R E S E A R C H
Open Access
Asymptotics and oscillation of
higher-order functional dynamic equations
with Laplacian and deviating arguments
Taher S Hassan
**Correspondence: [email protected] Department of Mathematics, Faculty of Science, University of Hail, Hail, 2440, Saudi Arabia
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
Abstract
In this paper, we deal with the asymptotics and oscillation of the solutions of
higher-order nonlinear dynamic equations with Laplacian and mixed nonlinearities of the form
rn–1(t)
φ
αn–1(
rn–2(t)(
· · ·(
r1(t)φ
α1x(t)
)
· · ·)
)
+
N
ν=0
pν(t)
φ
γν(
x(
gν(t)))
= 0on an above-unbounded time scale. By using a generalized Riccati transformation and integral averaging technique we study asymptotic behavior and derive some new oscillation criteria for the cases without any restrictions ong(t) and
σ
(t) and whennis even and odd. Our results obtained here extend and improve the results of Chen and Qu (J. Appl. Math. Comput. 44(1-2):357-377, 2014) and Zhanget al.(Appl. Math. Comput. 275:324-334, 2016).MSC: 34K11; 34N05; 39A10; 39A13; 39A21; 39A99
Keywords: asymptotic behavior; oscillation; higher order; dynamic equations; dynamic inequality; time scales
1 Introduction
We are concerned with the asymptotic and oscillatory behavior of the higher-order non-linear functional dynamic equation
rn–(t)φαn–
rn–(t)
· · ·r(t)φα
x(t)· · ·
+
N
ν=
pν(t)φγν
xgν(t)
= (.)
on an above-unbounded time scaleT, assuming without loss of generality thatt∈T. For
A⊂TandB⊂R, we denote byCrd(A,B) the space of right-dense continuous functions
fromAtoBand byC
rd(A,B) the set of functions inCrd(A,B) with right-dense continuous
-derivatives. We refer the readers to the books by Bohner and Peterson [, ] for an excellent introduction of calculus of time scales. Throughout this paper, we suppose that:
(i) n,N∈N,n≥, andφβ(u) :=|u|β–u,β> ;
(ii) ri∈Crd([t,∞)T, (,∞))fori= , , . . . ,n– are such that ∞
t r–/αi
i (τ)τ=∞; (.)
(iii) αi> ,i= , , . . . ,n– , andγν> ,ν= , , . . . ,N, are constants such that
γν>γ, ν= , , . . . ,l and γν<γ, ν=l+ ,l+ , . . . ,N; (.)
(iv) pν∈Crd([t,∞)T, [,∞)),ν= , , . . . ,N, are such that not all of thepν(t)vanish in
a neighborhood of infinity;
(v) gν:T→Tare rd-continuous functions such thatlimt→∞gν(t) =∞,ν= , , . . . ,N.
By a solution of equation (.) we mean a functionx∈C
rd([Tx,∞)T,R) for someTx≥
such thatx[i]∈C
rd([Tx,∞)T,R),i= , , . . . ,n– , that satisfies equation (.) on [Tx,∞)T,
where
x[i]:=riφαi
x[i–], i= , , . . . ,n, withrn= ,αn= , andx[]=x. (.)
A solutionx(t) of equation (.) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is nonoscillatory.
Oscillation criteria for higher-order dynamic equations on time scales have been studied by many authors. For instance, Graceet al.[] obtained sufficient conditions for oscillation for the higher-order nonlinear dynamic equation
xn(t) +p(t)xσg(t)γ= ,
whereγ is the quotient of positive odd integers, and whereg(t)≤t. In [], some compari-son criteria have been studied wheng(t)≤t, and some oscillation criteria are given when nis even andg(t) =t. The results in [] have been proved when
∞
t
∞
t
∞
s
p(u)ust=∞. (.)
