R E S E A R C H
Open Access
Existence of multiple positive solutions for
p-Laplacian multipoint boundary value
problems on time scales
Abdulkadir Dogan
**Correspondence:
[email protected] Department of Applied Mathematics, Faculty of Computer Sciences, Abdullah Gul University, Kayseri, 38039, Turkey
Abstract
In this paper, we considerp-Laplacian multipoint boundary value problems on time scales. By using a generalization of the Leggett-Williams fixed point theorem due to Bai and Ge, we prove that a boundary value problem has at least three positive solutions. Moreover, we study existence of positive solutions of a multipoint boundary value problem for an increasing homeomorphism and homomorphism on time scales. By using fixed point index theory, sufficient conditions for the existence of at least two positive solutions are provided. Examples are given to illustrate the results.
MSC: 34B15; 34B16; 34B18; 39A10
Keywords: time scales; boundary value problem;p-Laplacian; positive solutions; fixed point theorem
1 Introduction
The theory of dynamic equation on time scales (or measure chains) was initiated by Stefan Hilger in his PhD thesis in [] (supervised by Bernd Aulbach) as a means of unify-ing structure for the study of differential equations in the continuous case and study of finite difference equations in the discrete case. In recent years, it has gained a consider-able amount of interest and attracted the attention of many researchers, see, for example, [–]. It is still a new area, and the research in this area is rapidly growing. The study of time scales has led to several important applications,e.g., in the study of insect population models, heat transfer, neural networks, phytoremediation of metals, wound healing, and epidemic models.
p-Laplacian equations for boundary value problems (BVPs) with nonlinearity depend-ing on the first order derivative have been studied extensively, see [–] and references therein. However, there are few papers concerningp-Laplacian equations with nonlinear-ity depending on the first order derivative for BVPs on time scales, see [, ].
In this paper, we study the following three boundary value problems on time scales. () We are interested in the existence of at least three positive solutions to the following
p-Laplacian multipoint BVP on time scales
φp
u(t)+q(t)ft,u(t),u(t)= , t∈(,T)T, (.)
u() = n–
i=
αiu(ξi), φp
u(T)=
n–
i=
βiφp
u(ξ
i)
, (.)
whereφp(u) is ap-Laplacian operator,i.e.,φp(u) =|u|p–u, forp> , with (φp)–=φ
qand
p+
q= . The usual notation and terminology for time scales as can be found in [, ], will be used here. The interesting point is that the nonlinear termf is involved with the first order derivative explicitly and the main tool is a fixed point theorem due to Bai and Ge. The results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting.
The present work is motivated by the recent papers [, ]. In [], Yang and Xiao studied the existence of multiple positive solutions forφ-Laplacian multipoint BVPs
φx(t)+q(t)ft,x(t),x(t)= , t∈(, ),
x() = n–
i=
αix(ξi), φ
x()=
n–
i=
βiφ
x(ξi)
,
whereφ(·) is an odd and increasing homeomorphism,ξi∈(, ) with <ξ<ξ<· · ·<
ξn–< ,αiandβiare nonnegative constants andf(t,x(t),x(t)) is continuous and allowed to change sign.
() We consider the existence of at least three positive solutions to the following
p-Laplacian multipoint boundary value problem (BVP) on time scales
φp
u(t)∇+a(t)ft,u(t),u(t)= , t∈(,T)T, (.)
φp
u()= m–
i= aiφp
u(ξi), u(T) = m–
i=
biu(ξi), (.)
whereφp(u) isp-Laplacian operator,i.e.,φp(u) =|u|p–u, forp> , with (φp)–=φ
q and /p+ /q= .
Recently, much attention has been focused on the study of multipoint positive solutions of BVPs on time scales. When the nonlinear termf does not depend on the first order derivative, many researchers study multipoint boundary conditions on time scales, see [, , –]. However, little work has been done on the existence of positive solutions for multipoint BVP on time scales when the nonlinear term is involved in the first order derivative explicitly, see [].
