
s(t ) |x(t )|λ x∆ (t ) ∆t ,
then the combination of Theorems 2.4 and 2.5 gives us the following result. Theorem 2.6. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λδ > 0 and λ + δ > 1, and let r, s −1
b
be nonnegative rd-continuous functions on (a, b)T such that a r λ+δ−1 (t )∆t < ∞. If x : [a, b] ∩ T → R is delta differentiable with x(a) = 0 = x(b), (and x∆ does not change sign in (a, b)T ), then we have b
δ
s(t ) |x(t )|λ x∆ (t ) ∆t ≤ G(λ, δ)
b
a
λ+δ
r (t ) x∆ (t )
∆t .
(2.14)
a
To simplify the presentation of the results, we define
Λ(β) := sup µγ (t ) α≤t ≤β
Λ(α) := sup µγ (t ) α≤t ≤β
Rα (t ) :=
∆s
t
α
1
r (s) γ
|Q (t )| , r (t )
where Q (t ) =
|Q (t )| , r (t )
where Q (t ) =
,
β
q(s)∆s,
t
and Rβ (t ) :=
β
t
α
∆s 1
r ( s) γ
t
q(s)∆s,
.
Note that when T = R, we have Λ(α) = 0 = Λ(β) and when T = Z, we have
Λ(β) = sup
β−1 q ( s ) s=t r (t )
α≤t ≤β
,
t −1 q(s) s=α
Λ(α) = sup
and
α≤t ≤β
r (t )
.
(2.15)
Now, we are ready to state and prove the main results. Theorem 2.7. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign on (α, β)T . If x(α) = x∆ (β) = 0, then 22γ −2 Λ(β) +
2
3γ −2
(γ + 1)
1
γ +1
×
β
|Q (t )|
γ +1 γ
1
r (t )
α
γ
γ γ+1 (Rα (t ))γ ∆t
γ γ+1 β γ +1 1 |p(t )|γ +1 γ 1 γ × + ≥1− , (Rα (t )) ∆t γ 1+γ r (t ) c α β where Q (t ) = t q(s)∆s. If x∆ (α) = x(β) = 0, then
22γ −2 Λ(α) +
23γ −2 1
(γ + 1) γ +1
×
β
|Q (t )|
γ +1 γ
γ γ+1 γ Rβ (t ) ∆t
1
r γ (t )
α
(2.16)
γ γ+1 β γ +1 1 γ |p(t )|γ +1 γ 1 R β ( t ) ∆t + × ≥1− , γ 1+γ r (t ) c α t where Q (t ) = α q(s)∆s.
(2.17)
Proof. We prove (2.16). Without loss of generality we may assume that x(t ) ≥ 0 in [α, β]T . Multiplying (1.1) by xσ and integrating by parts, we have β
α
γ ∆
r (t ) x∆ (t )
β
+ α
xσ ( t ) ∆ t +
p(t )xσ (t ) x∆ (t )
γ
β
α
γ
p(t )xσ (t ) x∆ (t )
∆t = −
β
α
q(t ) (xσ (t ))
β γ ∆t = r (t ) x∆ (t ) x(t )α − γ +1
∆t .
β
α
γ +1
r (t ) x∆ (t )
∆t
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
Using the assumption that x(α) = x∆ (β) = 0, we have β
− α
r (t ) x∆ (t )
γ +1
This implies (note that Q (t ) = β
α
γ +1
r (t ) x∆ (t )
α
β
p(t )xσ (t ) x∆ (t )
γ
β
∆t = −
α
q(t ) (xσ (t ))
γ +1
∆t .
q(s)∆s) that
t
β
∆t =
β
∆t +
α
p(t )xσ (t ) x∆ (t )
γ
∆t −
β
α
γ +1
Q ∆ (t ) (xσ (t ))
∆t .
(2.18)
Integrating by parts (see (2.2)) the right hand side, we see that β
α
γ +1
r (t ) x∆ (t )
∆t =
β
α
p(t )xσ (t ) x∆ (t )
β ∆t − Q (t )(x(t ))γ +1 α +
γ
β
α
∆
Q (t ) xγ +1 (t )
∆t .
Again using the assumptions x(α) = 0 and Q (β) = 0, we obtain β
α
γ +1 r (t ) x (t ) dt =
∆
β
α
σ
∆
p(t )x (t ) x (t )
γ
∆t +
β
α
∆
Q (t ) xγ +1 (t )
∆t .
(2.19)
Applying the chain rule formula
xλ (t )
∆
=λ
1
λ−1
[hxσ (t ) + (1 − h)x(t )]
dhx∆ (t ),
for λ > 0,
(2.20)
0
which is a simple consequence of Keller’s chain rule [27, Theorem 1.90], and the inequality (see [29, p. 500]) aλ + bλ ≤ (a + b)λ ≤ 2λ−1 (aλ + bλ ),
if a, b ≥ 0, λ ≥ 1,
(2.21)
we see that
γ +1 ∆ ≤ (γ + 1 ) x ( t )
1
|hxσ (t ) + (1 − h)x(t )|γ dh x∆ (t )
0
1 σ 1 |hx (t )|γ dh + 2γ −1 (γ + 1) x∆ (t ) |(1 − h)x(t )|γ dh (γ + 1) x∆ (t ) 0 0 = 2γ −1 x∆ (t ) |xσ (t )|γ + 2γ −1 x∆ (t ) |x(t )|γ ≤ 2γ −1 |xσ (t ) + x(t )|γ x∆ (t ) . γ −1
≤2
(2.22)
This and (2.19) imply that β
α
γ +1
r (t ) x∆ (t )
∆t ≤
β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t + 2γ −1
β
α
|Q (t )| |x(t ) + xσ (t )|γ x∆ (t ) ∆t .
(2.23)
β Applying inequality (2.4) on the integral α |Q (t )| |x(t ) + xσ (t )|γ x∆ (t ) ∆t, with B(t ) = |Q (t )| , A(t ) = r (t ), λ = γ , δ = 1, we have
β
α
σ
γ
|Q (t )| |x(t ) + x (t )| x∆ (t ) ∆t ≤ K1 (α, β, γ , 1)
β
α
γ +1
r (t ) x∆ (t )
∆t ,
(2.24)
where
K1 (α, β, γ , 1) = 22γ −2 Λ(β) + 23γ −2
1
(γ + 1)
1
γ +1
β
α
|Q (x)|
γ +1 γ
1
r (x) γ
γ γ+1 (Rα (x))γ ∆x
.
Using that fact that xσ = x(t ) + µ(t )x∆ (t ), we see that β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t =
α
≤ α
β
β
γ |p(t )| x(t ) + µ(t )x∆ (t ) x∆ (t ) ∆t γ |p(t )| |x(t )| x∆ (t ) ∆t +
β
α
γ +1 µ(t ) |p(t )| x∆ (t ) ∆t .
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1783
γ Applying inequality (2.10) on the integral α |p(t )| |x(t )| x∆ (t ) ∆t with s(t ) = |p(t )|, λ = 1 and δ = γ , we see that β β ∆ γ γ +1 |p(t )| |x(t )| x (t ) ∆t ≤ G1 (α, β, 1, γ ) r (t ) x∆ (t ) ∆t , (2.25) β
α
α
where
G1 (α, β, 1, γ ) =
γ γ+1
γ 1+γ
β
× α
|p(t )|γ +1 (Rα (t ))γ ∆t (r (t ))γ
γ +1 1
.
Using the assumption that 0 ≤ p(t )µ(t ) ≤ r (t )/c, we see that β
α
γ
p(t ) |xσ (t )| x∆ (t ) ∆t ≤ G1 (α, β, 1, γ )
β
α
γ +1
r (t ) x∆ (t )
β
1
∆t +
c
α
γ +1
r (t ) x∆ (t )
∆t .
(2.26)
Substituting (2.24) and (2.26) into (2.23), we have
1−
1
β
c
α
γ +1
r (t ) x∆ (t )
∆t ≤ K1 (α, β, γ , 1)
β
α
+ G1 (α, β, 1, γ )
γ +1
r (t ) x∆ (t ) β
α
∆t
γ +1
r (t ) x∆ (t )
∆t .
