Electronic Journal of Differential Equations, Vol. 2006(2006), No. 113, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE OF SOLUTIONS FOR p-LAPLACIAN FUNCTIONAL
DYNAMIC EQUATIONS ON TIME SCALES
CHANGXIU SONG
Abstract. In this paper, the author studies boundary-value problems for pLaplacian functional dynamic equations on a time scale. By using the fixed
point theorem, sufficient conditions are established for the existence of positive
solutions.
1. Introduction
Let T be a closed nonempty subset of R, and let subspace have the topology
inherited from the Euclidean topology on R. In some of the current literature, T is
called a time scale (or measure chain). For notation, we shall use the convention
that, for each interval of J of R, J will denote time scales interval, that is, J := J ∩T.
In this paper, let T be a time scale such that −r, 0, T ∈ T. We are concerned
with the existence of positive solutions of the p-Laplacian dynamic equation, on a
time scale,
∇
[φp (x4 (t))] + λa(t)f (x(t), x(µ(t))) = 0,
x0 (t) = ψ(t),
t ∈ [−r, 0],
t ∈ (0, T ),
x(0) − B0 (x4 (0)) = 0,
x4 (T ) = 0,
(1.1)
where λ > 0 and φp (u) is the p-Laplacian operator, i.e., φp (u) = |u|p−2 u, p > 1,
(φp )−1 (u) = φq (u), p1 + 1q = 1. Also we assume the following:
(A) The function f : (R+ )2 → R+ is continuous;
(B) the function a : T → R+ is left dense continuous (i.e., a ∈ Cld (T, R+ )).
Here Cld (T, R+ ) denotes the set of all left dense continuous functions from
T to R+ ;
(C) ψ : [−r, 0] → R+ is continuous and r > 0;
(D) µ : [0, T ] → [−r, T ] is continuous, µ(t) ≤ t for all t;
(E) B0 : R → R is continuous and satisfies that there are β ≥ δ > 0 such that
δs ≤ B0 (s) ≤ βs for s ∈ R+ .
2000 Mathematics Subject Classification. 39K10, 34B15.
Key words and phrases. Time scale; boundary value problem; positive solutions;
p-Laplacian functional dynamic equations; fixed point theorem.
c
2006
Texas State University - San Marcos.
Submitted August 28, 2006. Published September 19, 2006.
Supported by grants 10571064 from NNSF of China, and 011471 from NSF of Guangdong.
1
2
C. SONG
EJDE-2006/113
p-Laplacian problems with two-, three-, m-point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example see [3, 5, 9, 11] and references therein. However, there are not
many concerning the p-Laplacian problems on time scales, especially for p-Laplacian
functional dynamic equations on time scales.
The motivations for the present work stems from many recent investigations in
[2, 7, 10] and references therein. Especially, Kaufmann and Raffoul [7] considered
a nonlinear functional dynamic equation on a time scale and obtained sufficient
conditions for the existence of positive solutions. In this paper, we apply the fixed
point theorem to obtain at least one positive solution of boundary value problem
(BVP for short) (1.1). We do not need the condition that f (x1 , x2 ) is increasing in
each xi , for xi > 0, i = 1, 2. And we claim the condition ψ ≡ 0 is not essential in
our results.
For convenience, we list the following well-known definitions which can be found
in [1, 4, 6] and the references therein.
Definition 1.1. For t < sup T and r > inf T, define the forward jump operator σ
and the backward jump operator ρ:
σ(t) = inf{τ ∈ T|τ > t} ∈ T,
ρ(r) = sup{τ ∈ T|τ < r} ∈ T
for all t, r ∈ T. If σ(t) > t, t is said to be right scattered, and if ρ(r) < r, r is said
to be left scattered. If σ(t) = t, t is said to be right dense, and if ρ(r) = r, r is
said to be left dense. If T has a right scattered minimum m, define Tκ = T − {m};
otherwise set Tκ = T. If T has a left scattered maximum M , define Tκ = T − {M };
otherwise set Tκ = T.
Definition 1.2. For x : T → R and t ∈ Tκ , we define the delta derivative of x(t),
x4 (t), to be the number (when it exists), with the property that, for any ε > 0,
there is a neighborhood U of t such that
|[x(σ(t)) − x(s)] − x4 (t)[σ(t) − s]| < ε|σ(t) − s|
for all s ∈ U . For x : T → R and t ∈ Tκ , we define the nabla derivative of x(t),
x∇ (t), to be the number (when it exists), with the property that, for any ε > 0,
there is a neighborhood V of t such that
|[x(ρ(t)) − x(s)] − x∇ (t)[ρ(t) − s]| < ε|ρ(t) − s|
for all s ∈ V . If T = R, then x4 (t) = x∇ (t) = x0 (t). If T = Z, then x4 (t) =
x(t + 1) − x(t) is forward difference operator while x∇ (t) = x(t) − x(t − 1) is the
backward difference operator.
