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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 796851, 13 pages
doi:10.1155/2008/796851
Research Article
Multiple Positive Solutions in the Sense of
Distributions of Singular BVPs on Time Scales and
an Application to Emden-Fowler Equations
Ravi P. Agarwal,
1
Victoria Otero-Espinar,
2
Kanishka Perera,
1
and Dolores R. Vivero
2
1
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
2
Departamento de An
´
alise Matem
´
atica, Facultade de Matem
´
aticas, Universidade de Santiago
de Compostela, 15782 Santiago de Compostela, Galicia, Spain
Correspondence should be addressed to Ravi P. Agarwal, agarwal@fit.edu
Received 21 April 2008; Accepted 17 August 2008
Recommended by Paul Eloe
This paper is devoted to using perturbation and variational techniques to derive some sufficient
conditions for the existence of multiple positive solutions in the sense of distributions to a singular
second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes
those problems related to the negative exponent Emden-Fowler equation.
Copyright q 2008 Ravi P. Agarwal et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The Emden-Fowler equation,
u
ΔΔ
tqt u
α
σt
0,t∈ 0, 1
T
, 1.1
arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic
mechanics, nuclear physics, and chemically reacting system see, e.g., 1 and the references
therein for the continuous model. The negative exponent Emden-Fowler equation α<0
has been used in modeling non-Newtonian fluids such as coal slurries 2. The physical
interest lies in the existence of positive solutions. We are interested in a broad class of singular
problem that includes those related with 1.1 and the more general equation
u
ΔΔ
tqt u
α
σt
g
t, u
σ
t
,t∈ 0, 1
T
. 1.2
Recently, existence theory for positive solutions of second-order boundary value
problems on time scales has received much attention see, e.g., 3–6 for general case, 7
for the continuous case, and 8 for the discrete case.
2 Advances in Difference Equations
In this paper, we consider the second-order dynamic equation with homogeneous
Dirichlet boundary conditions:
P
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
−u
ΔΔ
tF
t, u
σ
t
, Δ-a.e.t∈
D
κ
o
,
ut > 0,t∈ a, b
T
,
ua0 ub,
1.3
where we say that a property holds for Δ-a.e. t ∈ A ⊂ T or Δ-a.e. on A ⊂ T, Δ-a.e., whenever
there exists a set E ⊂ A with null Lebesgue Δ-measure such that this property holds for every
t ∈ A \ E, T is an arbitrary time scale, subindex T means intersection to T, a, b ∈ T are such
that a<ρb, D a, b
T
, D
κ
a, ρb
T
, D
κ
2
a, ρ
2
b
T
, D
o
a, b
T
, D
κ
o
a, ρb
T
,
and F : D × 0, ∞ →
R is an L
1
Δ
-Carath
´
eodory function on compact subintervals of 0, ∞,
that is, it satisfies the following conditions.
Ci For every x ∈ 0, ∞, F·,x is Δ-measurable in D
o
.
ii For Δ-a.e. t ∈ D
o
, Ft, · ∈ C0, ∞.
C
c
For every x
1
,x
2
∈ 0, ∞ with x
1
≤ x
2
, there exists m
x
1
,x
2
∈ L
1
Δ
D
o
such that
Ft, x
≤ m
x
1
,x
2
t for Δ-a.e.t∈ D
o
,x∈
x
1
,x
2
. 1.4
Moreover, in order to use variational techniques and critical point theory, we will
assume that F satisfy the following condition.
PM For every x ∈ 0, ∞, function P
F
: D × 0, ∞ → R defined for Δ-a.e. t ∈ D and
all x ∈ 0, ∞,as
P
F
t, x :
x
0
Ft, r dr, 1.5
satisfies that P
F
·,x is Δ-measurable in D
o
.
We consider the spaces
C
1
0,rd
D
κ
: C
1
rd
D
κ
∩ C
0
D,
C
1
c,rd
D
κ
: C
1
rd
D
κ
∩ C
c
D,
1.6
where C
1
rd
D
κ
is the set of all continuous functions on D such that they are Δ-differentiable
on D
κ
and their Δ-derivatives are rd-continuous on D
κ
, C
0
D is the set of all continuous
functions on D that vanish on the boundary of D,andC
c
D is the set of all continuous
functions on D with compact support on a, b
T
. We denote as ·
CD
the norm in CD,that
is, the supremum norm.