(2.27)
Then, we have from (2.27) that 1−
1 c
≤ K1 (α, β, γ , 1) + G1 (α, β, 1, γ ) 2
3γ −2
β
|Q (t )|
γ γ+1
γ +1 γ
(Rα (t ))γ ∆t 1 α r γ (t ) (γ + 1) γ +1 γ γ+1 β γ +1 1 |p(t )|γ +1 γ γ + , (Rα (t )) ∆t 1+γ r γ (t ) α
= 22γ −2 Λ(β) +
1
which is the desired inequality (2.16). The proof of (2.17) is similar to (2.16) by using (2.6) of Theorem 2.2, (2.7) instead of (2.5), (2.12) of Theorem 2.4 by using (2.13) instead of (2.11). The proof is complete. In Theorem 2.7 if r (t ) = 1, then we have the following result. Corollary 2.1. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then 2 γ −2
2
Λ(β) +
γ
+
1+γ
where Q (t ) = 2 γ −2
2
β t
β
γ +1 γ
γ
γ γ+1
|Q (t )| × (t − α) ∆t 1 α (γ + 1) γ +1 γ γ+1 β γ +1 1 1 γ +1 γ |p(t )| (t − α) ∆t × ≥1− , c
α
q(s)∆s. If x∆ (α) = x(β) = 0, then 23γ −2
Λ(α) +
γ
+
23γ −2
1+γ
β
γ +1 γ
γ
γ γ+1
|Q (t )| × (β − t ) ∆t 1 α (γ + 1) γ +1 γ γ+1 β γ +1 1 1 γ +1 γ |p(t )| (β − t ) ∆t × ≥1− , c
α
t
where Q (t ) = α q(s)∆s. As a special case of Theorem 2.7, when γ = 1, we have the following result. Corollary 2.2. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then
Λ(β) +
√
β
2 α
|Q (t )|2 rα (t )∆t r (t )
12
1
β
+√
2
α
p2 ( t ) r (t )
Rα (t )∆t
12
1
≥1− , c
1784
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
where Rα (t ) = α r∆(ss) and Q (t ) =
t
β t
q(s)∆s. If x∆ (α) = x(β) = 0, then
12 β 2 12 |Q (t )|2 p (t ) 1 1 Λ(α) + 2 rβ (t )∆t +√ Rβ (t )∆t ≥1− , r ( t ) r ( t ) c 2 α α β ∆s t where Rβ (t ) = t r (s) and Q (t ) = α q(s)∆s. √
β
As a special case of Corollary 2.2, when p(t ) = 0, we have the following result. Corollary 2.3. Suppose that x is a nontrivial solution of r (t )x∆ (t )
∆
+ q(t )xσ (t ) = 0,
t ∈ [α, β]T ,
(2.28)
and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then
√
β
|Q (t )|2 r (t )
2 α
where Q (t ) =
√
β t
β
α
α
∆t r (t )
∆t
12
+ Λ(β) ≥ 1,
q(s)∆s. If x∆ (α) = x(β) = 0, then
|Q (t )|2 r (t )
2
t
β
t
∆t r (t )
∆t
21
+ Λ(α) ≥ 1,
t
where Q (t ) = α q(s)∆s. Remark 1. Theorem 2.7 yields sufficient conditions for disfocality of (1.1), i.e., sufficient conditions so that there does not exist a nontrivial solution x satisfying x(α) = x∆ (β) = 0 or x∆ (α) = x(β) = 0. As a special case when T = R, it is known that Λ(α) = Λ(β) = 0 and then the results in Corollary 2.3, with r (t ) = 1, reduce to the following results obtained by Brown and Hinton [30] for the second order differential equation ′′
x + q(t )x(t ) = 0,
α ≤ t ≤ β.
(2.29)
Corollary 2.4 ([30]). If x is a solution of Eq. (2.29) such that x (α) = x′ (β) = 0, then β
2 α
Q 2 (s)(s − α)ds > 1,
where Q (t ) = β
2 α
β t
q(s)ds. If instead x′ (α) = x (β) = 0, then
Q 2 (s)(β − s)ds > 1,
t
where Q (t ) = α q(s)ds. On a time scale T, we note from the chain rule (2.20) that
∆ (t − a)λ+δ = (λ + δ)
1
[h(σ (t ) − a) + (1 − h)(t − a)]λ+δ−1 dh
0
≥ (λ + δ)
1
[h(t − a) + (1 − h)(t − a)]λ+δ−1 dh
0
= (λ + δ)(t − a)λ+δ−1 . This implies that τ
a
( t − a)
(λ+δ−1)
∆t ≤
τ
a
1
(λ + δ)
(t − a)λ+δ
∆
∆t =
(τ − a)λ+δ . (λ + δ)
(2.30)
Remark 2. By using the maximum of |Q | and |p| on [α, β]T and substituting (2.30) into the results of Corollary 2.1, we have the following results.
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1785
Corollary 2.5. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then 23γ −2 (β − α)γ
(γ + 1)
β
max
α≤t ≤β
t
q(s)∆s +
γ β γ γ +1 1 (β − α) max |p(t )| + 22γ −2 sup µγ (t ) q(s)∆s ≥ 1 − , α≤t ≤β γ +1 c α≤t ≤β t
and if x∆ (α) = x(β) = 0, then 23γ −2 (β − α)γ
(γ + 1)
γ t t γ γ +1 1 q(s)∆s + max (β − α) max |p(t )| + 22γ −2 sup µγ (t ) q(s)∆s ≥ 1 − . α≤t ≤β α≤t ≤β γ +1 c α≤t ≤β α α
As a special case when T = R, and p(t ) = 0, we have Λ(α) = Λ(β) = 0 and then the results in Corollary 2.5 reduce to the following results for the second order half-linear differential equation
(x′ (t ))γ
′
+ q(t )(x(t ))γ = 0,
α ≤ t ≤ β,
(2.31)
where γ ≥ 1 is a quotient of odd positive integers. Corollary 2.6. Suppose that x is a nontrivial solution of (2.31) and x′ does not change sign in (α, β). If x (α) = x′ (β) = 0, then
β q(s)ds ≥ 1. (β − α) max α≤t ≤β (γ + 1) t 23γ −2
γ
(2.32)
If instead x′ (α) = x (β) = 0, then 23γ −2
t (β − α) max q(s)ds ≥ 1. α≤ t ≤β (γ + 1) α γ
(2.33)
When γ = 1, the conditions (2.32) and (2.33) reduce to the conditions obtained by Harris and Kong [31] for second order differential equation (2.29). Corollary 2.7 ([31]). Suppose that x is a nontrivial solution of (2.29) and x′ does not change sign in (α, β). If x (α) = x′ (β) = 0, then
(β − α) max α≤t ≤β
β t
q(s)ds > 1.
(2.34)
If instead x′ (α) = x (β) = 0, then
t (β − α) max q(s)ds > 1. α≤t ≤β
(2.35)
α
As a special case when T = Z, we see that Λ(α) and Λ(β) are defined as in (2.15) and then the results in Corollary 2.5 reduce to the following results for the second order half-linear difference equation
∆(∆x(n))γ + p(n)(∆x(n))γ + q(n)(x(n + 1))γ = 0,
α ≤ n ≤ β,
(2.36)
where γ ≥ 1 is a quotient of odd positive integers and p(n) ≤ 1/c. Corollary 2.8. Suppose that x is a nontrivial solution of (2.36) and ∆x(n) does not change sign in (α, β)T . If x(α) = ∆x(β) = 0, then 1−
1 c
≤
23γ −2 (β − α)γ
(γ + 1)
β−1 β−1 γ γ γ+1 2γ −2 max q(s) + 2 sup q(s) + (β − α) max |p(n)| , α≤n≤β α≤n≤β γ +1 α≤n≤β s=n s=n
and if ∆x(α) = x(β) = 0, then 1−
1 c
≤
23γ −2 (β − α)γ
(γ + 1)
n−1 n−1 γ γ γ+1 2γ −2 max q(s) + 2 sup q(s) + (β − α) max |p(n)| . α≤n≤β α≤n≤β γ +1 α≤n≤β s=α s=α
Remark 3. One can apply the results in Theorem 2.7 to obtain some new results for Eqs. (1.4)–(1.6) as obtained for Eqs. (1.2) and (1.3) by using the integration formula (2.3). The details are left to the reader.
1786
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
If we apply the inequality
|a + b|λ ≤ 2λ−1 |a|λ + |b|λ ,
where a, b are real numbers and λ ≥ 1,
∆
with a = x(t ) and b = µ(t )hx (t ), then we have from (2.20) that
∆ γ +1 ∆ x (t ) ≤ (γ + 1) x (t )
1
x(t ) + µ(t )hx∆ (t )γ dh
0 1
|x(t )|γ dh + 2γ −1 (γ + 1) x∆ (t )
γ −1
(γ + 1) x∆ (t )
γ −1
γ +1 (γ + 1) x∆ (t ) |x(t )|γ + 2γ −1 µ(t ) x∆ (t ) .
≤2
0
=2
1
µ(t )hx∆ (t )γ dh
0
(2.37)
Substituting (2.37) into (2.19), we have that β
α
γ +1
r (t ) x∆ (t )
β
dt ≤ α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t + 2γ −1 (γ + 1)
+ 2γ −1
β
β
α
|Q (t |) x∆ (t ) |x(t )|γ ∆t
γ +1 µ(t ) |Q (t )| x∆ (t ) ∆t .
α
(2.38)
Using the inequality β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t ≤
β
α
γ |p(t )| |x(t )| x∆ (t ) ∆t +
β
α
γ +1 µ(t ) |p(t )| x∆ (t ) ∆t ,
we have from (2.38) that β
α
γ +1 r (t ) x∆ (t ) dt ≤
β
α
β
+ α
β
γ |p(t )| |x(t )| x∆ (t ) ∆t + 2γ −1 (γ + 1)
α
|Q (t |) x∆ (t ) |x(t )|γ ∆t
γ +1 ∆t . µ(t ) |p(t )| + 2γ −1 |Q (t )| x∆ (t )
(2.39)
We can now apply the inequalities in Theorems 2.4–2.5 to obtain new results different from the above results but in this case the condition µ(t ) |p(t )| ≤ r (t )/c is replaced by the new condition µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c. Applying inequality (2.10) on the term β
α
|Q (t )| x∆ (t ) |x(t )|γ ∆t ,
with s(t ) = |Q (t )| , λ = γ
δ = 1,
and
we have β
α
|Q (t |) |x(t )|γ x∆ (t ) ∆t ≤ K1∗ (α, β, γ , 1)
β
α
γ +1
r (t ) x∆ (t )
∆t ,
where K1∗ (α, β, γ , 1) =
γ +1 1
1
γ +1
β
|Q (t )|
γ +1 γ 1
(r (t ))
γ
γ |p(t )| |x(t )| x∆ (t ) ∆t ≤ G1 (α, β, 1, γ )
α
γ γ+1 Rγα (t )∆t
.