Wuet al.[] established Kamanev-type oscillation criteria for the higher-order nonlinear dynamic equation
rn–(t)
rn–(t)
· · ·r(t)x(t)
· · ·α
+ft,xg(t)= ,
whereαis the quotient of positive odd integers,g:T→Twithg(t) >tandlimt→∞g(t) =
∞, and there exists a positive rd-continuous functionp(t) such that f(utα,u)≥p(t) foru= . Sunet al.[] proved some criteria for oscillation and asymptotic behavior of the dynamic equation
rn–(t)
rn–(t)
· · ·r(t)x(t)
· · ·α
whereα≥ is the quotient of positive odd integers,g:T→Tis an increasing differen-tiable function withg(t)≤t,g◦σ=σ◦g, andlimt→∞g(t) =∞, and there exists a positive
rd-continuous functionp(t) such that f(utβ,u)≥p(t) foru= andβ≥ is the quotient of positive odd integers. Sunet al.[] studied quasilinear dynamic equations of the form
rn–(t)
rn–(t)
· · ·r(t)x(t)
· · ·α
+p(t)xβ(t) = ,
whereα,βare the quotients of positive odd integers. Also, the results obtained in [–] are presented when
∞
t rn–(t)
∞
t
rn–(s)
∞
s
p(u)u /α
s
t=∞. (.)
Hassan and Kong [] obtained asymptotics and oscillation criteria for thenth-order half-linear dynamic equation
x[n–](t) +p(t)φα[,n–]
xg(t)= ,
whereα[,n– ] :=α· · ·αn–, and Grace and Hassan [] further studied the asymptotics
and oscillation for the higher-order nonlinear dynamic equation
x[n–](t) +p(t)φγ
xσg(t)= .
However, the establishment of the results in [] requires the restriction on the time scale Tthatg∗◦σ=σ ◦g∗withg∗(t) =min{t,g(t)}, which is hardly satisfied. Hassan [] im-proved the results in [, ] and established oscillation criteria for the higher-order quasi-linear dynamic equation
x[n–](t) +p(t)φγ
xg(t)=
whennis even or odd and whenα>γ,α=γ, andα<γ withα=α· · ·αn–. Chen and Qu
[] considered the even-order advanced type dynamic equation with mixed nonlinearities
r(t)φγ
xn–(t)+
N
ν=
pν(t)φγν
xgν(t)
= , (.)
wheren≥ is even,γν> ,gν(t)≥t, andγ>· · ·>γl>γ>γl+>· · ·>γN> . Zhanget al.
[] studied the dynamic equation (.), wheren≥ is integer andg
ν(t) > , and obtained
some of the results in [] whenγ≥. Also, the results obtained in [, ] are given when ∞
t
∞
v
r–(s)
∞
s N
ν=
pν(τ)τ
/γ
s
v=∞. (.)
Huang [] extended the work in [] to the neutral advanced dynamic equation
r(t)φα
yn–(t)+
N
ν=
pν(t)φγν
xgν(t)
wheren≥ is integer,y(t) :=x(t) +p(t)x(g(t)),γν> ,g(t)≤t, andgν(t)≥t. For more
results on dynamic equations, we refer the reader to the papers [–].
In this paper, we will discuss the higher-order nonlinear dynamic equation (.) with mixed nonlinearities on a general time scale without any restrictions ong(t) andσ(t) and also without conditions (.), (.), and (.). The results in this paper improve the results in [, , –] on the oscillation of various dynamic equations.
2 Main results
We introduce the following notations:
k+:=max{k, }, k–:=max{–k, } for anyk∈R,
and
α[h,k] :=
αh· · ·αk, h≤k,
, h>k, (.)
withα=γ=α[,n– ] andβi=α[,i]. For anyt,s∈Tand for a fixedm∈ {, , . . . ,n– },
define the functionsRm,j(t,s),j= , , . . . ,m, andpˆj(t),j= , , . . . ,n– , by the following
recurrence formulas:
Rm,j(t,s) :=
, j= ,
t s[
Rm,j–(τ,s)
rm–j+(τ)]
/αm–j+τ, j= , , . . . ,m, (.)
and
ˆ
pj(t) :=
N
ν=pν(t), j= ,
[
rn–j(t) ∞
t pˆj–(τ)τ]
/αn–j
, j= , , . . . ,n– .
For a fixedm∈ {, . . . ,n– }, define the functionsp¯m,j(t,s),j= , , , . . . ,n– , by the
re-currence formula
¯
pm,j(t,s) :=
pm(t,s), j= ,
[r n–j(t)
∞
t p¯m,j–(τ,s)τ]
/αn–j
, j= , , . . . ,n– , (.)
with
ϕm,ν(t,t) :=
, gν(t)≥σ(t),
Rm,m(gν(t),t)
Rm,m(σ(t),t), gν(t)≤σ(t), and
pm(t,s) =p(t)φα
ϕm,(t,s)
+
N
ν=
pν(t)φγν(ϕm,ν(t,s))
ην
ην
such that
N
ν=
γνην=α and
N
ν=
where
δ(t,s) :=
[t∞p¯m,n–m–(τ,s)τ] /βm–
, <βm≤,
Rβm–
m,m (t,s), βm≥,
provided that the improper integrals involved are convergent.
