doi:10.1155/2010/809497
Research Article
Existence of Positive Solutions to a Nonlocal Boundary Value Problem with p-Laplacian on Time Scales
Ting-Ting Sun, Lin-Lin Wang, and Yong-Hong Fan
School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, China
Correspondence should be addressed to Lin-Lin Wang,wangll [email protected] Received 6 October 2009; Accepted 19 January 2010
Academic Editor: Paul Eloe
Copyrightq2010 Ting-Ting Sun 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.
The nonlocal boundary value problem, withp-Laplacian of the formΦputhtft, ut 0, t∈t1, tmT, ut1−n
j1θjuηj−m−2
i1 iuξi 0, utm 0, has been considered. Two existence criteria of at least one and three positive solutions are presented. The first one is based on the Four functionals fixed point theorem in the work of R. Avery et al.2008, and the second one is based on the Five functionals fixed point theorem. Meanwhile an example is worked out to illustrate the main result.
1. Introduction
Due to the unification of the theory of differential and difference equations, there have been many investigations working on the existence of positive solutions to boundary value problems for dynamic equations on time scales. Also there is much attention paid to the study of multipoint boundary value problem withp-Laplacian; see1–10 .
For convenience, throughout this paper we denoteΦpsas thep-Laplacian operator, that is,Φps |s|p−2s,p >1.Φp−1 Φq, where 1/p1/q1.
In 11 , the author discussed the positive solutions of a m-point boundary value problem for a second-order dynamic equation on a time scale
ut qtfut 0, t∈0, T T,
u0 m−2
i1
biuξi, uT m−2
i1
aiuξi, 1.1
whereai,bi ≥0 i1,2, . . . , m−2, andξi ∈0, ρTT with 0< ξ1 < ξ2 <· · ·< ξm−2 < ρT. And he got the existence of at least two positive solutions of the above problem by means of a fixed point theorem in a cone.
Zhao and Ge9 considered the following multi-point boundary value problem with one-dimensionalp-Laplacian:
Φp
xt
ft, xt 0, 0< t <1,
x0−m−1
i1
aixξi 0, x1 m−1
i1
βix ηi
0,
1.2
whereai,βi >0, 0<m−1
i1 aiξi≤1, 0<m−1
i1 βi1−ηi≤1,i1,2, . . . , m−1, 0ξ1 < ξ2<· · ·<
ξm−1 < η1 < η2 < · · ·< ηm−1 1. By using a fixed point theorem in a cone, they obtained the existence of at least one, two, or three positive solutions under some sufficient conditions.
Motivated by the above results, in this paper, we investigate the nonlocal boundary value problem withp-Laplacian
Φpu
t htft, ut 0, t∈t1, tmT,
ut1−n
j1
θju ηj
−m−2
i1
iuξi 0, utm 0, 1.3
where 0≤t1< ξ1< ξ2<· · ·< ξm−2< tmandt1< η1< η2 <· · ·< ηn< tm<∞.
For convenience, we list the following hypotheses:
H1i>0,i1,2, . . . , m−2,θj ≥0,j1,2, . . . , n,m−2
i1 iξin
j1θj <1;
H2ft, u∈Ct1, tm T×0,∞,0,∞andfis not identically zero on any compact subinterval oft1, tm T×0,∞;
H3ht ∈ Crdt1, tm T,0,∞ and h is not identically zero on any compact subinterval oft1, tm T, also it satisfies
Φq
tm
t1
hττ
<∞,
σtm
t1
Φq
tm
s
hττ
s <∞. 1.4
By using the Four functionals fixed point theorem and Five functionals fixed point theorem, we obtain the existence criteria of at least one positive solution and three positive solutions for the BVP 1.3. As an application, an example is worked out finally. The remainder of this paper is organized as follows.Section 2 is devoted to some preliminary discussions. We give and prove our main results inSection 3.
2. Preliminaries
The basic definitions and notations on time scales can be found in12,13 . In the following, we will provide some background materials on the theory of cones in Banach spaces. For more details, please refer to14,15 .
Definition 2.1. LetE be a Banach space. A nonempty, closed setP ⊂ E is said to be a cone provided that the following hypotheses are satisfied:
1ifx, y∈P,α, β≥0, thenαxβy∈P; 2ifx∈P,x /θ, thenx∈P.