Using the inequality β
α
β α
γ +1
r (t ) x∆ (t )
∆t ,
where G1 (α, β, 1, γ ) =
γ 1+γ
γ γ+1
β
× α
|p(t )|γ +1 γ R (t )∆t (r (t ))γ α
γ +1 1
and proceeding as in the proof of Theorem 2.7, we obtain the following results. Theorem 2.8. Assume that µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c where c is a positive constant such that c ≥ 1. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then γ −1
2
(γ + 1)
γ γ +1
β
α
|Q (t )| 1
γ +1 γ
r γ (t )
γ γ+1 Rγα (t )∆t
+
γ 1+γ
γ γ+1
β α
|p(t )|γ +1 γ Rα (t )∆t r γ (t )
γ +1 1
1
≥1− , c
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
where Q (t ) =
β t
1787
q(s)∆s. If x∆ (α) = x(β) = 0, then
2γ −1 (γ + 1)
γ γ +1
β
|Q (t )|
γ γ+1
γ +1 γ
γ
Rβ (t )∆t
1
r (t )
α
γ
+
γ 1+γ
γ γ+1
β α
|p(t )|γ +1 γ Rβ (t )∆t r γ (t )
γ +1 1
1
≥1− , c
t
where Q (t ) = α q(s)∆s. Remark 4. Note that when T = R the condition µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c will be removed since in this case
µ(t ) = 0.
In the following, we apply Theorems 2.3 and 2.6, to determine the lower bound for the distance between consecutive generalized zeros of solutions of (1.1). Note that the applications of the above results allow the use of arbitrary anti-derivative Q in the above arguments. In the following, we assume that Q ∆ (t ) = q(t ) and assume that there exists a unique h ∈ (α, β)T , such that R(h) := Rα (h) = Rβ (h).
(2.40)
Note that the best choice of h when r (t ) = 1 is h = (β + α) /2. In the following, we assume that K h (α, β, γ , 1) = Kh (α, β, γ , 1) < ∞,
(2.41)
where K h (α, β, γ , 1) =
1
3 γ −2
2
(γ + 1)
Λ := sup µγ (t )
γ +1
|Q (t )| , r (t )
β
1
(γ + 1) γ +1
Kh (α, β, γ , 1) =
α≤t ≤β
23γ −2
|Q (t )|
γ +1 γ
1
r γ (t )
α β
|Q (t )| 1
γ +1 γ
r (t )
α
γ
γ γ+1 Rγα (h)∆t
+ 22γ −2 Λ,
γ γ+1 γ
Rβ (h)∆t
+ 22γ −2 Λ,
where Q ∆ (t ) = q(t ),
and Gh (α, β, 1, γ ) = Gh (α, β, 1, γ ) < ∞,
(2.42)
where G (α, β, 1, γ ) = h
Gh (α, β, 1, γ ) =
γ 1+γ
γ γ+1
γ 1+γ
γ γ+1
β
|p(t )|γ +1 γ Rα (h)∆t r γ (t )
γ +1 1
β
|p(t )|γ +1 γ Rβ (h)∆t r γ (t )
γ +1 1
α
α
, .
Now, we assume that K (γ , 1) is the solution of the equation K (γ , 1) = K h (α, β, γ , 1) = Kh (α, β, γ , 1) and given by
K (γ , 1) =
2
3γ −2
(γ + 1)
1
γ +1
β
|Q (t )| 1
γ +1 γ
r (t )
α
γ
γ γ+1 Rγ (h)∆t
+ 22γ −2 Λ,
(2.43)
.
(2.44)
and similarly G(1, γ ) is given by G(1, γ ) =
γ 1+γ
γ γ+1
β α
|p(t )|γ +1 γ R (h)∆t r γ (t )
γ +1 1
Theorem 2.9. Assume that Q ∆ (t ) = q(t ) and suppose x is a nontrivial solution of (1.1). If x(α) = x(β) = 0, then K (γ , 1) + G(1, γ ) ≥ 1 −
1 c
,
where K (α, β) and K (α, β) are defined as in (2.43) and (2.44).
(2.45)
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
Proof. We multiply (1.1) by xσ (t ) and proceed as in Theorem 2.7 to obtain β
α
r (t ) x∆ (t )
γ +1
β
∆t =
γ
p(t )xσ (t ) x∆ (t )
α
∆t +
β
α
Q ∆ (t ) (xσ (t ))
γ +1
∆t .
Integrating by parts the right hand side (see (2.2)), we see that β
α
r (t ) x∆ (t )
γ +1
β
∆t =
β ∆t + Q (t )(x(t ))γ +1 α −
γ
p(t )xσ (t ) x∆ (t )
α
β
α
∆
Q (t ) xγ +1 (t )
∆t .
(2.46)
Using the facts that x(α) = 0 = x(β), we obtain β
α
γ +1
r (t ) x∆ (t )
β
dt ≤ α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t +
β
α
∆ |Q (t )| xγ +1 (t ) dt .
We proceed as in the proof of Theorem 2.7, to get β
α
γ +1 ∆ γ −1 |Q (t )| x (t ) ∆t ≤ 2
β α
|Q (t )| |x(t ) + xσ (t )|γ x∆ (t ) ∆t .
Applying inequality (2.9) with B(t ) = |Q (t )|, A(t ) = r (t ), λ = γ and δ = 1, we have β
α
∆ |Q (t )| xγ +1 (t ) dt ≤ 2γ −1 K (γ , 1)
β
α
γ +1
r (t ) x∆ (t )
∆t .
Also as in the proof of Theorem 2.7 by applying inequality (2.14), we obtain β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t ≤ G(1, γ )
From this inequality, after cancelling the proof.
β α
β
γ +1 1 r (t ) x∆ (t ) ∆t + c
α
γ +1
r (t ) x∆ (t )
β
α
γ +1
r (t ) x∆ (t )
∆t .
∆t, we get the desired inequality (2.45). This completes
In the following, we assume that (1.1) has a solution x which, together with x∆ is positive on (α, β)T and assume that (2.40) holds. From the chain rule (2.20), we see that
∆
xγ +1 (t )
= (γ + 1)
1
(hxσ (t ) + (1 − h)x(t ))γ dhx∆ (t )
0 1
≥ (γ + 1)x∆ (t )
(hxσ (t ))γ dh = x∆ (t ) (xσ (t ))γ .
0
Substituting this in (2.46), and assuming that x(α) = x(β) = 0, we have β
α
γ +1
r (t ) x∆ (t )
∆t ≤
β
α
β
≤ α
p(t ) (xσ (t ))
γ
x∆ (t ) ∆t −
β
α
γ
Q (t )x∆ (t ) (xσ (t )) ∆t
|Q (t ) − p(t )| (xσ (t ))γ x∆ (t )∆t .
(2.47)
Applying the inequality in Theorem 2.6 on the integral β
α
|Q (t ) − p(t )| (xσ (t ))γ x∆ (t )∆t ,
we have from (2.47) that G∗ (γ , 1) ≥ 1 − 1c , where G∗ (γ , 1) =
γ 1+γ
γ γ+1
β α
|Q (t ) − p(t )|γ +1 (R(h))γ ∆t r γ (t )
γ +1 1
.
This gives us the following results. Theorem 2.10. Assume that Q ∆ (t ) = q(t ) and suppose x(t ) is a nontrivial solution of (1.1). If x(α) = x(β) = 0, then G∗ (1, γ ) ≥ 1 −
1 c
,
where G∗ (1, γ ) is defined as in (2.48).
(2.48)
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1789
Remark 5. When r (t ) = 1, we see that (2.40) is satisfied if h = (β + α) /2. One can use this choice and the maximum values of |Q (t )| and |p(t )| to obtain new results different from the results obtained in Theorem 2.9. Due to the limited space, the details are left to the interested reader. From Theorem 2.10 one can also obtain new results when T = R and see that the obtained results will be different from the results established for differential equations with damping terms in [3–9] (note that in this time scale the right hand side (1 − 1/c ) of (2.45) changes to 1). Remark 6. The sufficient conditions for disconjugacy can be obtained from Theorems 2.9 and 2.10, i.e., sufficient conditions so that there does not exist a nontrivial solution x satisfying x(α) = x(β) = 0. In fact one can prove that Eq. (1.1) is disconjugate in [α, β]T if one of the following conditions holds K (γ , 1) + G(1, γ ) < 1 − G∗ (1, γ ) < 1 −
1 c
1 c
,
(2.49)
.
(2.50)
The proof is given by assuming that the equation is not disconjugate, and without loss of generality we can assume that x(α) = 0 = x(β) and follows the proof of Theorems 2.9 and 2.10 to get a contradiction to (2.49) or (2.50). Remark 7. We note that the condition µ(t ) |p(t )| ≤ r (t )/c and the condition µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c may be removed if one can prove an inequality of the form β
α
δ |p(t )| |x (t )| x∆ (t ) ∆t ≤ H (λ, δ) σ
λ
β
α
λ+δ |r (t )| x∆ (t ) ∆t ,
where H (λ, δ) is the coefficient of the inequality needs to be determined as in Theorems 2.1–2.6. This open question will be left to the interested reader. Problem 1. It would be interesting to extend the above technique to cover the equation r (t ) x∆ (t )
γ ∆
γ + p(t ) x∆ (t ) + q(t ) (xσ (t ))γ = 0,
when 0 < γ ≤ 1 (see [21]). Problem 2. It would be interesting to extend the above technique to cover the delay equation with oscillatory coefficients
r (t ) x∆ (t )
γ ∆
γ + p(t ) x∆ (t ) + q(t ) (x(τ (t )))γ = 0,
where the delay function τ (t ) satisfying τ (t ) < t and limt →∞ τ (t ) = ∞. Problem 3. It would be interesting to extend the above results to obtain the lower bound for the first eigenvalue of the boundary value problem
γ ∆ γ − r (t ) x∆ (t ) + p(t ) x∆ (t ) + q(t )xγ (σ (t )) = λxγ (σ (t )), . x(α) = x(β) = 0 This in fact needs an inequality of a Wirtinger’s type of the form β
α
γ +1 r (t ) x∆ (t ) ∆t ≥ W (a, b, γ )
β
α
s(t ) |x(t )|γ +1 ∆t ,
where W (a, b, γ ) is the coefficient of the inequality (see [16]). Acknowledgements The author is very grateful to the anonymous referee and the editor for valuable remarks and comments which significantly contributed to the quality of the paper. References [1] [2] [3] [4] [5] [6]
S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. S.H. Saker, Oscillation Theory of Dynamic Equations on Time Scales: Second and Third Orders, Lambert Academic Publishing, Germany, 2010. P. Hartman, A. Wintner, On an oscillation criterion of de la valee Poussion, Quart. Appl. Math. 13 (1955) 330–332. A.M. Fink, D.F. St. Mary, On an inequality of Nehari, Proc. Amer. Math. Soc. 21 (1969) 640–641. C. Ha, Eigenvalues of a Sturm–Liouville problem and inequalities of Lyapunov type, Proc. Amer. Math. Soc. 126 (1998) 1123–1129. B.G. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J. 4 (1997) 139–148.