In the sequel, we present conditions that guarantee the following conclusions:
(C) (i) every solution of equation (.) is oscillatory ifnis even;
(ii) every solution of equation (.) either is oscillatory or tends to zero eventually if
nis odd.
Theorem . Let conditions(i)-(v)hold.Furthermore,for each i∈ {, , . . . ,n– }and suf-ficiently large T,T∈[t,∞)T,one of the following conditions is satisfied:
(a) eitherT∞p¯i,n–i–(τ,T)τ=∞,or
∞
T p¯i,n–i–(τ,T)τ<∞and either
lim sup t→∞ R
βi
i,i(t,T) ∞
t
¯
pi,n–i–(τ,T)τ>
or
lim sup t→∞
Ri,i(t,T) ∞
t
¯
pi,n–i–(τ,T)τ
/βi > ;
(b) there existsρi∈Crd([t,∞)T, (,∞))such that
lim sup t→∞
t
T
ρi(τ)p¯i,n–i–(τ,T) –
(ρ
i (τ))+
Rβi
i,i(σ(τ),T)
τ=∞; (.)
(c) there existsρi∈Crd([t,∞)T, (,∞))such that
lim sup t→∞
t
T
ρi(τ)p¯i,n–i–(τ,T)
–
ρβi
i (τ)
(ρ
i (τ))+
+βi
+βi r
(τ)
Ri,i–(τ,T)
βi/α
τ=∞; (.)
(d) there existρi∈Crd([t,∞)T, (,∞))andHi,hi∈Crd(D,R),where
D≡ {(t,τ) :t≥τ ≥t},such that
Hi(t,t) = , t≥t, Hi(t,τ) > , t>τ ≥t, (.)
andHihas a nonpositive continuous-partial derivativeHiτ(t,τ)with respect to the second variable and satisfies
Hτ
i (t,τ) +Hi(t,τ) ρi (τ)
ρσ
i (τ)
= –hi(t,τ)
ρσ
i (τ)
Hβi/(+βi)
and
lim sup t→∞
Hi(t,T)
t
T
ρi(τ)p¯i,n–i–(τ,T)Hi(t,τ)
–
ρβi
i (τ)
(hi(t,τ))–
+βi
+βi r
(τ)
Ri,i–(τ,T)
βi/α
τ=∞; (.)
(e) there existsρi∈Crd([t,∞)T, (,∞))such that
lim sup t→∞
t
T
ρi(τ)p¯i,n–i–(τ,T)
– (ρ
i (τ))
βiρi(τ)δσ(τ,T)
r(τ)
Ri,i–(τ,T)
/α
τ=∞; (.)
(f ) there existρi∈Crd([t,∞)T, (,∞))andHi,hi∈Crd(D,R),where
D≡ {(t,τ) :t≥τ ≥t},such that(.)holds andHihas a nonpositive continuous -partial derivativeHτ
i (t,τ)with respect to the second variable and satisfies
Hτ
i (t,τ) +Hi(t,τ) ρ
i (τ) ρiσ(τ)= –
hi(t,τ) ρiσ(τ)
Hi(t,τ) (.)
and
lim sup t→∞
Hi(t,T)
t
T
ρi(τ)p¯i,n–i–(τ,T)Hi(t,τ)
– [(hi(t,τ))–]
βiρi(τ)δσ(τ,T)
r(τ)
Ri,i–(τ,T)
/α
τ=∞. (.)
Moreover,for the case where n is odd,assume that,for an integer j∈ {, , . . . ,n– },
∞
T
ˆ
pj(τ)τ=∞. (.)
Then conclusions(C)hold.
Example . Consider the higher-order nonlinear dynamic equation (.), where βi= α[,i]≤ andr(t) :=t
ξ
β with
ξ=
> ifnis even,
≤ ifnis odd, and where
ri(t) :=
tαi
βi
, i= , . . . ,n– and p(t) :=
ζ
tα+φ
α(ϕi,(t,t))
withζ> .