On the other hand, we consider the first-order Sobolev spaces
H
1
Δ
D :
v : D −→ R : v ∈ ACD,v
Δ
∈ L
2
Δ
D
o
,
H H
1
0,Δ
D :
v : D −→ R : v ∈ H
1
Δ
D,va0 vb
,
1.7
Ravi P. Agarwal et al. 3
where ACD is the set of all absolutely continuous functions on D. We denote as
t
2
t
1
fsΔs
t
1
,t
2
T
fsΔs for t
1
,t
2
∈ D, t
1
<t
2
,f∈ L
1
Δ
t
1
,t
2
T
. 1.8
The set H is endowed with the structure of Hilbert space together with the inner
product ·, ·
H
: H × H → R given for every v, w ∈ H × H by
v, w
H
:
v
Δ
,w
Δ
L
2
Δ
:
b
a
v
Δ
s · w
Δ
sΔs; 1.9
we denote as ·
H
its induced norm.
Moreover, we consider the sets
H
0,loc
: H
1
loc,Δ
D ∩ C
0
D,
H
c,loc
: H
1
loc,Δ
D ∩ C
c
D,
1.10
where H
1
loc,Δ
D is the set of all functions such that their restriction to every closed subinterval
J of a, b
T
belong to the Sobolev space H
1
Δ
J.
We refer the reader to 9–11 for an introduction to several properties of Sobolev spaces
and absolutely continuous functions on closed subintervals of an arbitrary time scale, and to
12 for a broad introduction to dynamic equations on time scales.
Definition 1.1. u is said to be a solution in the sense of distributions to P if u ∈ H
0,loc
, u>0
on a, b
T
, and equality
b
a
u
Δ
s · ϕ
Δ
s − F
s, u
σ
s
· ϕ
σ
s
Δs 0 1.11
holds for all ϕ ∈ C
1
c,rd
D
κ
.
From the density properties of the first-order Sobolev spaces proved in 9, Seccion 3.2,
we deduce that if u is solution in the sense of distributions, then, 1.11 holds for all ϕ ∈ H
c,loc
.
This paper is devoted to prove the existence of multiple positive solutions to P by
using perturbation and variational methods.
This paper is organized as follows. In Section 2, we deduce sufficient conditions for
the existence of solutions in the sense of distributions to P . Under certain hypotheses,
we approximate solutions in the sense of distributions to problem P by a sequence of
weak solutions to weak problems. In Section 3, we derive some sufficient conditions for the
existence of at least one or two positive solutions to P.
These results generalize those given in 7 for T 0, 1, where problem P is defined
on the whole interval 0, 1 ∩ T and the authors assume that F ∈ C0, 1 × 0, ∞, R instead
of C and PM.Thesufficient conditions for the existence of multiple positive solutions
obtained in this paper are applied to a great class of bounded time scales such as finite union
of disjoint closed intervals, some convergent sequences and their limit points, or Cantor sets
among others.
4 Advances in Difference Equations
2. Approximation to P by weak problems
In this section, we will deduce sufficient conditions for the existence of solutions in the sense
of distributions to P, where F f g and f, g : D × 0, ∞ →
R satisfy C and PM, f
satisfies C
c
,andg satisfies the following condition.
C
g
For every p ∈ 0, ∞, there exists M
p
∈ L
1
Δ
D
o
such that
gt, x
≤ M
p
t for Δ-a.e.t∈ D
o
,x∈ 0,p. 2.1
Under these hypotheses, we will be able to approximate solutions in the sense of
distributions to problem P by a sequence of weak solutions to weak problems.
First of all, we enunciate a useful property of absolutely continuous functions on
Dwhose proof we omit because of its simplicity.
Lemma 2.1. If v ∈ ACD,thenv
±
: max{± v,0}∈ACD,
v
Δ
− v
Δ
·
v
Δ
≤ 0,
v
−
Δ
v
Δ
·
v
−
Δ
≤ 0, 2.2
Δ-a.e. on D
o
.
We fix {ε
j
}
j≥1
a sequence of positive numbers strictly decreasing to zero; f or every
j ≥ 1, we define f
j
: D × 0, ∞ → R as
f
j
t, xf
t, max
x, ε
j
for every t, x ∈ D × 0, ∞. 2.3
Note that f
j
satisfies C and C
g
; consider the following modified weak problem
P
j
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−u
ΔΔ
tf
j
t, u
σ
t
g
t, u
σ
t
, Δ-a.e.t∈
D
κ
o
,
ut > 0,t∈ a, b
T
,
ua0 ub.