Every coneP ⊂Einduces a partial ordering “≤” onEdefined byx≤yif and only if y−x∈P.
Definition 2.2. A mapαis said to be a nonnegative continuous concave functional on a coneP of a real Banach spaceEifα:P → 0,∞is continuous andαtx1−ty≥tαx1−tαy for allx, y ∈P andt ∈ 0,1 . Similarly, we say that the mapβis a nonnegative continuous convex functional on a conePof a real Banach spaceEifβ:P → 0,∞is continuous and βtx 1−ty≤tβx 1−tβyfor allx, y∈Pandt∈0,1 .
LetαandΨbe nonnegative continuous concave functionals onP, and letβandθbe nonnegative continuous convex functionals onP; then for positive numbersr, ι, υ, andR, define the sets
Q
α, β, r, R
x∈P :r≤αx, βx≤R , UΨ, ι x∈Q
α, β, r, R
:ι≤Ψx , Vθ, υ x∈Q
α, β, r, R
:θx≤υ .
2.1
The following lemma can be found in16 .
Lemma 2.3four functionals fixed point theorem. If P is a cone in a real Banach space E,αand Ψare nonnegative continuous concave functionals onP,β,andθare nonnegative continuous convex functionals onP, and there exist nonnegative positive numbersr, ι, υ,andR, such that
A:Q
α, β, r, R
−→P 2.2
is a completely continuous operator, andQα, β, r, Ris a bounded set. If i{x∈UΨ, ι:βx< R} ∩ {x∈Vθ, υ:r < αx}/∅,
iiαAx≥r, for allx∈Qα, β, r, R,withαx randυ < θAx, iiiαAx≥r, for allx∈Vθ, υ,withαx r,
ivβAx≤R, for allx∈Qα, β, r, R,withβx RandΨAx< ι, vβAx≤R, for allx∈UΨ, ι,withβx R,
thenAhas a fixed pointxinQα, β, r, R.
We are now in a position to present the Five functionals fixed point theorem see 17 . Let γ, β, θ be nonnegative continuous convex functionals onP and α, ϕ nonnegative continuous concave functionals onP. For nonnegative numbersh, a, b, d,and c,define the following convex sets:
P γ, c
x∈P :γx< c , P
γ, α, a, c
x∈P :a≤αx, γx≤c , Q
γ, β, d, c
x∈P:βx≤d, γx≤c , P
γ, θ, α, a, b, c
x∈P :a≤αx, θx≤b, γx≤c , Q
γ, β, ϕ, h, d, c
x∈P :h≤ϕx, βx≤d, γx≤c .
2.3
Lemma 2.4 five functionals fixed point theorem. LetP be a cone in a real Banach space E.
Suppose that there exist nonnegative numberscandM, nonnegative continuous concave functionals αandϕonP, and nonnegative continuous convex functionalsγ, β,andθonP, with
αx≤βx, x ≤Mγx ∀x∈P γ, c
. 2.4
Suppose thatA:Pγ, c → Pγ, cis completely continuous and there exist nonnegative numbers h, a, k, b,with 0< a < bsuch that
i{x∈Pγ, θ, α, b, k, c:αx> b}/∅andαAx> bforx∈Pγ, θ, α, b, k, c, ii{x∈Qγ, β, ϕ, h, a, c:βx< a}/∅andβAx< aforx∈Qγ, β, ϕ, h, a, c, iiiαAx> bforx∈Pγ, α, b, cwithθAx> k,
ivβAx< aforx∈Qγ, β, a, cwithϕAx< h, then A has at least three fixed pointsx1, x2, x3∈Pγ, csuch that
βx1< a, αx2> b, βx3> a with αx3< b. 2.5
Consider the Banach space E Ct1, σtm T equipped with the norm u maxt∈t1,σtm T|ut|. Suppose,η∈Twitht1< < η < σtm. For the sake of convenience, we take the notations
m−2
i1
i, h0 Φq
tm
t1
hττ
, M0
t1
Φq
η
hττ
s,
M
t1
Φq
tm
s
hττ
s, Mη
η
t1
Φq
tm
s
hττ
s, Mσtm
σtm
t1
Φq
tm
s
hττ
s.