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790 Chung-Fen Lee, C.C. Yeh, C.H. Hong, R.P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett. 17 (2004) 847–853. S.H. Saker, Lyapunov type inequalities for a second order differential equations with a damping term, Ann. Polon. Math. 103 (2012) 37–57. A. Tiryaki, M. Ünal, D. Çakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl. 332 (2007) 497–511. M. Bohner, S. Clark, J. Ridenhour, Lyapunov inequalities for time scales, J. Inequal Appl. 7 (2002) 61–77.L. A.M. Lyapunov, Probleme General de la Stabilitie du Movement, in: Ann. of Math. Stud., vol. 17, Preinceton Univ. Press, Princeton, NJ, 1949. S.H. Saker, Opial’s type inequalities on time scales and some applications, Ann. Polon. Math. 104 (2012) 243–260. S.H. Saker, Lyapunov inequalities for half-linear dynamic equations on time scales and disconjugacy, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 18 (2011) 149–161. S.H. Saker, Some new inequalities of Opial’s type on time scales, Abstr. Appl. Anal. (2012) Art. no. 683136. S.H. Saker, Some Opial-type inequalities on time scales, Abstr. Appl. Anal. 2011 (2011) 19. Art. no. 265316. R.P. Agarwal, M. Bohner, D. O’Regan, S.H. Saker, Some Wirtinger-type inequalities on time scales and their applications, Pacific J. Math. 252 (2011) 1–18. W.N. Li, Some Pachpatte type inequalities on time scales, Comput. Math. Appl. 57 (2009) 275–282. W.N. Li, Some new dynamic inequalities on time scales, J. Math. Anal. Appl. 319 (2007) 802–814. W.N. Li, M. Han, Bounds for certain nonlinear dynamic inequalities on time scales, Discrete Dyn. Nat. Soc. 2009 (2009) 14. ID 897087. S.H. Saker, Some nonlinear dynamic inequalities on time scales and applications, J. Math. Inequal. 4 (2010) 561–579. S.H. Saker, New inequalities of Opial’s type on time scales and some of their applications, Discrete Dyn. Nat. Soc. (2012) Art. no. 362526. S.H. Saker, Some nonlinear dynamic inequalities on time scales, Math. Inequal. Appl. 14 (2011) 633–645. S.H. Saker, Nonlinear dynamic inequalities of Gronwall–Bellman type on time scales, EJQTDE 2011 (86) (2011) 1–26. M. Bohner, B. Kaymakçalan, Opial inequalities on time scales, Ann. Polon. Math. 77 (1) (2001) 11–20. V. Kac, P. Cheung, Quantum Calculus, Springer, New York, 2001. B. Karpuz, B. Kaymakçalan, Ö. Öclan, A generalization of Opial’s inequality and applications to second order dynamic equations, Differ. Equ. Dyn. Syst. 18 (2010) 11–18. M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. D.S. Mitinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalties in Analysis, Kluwer Academic Publisher, 1993. R.C. Brown, D.B. Hinton, Opial’s inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997) 1123–1129. B.J. Harris, Q. Kong, On the oscillation of differential equations with an oscillatory coefficient, Trans. Amer. Math. Soc. 347 (1995) 1831–1839.
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Applications of Opial inequalities on time scales on dynamic equations with damping terms S.H. Saker Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
article
info
Article history: Received 21 November 2011 Received in revised form 7 April 2013 Accepted 30 April 2013 Keywords: Opial’s inequality Disfocality Disconjugacy Oscillation Dynamic equations Time scales
abstract In this paper, we will employ some dynamic inequalities of Opial’s type on time scales to prove several results related to the spacing between consecutive zeros of a solution of a second order dynamic equation with a damping term. We also obtain several results related to the spacing between a zero of the solution and/or a zero of its derivative. As a special case, we will establish some new results for differential and difference equations with damping terms. For illustration, we will derive some well-known results obtained for differential equations without damping terms. The results yield conditions for disfocality and disconjugacy for dynamic equations with damping terms on time scales. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The study of dynamic equations on time scales which goes back to its founder Stefan Hilger [1] becomes an area of mathematics and recently has received a lot of attention. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale T, which may be an arbitrary closed subset of the real numbers R. In this paper, we consider the second-order half-linear dynamic equation with a damping term
r (t ) x∆ (t )
γ ∆
γ + p(t ) x∆ (t ) + q(t ) (xσ (t ))γ = 0,
t ∈ [α, β]T ,
(1.1)
where T is an arbitrary time scale and σ (t ) is the forward jump operator on T which is defined by σ (t ) := inf{s ∈ T : s > t }. The main aim is to apply some dynamic inequalities of Opial’s type on time scales to prove several results related to the problems: (i) obtain lower bounds for the spacing β − α where x is a solution of (1.1) and satisfies x(α) = x∆ (β) = 0, or x∆ (α) = x(β) = 0, (ii) obtain lower bounds for the spacing of zeros of a solution of (1.1). Throughout the paper, we will assume that γ ≥ 1 is a quotient of odd positive integers, r, p and q are real rd-continuous functions defined on T with r (t ) > 0 and µ(t ) |p(t )| ≤ r (t )/c where c is a positive constant such that c ≥ 1. We also assume that sup T = ∞, and define the time scale interval [a, b]T by [a, b]T := [a, b] ∩ T. We say that a solution x of (1.1) has a generalized zero at t if x(t ) = 0, and has a generalized zero in (t, σ (t )) in case x(t )xσ (t ) < 0 and µ(t ) > 0. Eq. (1.1) is disconjugate on the interval [t0 , b]T , if there is no nontrivial solution of (1.1) with two (or more) generalized zeros in [t0 , b]T . The solution x(t ) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Eq. (1.1) is said to be oscillatory if all its solutions are oscillatory. For
E-mail address: [email protected]. URL: http://www.mans.edu.eg/pcvs/30040/. 0895-7177/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mcm.2013.04.006
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
more details of oscillation theory of dynamic equations on time scales, we refer to the book [2]. We say that (1.1) is right disfocal (left disfocal) on [α, β]T if the solutions of (1.1) such that x∆ (α) = 0 (x∆ (β) = 0) have no generalized zeros in [α, β]T . For Eq. (1.1) the point β > α is called a right focal point of α if the solution of (1.1) with initial conditions x(α) ̸= 0, x∆ (α) = 0 satisfies x(β) = 0. Left focal point is defined similarly. We note that, Eq. (1.1) in its general form covers several different types of differential and difference equations depending on the choice of the time scale T. For example, when T = R, we have σ (t ) = t, µ(t ) = 0, x∆ (t ) = x′ (t ) and (1.1) becomes the second-order differential equation
(r (t )(x′ (t ))γ )′ + p(t )(x (t ))γ + q(t )xγ (t ) = 0. ′
(1.2)
∆
When T = Z, we have σ (t ) = t + 1, µ(t ) = 1, x (t ) = ∆x(t ) = x(t + 1) − x(t ) and (1.1) becomes the second-order difference equation
∆(r (t ) (∆x(t ))γ ) + p(t ) (∆x(t ))γ + q(t )xγ (t + 1) = 0.
(1.3)
When T = hZ, h > 0, we have σ (t ) = t + h, µ(t ) = h, x∆ (t ) = ∆h x(t ) =
x(t + h) − x(t ) h
,
and (1.1) becomes the second-order difference equation
∆h (r (t )(∆h x(t ))γ ) + p(t ) (∆h x(t ))γ + q(t )xγ (t + h) = 0.
(1.4)
When T = q = {t : t = q , k ∈ N, q > 1}, we have σ (t ) = q t, µ(t ) = (q − 1)t , k
N
x∆ (t ) = ∆q x(t ) =
x(qt ) − x(t )
(q − 1)t
,
and (1.1) becomes the second-order q-difference equation
γ ∆q (r (t )(∆q x(t ))γ ) + p(t ) ∆q x(t ) + q(t )xγ (qt ) = 0. √ √ When T = N20 = {t 2 : t ∈ N0 }, we have σ (t ) = ( t + 1)2 and µ(t ) = 1 + 2 t, √ x(( t + 1)2 ) − x(t ) ∆ N x( t ) = , √ 1+2 t
(1.5)
and (1.1) becomes the second-order difference equation
√ ∆N (r (t )(∆N x(t ))γ ) + p(t ) (∆N x(t ))γ + q(t )xγ (( t + 1)2 ) = 0.