2.4
Definition 2.2. u is said to be a weak solution to P
j
if u ∈ H, u>0ona, b
T
, and equality
b
a
u
Δ
s · ϕ
Δ
s −
f
j
s, u
σ
s
g
s, u
σ
s
· ϕ
σ
s
Δs 0 2.5
holds for all ϕ ∈ C
1
0,rd
D
κ
.
u
is said to be a weak lower solution to P
j
if u ∈ Hu> 0ona, b
T
, and inequality
b
a
u
Δ
s · ϕ
Δ
s −
f
j
s, u
σ
s
g
s, u
σ
s
· ϕ
σ
s
Δs ≤ 0 2.6
holds for all ϕ ∈ C
1
0,rd
D
κ
such that ϕ ≥ 0onD.
Ravi P. Agarwal et al. 5
The concept of weak upper solution to P
j
is defined by reversing the previous
inequality.
We remark that the density properties of the first-order Sobolev spaces proved in 9,
Seccion 3.2 allows to assert that relations in Definition 2.2 are valid for all ϕ ∈ H and for all
ϕ ∈ H such that ϕ ≥ 0onD, respectively.
By standard arguments, we can prove the following result.
Proposition 2.3. Assume that f, g : D × 0, ∞ →
R satisfy (C and (PM, f satisfies (C
c
, and
g satisfies (C
g
.
Then,ifforsomej ≥ 1 there exist u
j
and u
j
as a lower and an upper weak solution, respectively,
to P
j
such that u
j
≤ u
j
on D,thenP
j
has a weak solution u
j
∈ u
j
, u
j
: {v ∈ H : u
j
≤ v ≤
u
j
on D}.
Next, we will deduce the existence of one solution in the sense of distributions to
P from the existence of a sequence of weak solutions to P
j
. In order to do this, we fix
{a
k
}
k≥1
, {b
k
}
k≥1
⊂ D two sequences such that {a
k
}
k≥1
⊂ a, a b/2
T
is strictly decreasing
to a if a σa, a
k
a for all k ≥ 1ifa<σa and {b
k
}
k≥1
⊂ a b/2,b
T
is strictly
increasing to b if ρbb, b
k
b for all k ≥ 1ifρb <b. We denote that D
k
:a
k
,b
k
T
,
k ≥ 1. Moreover, we fix {δ
k
}
k≥1
a sequence of positive numbers strictly decreasing to zero
such that
σ
a
k
,ρb
k
T
⊂
a δ
k
,b− δ
k
T
,δ
k
≤
b − a
2
for k ≥ 1. 2.7
Proposition 2.4. Suppose that F f g and f, g : D × 0, ∞ →
R satisfy (C and (PM, f
satisfies (C
c
, and g satisfies (C
g
.
Then, if for every j ≥ 1, u
j
∈ H is a weak solution to P
j
and
ν
δ
: inf
j≥1
min
aδ,b−δ
T
,u
j
> 0 ∀δ ∈
0,
b − a
2
,
2.8
M : sup
j≥1
max
D
u
j
< ∞,
2.9
then a subsequence of {u
j
}
j≥1
converges pointwise in D to a solution in the sense of distributions u
1
to P.
Proof. Let k ≥ 1 be arbitrary; we deduce, from 2.2, 2.7, 2.8,and2.9, that there exists a
constant K
k
≥ 0 such that for all j ≥ 1,
b
k
a
k
u
Δ
j
s
2
Δs
u
Δ
j
a
k
2
· μ
a
k
u
Δ
j
ρb
k
2
· μ
ρ
b
k
ρb
k
σa
k
u
Δ
j
s ·
u
j
− ν
δ
k
Δ
sΔs
≤ K
k
u
j
,
u
j
− ν
δ
k
H
.
2.10
6 Advances in Difference Equations
Therefore, for all j ≥ 1 so large that ε
j
<ν
δ
1
,asu
j
is a weak solution to P
j
, by taking
ϕ
1
:u
j
− ν
δ
1
∈ H as the test function in 2.5,from2.9, C
c
and C
g
, we can assert that
there exists l ∈ L
1
Δ
D
o
such that
b
1
a
1
u
Δ
j
s
2
Δs ≤ K
1
b
a
F
s, u
σ
j
s
· ϕ
σ
1
sΔs
≤ K
1
M
b
a
lsΔs,
2.11
that is, {u
j
}
j≥1
is bounded in H
1
Δ
D
1
and hence, there exists a subsequence {u
1
j
}
j≥1
which
converges weakly in H
1
Δ
D
1
and strongly in CD
1
to some u
1
∈ H
1
Δ
D
1
.
For every k ≥ 1, by considering for each j ≥ 1 the weak solution to P
k
j
u
k
j
and
by repeating the previous construction, we obtain a sequence {u
k1
j
}
j≥1
which converges
weakly in H
1
Δ
D
k1
and strongly in CD
k1
to some u
k1
∈ H
1
Δ
D
k1
with {u
k1
j
}
j≥1
⊂
{u
k
j
}
j≥1
. By definition, we know that for all k ≥ 1, u
k1
|
D
k
u
k
.