Rt
Definition 1.3. If F 4 (t) = f (t), then we define the delta integral by a f (s)4s =
Rt
F (t) − F (a). If Φ∇ (t) = f (t), then we define the nabla integral by a f (s)∇s =
Φ(t) − Φ(a).
In the following, we provide the definition of cones in Banach spaces, and we
then state the fixed-point theorem for a cone preserving operator.
Definition 1.4. Let X be a real Banach space. A nonempty, closed, convex set
K ∈ X is called a cone, if it satisfies the following two conditions:
(i) x ∈ K, λ ≥ 0 implies λx ∈ K;
(ii) x and −x in K implies x = 0.
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EXISTENCE OF SOLUTIONS
3
Every cone K ⊂ X induces an ordering in X given by x ≤ y if and only if
y − x ∈ K.
Lemma 1.5 ([8]). Assume that X is a Banach space and K ⊂ X is a cone
in X; Ω1 , Ω2 are open subsets of X, and 0 ∈ Ω1 ⊂ Ω2 . Furthermore, let F :
K ∩ (Ω2 \ Ω1 ) → K be a completely continuous operator satisfying one of the
following conditions:
(i) kF (x)k ≤ kxk for all x ∈ K ∩ ∂Ω1 , kF (x)k ≥ kxk for all x ∈ K ∩ ∂Ω2 ;
(i) kF (x)k ≤ kxk for all x ∈ K ∩ ∂Ω2 , kF (x)k ≥ kxk for all x ∈ K ∩ ∂Ω1 .
Then there is a fixed point of F in K ∩ (Ω2 \ Ω1 ).
2. Main results
We note that x(t) is a solution of (1.1) if and only if
 RT

B
φ
λa(r)f
(x(r),
x(µ(r)))∇r
0
q

 R
R0
x(t) = + 0t φq sT λa(r)f (x(r), x(µ(r)))∇r 4s, t ∈ [0, T ],



ψ(t),
t ∈ [−r, 0].
(2.1)
Let X = Cld ([0, T ], R) be endowed with the norm kxk = maxt∈[0,T ] |x(t)| and
δ
kxk for t ∈ [0, T ]}.
T +β
Clearly, X is a Banach space with the norm kxk and K is a cone in X. For each
x ∈ X, extend x(t) to [−r, T ] with x(t) = ψ(t) for t ∈ [−r, 0].
For t ∈ [0, T ], define F : P → X as
Z T
F x(t) = B0 φq
λa(r)f (x(r), x(µ(r)))∇r
0
(2.2)
Z t Z T
K = {x ∈ X : x(t) ≥
+
λa(r)f (x(r), x(µ(r)))∇r 4s,
φq
0
s
We seek a fixed point, x1 , of F in the cone P . Define
(
x1 (t), t ∈ [0, T ],
x(t) =
ψ(t), t ∈ [−r, 0].
Then x(t) denotes a positive solution of (1.1). It follows from (2.2) that
kF xk = (F x)(T )
Z
= B0 φq
T
λa(r)f (x(r), x(µ(r)))∇r
0
Z
+
T
φq
T
Z
0
≤ (T + β)λ
λa(r)f (x(r), x(µ(r)))∇r 4s
s
q−1
φq
Z
T
a(r)f (x(r), x(µ(r))) .
0
From (2.2) and (2.3), we have the following lemma.
Lemma 2.1. Let F be defined by (2.2). If x ∈ K, then
(i) F (K) ⊂ K.
(ii) F : K → K is completely continuous.
(2.3)
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C. SONG
(iii) x(t) ≥
δ
T +β kxk,
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t ∈ [0, T ].
We need to define subsets of [0, T ] with respect to the delay µ. Set
Y1 := {t ∈ [0, T ] : µ(t) < 0};
Y2 := {t ∈ [0, T ] : µ(t) ≥ 0}.
R
Throughout this paper, we assume Y1 is nonempty and Y1 a(r)∇r > 0. Let
λ1−q
RT
l =:
(T + β)φq
a(r)∇r
0
e
l =:
,
(T + β)φq
m =:
1
RT
0
(T + β)λ1−q
,
R
a(r)∇r
Y1
δ 2 φq
a(r)∇r
.