2.6
Define a cone
P
⎧⎨
⎩u∈E:ut≥0, ut1−n
j1
θju ηj
−m−2
i1
iuξi 0,
for t∈t1, σtm T andut≤0, ut≥0 for t∈t1, tmT, utm 0
2.7
and an operatorA:P → Eby
Au Φq
tm
t1hτfτ, uττ
−
m−2
i1 i
ξi
t1Φq
tm
s hτfτ, uττ
s
− n
j1θjΦq
tm
ηjhτfτ, uττ
t
t1
Φq
tm
s
hτfτ, uττ
s.
2.8
Lemma 2.5. A:P → P.
Proof. Foru∈P,t∈t1, σtm T,
Aut≥ 1−n
j1θj−m−2
i1 iξi
Φq
tm
t1
hτfτ, uττ
t
t1
Φq
tm
s
hτfτ, uττ
s
≥0.
2.9
From the definition ofA, it is clear that
Aut Φq
tm
t
hsfs, uss
≥0, t∈t1, tm T, 2.10
is continuous,Aut1−n
j1θjAuηj−m−2
i1 iAuξi 0,and Auσtmis the maximum value ofAutont1, σtm T.
Letgt tm
t hsfs, uss, theng : R → Ris continuous,g : T → Ris delta differentiable ont1, tm Tk, andΦq:R → Ris continuously differentiable. MoreoverΦqsis monotonically increasing fors≥0 andgt −htft, ut≤0. Then by the chain rule12, Theorem 1.87, page 31 , we obtain
Aut Φq
gc
gt≤0, 2.11
wherecis in the intervalt, σt . So,A:P → P. This completes the proof.
3. Main Results and an Example
Theorem 3.1. Assume that (H1), (H2), and (H3) hold, if there exist constantsr,ι,υ,RwithR >
max{h0Mσtm/M0,σtm−t1/−t1}ι,υ≥max{h0Mη/Mι,η−t1/− t1r},r < ιand suppose thatft, usatisfies the following conditions:
A1ft, u≤ΦpR/h0Mσtmfor allt, u∈t1, tm T×0, R , A2ft, u≥Φpr/M0for allt, u∈, η T×r, υ ,
then the BVP1.3has a fixed pointu∈Psuch that
t∈,η minT
ut≥r, max
t∈t1,σtm T
ut≤R. 3.1
Define maps
αu Ψu min
t∈,η T
ut, θu max
t∈,η T
ut, βu max
t∈t1,σtm T
ut, 3.2
and letQα, β, r, R,UΨ, ιandVθ, υbe defined by2.1.
In order to complete the proof ofTheorem 3.1, we first need to prove the following lemma.
Lemma 3.2. Qα, β, r, Ris bounded andA:Qα, β, r, R → Pis completely continuous.
Proof. For all u ∈ Qα, β, r, R, u maxt∈t1,σtm T|ut| βu ≤ R, which means that Qα, β, r, Ris a bounded set.
According toLemma 2.5, it is clear thatA:Qα, β, r, R → P.
In view of the continuity off, there exists a constantC >0 such thatft, u<ΦpC, for allt∈t1, σtm T,u∈Qα, β, r, R. Consider
AuAuσtm≤ h0
Mσtm
C, 3.3
which means thatAQα, β, r, Ris uniformly bounded.
In addition, for allt1≤t≤t∗≤σtm, we have Au
t
−Aut∗
t∗
t
Φq
tm
s
hτfτ, uττ
s
≤Ch0t−t∗. 3.4
Applying the Arzel`a-Ascoli theorem on time scales18 , one can show thatAQα, β, r, Ris relatively compact.
Now we prove thatA : Qα, β, r, R → P is continuous. Let{un}n∈Nbe a sequence in Qα, β, r, R which converges to u0 ∈ Qα, β, r, R uniformly on t1, σtm T. Because
AQα, β, r, R is relatively compact, the sequence {Aun} admits a subsequence {Aunm} converging tovtuniformly ont1, σtm T. In addition,
0≤Aunt≤C h0
ω Mσtm
. 3.5
Observe that
Aunt Φq
tm
t1hτfτ, unττ
−
m−2
i1 i
ξi
t1Φq
tm
s hτfτ, unττ s
− n
j1θjΦq
tm
ηjhτfτ, unττ
t
t1
Φq
tm
s
hτfτ, unττ
s.