Let u
1
: D → R be given by u
1
: u
k
on D
k
for all k ≥ 1andu
1
a : 0 : u
1
b so that
u
1
> 0ona, b
T
, u
1
∈ H
1
loc,Δ
D ∩ Ca, b
T
, u
1
is continuous in every isolated point of the
boundary of D,and{u
k
k
}
k≥1
converges pointwise in D to u
1
.
We will show that u
1
∈ C
0
D; we only have to prove that u
1
is continuous in every
dense point of the boundary of D.Let0<ε<Mbe arbitrary, it follows from C
c
and C
g
that there exist m
ε
∈ L
1
Δ
D
o
such that m
ε
≥ 0onD
o
and Ft, x ≤ m
ε
t for Δ-a.e. t ∈ D
o
and
all x ∈ ε, M;letϕ
ε
∈ H be the weak solution to
−ϕ
ΔΔ
ε
tm
ε
t, Δ-a.e.t∈
D
κ
o
,ϕ
ε
a0 ϕ
ε
b; 2.12
we know see 4 that ϕ
ε
> 0ona, b
T
.
For all k ≥ 1 so large that ε
k
k
<ε,sinceu
k
k
and ϕ
ε
are weak solutions to some problems,
by taking ϕ
2
u
k
k
− ε − ϕ
ε
∈ H as the test function in their respective problems, we obtain
u
k
k
, ϕ
2
H
b
a
F
s, u
σ
k
k
s
· ϕ
σ
2
sΔs
≤
b
a
m
ε
s · ϕ
σ
2
sΔs
ϕ
ε
, ϕ
2
H
;
2.13
thus, 2.2 yields to
ϕ
2
2
H
≤
u
k
k
− ϕ
ε
, ϕ
2
H
≤ 0, 2.14
which implies that 0 ≤ u
k
k
≤ ε ϕ
ε
on D and so 0 ≤ u
1
≤ ε ϕ
ε
on D. Thereby, the continuity
of ϕ
ε
in every dense point of the boundary of D and the arbitrariness of ε guarantee that
u
1
∈ C
0
D.
Finally, we will see that 1.11 holds for every test function ϕ ∈ C
1
c,rd
D
κ
;fixoneof
them.
For all k ≥ 1 so large that supp ϕ ⊂ a
k
,b
k
T
and all j ≥ 1 so large that ε
k
j
<ν
δ
k
,asu
k
j
is a weak solution to P
k
j
, by taking ϕ ∈ C
1
c,rd
D
κ
⊂ C
1
0,rd
D
κ
as the test function in 2.5
and bearing in mind 2.7, we have
b
k
a
k
u
Δ
k
j
s · ϕ
Δ
sΔs
u
k
j
,ϕ
H
b
k
a
k
F
s, u
σ
k
j
s
· ϕ
σ
sΔs, 2.15
Ravi P. Agarwal et al. 7
whence it follows, by taking limits, that
b
k
a
k
u
k
Δ
s · ϕ
Δ
s − F
s,
u
k
σ
s
· ϕ
σ
s
Δs 0, 2.16
which is equivalent because u
1
|
D
k
u
k
and ϕ 0 ϕ
σ
on D
o
\ D
o
k
to
b
a
u
Δ
1
s · ϕ
Δ
s − F
s, u
σ
1
s
· ϕ
σ
s
Δs 0, 2.17
and the proof is therefore complete.
Propositions 2.3 and 2.4 lead to the following sufficient condition for the existence of
at least one solution in the sense of distributions to problem P.
Corollary 2.5. Let F f g be such that f,g : D × 0, ∞ →
R satisfy (C and (PM, f satisfies
(C
c
, and g satisfies (C
g
.
Then, if for each j ≥ 1 there exist u
j
and u
j
a lower and an upper weak solution, respectively,
to P
j
such that u
j
≤ u
j
on D and
inf
j≥1
min
aδ, b−δ
T
u
j
> 0 ∀δ ∈
0,
b − a
2
, sup
j≥1
max
D
u
j
< ∞, 2.18
then P has a solution in the sense of distributions u
1
.
Finally, fixed u
1
∈ H
0,loc
is a solution in the sense of distributions to P with F f g,
we will derive the existence of a second solution in the sense of distributions to Pgreater
than or equal to u
1
on D. For every k ≥ 1, consider the weak problem
P
k
⎧
⎨
⎩
−v
ΔΔ
tF
t,
u
1
v
σ
t
− F
t, u
σ
1
t
, Δ-a.e.t∈
D
κ
k
o
,
v
a
k
0 v
b
k
.