In recent papers, the authors in [, ] have investigated the existence of positive solu-tions of the following BVP on time scales:
φu(t)∇+a(t)ft,u(t)= , t∈[,T],
φu()= m–
i= aiφ
u(ξi), u(T) = m–
i= biu(ξi),
whereφ:R→Ris an increasing homeomorphism and a homomorphism andφ() = and
φu(t)∇+a(t)ft,u(t)= , t∈(,T),
u() = m–
i=
aiu(ξi), φ
u(T)= m–
i= biφ
whereφ :R→Ris an increasing homeomorphism and a positive homomorphism and
φ() = .
All the above-mentioned works about positive solutions were done under the assump-tion thatfis allowed to depend just onu, while the first order derivativeuis not involved
explicitly in the nonlinear termf.
Motivated by all the works above, our main results will depend on an application of a generalization of the Leggett-Williams fixed point theorem due to Bai and Ge. Here, the emphasis is that the nonlinear term is involved explicitly with the first order derivative. We shall prove that the BVP (.) and (.) has at least three positive solutions.
() We will be concerned with proving the existence of positive solutions to the boundary value problem on time scaleTgiven by
φu(t)∇+q(t)ft,u(t)= , t∈(,T)T, (.)
φu()= , u(T) = m–
i=
aiu(ξi), (.)
whereφ:R→Ris an increasing homeomorphism and a homomorphism andφ() = . A projectionφ:R→Ris called an increasing homeomorphism and a homomorphism if the following conditions are satisfied:
(i) ifx≤y, thenφ(x)≤φ(y),∀x,y∈R;
(ii) φis a continuous bijection and its inverse mapping is also continuous; (iii) φ(xy) =φ(x)φ(y),∀x,y∈R.
In recent years, much attention has been paid to the existence of positive solutions for nonlinear boundary value problems on time scales, see [–, –] and the ref-erence therein. On the other hand, multipoint nonlinear boundary value problems with
p-Laplacian operators on time scales have also been studied extensively in the literature [, –]. However, to the best of our knowledge, there are few works on the increasing homeomorphism and homomorphism on time scales [].
Su et al.[] considered the followingm-point singularp-Laplacian boundary value problem on time scales of the form
ϕp
u(t)∇+q(t)ft,u(t)= , t∈(,T)T,
u() = m–
i=
αiu(ξi), u(T) – m–
i=
ψi
u(ξi)
= ,
whereϕp(u) =|u|p–u, forp> .ψi:R→Ris continuous and nondecreasing <ξ<ξ< · · ·<ξm–<ρ(T). By using the well-known Schauder fixed point theorem and upper and
lower solution method, they obtained some new existence criteria for positive solutions of the boundary value problem.
In [], Liang and Zhang studied the existence of positive solutions of boundary value problems on time scales:
ϕu(t)∇+a(t)fu(t)= , t∈[,T]T,
u() = m–
i=
whereϕ:R→Ris an increasing homeomorphism and a positive homomorphism and
ϕ() = . They obtained the countably many positive solutions by using a fixed point index theory and fixed point theorem.
This work is motivated by recent papers [, ]. Existence of at least two positive solu-tions to BVP (.) and (.) are established by means of fixed point index theory. We also point out that whenT=R,p= , (.) and (.) becomes a boundary value problem of dif-ferential equations and is just the problem considered in []. Our main results improve and extend the main results of [, ].
2 Preliminaries
In this section, we provide some background materials from the theory of cones in Banach spaces. The following definitions can be found in the book by Deimling [], as well as in the book by Guo and Lakshmikantham [].
Definition . LetEbe a real Banach space. A nonempty, closed, convex setP⊂Eis a cone if it satisfies the following two conditions:
(i) x∈P,λ≥imply thatλx∈P; (ii) x∈P,–x∈Pimply thatx= .
Every coneP⊂Einduces an ordering inEgiven byx≤yif and only ify–x∈P.
Definition . A mapψis said to be a nonnegative continuous concave functional on a conePof a real Banach spaceEifψ:P→[,∞) is continuous and
ψtx+ ( –t)y≥tψ(x) + ( –t)ψ(y) for allx,y∈Pandt∈[, ].
Similarly, we say the mapαis a nonnegative continuous convex functional on a coneP
of a real Banach spaceEifα:P→[,∞) is continuous and
αtx+ ( –t)y≤tα(x) + ( –t)α(y) for allx,y∈Pandt∈[, ].