Choose ann-tuple (η,η, . . . ,ηn) with <ηj< satisfying (.). It is clear that conditions
(.) hold since
∞
t r–/α
(τ)τ=β /β
∞
t
τ
τξ/α =∞ and
∞
t r–/αi
i (τ)τ=β
/αi
i
∞
t
and hence (.) holds. Also,
ˆ
pn–(t) =
r(t)
∞
t
ˆ
pn–(τ)τ
/α
≥ζ/α
α
tξ
∞
t
τα+τ /α
≥ζ/α
tξ
∞
t
–
τα
τ
/α = ζ
/α
t+ξ/α.
Ifnis odd, then
∞
T
ˆ
pn–(τ)τ=ζ/α ∞
T τ τ+ξ/α =∞,
so that condition (.) holds. Then, by Theorem .(c) conclusions (C) hold if
ζ/α[i+,n–]>
α
βi/α β
i
+βi
+βi .
3 Lemmas
In order to prove the main results, we need the following lemmas. The first two lemmas are extensions of Lemmas and in [] to the nonlinear equation (.) with exactly the same proof.
Lemma . Let x(t)∈Cn
rd(T, [,∞)).Assume that(x[n–])
(t)is of eventually one sign and
not identically zero.Then there exists an integer m∈ {, , . . . ,n– }with m+n odd for (x[n–])(t)≤or with m+n even for(x[n–])(t)≥such that
x[k](t) > for k= , , . . . ,m (.)
and
(–)m+kx[k](t) > for k=m,m+ , . . . ,n– (.)
eventually.
Lemma . Assume that equation(.)has an eventually positive solution x(t)and m∈
{, , . . . ,n– }is given in Lemma.such that(.)and(.)hold for t∈[t,∞)Tfor some
t∈[t,∞)T.Then the following hold for t∈(t,∞)T: (a) fori= , , . . . ,m,
x[m–i](t)
Rm,i(t,t)
is strictly decreasing; (.)
(b) fori∈ {, , . . . ,m}andj= , , . . . ,m–i,
x[j](t)≥φα–[j+,m–i]
x[m–i](t) Rm,i(t,t)
Lemma . Assume that equation(.)has an eventually positive solution x(t)and m is given in Lemma.such that m∈ {, , . . . ,n– }and(.)and(.)hold for t≥t∈
[t,∞)T.Then,for t∈[t,∞)T,where gν(t) >tfor t≥t,and for j=m,m+ , . . . ,n– , ∞
t
¯
pm,n–j–(τ,t)τ<∞
and
(–)m+jx[j](t)≥φα[,j]
xσ(t) ∞
t
¯
pm,n–j–(τ,t)τ. (.)
Proof We show it by a backward induction. By Lemma . withm≥ we see thatx(t) is strictly increasing on [t,∞)T. As a result, (.) and (.) hold for t∈[t,∞)T. Let
t∈[t,∞)Tbe fixed. Then, forν= , , . . . ,N, ifgν(t)≥σ(t), thenx(gν(t))≥x(t) by the
fact thatx(t) is strictly increasing. Now consider the case wheregν(t)≤σ(t). In view of
Lemma .(a), we see that fori=m, R x(t)
m,m(t,t)is decreasing on (t,∞)Tand that there exists t≥tsuch thatgν(t) >tfort≥t, so that
xgν(t)
≥Rm,m(gν(t),t)
Rm,m(σ(t),t)
xσ(t) fort∈[t
,∞)T.
In both cases, we have
xgν(t)
≥ϕm,ν(t,t)xσ(t) fort∈[t,∞)T.
Therefore,
N
ν=
pν(t)φγν
xgν(t)
≥
N
ν=
pν(t)φγν
ϕm,ν(t,t)
xσ(t)γν
=φα
xσ(t)
N
ν=
pν(t)φγν
ϕm,ν(t,t)
xσ(t)γν–α.
Using the arithmetic-geometric mean inequality (see [], p.), we have
N
ν= ηνvν≥
N
ν=
vην
ν for anyvν≥,ν= , . . . ,N.
Then, fort≥T,
N
ν=
pν(t)φγν
ϕm,ν(t,t)
xσ(t)γν–α
=p(t)φα
ϕm,(t,t)
+
N
ν= ην
pν(t)φγν(ϕm,ν(t,t)) ην
xσ(t)γν–α
≥p(t)φα
ϕm,(t,t)
+
N
ν=
pν(t)φγν(ϕm,ν(t,t))
ην
ην
In view of (.), we have
N
ν=
γνην–α
N
ν= ην= .