2.19
For every k ≥ 1, consider H
k
: H
1
0,Δ
D
k
as a subspace of H by defining it for every
v ∈ H
k
as v 0onD \ D
k
and define the functional Φ
k
: H
k
⊂ H → R for every v ∈ H
k
as
Φ
k
v :
1
2
v
2
H
−
b
k
a
k
G
s,
v
σ
s
Δs, 2.20
where function G : D × 0, ∞ →
R is defined for Δ-a.e. t ∈ D and all x ∈ 0, ∞ as
Gt, x :
x
0
F
t, u
σ
1
tr
− F
t, u
σ
1
t
dr. 2.21
As a consequence of Lemma 2.1, we deduce that every weak solution to
P
k
is
nonnegative on D
k
and by reasoning as in 4,Section3, one can prove that Φ
k
is weakly
lower semicontinuous, Φ
k
is continuously differentiable in H
k
, for every v, w ∈ H
k
,
Φ
k
vwv, w
H
−
b
k
a
k
F
s,
u
1
v
σ
s
− F
s, u
σ
1
s
· w
σ
sΔs, 2.22
and weak solutions to
P
k
match up to the critical points of Φ
k
.
Next, we will assume the following condition.
NI For Δ-a.e. t ∈ D
o
, ft, · is nonincreasing on 0, ∞.
8 Advances in Difference Equations
Proposition 2.6. Suppose that F f g is such that f, g : D × 0, ∞ →
R satisfy (C and ( PM,
f satisfies (C
c
and (NI, and g satisfies (C
g
.
If {v
k
}
k≥1
⊂ H, v
k
∈ H
k
is a bounded sequence in H such that
inf
k≥1
Φ
k
v
k
> 0, lim
k→∞
Φ
k
v
k
H
∗
k
0, 2.23
then {v
k
}
k≥1
has a subsequence convergent pointwise in D to a nontrivial function v ∈ H such that
v ≥ 0 in D and u
2
: u
1
v is a solution in the sense of distributions to P.
Proof. Since {v
k
}
k≥1
is bounded in H, it has a subsequence which converges weakly in H and
strongly in C
0
D to some v ∈ H.
For every k ≥ 1, by 2.2,weobtain
v
−
k
H
≤
Φ
k
v
k
H
∗
k
, 2.24
which implies, from 2.23,thatv ≥ 0onD and so u
2
: u
1
v>0ona, b
T
.
In order to show t hat u
2
: u
1
v ∈ H
0,loc
is a solution in the sense of distributions
to P,fixϕ ∈ C
1
c,rd
D
k
arbitrary and choose k ≥ 1 so large that supp ϕ ⊂ a
k
,b
k
T
, bearing
in mind that u
1
is a solution in the sense of distributions to P, and the pass to the limit in
2.22 with v v
k
and w ϕ yields to
0
b
a
v
Δ
s · ϕ
Δ
s −
F
s,
u
1
v
σ
s
− F
s, u
σ
1
s
· ϕ
σ
s
Δs
b
a
u
Δ
2
s · ϕ
Δ
s − F
s, u
σ
2
s
· ϕ
σ
s
Δs;
2.25
thus, u
2
is a solution in the sense of distributions to P.
Finally, we will see that v is not the trivial function; suppose that v 0onD. Condition
NIensures that function G defined in 2.21 satisfies for every k ≥ 1andΔ-a.e. s ∈ D
o
,
G
s,
v
k
σ
s
≥
f
s,
u
1
v
k
σ
s
− f
s, u
σ
1
s
·
v
k
σ
s
v
k
σ
s
0
g
s, u
σ
1
sr
− g
s, u
σ
1
s
dr,
2.26
so that, by 2.20 and 2.22, we have, for every k ≥ 1,
Φ
k
v
k
≤
1
2
v
k
2
H
−
v
k
,v
k
H
Φ
k
v
k
v
k
−
b
a
g
s,
u
1
v
k
σ
s
− g
s, u
σ
1
s
·
v
k
σ
sΔs
b
a
v
k
σ
s
0
g
s, u
σ
1
sr
− g
s, u
σ
1
s
dr
Δs;
2.27
moreover, as we know that v
k
≤ p on D for some p>0, it follows from C
g
that there exists
m ∈ L
1
Δ
D
o
such that
Φ
k
v
k
≤
1
2
v
−
k
2
H
−
v
k
2
H
Φ
k
v
k
v
k
2
b
a
ms ·
v
k
σ
sΔs
≤
1
2
v
−
k
2
H
Φ
k
v
k
H
∗
k
·
v
k
H
2
b
a
ms ·
v
k
σ
sΔs,
2.28
Ravi P. Agarwal et al. 9
and hence, since {v
k
}
k≥1
is bounded in H and converges pointwise in D to the trivial function
v, we deduce, from the second relation in 2.23 and 2.24, that lim
k→∞
Φ
k
v
k
≤ 0 which
contradicts the first relation in 2.23. Therefore, v is a nontrivial function.