(1.6)
Some special cases of Eq. (1.2) have been studied by some authors, we refer to the papers by Hartman and Wintner [3], Fink and Mary [4], Ha [5], Pachpatte [6], Lee et al. [7], Saker [8] and Tiryaki, Ünal and Çakmak [9]. In these papers the authors established some inequalities of Lyapunov’s type. These inequalities give implicit lower bounds on the distance between zeros of a nontrivial solution and also lower bounds for the spacing between zeros of a solution and/ or its derivative. On time scales, the best known existence results of Lyapunov’s type for a special case of (1.1) (when γ = 1 and r (t ) = 1 and p(t ) = 0) are due to Bohner et al. [10]. They extended the Lyapunov inequality obtained for differential equations in [11] to the dynamic equation x∆∆ (t ) + q(t )xσ (t ) = 0,
(1.7)
where q(t ) is a positive rd-continuous function defined on T. The results in [10] have been proved by employing the quadratic b functional equation z(x) := a (x∆ (t ))2 − q(t ) (xσ )2 ∆t = 0. In [12] the author proved some new Opial type inequalities on time scales and applied these inequalities to establish new Lyapunov type inequalities for the equation
∆
r ( t ) x∆ ( t )
+ q(t )xσ (t ) = 0,
t ∈ [α, β]T ,
(1.8)
on an arbitrary time scale T, where r, q are rd-continuous functions satisfying β
α
1/r (t )∆t < ∞,
and α
β
|q(t )| dt < ∞.
(1.9)
In [13] the author considered the half-linear dynamic equation
∆
r (t )ϕ(x∆ )
+ p(t )ϕ(xσ (t )) = 0,
t ∈ [a, b]T , γ −1
(1.10)
on an arbitrary time scale T, where ϕ(u) = |u| u, γ > 0 is a positive constant, r and p are real rd-continuous positive functions and established some Lyapunov type inequalities by employing Hölder’s inequality.
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
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In this paper, we will employ a technique depends on the applications of some dynamic inequalities of Opial’s type on time scales established by the author in [14,15] to prove several results related to the problems (i)–(ii) above. The technique that we will apply in this paper is not only different from the techniques employed in the above mentioned papers, when T = R, but also give new results for Eqs. (1.2)–(1.6) with damping terms. To illustrate the main results, as special cases, we will derive some results obtained for differential equations without damping terms. The results yield conditions for disfocality and disconjugacy for dynamic equations with damping terms. Of particular interest in this paper is when q is oscillatory. To the best of the author’s knowledge nothing is known regarding the distribution of zeros of the dynamic equation (1.1) with a damping term on time scales, so the paper initiates the study. 2. Main results During the past decade a number of dynamic inequalities have been established by some authors which are motivated by some applications. For contributions, we refer the reader to [16–20,14,21–26] and the references cited therein. In this section, to prove the main results we will need some dynamic inequalities of Opial’s type that has been established in [14,15]. These inequalities are presented below in Theorems 2.1–2.6, but before we present these inequalities, for completeness, we recall the following concepts related to the notion of time scales. A function f : T → R is said to be right-dense continuous (rd-continuous) provided that f is continuous at right-dense points and at left-dense points in T, the left hand limits exist and are finite. The set of all such rd-continuous functions is denoted by Crd (T). The graininess function µ for a time scale T is defined by µ(t ) := σ (t ) − t, and for any function f : T → R the notation f σ (t ) denotes f (σ (t )). We will make use of the following product and quotient rules for the derivative of the product fg and the quotient f /g (where gg σ ̸= 0, here g σ = g ◦ σ ) of two differentiable functions f and g ∆
∆
σ ∆
∆
∆ σ
∆
(fg ) = f g + f g = fg + f g ,
f
and
=
g
f ∆ g − fg ∆ gg σ
.
(2.1)
The integration by parts formula is given by b
∆
f (t )g (t )∆t = [f (t )g (t )
]ba
−
a
b
f ( t ) ∆t = a
b a
f (t )∆t =
(2.2)
b a
f (t )dt and the integration formula on a discrete time scale is defined by
f (t )µ(t ).
(2.3)
t ∈(a,b)
For example, if T = Z, then
b a
f (t )∆t =
f (n) and if T = qN0 , where q > 1, then on the interval [a, b/q] we have
b−1 n=a
logq (b)−1
b
f ∆ (t )g σ (t )∆t .
a
Note that if T = R, then
b
f (t )∆t = (q − 1) a
qn f (qn ).
n=logq (a)
The books on the subject of time scales by Bohner and Peterson [27,28] summarize and organize much of time scale calculus and contain some results for dynamic equations on time scales. Now, we are ready to state the main inequalities that will be needed in the proofs of the main results. We begin with the inequalities proved in [14]. Theorem 2.1. Let T be a time scale with a, τ ∈ T and λ, δ be positive real numbers such that λ ≥ 1, and let A, B be nonnegative
τ
−1
rd-continuous functions on (a, τ )T such that a A λ+δ−1 (t )∆t < ∞. If x : [a, τ ] ∩ T → R+ is delta differentiable with x(a) = 0, (and x∆ does not change sign in (a, τ )T ), then we have τ
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ≤ K1 (a, τ , λ, δ)
τ
a
λ+δ
A(t ) x∆ (t )
∆t ,
(2.4)
a
where λ−1
K1 (a, τ , λ, δ) := 2
sup a≤t ≤τ
τ
×
λ
µ (t )
(B(t ))
λ+δ λ
B(t )
+2
A(t )
− λδ
(A(t ))
a
2λ−1
t
δ λ+δ
−1 A λ+δ−1
δ λ+δ
(t )∆t
(λ+δ−1)
λ λ+δ
∆t
.
(2.5)
a
Theorem 2.2. Let T be a time scale with τ , b ∈ T and λ, δ be positive real numbers such that λ ≥ 1, and let A, B be nonnegative
b
−1
rd-continuous functions on (t , b)T such that τ A λ+δ−1 (t )∆t < ∞. If x : [τ , b] ∩ T → R+ is delta differentiable with x(b) = 0,
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
(and x∆ does not change sign in (τ , b)T ), then we have b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ≤ K2 (t , b, λ, δ)
τ
b
λ+δ
A(t ) x∆ (t )
τ
∆t ,
(2.6)
where λ−1
K2 (τ , b, λ, δ) := 2
sup
τ ≤t ≤b
λ
µ (t )
b
(B(t ))
× τ
λ+δ λ
B(t )
2λ−1
+2
A(t )
(A(t ))
b
− λδ
δ λ+δ
−1 A λ+δ−1
δ λ+δ
(t )∆t
λ λ+δ
(λ+δ−1)
∆t
.
(2.7)
t
In the following, we assume that there exists τ ∈ (a, b) which is the unique solution of the equation K (λ, δ) = K1 (a, τ , λ, δ) = K2 (τ , b, λ, δ) < ∞,
(2.8)
where K1 (a, τ , λ, δ) and K2 (τ , b, λ, δ) are defined as in Theorems 2.1 and 2.2. Note that since b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t =
τ
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t
a
a
b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ,
+ τ
then the combination of Theorems 2.1 and 2.2 gives us the following result. Theorem 2.3. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λ ≥ 1, and let A, B be nonnegative −1
b
rd-continuous functions on (a, b)T such that a A λ+δ−1 (t )∆t < ∞. If x : [a, b] ∩ T → R+ is delta differentiable with x(a) = 0 = x(b), (and x∆ does not change sign in (a, b)T ), then we have b
δ λ B(t ) |x(t ) + xσ (t )| x∆ (t ) ∆t ≤ K (λ, δ)
b
λ+δ
A(t ) x∆ (t )
∆t .
(2.9)
a
a
In the following, we present the inequalities that have been proved in [15]. Theorem 2.4. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λ + δ > 1, and let r, s be
τ
−1
nonnegative rd-continuous functions on (a, τ )T such that a r λ+δ−1 (t )∆t < ∞. If x : [a, τ ] ∩ T → R is delta differentiable with x(a) = 0 (and x∆ does not change sign in (a, τ )T ), then τ
δ s(t ) |x(t )|λ x∆ (t ) ∆t ≤ G1 (a, τ , λ, δ)
a
τ
λ+δ
r (t ) x∆ (t )
∆t ,
(2.10)
a
where G1 (a, τ , λ, δ) =
δ λ+δ
δ λ+δ
τ
(s(t ))
λ+δ λ
(r (t ))
− λδ
t
r
a
−1 λ+δ−1
(t )∆t
(λ+δ−1)
λ λ+δ
.
∆t
(2.11)
a
Theorem 2.5. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λ + δ > 1, and let r , s be
b
−1
nonnegative rd-continuous functions on (b, τ )T such that τ r λ+δ−1 (t )∆t < ∞. If x : [τ , b] ∩ T → R is delta differentiable with x(b) = 0, (and x∆ does not change sign in (τ , b)T ), then we have b
τ
δ s(t ) |x(t )| x∆ (t ) ∆t ≤ G2 (τ , b, λ, δ) λ
b
λ+δ
r (t ) x∆ (t )
τ
∆t ,
(2.12)
where G2 (τ , b, λ, δ) =
δ λ+δ
δ λ+δ
b
τ
(s(t ))
λ+δ λ
(r (t ))
− λδ
b
r
−1 λ+δ−1
(t )∆t
(λ+δ−1)
λ λ+δ
∆t
.
t
In the following, we assume that there exists τ ∈ (a, b) which is the unique solution of the equation G(λ, δ) = G1 (a, τ , λ, δ) = G2 (τ , b, λ, δ) < ∞,
(2.13)
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1781
where G1 (a, τ , λ, δ) and G2 (τ , b, λ, δ) are defined as in Theorems 2.4 and 2.5. Note that since b
δ
s(t ) |x(t )|λ x∆ (t ) ∆t =
a
τ
δ
s(t ) |x(t )|λ x∆ (t ) ∆t +
b
τ
a
δ
s(t ) |x(t )|λ x∆ (t ) ∆t ,
then the combination of Theorems 2.4 and 2.5 gives us the following result. Theorem 2.6. Let T be a time scale with a, b ∈ T and λ, δ be positive real numbers such that λδ > 0 and λ + δ > 1, and let r, s −1
b
be nonnegative rd-continuous functions on (a, b)T such that a r λ+δ−1 (t )∆t < ∞. If x : [a, b] ∩ T → R is delta differentiable with x(a) = 0 = x(b), (and x∆ does not change sign in (a, b)T ), then we have b
δ
s(t ) |x(t )|λ x∆ (t ) ∆t ≤ G(λ, δ)
b
a
λ+δ
r (t ) x∆ (t )
∆t .