Hence,
N
ν=
pν(t)φγν
ϕm,ν(t,t)
xσ(t)γν–α
≥p(t)φα
ϕm,(t,t)
+
N
ν=
pν(t)φγν(ϕm,ν(t,t)) ην
ην
=p(t,t).
This, together with (.), shows that, fort∈[t,∞)T,
–x[n–](t)≥p(t,t)φα
xσ(t)=p¯
m,(t,t)φα
xσ(t). (.)
Replacingtbyτ in (.), integrating fromt∈[t,∞)Ttov∈[t,∞)T, and using (.), we
have
x[n–](t) > –x[n–](v) +x[n–](t)≥
v
t
¯
pm,(τ,t)φα
xσ(τ)τ
≥φα
xσ(t)
v
t
¯
pm,(τ,t)τ.
Hence, by taking limits asv→ ∞we obtain that
x[n–](t)≥φα
xσ(t) ∞
t ¯
pm,(τ,t)τ.
This shows that t∞p¯m,(τ,t)τ < ∞ and (.) holds for j = n – . Assume that
∞
t p¯m,n–j–(τ,t)τ <∞and (.) holds for somej∈ {m+ ,m+ , . . . ,n– }. Then, for
(.),
(–)m+jx[j–](t) = (–)m+jφα–
j
x[j](t)
rj(t)
≥φ–α
j
φα[,j]
xσ(t) rj(t)
∞
t
¯
pm,n–j–(τ,t)τ
/αj
=φα[,j–]
xσ(t)p¯m,n–j(t,t).
Replacingtbyτ and then integrating it fromt∈[t,∞)Ttov∈[t,∞)T, we have
(–)m+j–x[j–](t) > (–)m+jx[j–](v) –x[j–](t)
≥ v
t
φα[,j–]
xσ(τ)p¯
m,n–j(τ,t)τ
≥φα[,j–]
xσ(t)
v
t
¯
Taking limits asv→ ∞, we obtain that
(–)m+j–x[j–](t)≥φα[,j–]
xσ(t)
∞
t
¯
pm,n–j(τ,t)τ.
This shows thatt∞p¯m,n–j(τ,t)τ<∞and (.) holds forj– . Therefore, the conclusion
holds.
The following lemma improves [], Lemma ; also see [–].
Lemma . Let(.)hold.Then,there exists an N -tuple(η,η, . . . ,ηN)withην>
satis-fying(.).
Lemma .(see []) Letω(u) =au–bu+/β,where a,u≥and b,β> .Then
ω(u)≤
β
b β
a +β
+β
.
4 Proofs of main results
Proof of Theorem. Assume that equation (.) has a nonoscillatory solutionx(t). Then, without loss of generality, assume thatx(t) > andx(gν(t)) > fort∈[t,∞)T. It follows
from Lemma . that there exists an integerm∈ {, , . . . ,n– }withm+nodd such that (.) and (.) hold fort∈[t,∞)Tfor somet∈[t,∞)T. Lett≥tbe such thatgν(t) >t
fort∈[t,∞)T.
(i) Assume thatm≥.
Part I: Assume that (a) holds. By Lemma . we have that, forj=m,
∞
t
¯
pm,n–m–(τ,t)τ<∞,
which contradictst∞p¯m,n–m–(τ,t)τ =∞. If
∞
t p¯m,n–m–(τ,t)τ<∞, then by
Lem-ma . we have that, forj=m,
x[m](t)≥φα[,m]
xσ(t)
∞
t
¯
pm,n–m–(τ,t)τ
≥φβm
x(t)
∞
t
¯
pm,n–m–(τ,t)τ. (.)
By Lemma .(b) withi= andj= we get
x(t)≥φα–[,m]x[m](t)Rm,m(t,t)
=φβ–mx[m](t)Rm,m(t,t). (.)
Substituting (.) into (.), we obtain that
≥Rβm
m,m(t,t) ∞
t
¯
which contradictslim supt→∞Rβm
m,m(t,t)
∞
t p¯m,n–m–(τ,t)τ > . Substituting (.) into
(.), we obtain that
≥Rm,m(t,t) ∞
t
¯
pm,n–m–(τ,t)τ
/βm ,
which contradictslim supt→∞Rm,m(t,t)(
∞
t p¯m,n–m–(τ,t)τ)/
βm> .