3. Results on the existence and uniqueness of solutions
In this section, we will derive the existence of solutions in the sense of distributions to P
where F f g
0
ηg
1
, η ≥ 0 is a small parameter, and f, g
0
,g
1
: D × 0, ∞ → R satisfy C,
PM as well as the following conditions.
H
1
There exists a constant x
0
∈ 0, ∞ and a nontrivial function f
0
∈ L
1
Δ
D
o
such
that f
0
≥ 0 Δ-a.e. on D
o
and
ft, x ≥ f
0
t,g
0
t, x,g
1
t, x ≥ 0forΔ-a.e.t∈ D
o
,x∈
0,x
0
. 3.1
H
2
For every p ∈ 0, ∞, there exist m
p
∈ L
1
Δ
D
o
and K
p
≥ 0 such that
ft, x
≤ m
p
t for Δ-a.e.t∈ D
o
,x∈ p, ∞,
g
1
t, x
≤ K
p
for Δ-a.e.t∈ D
o
,x∈ 0,p.
3.2
H
3
There are m
0
∈ L
2
Δ
D
o
such that
g
0
t, x
≤ λx m
0
t for Δ-a.e.t∈ D
o
,x∈ 0, ∞, 3.3
for some λ<λ
1
, where λ
1
is the smallest positive eigenvalue of problem
−u
ΔΔ
tλu
σ
t,t∈ D
κ
2
,
ua0 ub.
3.4
3.1. Existence of one solution. Uniqueness
Theorem 3.1. Suppose that f,g
0
,g
1
: D × 0, ∞ → R satisfy (C,(PM, and ( H
1
–H
3
. Then,
there exists a η
0
> 0 such that for every η ∈ 0,η
0
, problem P with F f g
0
ηg
1
has a solution
in the sense of distributions u
1
.
Proof. Let η ≥ 0 be arbitrary; conditions H
1
–H
3
guarantee that g : g
0
ηg
1
satisfies
C
g
. We will show that there exists a η
0
> 0 such that for every η ∈ 0,η
0
, hypotheses in
Corollary 2.5 are satisfied.
Let x
0
and f
0
be given in H
1
,weknow,from4, Proposition 2.7, that we can choose
ε ∈ 0, 1 so small that the weak solution u
∈ H to
−u
ΔΔ
tεf
0
t, Δ-a.e.t∈
D
κ
o
,ua0 ub, 3.5
satisfies that u
> 0ona, b
T
and u ≤ x
0
on D.
Let j ≥ 1 be so large that ε
j
<x
0
, we obtain, by H
1
,that
−u
ΔΔ
t ≤ f
0
t ≤ f
j
t, u
σ
t
g
t, u
σ
t
, Δ-a.e.t∈ D
o
, 3.6
whence it follows that u
is a weak lower solution to P
j
.
10 Advances in Difference Equations
As a consequence of C, PM, and H
1
–H
3
, by reasoning as in 4, Theorem 4.2,
we deduce that problem
−u
ΔΔ
tf
j
t, u
σ
t
g
0
t, u
σ
t
1, Δ-a.e.t∈
D
κ
o
,
ut > 0,t∈ a, b
T
,
ua0 ub
3.7
has some weak solution
u
j
∈ H which, from Lemma 2.1 and H
1
,satisfiesthatu ≤ u
j
on
D.Wewillseethat{
u
j
}
j≥1
is bounded in C
0
D, by taking ϕ
j
:u
j
− x
0
∈ H as the test
function, we know from 2.2, H
2
, and H
3
that there exist m
x
0
∈ L
2
Δ
D
o
such that
ϕ
j
2
H
≤
u
j
− x
0
,ϕ
j
H
b
a
f
j
s,
u
σ
j
s
g
0
s,
u
σ
j
s
1
· ϕ
σ
j
sΔs
≤
b
a
λ
u
σ
j
sm
x
0
sm
0
s1
· ϕ
σ
j
sΔs;
3.8
so that, it follows from the fact that the immersion from H into C
0
D is compact, see 9,
Proposition 3.7, Wirtinger’s inequality 10, Corollary 3.2 and relation λ<λ
1
that {ϕ
j
}
j≥1
is
bounded in H and, hence, {
u
j
}
j≥1
is bounded in C
0
D. Thereby, condition H
2
allows to
assert that there exists η
0
≥ 0, such that for all η ∈ 0,η
0
−
u
ΔΔ
j
t ≥ f
j
t,
u
σ
j
t
g
0
t,
u
σ
j
t
ηg
1
t,
u
σ
j
t
, Δ-a.e.t∈ D
o
, 3.9
holds, which implies that
u
j
is a weak upper solution to P
j
.