3.6
Hence, by the Lebesgue’s dominated convergence theorem on time scales19 , insertunm into the above equality and then letm → ∞, we obtain
vt Φq
tm
t1hτfτ, u0ττ
−
m−2
i1 iξi
t1Φq
tm
s hτfτ, u0ττ s
− n
j1θjΦq
tm
ηjhτfτ, u0ττ
t
t1
Φq
tm
s
hτfτ, u0ττ
s.
3.7
From the definition of A, we know that vt Au0t on t1, σtm T. This shows that each subsequence of {Aunt}∞n1 uniformly converges to Au0t. Therefore the sequence {Aunt}∞n1 uniformly converges to Au0t. This means that A is continuous at u0 ∈ Qα, β, r, R. So,Ais continuous onQα, β, r, Rsinceu0is arbitrary. Thus,A:Qα, β, r, R → Pis completely continuous. This completes the proof.
Proof ofTheorem 3.1. Let
u0 ι M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 iξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− ι M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ ι M
t
t1
Φq
tm
s
hττ
s.
3.8
Clearly,u0∈P. By direct calculation,
Ψu0
ι M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− ι M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ ι M
t1
Φq
tm
s
hττ
s
≥ ι M
t1
Φq
tm
s
hττ
sι,
βu0
ι M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− ι M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ ι M
σtm
t1
Φq
tm
s
hττ
s
< ι M
⎛
⎜⎝Φq
tm
t1hτΔτ
σtm
t1
Φq
tm
s
hττ
s
⎞
⎟⎠≤R,
θu0
ι M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− ι M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ ι M
η
t1
Φq
tm
s
hττ
s
≤ ι M
⎛
⎜⎝Φq
tm
t1hττ
η
t1
Φq
tm
s
hττ
s
⎞
⎟⎠≤υ,
αu0 ι M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 iξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− ι M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ ι M
t1
Φq
tm
s
hττ
s
> ι M
t1
Φq
tm
s
hττ
s > r.
3.9
So,u0∈ {u∈UΨ, ι:βu< R}∩{u∈Vθ, υ:r < αu},which means thatiinLemma 2.3 is satisfied.
For allu ∈ Qα, β, r, R,withαu r andυ < θAu, we haveαAu Au ≥ −t1/η−t1Auη −t1/η−t1θAu > −t1/η−t1υ > r, and for all u∈Qα, β, r, R,withβu RandΨAu< ι, we obtain thatβAu Auσtm≤σtm− t1/−t1Au σtm−t1/−t1ΨAu < σtm−t1/−t1ι < R. Hence,ii andivinLemma 2.3are fulfilled.
For anyu∈Vθ, υ,withαu r,
αAu Φq
tm
t1hτfτ, uττ
−m−2
i1 i
ξi
t1Φq
tm
s hτfτ, uττ s
− n−2
j1θjΦq
tm
ηjhτfτ, uττ
t1
Φq
tm
s
hτfτ, uττ
s
≥
t1
Φq
η
hτfτ, uττ
s≥
t1
Φq
η
hτΦp
r M0
τ
sr,
3.10
and for allu∈UΨ, ι,withβu R,
βAu Φq
tm
t1hτfτ, uττ
−m−2
i1 i
ξi
t1Φq
tm
s hτfτ, uττ s
− n−2
j1 θjΦq
tm
ηjhτfτ, uττ
η
t1
Φq
tm
s
hτfτ, uττ
s
≤ Φq
tm
t1hτΦp
R/
h0Mσtm
τ
σtm
t1
Φq
tm
s
hτΦp
R h0Mσtm
τ
s R.
3.11
ThusiiiandvinLemma 2.3hold true. So, byLemma 2.3, the BVP1.3has a fixed point uinQα, β, r, R. This completes the proof.