In additions to Conditions (A)–(E), we shall also consider the following:
(H1) limx→0+ f (x,ψ(s))
< lp−1 , uniformly in s ∈ [−r, 0];
xp−1
f (x1 ,x2 )
(H2) limx1 →0+ ;x2 →0+ max{x
< lp−1 ;
p−1 p−1
,x
}
1
(H3) limx→∞
f (x,ψ(s))
xp−1
2
> mp−1 , uniformly in s ∈ [−r, 0].
Theorem 2.2. Assume Conditions (A)–(E), (H1)–(H3) are satisfied. Then, for
each 0 < λ < ∞, BVP (1.1) has at least a positive solution.
Proof. Apply Condition (H1) and set ε1 > 0 such that if 0 < x ≤ ε1 , then
f (x, ψ(s)) < (lx)p−1 ,
for each s ∈ [−r, 0].
Apply Condition (H2) and set ε2 > 0 such that if 0 < x1 ≤ ε2 , 0 < x2 ≤ ε2 , then
f (x1 , x2 ) < max{x1p−1 , x2p−1 }lp−1 .
Set ρ1 = min{ε1 , ε2 }. Then, for any x ∈ K with kxk = ρ1 , from (2.3), we have
kF xk
≤ (T + β)λq−1 φq
Z
T
a(r)f (x(r), x(µ(r)))∇r
0
= (T + β)λ
q−1
h
φq
Z
Z
Y1
Z
t∈[0,T ]
= l(T + β)λ
kxkφq
i
Y2
≤ l(T + β)λq−1 max {x(t)}φq
q−1
a(r)f (x(r), x(µ(r)))∇r
a(r)f (x(r), ψ(µ(r)))∇r +
T
a(r)∇r
0
Z
T
a(r)∇r
0
= kxk for x ∈ K ∩ ∂Ω1 ,
(2.4)
where Ω1 = {x ∈ K : kxk < ρ1 }. On the other hand, apply Condition (H3) and set
ρ2
ρ2 > ρ1 such that if x ≥ T +1+β
, then
f (x, ψ(s)) > (mx)p−1 ,
for each s ∈ [−r, 0].
Define Ω2 = {x ∈ K : kxk < ρ2 }. For x ∈ K with kxk = ρ2 , we have
x(t) ≥
δ
kxk,
T +β
t ∈ [0, T ],
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EXISTENCE OF SOLUTIONS
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Thus, we have
kF xk = (F x)(T )
Z T
≥ δφq
λa(r)f (x(r), x(µ(r)))∇r
0
Z
q−1
≥ δλ φq
a(r)f (x(r), ψ(µ(r)))∇r
Y1
Z
q−1
≥ mδλ
min{x(t)}φq
a(r)∇r
t∈Y1
(2.5)
Y1
2 q−1
Z
mδ λ
kxkφq
a(r)∇r
≥
T +β
Y1
= kxk for x ∈ K ∩ ∂Ω2 .
Applying Condition (i) of Lemma 1.5, the proof is complete.
Note that Theorem 2.2 is useful, but it does not apply if
f (x1 , x2 ) = x21 + x22 ,
and if µ(t) < 0 for some t satisfying ψ(µ(t)) > l and p = 2, for example. That is,
Condition (H1) is not satisfied in this case.
We now provide a second theorem to address the above case. Firstly, we assume
(H2’)
lim
x1 →0+ ;x2 →0+
f (x1 , x2 )
<e
lp−1 .
max{x1p−1 , x2p−1 }
Theorem 2.3. Assume Conditions (A)–(E), (H2’) and (H3) are satisfied. Then,
there exists L > 0 such that for each 0 < λ < L, BVP (1.1) has at least a positive
solution.
Proof. We outline the proof as a modification of the proof of Theorem 2.2. Only
the argument in the construction of Ω1 is modified. As in the proof of Theorem 2.2,
apply Condition (H2’); this time set ε2 > 0 such that if 0 < x1 ≤ ε2 , 0 < x2 ≤ ε2 ,
then
p−1
l
.
f (x1 , x2 ) < max xp−1 , xp−1 e
1
2
Now, set ρ1 = ε2 and define
Ω1 = {x ∈ K : kxk < ρ1 }
In particular, note that ρ1 is independent of λ. Let
R
n max
op−1
x∈Ω1 Y1 f (x(r), ψ(µ(r))∇r)
D=
RT
(T + β)φq 0 a(r)∇r
Recall that f is continuous so D is well defined. Assume
n
o
ρ1
R
λq−1 ≤ min 1,
:= Lq−1 .
maxx∈Ω1 Y1 f (x(r), ψ(µ(r))∇r)
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C. SONG
EJDE-2006/113
Then we have
kF xk
≤ (T + β)λq−1 φq
Z
T
a(r)f (x(r), x(µ(r)))∇r
0
= (T + β)λ
q−1
h
φq
Z
Z
a(r)f (x(r), ψ(µ(r)))∇r +
Y1
a(r)f (x(r), x(µ(r)))∇r
i
Y2
≤ (T + β)λq−1 max Dq−1 , e
l max{x(t)} φq
Z
t∈Y2
T
a(r)∇r
0
≤ kxk for x ∈ K ∩ ∂Ω1 .