Therefore, for every j ≥ 1 so large, we have a lower and an upper solution to P
j
,
respectively, such that 2.2 is satisfied and so, Corollary 2.5 guarantees that problem P has
at least one solution in the sense of distributions u
1
.
Theorem 3.2. If f : D × 0, ∞ → R satisfies (C,(C
c
, and (NI, then, P with F f has at
most one solution in the sense of distributions.
Proof. Suppose that P has two solutions in the sense of distributions u
1
,u
2
∈ H
0,loc
.Letε>0
be arbitrary, take ϕ u
1
− u
2
− ε
∈ H
c,loc
as the test function in 1.11,by2.2 and NI,
we have
ϕ
2
H
≤
u
1
− u
2
− ε, ϕ
H
b
a
f
s, u
σ
1
s
− f
s, u
σ
2
s
· ϕ
σ
sΔs ≤ 0, 3.10
thus, u
1
≤ u
2
ε on D. The arbitrariness of ε leads to u
1
≤ u
2
on D and by interchanging u
1
and u
2
, we conclude that u
1
u
2
on D.
Corollary 3.3. If f : D × 0, ∞ → R satisfies (C,(PM,(NI, and (H
1
-(H
2
with g
0
0 g
1
,
then P with F f has a unique solution in the sense of distributions.
Ravi P. Agarwal et al. 11
3.2. Existence of two ordered solutions
Next, by using Theorem 3.1 which ensures the existence of a solution in the sense of
distributions to P, we will deduce, by applying Proposition 2.6, the existence of a second
one greater than or equal to the first one on the whole interval D; in order to do this, we will
assume that f, g
0
,g
1
: D × 0, ∞ → R satisfy C, PM, H
1
–H
3
, as well as the following
conditions.
H
4
For Δ-a.e. t ∈ D
o
, ft, · is nonincreasing and convex on 0,x
0
with x
0
given in
H
1
.
H
5
There are constants θ>2, C
1
,C
2
≥ 0andx
1
> 0 such that
g
1
t, x
≤ C
1
x
θ−1
C
2
for Δ-a.e.t∈ D
o
,x∈ 0, ∞,
0 <
x
0
g
1
t, rdr ≤
1
θ
xg
1
t, x for Δ-a.e.t∈ D
o
,x∈
x
1
, ∞
.
3.11
We will use the following variant of the mountain pass, see 13.
Lemma 3.4. If Φ is a continuously differentiable functional defined on a Banach space H and there
exist v
0
,v
1
∈ H such that
c : inf
γ∈Γ
max
v∈γ0,1
Φv > Φv
0
, Φv
1
, 3.12
where Γ is the class of paths in H joining v
0
and v
1
, then there is a sequence {v
k
}
k≥1
⊂ H such that
lim
k→∞
Φ
v
k
c, lim
k→∞
1
v
k
H
Φ
v
k
H
∗
0. 3.13
Theorem 3.5. Let f,g
0
,g
1
: D × 0, ∞ → R be such that (C,(PM, and H
1
–H
5
hold. Then,
there exists an η
0
> 0 such that for every η ∈ 0,η
0
, problem P with F f g
0
ηg
1
has two
solutions in the sense of distributions u
1
,u
2
such that u
1
≤ u
2
on D and u
2
− u
1
∈ H.
Proof. Conditions H
1
–H
4
allow to suppose that for Δ-a.e. t ∈ D
o
, ft, · is nonnegative,
nonincreasing, and convex on 0, ∞ because these conditions can be obtained by simply
replacing on D × x
0
, ∞f and g
0
with ft, x
0
and g
0
t, xft, x − ft, x
0
, respectively.
Let u
1
be a solution in the sense of distributions to P, its existence is guaranteed by
Theorem 3.1,andletη>0 be arbitrary; it is clear that F f g with g : g
0
ηg
1
satisfies
hypothesis in Proposition 2.6; we will derive the existence of an η
0
> 0 such that for every η ∈
0,η
0
, we are able to construct a sequence {v
k
}
k≥1
⊂ H in the conditions of Proposition 2.6.