(2.14)
a
To simplify the presentation of the results, we define
Λ(β) := sup µγ (t ) α≤t ≤β
Λ(α) := sup µγ (t ) α≤t ≤β
Rα (t ) :=
∆s
t
α
1
r (s) γ
|Q (t )| , r (t )
where Q (t ) =
|Q (t )| , r (t )
where Q (t ) =
,
β
q(s)∆s,
t
and Rβ (t ) :=
β
t
α
∆s 1
r ( s) γ
t
q(s)∆s,
.
Note that when T = R, we have Λ(α) = 0 = Λ(β) and when T = Z, we have
Λ(β) = sup
β−1 q ( s ) s=t r (t )
α≤t ≤β
,
t −1 q(s) s=α
Λ(α) = sup
and
α≤t ≤β
r (t )
.
(2.15)
Now, we are ready to state and prove the main results. Theorem 2.7. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign on (α, β)T . If x(α) = x∆ (β) = 0, then 22γ −2 Λ(β) +
2
3γ −2
(γ + 1)
1
γ +1
×
β
|Q (t )|
γ +1 γ
1
r (t )
α
γ
γ γ+1 (Rα (t ))γ ∆t
γ γ+1 β γ +1 1 |p(t )|γ +1 γ 1 γ × + ≥1− , (Rα (t )) ∆t γ 1+γ r (t ) c α β where Q (t ) = t q(s)∆s. If x∆ (α) = x(β) = 0, then
22γ −2 Λ(α) +
23γ −2 1
(γ + 1) γ +1
×
β
|Q (t )|
γ +1 γ
γ γ+1 γ Rβ (t ) ∆t
1
r γ (t )
α
(2.16)
γ γ+1 β γ +1 1 γ |p(t )|γ +1 γ 1 R β ( t ) ∆t + × ≥1− , γ 1+γ r (t ) c α t where Q (t ) = α q(s)∆s.
(2.17)
Proof. We prove (2.16). Without loss of generality we may assume that x(t ) ≥ 0 in [α, β]T . Multiplying (1.1) by xσ and integrating by parts, we have β
α
γ ∆
r (t ) x∆ (t )
β
+ α
xσ ( t ) ∆ t +
p(t )xσ (t ) x∆ (t )
γ
β
α
γ
p(t )xσ (t ) x∆ (t )
∆t = −
β
α
q(t ) (xσ (t ))
β γ ∆t = r (t ) x∆ (t ) x(t )α − γ +1
∆t .
β
α
γ +1
r (t ) x∆ (t )
∆t
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
Using the assumption that x(α) = x∆ (β) = 0, we have β
− α
r (t ) x∆ (t )
γ +1
This implies (note that Q (t ) = β
α
γ +1
r (t ) x∆ (t )
α
β
p(t )xσ (t ) x∆ (t )
γ
β
∆t = −
α
q(t ) (xσ (t ))
γ +1
∆t .
q(s)∆s) that
t
β
∆t =
β
∆t +
α
p(t )xσ (t ) x∆ (t )
γ
∆t −
β
α
γ +1
Q ∆ (t ) (xσ (t ))
∆t .
(2.18)
Integrating by parts (see (2.2)) the right hand side, we see that β
α
γ +1
r (t ) x∆ (t )
∆t =
β
α
p(t )xσ (t ) x∆ (t )
β ∆t − Q (t )(x(t ))γ +1 α +
γ
β
α
∆
Q (t ) xγ +1 (t )
∆t .
Again using the assumptions x(α) = 0 and Q (β) = 0, we obtain β
α
γ +1 r (t ) x (t ) dt =
∆
β
α
σ
∆
p(t )x (t ) x (t )
γ
∆t +
β
α
∆
Q (t ) xγ +1 (t )
∆t .
(2.19)
Applying the chain rule formula
xλ (t )
∆
=λ
1
λ−1
[hxσ (t ) + (1 − h)x(t )]
dhx∆ (t ),
for λ > 0,
(2.20)
0
which is a simple consequence of Keller’s chain rule [27, Theorem 1.90], and the inequality (see [29, p. 500]) aλ + bλ ≤ (a + b)λ ≤ 2λ−1 (aλ + bλ ),
if a, b ≥ 0, λ ≥ 1,
(2.21)
we see that
γ +1 ∆ ≤ (γ + 1 ) x ( t )
1
|hxσ (t ) + (1 − h)x(t )|γ dh x∆ (t )
0
1 σ 1 |hx (t )|γ dh + 2γ −1 (γ + 1) x∆ (t ) |(1 − h)x(t )|γ dh (γ + 1) x∆ (t ) 0 0 = 2γ −1 x∆ (t ) |xσ (t )|γ + 2γ −1 x∆ (t ) |x(t )|γ ≤ 2γ −1 |xσ (t ) + x(t )|γ x∆ (t ) . γ −1
≤2
(2.22)
This and (2.19) imply that β
α
γ +1
r (t ) x∆ (t )
∆t ≤
β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t + 2γ −1
β
α
|Q (t )| |x(t ) + xσ (t )|γ x∆ (t ) ∆t .
(2.23)
β Applying inequality (2.4) on the integral α |Q (t )| |x(t ) + xσ (t )|γ x∆ (t ) ∆t, with B(t ) = |Q (t )| , A(t ) = r (t ), λ = γ , δ = 1, we have
β
α
σ
γ
|Q (t )| |x(t ) + x (t )| x∆ (t ) ∆t ≤ K1 (α, β, γ , 1)
β
α
γ +1
r (t ) x∆ (t )
∆t ,
(2.24)
where
K1 (α, β, γ , 1) = 22γ −2 Λ(β) + 23γ −2
1
(γ + 1)
1
γ +1
β
α
|Q (x)|
γ +1 γ
1
r (x) γ
γ γ+1 (Rα (x))γ ∆x
.
Using that fact that xσ = x(t ) + µ(t )x∆ (t ), we see that β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t =
α
≤ α
β
β
γ |p(t )| x(t ) + µ(t )x∆ (t ) x∆ (t ) ∆t γ |p(t )| |x(t )| x∆ (t ) ∆t +
β
α
γ +1 µ(t ) |p(t )| x∆ (t ) ∆t .
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1783
γ Applying inequality (2.10) on the integral α |p(t )| |x(t )| x∆ (t ) ∆t with s(t ) = |p(t )|, λ = 1 and δ = γ , we see that β β ∆ γ γ +1 |p(t )| |x(t )| x (t ) ∆t ≤ G1 (α, β, 1, γ ) r (t ) x∆ (t ) ∆t , (2.25) β
α
α
where
G1 (α, β, 1, γ ) =
γ γ+1
γ 1+γ
β
× α
|p(t )|γ +1 (Rα (t ))γ ∆t (r (t ))γ
γ +1 1
.
Using the assumption that 0 ≤ p(t )µ(t ) ≤ r (t )/c, we see that β
α
γ
p(t ) |xσ (t )| x∆ (t ) ∆t ≤ G1 (α, β, 1, γ )
β
α
γ +1
r (t ) x∆ (t )
β
1
∆t +
c
α
γ +1
r (t ) x∆ (t )
∆t .
(2.26)
Substituting (2.24) and (2.26) into (2.23), we have
1−
1
β
c
α
γ +1
r (t ) x∆ (t )
∆t ≤ K1 (α, β, γ , 1)
β
α
+ G1 (α, β, 1, γ )
γ +1
r (t ) x∆ (t ) β
α
∆t
γ +1
r (t ) x∆ (t )
∆t .
(2.27)
Then, we have from (2.27) that 1−
1 c
≤ K1 (α, β, γ , 1) + G1 (α, β, 1, γ ) 2
3γ −2
β
|Q (t )|
γ γ+1
γ +1 γ
(Rα (t ))γ ∆t 1 α r γ (t ) (γ + 1) γ +1 γ γ+1 β γ +1 1 |p(t )|γ +1 γ γ + , (Rα (t )) ∆t 1+γ r γ (t ) α
= 22γ −2 Λ(β) +
1
which is the desired inequality (2.16). The proof of (2.17) is similar to (2.16) by using (2.6) of Theorem 2.2, (2.7) instead of (2.5), (2.12) of Theorem 2.4 by using (2.13) instead of (2.11). The proof is complete. In Theorem 2.7 if r (t ) = 1, then we have the following result. Corollary 2.1. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then 2 γ −2
2
Λ(β) +
γ
+
1+γ
where Q (t ) = 2 γ −2
2
β t
β
γ +1 γ
γ
γ γ+1
|Q (t )| × (t − α) ∆t 1 α (γ + 1) γ +1 γ γ+1 β γ +1 1 1 γ +1 γ |p(t )| (t − α) ∆t × ≥1− , c
α
q(s)∆s. If x∆ (α) = x(β) = 0, then 23γ −2
Λ(α) +
γ
+
23γ −2
1+γ
β
γ +1 γ
γ
γ γ+1
|Q (t )| × (β − t ) ∆t 1 α (γ + 1) γ +1 γ γ+1 β γ +1 1 1 γ +1 γ |p(t )| (β − t ) ∆t × ≥1− , c
α
t
where Q (t ) = α q(s)∆s. As a special case of Theorem 2.7, when γ = 1, we have the following result. Corollary 2.2. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then
Λ(β) +
√
β
2 α
|Q (t )|2 rα (t )∆t r (t )
12
1
β
+√
2
α
p2 ( t ) r (t )
Rα (t )∆t
12
1
≥1− , c
1784
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
where Rα (t ) = α r∆(ss) and Q (t ) =
t
β t
q(s)∆s. If x∆ (α) = x(β) = 0, then
12 β 2 12 |Q (t )|2 p (t ) 1 1 Λ(α) + 2 rβ (t )∆t +√ Rβ (t )∆t ≥1− , r ( t ) r ( t ) c 2 α α β ∆s t where Rβ (t ) = t r (s) and Q (t ) = α q(s)∆s. √
β
As a special case of Corollary 2.2, when p(t ) = 0, we have the following result. Corollary 2.3. Suppose that x is a nontrivial solution of r (t )x∆ (t )
∆
+ q(t )xσ (t ) = 0,
t ∈ [α, β]T ,
(2.28)
and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then
√
β
|Q (t )|2 r (t )
2 α
where Q (t ) =
√
β t
β
α
α
∆t r (t )
∆t
12
+ Λ(β) ≥ 1,
q(s)∆s. If x∆ (α) = x(β) = 0, then
|Q (t )|2 r (t )
2
t
β
t
∆t r (t )
∆t
21
+ Λ(α) ≥ 1,
t
where Q (t ) = α q(s)∆s. Remark 1. Theorem 2.7 yields sufficient conditions for disfocality of (1.1), i.e., sufficient conditions so that there does not exist a nontrivial solution x satisfying x(α) = x∆ (β) = 0 or x∆ (α) = x(β) = 0. As a special case when T = R, it is known that Λ(α) = Λ(β) = 0 and then the results in Corollary 2.3, with r (t ) = 1, reduce to the following results obtained by Brown and Hinton [30] for the second order differential equation ′′
x + q(t )x(t ) = 0,
α ≤ t ≤ β.