Part II: Assume that (b) holds. Define
wm(t) :=ρm(t)
x[m](t)
xβm(t). (.)
By the product rule and the quotient rule we have
wm(t) =ρm(t)
x[m](t) xβm(t)
+ρm(t)
x[m](t) xβm(t)
σ
=ρm(t)
xβm(t)(x[m](t))– (xβm(t))x[m](t)
(xβm(t))σxβm(t)
+ρm(t)
x[m](t)
xβm(t) σ
=ρm(t)
(x[m](t))
(xβm(t))σ –ρm(t)
(xβm(t))
(xβm(t))σ
x[m](t)
xβm(t)+ρ
m(t)
x[m](t)
xβm(t) σ
. (.)
From Lemma . withj=m+ we have
–x[m+](t)≥φα[,m+]
xσ(t)
∞
t
¯
pm,n–m–(τ,t)τ, (.)
which, together with (.), implies that, fort∈[t,∞)T,
–x[m](t)≥φα[,m]
xσ(t) rm+(t)
∞
t
¯
pm,n–m–(τ,t)τ
/αm+
=φβm
xσ(t)p¯
m,n–m–(t,t). (.)
Substituting (.) into (.), we obtain
wm(t)≤–ρm(t)p¯m,n–m–(t,t) +ρm(t)
x[m](t) xβm(t)
σ
–ρm(t)
(xβm(t))
(xβm(t))σ x[m](t) xβm(t).
When <βm≤, sincex(t) is strictly increasing, by Pötzsche chain rule ([], Thm. .)
we obtain
xβm(t) =β
m
x(t) +hμ(t)x(t)βm–dh x(t)
=βm
( –h)x(t) +h xσ(t)βm–dh x(t)
≥βm
Hence,
wm(t)≤–ρm(t)p¯m,n–m–(t,t) +ρm(t)
x[m](t)
xβm(t) σ
–βmρm(t)
x(t)
xσ(t)
x[m](t)
xβm(t) σ
≤–ρm(t)p¯m,n–m–(t,t) +ρm(t)
x[m](t)
xβm(t) σ
. (.)
Whenβm≥, sincex(t) is strictly increasing, again by Pötzsche chain rule we obtain
xβm(t) =β
m
x(t) +hμ(t)x(t)βm–dh x(t)
=βm
( –h)x(t) +h xσ(t)βm–
dh x(t)
≥βm
x(t)βm– x(t).
Therefore,
w
m(t)≤–ρm(t)p¯m,n–m–(t,t) +ρm(t)
x[m](t)
xβm(t) σ
–βmρm(t)
x(t)
x(t)
x[m](t)
xβm(t) σ
≤–ρm(t)p¯m,n–m–(t,t) +ρm(t)
x[m](t)
xβm(t) σ
. (.)
Then, forβm> ,
wm(t)≤–ρm(t)p¯m,n–m–(t,t) +ρm(t)
x[m](t) xβm(t)
σ
. (.)
By using Lemma . (b) withi= andj= we see that
x(t)≥φα–[,m]x[m](t)Rm,m(t,t),
which implies
x[m](t)
xβm(t) ≤ Rβm
m,m(t,t)
. (.)
Substituting (.) into (.), we get
wm(t)≤–ρm(t)p¯m,n–m–(t,t) +
ρm(t) Rβm
m,m(σ(t),t)
≤–ρm(t)p¯m,n–m–(t,t) +
(ρm(t))+
Rβm
m,m(σ(t),t)
fort∈[t,∞)T.