Theorem 3.3. Assume that (H1), (H2), and (H3) hold. If there exist constantsh, a, b, c, k, withb <
aM0/h0Mη,c >h0Mσtm/h0Mηa,k <h0Mη/h0Mσtmc, k≥max{η−t1/−t1,h0Mη/M}b,a≥max{η−t1/−t1,h0Mη/M}h, further suppose thatft, usatisfies the following conditions:
B1ft, u<Φpa/h0Mηfor allt, u∈t1, tm T×0, c , B2ft, u≥Φpb/M0for allt, u∈, η T×b, k ,
then the BVP1.3has at least three positive solutionsu1,u2andu3such that
t∈,η maxT
u1t< a < max
t∈,η T
u3t, min
t∈,η Tu3t< b < min
t∈,η Tu2t. 3.12
Proof. Define these maps
αu ϕu min
t∈,η ut, βu θu max
t∈,η ut, γu max
t∈t1,σtm
ut, 3.13
and let Pγ, c, Pγ, α, b, c,Qγ, β, a, c,Pγ, θ, α, b, k, c and Qγ, β, ϕ, h, a, cbe defined by 2.3. It is clear that
αu≤βu, u ≤γu, ∀u∈Pγ, c. 3.14
Using similar methods as those in Lemma 3.2, we obtain thatA : Pγ, c → P is completely continuous. Thus, we only need to show that A : Pγ, c → Pγ, c. Letu ∈ Pγ, c, then
γAu Φq
tm
t1hτfτ, uττ
−
m−2
i1 iξi
t1Φq
tm
s hτfτ, uττ s
− n
j1θjΦq
tm
ηjhτfτ, uττ
σtm
t1
Φq
tm
s
hτfτ, uττ
s
≤ Φq
tm
t1hτfτ, uττ
σtm
t1
Φq
tm
s
hτfτ, uττ
s
≤ Φq
tm
t1hτΦp
a/
h0Mη
τ
σtm
t1
Φq
tm
s
hτΦp
a h0Mη
τ
s
≤c,
3.15
which implies thatAPγ, c⊂Pγ, c.
LetNh0/Mηand
u1 b M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 iξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− b M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ b M
t
t1
Φq
tm
s
hττ
s,
u2 a N
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− a N
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ a N
t
t1
Φq
tm
s
hττ
s,
3.16
we can verify thatu1, u2∈P. By calculation,
αu1 b
M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− b M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ b M
t1
Φq
tm
s
hττ
s
≥ b M
⎛
⎝1−n
j1θj−m−2
i1 iξi
Φq
tm
t1
hττ
t1
Φq
tm
s
hττ
s
⎞
⎠
> b M
t1
Φq
tm
s
hττ
sb,
θu1
b M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− b M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ b M
η
t1
Φq
tm
s
hττ
s
≤ b M
⎛
⎜⎝Φq
tm
t1hττ
η
t1
Φq
tm
s
hττ
s
⎞
⎟⎠≤k,
γu1
b M
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 i
ξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− b M
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ b M
σtm
t1
Φq
tm
s
hττ
s
≤ b M
⎛
⎜⎝Φq
tm
t1hττ
σtm
t1
Φq
tm
s
hττ
s
⎞
⎟⎠≤c,
βu2 a N
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 iξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− a N
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ a N
η
t1
Φq
tm
s
hττ
s
< a N
⎛
⎜⎝Φq
tm
t1hττ
η
t1
Φq
tm
s
hττ
s
⎞
⎟⎠a,
ϕu2
a N
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 iξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− a N
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ a N
t1
Φq
tm
s
hττ
s
≥ a N
⎛
⎝1−n
j1θj−m−2
i1 iξi
Φq
tm
t1
hττ
t1
Φq
tm
s
hττ
s
⎞
⎠
> a N
t1
Φq
tm
s
hττ
s > h,
γu2 a N
⎛
⎜⎝Φq
tm
t1hττ
−m−2
i1 iξi
t1Φq
tm
s hττ
s
⎞
⎟⎠
− a N
⎛
⎜⎝ n
j1θjΦq
tm
ηjhττ
⎞
⎟⎠ a N
σtm
t1
Φq
tm
s
hττ
s
≤ a N
⎡
⎢⎣Φq
tm
t1hττ
σtm
t1
Φq
tm
s
hττ
s
⎤
⎥⎦< c.