For every k ≥ 1andv ∈ H
k
, as a straight-forward consequence of NI, H
3
, H
5
,
and the compact immersion from H into C
0
D, we deduce that there exist two constants
C
3
,C
4
≥ 0 such that function G, defined in 2.21, satisfies for Δ-a.e. s ∈ D
o
,
G
s,
v
σ
s
≤
λ
2
v
σ
2
sC
3
m
0
s1
v
H
ηC
4
1 v
H
θ−1
v
H
, 3.14
which implies, by 2.20 and Wirtinger’s inequality 10, Corollary 3.2, that there exists a
constant C
5
≥ 0 such that
Φ
k
v
1
2
v
2
H
−
b
k
a
k
G
s,
v
σ
s
Δs
≥
1
2
1 −
λ
λ
1
v
2
H
− C
5
1 η
1 v
H
θ−1
v
H
.
3.15
12 Advances in Difference Equations
Thereby, as λ<λ
1
, there exist constants R, η
0
,c
0
> 0 such that
inf
v∈H
k
v
H
R
Φ
k
v ≥ c
0
> 0 ∀ k ≥ 1,η∈
0,η
0
. 3.16
Let η ∈ 0,η
0
be arbitrary. From the second relation in H
5
,weobtainthat
g
1
t, x ≥ C
6
x
θ−1
for Δ-a.e.t∈ D
o
,x∈
x
1
, ∞
, 3.17
for some constant C
6
> 0; thus, it is not difficult to prove that there is a v
1
∈ H
1
such that
v
1
> 0ona, b
T
, v
1
H
>Rand Φ
1
v
1
< 0 and hence, since Φ
1
00, by denoting as Γ
1
the
class of paths in H
1
joining 0 and v
1
, it follows from 3.16 that
c
1
: inf
γ∈Γ
1
max
v∈γ0,1
Φ
1
v ≥ c
0
> Φ
1
0, Φ
1
v
1
, 3.18
hence, Lemma 3.4 establishes the existence of a sequence {v
k
}
k≥1
⊂ H
1
such that
lim
k→∞
Φ
1
v
k
c
1
, lim
k→∞
1
v
k
H
Φ
1
v
k
H
∗
1
0. 3.19
Consequently, bearing in mind that H
1
⊂ H
k
and Φ
k
|
H
1
Φ
1
for all k ≥ 1andby
removing a finite number of terms if it is necessary, we obtain a sequence {v
k
}
k≥1
⊂ H such
that v
k
∈ H
k
for every k ≥ 1and
0 <
c
0
2
≤ Φ
k
v
k
≤ k ≥ 1, lim
k→∞
1
v
k
H
Φ
k
v
k
H
∗
k
0, 3.20
we will show that this sequence is bounded in H.
From 2.2, we deduce that
0 ≤ lim
k→∞
v
−
k
H
≤ lim
k→∞
Φ
k
v
k
H
∗
k
0, 3.21
For every k ≥ 1, from 2.2, 2.20,and2.22, we have that
Φ
k
v
k
−
1
2
Φ
k
v
k
v
k
≥
1
2
v
−
k
2
H
b
a
H
F
s,
v
k
σ
s
Δs, 3.22
where, for Δ-a.e. s ∈ D
o
,
H
F
s,
v
k
σ
s
1
2
F
s,
u
1
v
k
σ
s
F
s, u
σ
1
s
·
v
k
σ
s −
u
1
v
k
σ
s
u
σ
1
s
Fs, r dr;
3.23
as a straight-forward consequence of the convexity of f and conditions H
2
, H
3
, H
5
,and
3.17, we deduce that there exist constants C
7
> 0andC
8
,C
9
≥ 0 such that
b
a
H
F
s,
v
k
σ
s
Δs ≥ C
7
v
k
σ
θ
L
θ
Δ
− C
8
v
k
σ
2
L
2
Δ
1
− C
9
. 3.24
Therefore, relations 3.20, 3.21, 3.22,and3.24 allow to assert that sequence
{v
k
σ
}
k≥1
is bounded in L
θ
Δ
D
o
and so, as for every k ≥ 1,
1
2
v
k
2
H
≤ Φ
k
v
k
b
a
v
k
σ
s
0
g
s, u
σ
1
sr
− g
s, u
σ
1
s
dr
Δs. 3.25
We conclude by 3.20, H
3
,andH
5
that {v
k
}
k≥1
is bounded in H and Proposition 2.6 leads
to the result.
Ravi P. Agarwal et al. 13
Acknowledgments
This research is partially supported by MEC and F.E.D.E.R. Project MTM2007-61724, and by
Xunta of Galicia and F.E.D.E.R. Project PGIDIT05PXIC20702PN, Spain.
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