Lemmas Lemma 1. Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
established the existence of one positive solution for BVPs 1.3 by applying the fixed point theorem on cones duo to Krasnosel’skii and Guo.
Motivated by [2] and [5], the purpose of this paper is to establish the existence of positive solutions for BVP 1.1 by using Krasnosel’skii fixed point theorem in cones.
The rest of the paper is organized as follows. In Section 2, we give some Lemmas. In Section 3, the main result of this paper for the existence of at least one positive
solution of BVP 1.1 is established.
2. Lemmas Lemma 2.1.
If
n ≥ 2
and
yt ∈ C[0, 1]
, then the boundary value problem
u
n
t + yt = 0, 0 t 1,
u0 = h R
1
utdζt ,
u
′
0 = 0, · · · , u
n−2
0 = 0, u1 = g
R
1
utdθt ,
2.1
has a unique solution
ut = R
1
Gt, sysds + t
n−1
g R
1
usdθs + 1 − t
n−1
h R
1
usdζs ,
where
Gt, s =
1 − s
n−1
t
n−1
− t − s
n−1
n − 1 ,
0 ≤ s ≤ t ≤ 1, 1 − s
n−1
t
n−1
n − 1 ,
0 ≤ t ≤ s ≤ 1.
Proof. The proof follows by direct calculations, we omitted here. Lemma 2.2. Gt, s
has the following properties
i 0 ≤ Gt, s ≤ ks
,
t, s ∈ [0, 1]
, where
ks = s1 − s
n−1
n − 2 ;
ii Gt, s ≥ γtks
,
t, s ∈ [0, 1]
, where
γt = min t
n−1
n − 1 ,
1 − tt
n−2
n − 1 =
t
n−1
n−1
, 0 ≤ t ≤
1 2
,
1−tt
n−2
n−1
,
1 2
≤ t ≤ 1.
Proof. It is obvious that Gt, s is nonnegative. Moreover,
EJQTDE, 2008 No. 7, p. 3
Gt, s =
t1 − s
n−1
− t − s
n−1
n − 1 ,
0 ≤ s ≤ t ≤ 1, 1 − s
n−1
t
n−1
n − 1 ,
0 ≤ t s ≤ 1 =
1 n − 1
s1 − t[t1 − s
n−2
+ t1 − s
n−3
t − s + · · · +t1 − st − s
n−3
+ t − s
n−2
], 0 ≤ s ≤ t ≤ 1,
1 − s
n−1
t
n−1
, 0 ≤ t s ≤ 1
≤ 1
n − 1 n − 1s1 − s
n−1
, 0 ≤ s ≤ t ≤ 1,
1 − s
n−1
s
n−1
, 0 ≤ t s ≤ 1
≤ s1 − s
n−1
n − 2 = ks,
t, s ∈ [0, 1], that is, i holds.
If s = 0 or s = 1, it is easy to know that ii holds. If s ∈ 0, 1 and t ∈ [0, 1], then we have
Gt, s ks
=
1 − s
n−1
t
n−1
− t − s
n−1
n − 1s1 − s
n−1
, s ≤ t,
1 − s
n−1
t
n−1
n − 1s1 − s
n−1
, t s
=
s1 − t[t1 − s
n−2
+ t1 − s
n−3
t − s + · · · + t − s
n−2
] n − 1s1 − s
n−1
, s ≤ t,
t
n−1
n − 1s ,
t s
≥
s1 − tt
n−2
1 − s
n−2
n − 1s1 − s
n−1
, s ≤ t,
t
n−1
n − 1s ,
t s
≥
1 − tt
n−2
n − 1 ,
s ≤ t, t
n−1
n − 1 ,
t s, which implies that
Gt, s ks
≥ γt,
for s ∈ 0, 1 and t ∈ [0, 1]. Thus, ii holds.
Lemma 2.3 [8].
Let
E = E, k · k
be a Banach space and let
K ⊂ E
be a cone in
E
. Assume
Ω
1
and
Ω
2
are open subsets of
E
with
0 ∈ Ω
1
and
Ω
1
⊂ Ω
2
and let
EJQTDE, 2008 No. 7, p. 4
A : K ∩ Ω
2
\ Ω
1
→ K
be a continuous and completely continuous. In addition, suppose either
1 kAuk ≤ kuk
,
u ∈ K ∩ ∂Ω
1
, and
k Auk ≥ kuk
,
u ∈ K ∩ ∂Ω
2
, or
2 kAuk ≥ kuk
,
u ∈ K ∩ ∂Ω
1
, and
k Auk ≤ kuk
,
u ∈ K ∩ ∂Ω
2
. Then
A
has a fixed point in
K ∩ Ω
2
\ Ω
1
.