The remainder of the proof of Theorem 2.2 carries over verbatim.
We now consider analogous conditions:
> mp−1 , uniformly in s ∈ [−r, 0];
(H4) limx→0+ f (x,ψ(s))
xp−1
f (x,ψ(s))
(H5) limx→∞ xp−1 < lp−1 , uniformly in s ∈ [−r, 0];
f (x1 ,x2 )
< lp−1 .
(H6) limx1 →∞;x2 →∞ max{x
p−1 p−1
,x
}
1
2
Theorem 2.4. Assume Conditions (A)–(E), (H4)–(H6) are satisfied. Then, for
each 0 < λ < ∞, BVP (1.1) has at least a positive solution.
Proof. Apply Condition (H4) and set ρ1 > 0 such that if 0 < x ≤ ρ1 , then
f (x, ψ(s)) > (mx)p−1 .
Define Ω1 = {x ∈ K : kxk < ρ1 }. For x ∈ K with kxk = ρ1 , we have
δ
x(t) ≥
kxk, t ∈ [0, T ],
T +β
Thus,
kF xk = (F x)(T )
Z T
≥ δφq
λa(r)f (x(r), x(µ(r)))∇r
0
Z
q−1
a(r)f (x(r), ψ(µ(r)))∇r
≥ δλ φq
Y1
Z
≥ mδλq−1 min{x(t)}φq
a(r)∇r
t∈Y1
2 q−1
Y1
Z
mδ λ
kxkφq
a(r)∇r
T +β
Y1
= kxk for x ∈ K ∩ ∂Ω1 .
≥
To construct Ω2 , we consider two cases, f bounded and f unbounded: When f is
bounded, the construction is straightforward. If f (x1 , x2 ) is bounded by N p−1 > 0,
set
Z T
ρ2 = max 2ρ1 , N (T + β)φq
a(r)∇r .
0
Then define Ω2 = {x ∈ K : kxk < ρ2 }. For x ∈ K with kxk = ρ2 , we have
Z T
kF xk ≤ N (T + β)φq
a(r)∇r ≤ ρ2 .
0
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EXISTENCE OF SOLUTIONS
7
Assume f is unbounded. Apply Condition (H5) and set ε1 > 0 such that if x > ε1 ,
then
f (x, ψ(s)) < (lx)p−1 , for each s ∈ [−r, 0].
Apply Condition (H6) and set ε2 > 0 such that if x1 ≥ ε2 , x2 ≥ ε2 , then
f (x1 , x2 ) < max x1p−1 , x2p−1 lp−1 .
Set ρ2 = max{2ρ1 , ε1 , ε2 }. Then, for any x ∈ K with kxk = ρ2 , from (2.3), we have
kF xk
≤ (T + β)λq−1 φq
T
Z
a(r)f (x(r), x(µ(r)))∇r
0
= (T + β)λ
q−1
h
φq
Z
Z
a(r)f (x(r), ψ(µ(r)))∇r +
Y1
i
Y2
≤ l(T + β)λq−1 max {x(t)}φq
Z
t∈[0,T ]
= l(T + β)λq−1 kxkφq
a(r)f (x(r), x(µ(r)))∇r
T
a(r)∇r
0
T
Z
a(r)∇r
0
= kxk for x ∈ K ∩ ∂Ω2 ,
where Ω2 = {x ∈ K : kxk < ρ2 }. Apply Condition (ii) of Lemma 1.5, the proof is
complete.
Similarly, assuming
(H6’)
f (x1 , x2 )
ep−1 ,
p−1 < l
max{xp−1
,
x
}
1
2
we have the following theorem which is analogous to Theorem 2.3.
lim
x1 →∞;x2 →∞
Theorem 2.5. Assume Conditions (A)–(E), (H5) and (H6’) are satisfied. Then,
there exists L > 0 such that for each 0 < λ < L, BVP (1.1) has at least one positive
solution.
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Changxiu Song
School of Mathematical Sciences, South China Normal University, Guangzhou 510631,
China
School of Applied Mathematics, Guangdong University of Technology, Guangzhou
510006, China
E-mail address: scx168@sohu.com