3.17
So, u1 ∈ Pγ, θ, α, b, k, c, αu1 > b, u2 ∈ Qγ, β, ϕ, h, a, c, βu2 < a,which means that {u∈Pγ, θ, α, b, k, c:αu> b}and{u∈Qγ, β, ϕ, h, a, c:βu< a}are not empty.
Foru∈Pγ, θ, α, b, k, c,
αAu Φq
tm
t1hτfτ, uττ
−m−2
i1 iξi
t1Φq
tm
s hτfτ, uττ
s
− n
j1θjΦq
tm
ηjhτfτ, uττ
t1
Φq
tm
s
hτfτ, uττ
s
≥ 1−n
j1θj−m−2
i1 iξi
Φq
tm
t1
hτfτ, uττ
t1
Φq
tm
s
hτfτ, uττ
s
>
t1
Φq
η
hτfτ, uττ
s
≥
t1
Φq
η
hτΦp
b M0
τ
sb,
3.18
and foru∈Qγ, β, ϕ, h, a, c,
βAu Φq
tm
t1hτfτ, uττ
−m−2
i1 i
ξi
t1Φq
tm
s hτfτ, uττ s
− n
j1θjΦq
tm
ηjhτfτ, uττ
η
t1
Φq
tm
s
hτfτ, uττ
s
≤ Φq
tm
t1hτfτ, uττ
η
t1
Φq
tm
s
hτfτ, uττ
s
< Φq
tm
t1hτΦp
a/
h0Mη
τ
η
t1
Φq
tm
s
hτΦp
a h0Mη
τ
sa.
3.19
ThusiandiiinLemma 2.4hold.
On the other hand, foru ∈ Pγ, α, b, cwith θAu > k, we haveαAu Au ≥ −t1/η−t1Auη −t1/η−t1θAu > −t1/η−t1k > b. And for u∈Pγ, β, a, cwithϕAu< h, we can obtainβAu Auη≤ η−t1/−t1Au η−t1/−t1ϕAu<η−t1/−t1h < a.Thus,iiiandivinLemma 2.4hold.
So, byLemma 2.4, we obtain that the BVP1.3has at least three positive solutions u1, u2, u3∈Pγ, csuch that
t∈,η maxTu1t< a < max
t∈,η Tu3t, min
t∈,η Tu3t< b < min
t∈,η Tu2t. 3.20
This completes the proof.
Remark 3.4. LetR c, r b,υ k, we can find that the conditions of Theorem 3.1are contained inTheorem 3.3.
Example 3.5. LetT{0.1,0.18} ∪0.2,1 ∪ {1.2} ∪1.5,2 ,p2, consider the following eight- point BVP:
Φpu
t htft, ut 0, t∈0.1,2T,
u0.1−3
j1
θju ηj
−3
i1
iuξi 0, u2 0, 3.21
whereht tσt,θ11/12,θ2 1/7,θ3 1/42,1 1/6,2 1/24,3 1/8,ξ1 0.33, ξ20.45,ξ31.65,η10.88,η21.86,η31.95, for allt∈T, and
ft, u
⎧⎨
⎩
0.00015, 0.4,1.8 T×0.0001,0.055 ,
gu, other, 3.22
whereguis continuous, 0≤gu≤0.0026, andg0.0001 g0.055 0.00015.
Set0.4,η1.8, by calculation, 3
j1
θj 1 4,
3 i1
i 1
3, h03.99, M0 0.924,
M 3.539656
3 , Mη 15.050656
3 , Mσtm 15.282656
3 ,
3.23
and letb0.0001,k0.055,c0.102 585312,a0.045,h0.000125. Clearly, we can verify that the conditions inTheorem 3.3are fulfilled. Thus, byTheorem 3.3, the BVP3.21has at least three positive solutionsu1,u2andu3such that
t∈,η maxTu1t<0.045< max
t∈,η Tu3t, min
t∈,η T
u3t<0.0001< min
t∈,η T
u2t. 3.24
Remark 3.6. If we letR 0.102 585312,r 0.0001,ι r10−5,υ 0.055, we can also verify that the conditions inTheorem 3.1are satisfied.