(2.29)
Corollary 2.4 ([30]). If x is a solution of Eq. (2.29) such that x (α) = x′ (β) = 0, then β
2 α
Q 2 (s)(s − α)ds > 1,
where Q (t ) = β
2 α
β t
q(s)ds. If instead x′ (α) = x (β) = 0, then
Q 2 (s)(β − s)ds > 1,
t
where Q (t ) = α q(s)ds. On a time scale T, we note from the chain rule (2.20) that
∆ (t − a)λ+δ = (λ + δ)
1
[h(σ (t ) − a) + (1 − h)(t − a)]λ+δ−1 dh
0
≥ (λ + δ)
1
[h(t − a) + (1 − h)(t − a)]λ+δ−1 dh
0
= (λ + δ)(t − a)λ+δ−1 . This implies that τ
a
( t − a)
(λ+δ−1)
∆t ≤
τ
a
1
(λ + δ)
(t − a)λ+δ
∆
∆t =
(τ − a)λ+δ . (λ + δ)
(2.30)
Remark 2. By using the maximum of |Q | and |p| on [α, β]T and substituting (2.30) into the results of Corollary 2.1, we have the following results.
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1785
Corollary 2.5. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then 23γ −2 (β − α)γ
(γ + 1)
β
max
α≤t ≤β
t
q(s)∆s +
γ β γ γ +1 1 (β − α) max |p(t )| + 22γ −2 sup µγ (t ) q(s)∆s ≥ 1 − , α≤t ≤β γ +1 c α≤t ≤β t
and if x∆ (α) = x(β) = 0, then 23γ −2 (β − α)γ
(γ + 1)
γ t t γ γ +1 1 q(s)∆s + max (β − α) max |p(t )| + 22γ −2 sup µγ (t ) q(s)∆s ≥ 1 − . α≤t ≤β α≤t ≤β γ +1 c α≤t ≤β α α
As a special case when T = R, and p(t ) = 0, we have Λ(α) = Λ(β) = 0 and then the results in Corollary 2.5 reduce to the following results for the second order half-linear differential equation
(x′ (t ))γ
′
+ q(t )(x(t ))γ = 0,
α ≤ t ≤ β,
(2.31)
where γ ≥ 1 is a quotient of odd positive integers. Corollary 2.6. Suppose that x is a nontrivial solution of (2.31) and x′ does not change sign in (α, β). If x (α) = x′ (β) = 0, then
β q(s)ds ≥ 1. (β − α) max α≤t ≤β (γ + 1) t 23γ −2
γ
(2.32)
If instead x′ (α) = x (β) = 0, then 23γ −2
t (β − α) max q(s)ds ≥ 1. α≤ t ≤β (γ + 1) α γ
(2.33)
When γ = 1, the conditions (2.32) and (2.33) reduce to the conditions obtained by Harris and Kong [31] for second order differential equation (2.29). Corollary 2.7 ([31]). Suppose that x is a nontrivial solution of (2.29) and x′ does not change sign in (α, β). If x (α) = x′ (β) = 0, then
(β − α) max α≤t ≤β
β t
q(s)ds > 1.
(2.34)
If instead x′ (α) = x (β) = 0, then
t (β − α) max q(s)ds > 1. α≤t ≤β
(2.35)
α
As a special case when T = Z, we see that Λ(α) and Λ(β) are defined as in (2.15) and then the results in Corollary 2.5 reduce to the following results for the second order half-linear difference equation
∆(∆x(n))γ + p(n)(∆x(n))γ + q(n)(x(n + 1))γ = 0,
α ≤ n ≤ β,
(2.36)
where γ ≥ 1 is a quotient of odd positive integers and p(n) ≤ 1/c. Corollary 2.8. Suppose that x is a nontrivial solution of (2.36) and ∆x(n) does not change sign in (α, β)T . If x(α) = ∆x(β) = 0, then 1−
1 c
≤
23γ −2 (β − α)γ
(γ + 1)
β−1 β−1 γ γ γ+1 2γ −2 max q(s) + 2 sup q(s) + (β − α) max |p(n)| , α≤n≤β α≤n≤β γ +1 α≤n≤β s=n s=n
and if ∆x(α) = x(β) = 0, then 1−
1 c
≤
23γ −2 (β − α)γ
(γ + 1)
n−1 n−1 γ γ γ+1 2γ −2 max q(s) + 2 sup q(s) + (β − α) max |p(n)| . α≤n≤β α≤n≤β γ +1 α≤n≤β s=α s=α
Remark 3. One can apply the results in Theorem 2.7 to obtain some new results for Eqs. (1.4)–(1.6) as obtained for Eqs. (1.2) and (1.3) by using the integration formula (2.3). The details are left to the reader.
1786
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
If we apply the inequality
|a + b|λ ≤ 2λ−1 |a|λ + |b|λ ,
where a, b are real numbers and λ ≥ 1,
∆
with a = x(t ) and b = µ(t )hx (t ), then we have from (2.20) that
∆ γ +1 ∆ x (t ) ≤ (γ + 1) x (t )
1
x(t ) + µ(t )hx∆ (t )γ dh
0 1
|x(t )|γ dh + 2γ −1 (γ + 1) x∆ (t )
γ −1
(γ + 1) x∆ (t )
γ −1
γ +1 (γ + 1) x∆ (t ) |x(t )|γ + 2γ −1 µ(t ) x∆ (t ) .
≤2
0
=2
1
µ(t )hx∆ (t )γ dh
0
(2.37)
Substituting (2.37) into (2.19), we have that β
α
γ +1
r (t ) x∆ (t )
β
dt ≤ α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t + 2γ −1 (γ + 1)
+ 2γ −1
β
β
α
|Q (t |) x∆ (t ) |x(t )|γ ∆t
γ +1 µ(t ) |Q (t )| x∆ (t ) ∆t .
α
(2.38)
Using the inequality β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t ≤
β
α
γ |p(t )| |x(t )| x∆ (t ) ∆t +
β
α
γ +1 µ(t ) |p(t )| x∆ (t ) ∆t ,
we have from (2.38) that β
α
γ +1 r (t ) x∆ (t ) dt ≤
β
α
β
+ α
β
γ |p(t )| |x(t )| x∆ (t ) ∆t + 2γ −1 (γ + 1)
α
|Q (t |) x∆ (t ) |x(t )|γ ∆t
γ +1 ∆t . µ(t ) |p(t )| + 2γ −1 |Q (t )| x∆ (t )
(2.39)
We can now apply the inequalities in Theorems 2.4–2.5 to obtain new results different from the above results but in this case the condition µ(t ) |p(t )| ≤ r (t )/c is replaced by the new condition µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c. Applying inequality (2.10) on the term β
α
|Q (t )| x∆ (t ) |x(t )|γ ∆t ,
with s(t ) = |Q (t )| , λ = γ
δ = 1,
and
we have β
α
|Q (t |) |x(t )|γ x∆ (t ) ∆t ≤ K1∗ (α, β, γ , 1)
β
α
γ +1
r (t ) x∆ (t )
∆t ,
where K1∗ (α, β, γ , 1) =
γ +1 1
1
γ +1
β
|Q (t )|
γ +1 γ 1
(r (t ))
γ
γ |p(t )| |x(t )| x∆ (t ) ∆t ≤ G1 (α, β, 1, γ )
α
γ γ+1 Rγα (t )∆t
.
Using the inequality β
α
β α
γ +1
r (t ) x∆ (t )
∆t ,
where G1 (α, β, 1, γ ) =
γ 1+γ
γ γ+1
β
× α
|p(t )|γ +1 γ R (t )∆t (r (t ))γ α
γ +1 1
and proceeding as in the proof of Theorem 2.7, we obtain the following results. Theorem 2.8. Assume that µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c where c is a positive constant such that c ≥ 1. Suppose that x is a nontrivial solution of (1.1) and x∆ does not change sign in (α, β)T . If x(α) = x∆ (β) = 0, then γ −1
2
(γ + 1)
γ γ +1
β
α
|Q (t )| 1
γ +1 γ
r γ (t )
γ γ+1 Rγα (t )∆t
+
γ 1+γ
γ γ+1
β α
|p(t )|γ +1 γ Rα (t )∆t r γ (t )
γ +1 1
1
≥1− , c
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
where Q (t ) =
β t
1787
q(s)∆s. If x∆ (α) = x(β) = 0, then
2γ −1 (γ + 1)
γ γ +1
β
|Q (t )|
γ γ+1
γ +1 γ
γ
Rβ (t )∆t
1
r (t )
α
γ
+
γ 1+γ
γ γ+1
β α
|p(t )|γ +1 γ Rβ (t )∆t r γ (t )
γ +1 1
1
≥1− , c
t
where Q (t ) = α q(s)∆s. Remark 4. Note that when T = R the condition µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c will be removed since in this case
µ(t ) = 0.