Integrating both sides fromttotwe get
t
t
ρm(τ)p¯m,n–m–(τ,t) –
(ρ
m(τ))+
Rβm
m,m(σ(τ),t)
τ≤wm(t) –wm(t)≤wm(t),
Part III: Assume that (c) holds. When <βm≤, by the definition ofwm(t), sincex(t) is
strictly increasing, (.) can be written as
Substituting (.) into (.), we get, forβm≥,
Using Lemma . with
a:=ρm(t)+, b:=βmρm(t)
From this and from (.) we have
wm(t)≤–ρm(t)p¯m,n–m–(t,t) +
Integrating both sides fromttot, we get
Part IV: Assume that (d) holds. Multiplying both sides of (.), withtreplaced byτ, by
Integrating by parts and using (.) and (.), we obtain
t
Using Lemma . with
From this last inequality and from (.) we have
Part V: Assume that (e) holds. From (.) we have
–
Integrating both sides fromttot, we get
t
Part VI: Assume that (f ) holds. Multiplying both sides of (.), withtreplaced byτ, by Hm(t,τ) and integrating with respect toτfromttot∈[t,∞)T, we have
Integrating by parts and using (.) and (.), we obtain
Now,
βmρm(τ)Hm(t,τ)δσ(τ,t)
Rm,m–(τ,t)
r(τ)
/αw
m(τ) ρm(τ)
σ
–hm(t,τ)
–
Hm(t,τ)
wm(τ) ρm(τ)
σ
=
βmρm(τ)Hm(t,τ)δσ(τ,t)
Rm,m–(τ,t)
r(τ)
/αw
m(τ) ρm(τ)
σ
– (hm(t,τ))–
βmρm(τ)δσ(τ,t)[Rm,m–r(τ()τ,t)]/α
– [(hm(t,τ))–]
βmρm(τ)δσ(τ,t)
r(τ)
Rm,m–(τ,t)
/α
≥ – [(hm(t,τ))–]
βmρm(τ)δσ(τ,t)
r(τ)
Rm,m–(τ,t)
/α .
Consequently,
Hm(t,t)
t
t
ρm(τ)p¯m,n–m–(τ,t)Hm(t,τ)
– [(hm(t,τ))–]
βmρm(τ)δσ(τ,t)
r(τ)
Rm,m–(τ,t)
/α
τ≤wm(t),
which contradicts assumption (.).
(ii) We show that ifm= , thenlimt→∞x(t) = . In fact, from Lemma . we see that it is only possible whennis odd. In this case,
(–)kx[k](t) > and
(–)kx[k](t)< fort∈[t
,∞)Tandk= , , . . . ,n– .
(.)
Hence,
lim t→∞(–)
kx[k](t) =l
k≥ fork= , , . . . ,n– .
We claim that limt→∞x(t) =l= . Assume thatl> . Then, for sufficiently larget∈
[t,∞)T, we havex(gν(t))≥lfort≥t. It follows that
φγν
xgν(t)
≥lγν
≥L fort∈[t,∞)T,
whereL:=minNν={lγν
}> . Then from (.) we obtain
–x[n–](t)≥L
N
ν=
Integrating this fromttov∈[t,∞)T, we get
–x[n–](v) +x[n–](t)≥L
v
t
ˆ
p(τ)τ,
and by (.) we see thatx[n–](v) > . Hence, by taking limits asv→ ∞we have
x[n–](t)≥L
∞
t ˆ
p(τ)τ.
Ift∞pˆ(τ)τ=∞, then we have reached a contradiction. Otherwise,
x[n–](t)≥L/αn–
rn–(t)
∞
t
ˆ
p(τ)τ
/αn–
=L/αn–pˆ
(t).
Integrating this fromttov∈[t,∞)Tand lettingv→ ∞, by (.) we get
–x[n–](t)≥L/αn–
∞
t
ˆ
p(τ)τ.
Ift∞pˆ(τ)τ=∞, then we have reached a contradiction. Otherwise,
–x[n–](t)≥L/α[n–,n–]
rn–(t)
∞
t
ˆ
p(τ)τ
/αn–
=L/α[n–,n–]pˆ(t).
Continuing this process, we get
–x[](t)≥L/α[,n–]
∞
t
ˆ
pn–(τ)τ.
Ift∞pˆn–(τ)τ=∞, then we have reached a contradiction. Otherwise,
–x(t)≥L/α[,n–]
r(t)
∞
t
ˆ
pn–(τ)τ
/α
=L/αpˆn–(t).
Again, integrating fromttot∈[t,∞)T, we get
–x(t) +x(t)≥L/α
t
t
ˆ
pn–(τ)τ.
Ift∞pˆn–(τ)τ=∞, then we havelimt→∞x(t) = –∞, which contradicts the assumption
thatx(t) > eventually. This shows that ifm= , thenlimt→∞x(t) = . This completes the
proof.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This work was supported by Research Deanship of Hail University under grant No. 0150287.
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