In the following, we apply Theorems 2.3 and 2.6, to determine the lower bound for the distance between consecutive generalized zeros of solutions of (1.1). Note that the applications of the above results allow the use of arbitrary anti-derivative Q in the above arguments. In the following, we assume that Q ∆ (t ) = q(t ) and assume that there exists a unique h ∈ (α, β)T , such that R(h) := Rα (h) = Rβ (h).
(2.40)
Note that the best choice of h when r (t ) = 1 is h = (β + α) /2. In the following, we assume that K h (α, β, γ , 1) = Kh (α, β, γ , 1) < ∞,
(2.41)
where K h (α, β, γ , 1) =
1
3 γ −2
2
(γ + 1)
Λ := sup µγ (t )
γ +1
|Q (t )| , r (t )
β
1
(γ + 1) γ +1
Kh (α, β, γ , 1) =
α≤t ≤β
23γ −2
|Q (t )|
γ +1 γ
1
r γ (t )
α β
|Q (t )| 1
γ +1 γ
r (t )
α
γ
γ γ+1 Rγα (h)∆t
+ 22γ −2 Λ,
γ γ+1 γ
Rβ (h)∆t
+ 22γ −2 Λ,
where Q ∆ (t ) = q(t ),
and Gh (α, β, 1, γ ) = Gh (α, β, 1, γ ) < ∞,
(2.42)
where G (α, β, 1, γ ) = h
Gh (α, β, 1, γ ) =
γ 1+γ
γ γ+1
γ 1+γ
γ γ+1
β
|p(t )|γ +1 γ Rα (h)∆t r γ (t )
γ +1 1
β
|p(t )|γ +1 γ Rβ (h)∆t r γ (t )
γ +1 1
α
α
, .
Now, we assume that K (γ , 1) is the solution of the equation K (γ , 1) = K h (α, β, γ , 1) = Kh (α, β, γ , 1) and given by
K (γ , 1) =
2
3γ −2
(γ + 1)
1
γ +1
β
|Q (t )| 1
γ +1 γ
r (t )
α
γ
γ γ+1 Rγ (h)∆t
+ 22γ −2 Λ,
(2.43)
.
(2.44)
and similarly G(1, γ ) is given by G(1, γ ) =
γ 1+γ
γ γ+1
β α
|p(t )|γ +1 γ R (h)∆t r γ (t )
γ +1 1
Theorem 2.9. Assume that Q ∆ (t ) = q(t ) and suppose x is a nontrivial solution of (1.1). If x(α) = x(β) = 0, then K (γ , 1) + G(1, γ ) ≥ 1 −
1 c
,
where K (α, β) and K (α, β) are defined as in (2.43) and (2.44).
(2.45)
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S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
Proof. We multiply (1.1) by xσ (t ) and proceed as in Theorem 2.7 to obtain β
α
r (t ) x∆ (t )
γ +1
β
∆t =
γ
p(t )xσ (t ) x∆ (t )
α
∆t +
β
α
Q ∆ (t ) (xσ (t ))
γ +1
∆t .
Integrating by parts the right hand side (see (2.2)), we see that β
α
r (t ) x∆ (t )
γ +1
β
∆t =
β ∆t + Q (t )(x(t ))γ +1 α −
γ
p(t )xσ (t ) x∆ (t )
α
β
α
∆
Q (t ) xγ +1 (t )
∆t .
(2.46)
Using the facts that x(α) = 0 = x(β), we obtain β
α
γ +1
r (t ) x∆ (t )
β
dt ≤ α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t +
β
α
∆ |Q (t )| xγ +1 (t ) dt .
We proceed as in the proof of Theorem 2.7, to get β
α
γ +1 ∆ γ −1 |Q (t )| x (t ) ∆t ≤ 2
β α
|Q (t )| |x(t ) + xσ (t )|γ x∆ (t ) ∆t .
Applying inequality (2.9) with B(t ) = |Q (t )|, A(t ) = r (t ), λ = γ and δ = 1, we have β
α
∆ |Q (t )| xγ +1 (t ) dt ≤ 2γ −1 K (γ , 1)
β
α
γ +1
r (t ) x∆ (t )
∆t .
Also as in the proof of Theorem 2.7 by applying inequality (2.14), we obtain β
α
γ |p(t )| |xσ (t )| x∆ (t ) ∆t ≤ G(1, γ )
From this inequality, after cancelling the proof.
β α
β
γ +1 1 r (t ) x∆ (t ) ∆t + c
α
γ +1
r (t ) x∆ (t )
β
α
γ +1
r (t ) x∆ (t )
∆t .
∆t, we get the desired inequality (2.45). This completes
In the following, we assume that (1.1) has a solution x which, together with x∆ is positive on (α, β)T and assume that (2.40) holds. From the chain rule (2.20), we see that
∆
xγ +1 (t )
= (γ + 1)
1
(hxσ (t ) + (1 − h)x(t ))γ dhx∆ (t )
0 1
≥ (γ + 1)x∆ (t )
(hxσ (t ))γ dh = x∆ (t ) (xσ (t ))γ .
0
Substituting this in (2.46), and assuming that x(α) = x(β) = 0, we have β
α
γ +1
r (t ) x∆ (t )
∆t ≤
β
α
β
≤ α
p(t ) (xσ (t ))
γ
x∆ (t ) ∆t −
β
α
γ
Q (t )x∆ (t ) (xσ (t )) ∆t
|Q (t ) − p(t )| (xσ (t ))γ x∆ (t )∆t .
(2.47)
Applying the inequality in Theorem 2.6 on the integral β
α
|Q (t ) − p(t )| (xσ (t ))γ x∆ (t )∆t ,
we have from (2.47) that G∗ (γ , 1) ≥ 1 − 1c , where G∗ (γ , 1) =
γ 1+γ
γ γ+1
β α
|Q (t ) − p(t )|γ +1 (R(h))γ ∆t r γ (t )
γ +1 1
.
This gives us the following results. Theorem 2.10. Assume that Q ∆ (t ) = q(t ) and suppose x(t ) is a nontrivial solution of (1.1). If x(α) = x(β) = 0, then G∗ (1, γ ) ≥ 1 −
1 c
,
where G∗ (1, γ ) is defined as in (2.48).
(2.48)
S.H. Saker / Mathematical and Computer Modelling 58 (2013) 1777–1790
1789
Remark 5. When r (t ) = 1, we see that (2.40) is satisfied if h = (β + α) /2. One can use this choice and the maximum values of |Q (t )| and |p(t )| to obtain new results different from the results obtained in Theorem 2.9. Due to the limited space, the details are left to the interested reader. From Theorem 2.10 one can also obtain new results when T = R and see that the obtained results will be different from the results established for differential equations with damping terms in [3–9] (note that in this time scale the right hand side (1 − 1/c ) of (2.45) changes to 1). Remark 6. The sufficient conditions for disconjugacy can be obtained from Theorems 2.9 and 2.10, i.e., sufficient conditions so that there does not exist a nontrivial solution x satisfying x(α) = x(β) = 0. In fact one can prove that Eq. (1.1) is disconjugate in [α, β]T if one of the following conditions holds K (γ , 1) + G(1, γ ) < 1 − G∗ (1, γ ) < 1 −
1 c
1 c
,
(2.49)
.
(2.50)
The proof is given by assuming that the equation is not disconjugate, and without loss of generality we can assume that x(α) = 0 = x(β) and follows the proof of Theorems 2.9 and 2.10 to get a contradiction to (2.49) or (2.50). Remark 7. We note that the condition µ(t ) |p(t )| ≤ r (t )/c and the condition µ(t )(|p(t )| + 2γ −1 |Q (t )|) ≤ r (t )/c may be removed if one can prove an inequality of the form β
α
δ |p(t )| |x (t )| x∆ (t ) ∆t ≤ H (λ, δ) σ
λ
β
α
λ+δ |r (t )| x∆ (t ) ∆t ,
where H (λ, δ) is the coefficient of the inequality needs to be determined as in Theorems 2.1–2.6. This open question will be left to the interested reader. Problem 1. It would be interesting to extend the above technique to cover the equation r (t ) x∆ (t )
γ ∆
γ + p(t ) x∆ (t ) + q(t ) (xσ (t ))γ = 0,
when 0 < γ ≤ 1 (see [21]). Problem 2. It would be interesting to extend the above technique to cover the delay equation with oscillatory coefficients
r (t ) x∆ (t )
γ ∆
γ + p(t ) x∆ (t ) + q(t ) (x(τ (t )))γ = 0,
where the delay function τ (t ) satisfying τ (t ) < t and limt →∞ τ (t ) = ∞. Problem 3. It would be interesting to extend the above results to obtain the lower bound for the first eigenvalue of the boundary value problem
γ ∆ γ − r (t ) x∆ (t ) + p(t ) x∆ (t ) + q(t )xγ (σ (t )) = λxγ (σ (t )), . x(α) = x(β) = 0 This in fact needs an inequality of a Wirtinger’s type of the form β
α
γ +1 r (t ) x∆ (t ) ∆t ≥ W (a, b, γ )
β
α
s(t ) |x(t )|γ +1 ∆t ,
where W (a, b, γ ) is the coefficient of the inequality (see [16]). Acknowledgements The author is very grateful to the anonymous referee and the editor for valuable remarks and comments which significantly contributed to the quality of the paper. References [1] [2] [3] [4] [5] [6]
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