Letψbe a nonnegative continuous concave functional onP, andαandβbe nonnegative continuous convex functionals onP.
For nonnegative real numbersr,aandl, we define the following convex sets
P= (α,r;β,l) =u∈P:α(u) <r,β(u) <l,
¯
P= (α,r;β,l) =u∈P:α(u)≤r,β(u)≤l,
P= (α,r;β,l;ψ,a) =u∈P:α(u) <r,β(u) <l,ψ(u) >a,
¯
P= (α,r;β,l;ψ,a) =u∈P:α(u)≤r,β(u)≤l,ψ(u)≥a.
To prove our main results, we need the following fixed point theorem, which comes from Bai and Ge in [].
(A) there existsM> such thatu ≤Mmax{α(u),β(u)}for allu∈P; (A) P(α,r;β,l)=∅for anyr> andl> ;
(A) ψ(u)≤α(u)for allu∈ ¯P(α,r;β,l);
and if F:P¯(α,r;β,l)→(α,r;β,l)is completely continuous operator,which satisfies
(B) {u∈ ¯P(α,d;β,l;ψ,b) :ψ(u) >b} =∅,ψ(Fu) >bforu∈ ¯P(α,d;β,l;ψ,b);
(B) α(Fu) <r,β(Fu) <lforu∈ ¯P(α,r;β,l);
(B) ψ(Fu) >bforu∈ ¯P(α,r;β,l;ψ,b)withα(Fu) >d.
Then F has at least three different fixed points u,uand uinP¯(α,r;β,l)with u∈P(α,r;β,l), u∈
¯
P(α,r;β,l;ψ,b) :ψ(u) >b
;
and
u∈ ¯P(α,r;β,l)\ ¯
P(α,r;β,l;ψ,b)∪ ¯P(α,r;β,l)
.
Let the Banach space
E=Cld[,T]T
=u|[,T]T→R|uis-differentiable on [,T]T,
anduis ld-continuous on [,T]T
be endowed with the norm
u=max sup t∈[,T]T
u(t), sup t∈[,T]T
u(t).
Define
P=u∈E|u(t)≥,u(t)≥, andu(t) is concave on [,T]T.
Clearly,Pis a cone.
3 Existence of triple positive solutions to (1.1) and (1.2)
Throughout the section, we suppose that the following conditions are satisfied. (H) ,T∈T, <ξ<ξ<· · ·<ξn–<ρ(T),ξi∈T,αi,βi∈[,∞)satisfy
≤ni=–αi< and≤
n–
i= βi< ; (H) η=min{t∈T: T ≤t<T}exists;
(H) q(t)∈Cld([,T]T, [,∞))with <ηTq(t)t<∞;
(H) f : (,T)T×[,∞)×R→[,∞)is continuous;
(H) q(t)f(t, , )≡,f(t, , )≥on[,T]T.
Lemma . Ifni=–αi= and
n–
i= βi= ,then for h∈Cld[,T]T,
φp
u(t)+h(t) = , t∈(,T)T, (.)
u() = n–
i=
αiu(ξi), φp
u(T)= n–
i=
βiφp
has the unique solution
u(t) =
t
φq
A+
T
s
h(τ)τ
s+B, (.)
where
A=
n–
i= βi
T
ξi h(τ)τ
–ni=–βi
, B=
n–
i= αi –ni=–αi
ξi
φq
A+
T
s
h(τ)τ
s.
Lemma . The solution of BVP(.)and(.)satisfies u≥,for t∈[,T]T.
Lemma . Suppose(H)holds,if h∈Cld[,T]and h≥,then the unique solution u of
(.)and(.)satisfies
inf t∈[,T]Tu(t)≥
γu,
where
γ =
n–
i= αiξi
T–ni=–αi(T–ξi).
Proof Let
ϕ(s) =φq
n–
i= βi
T
ξi h(τ)τ
–ni=–βi +
T
s
h(τ)τ
.
Clearlyu(t) =ϕ(t)≥. This implies that
min
t∈[,T]u(t) =u(), u=u(T).
It is easy to see thatu(t
)≤u(t) for anyt,t∈[,T] witht≤t. Henceu(t) is a
decreasing function on [,T]. This means that the graph ofu(t) is concave down on (,T). For eachi∈ {, , . . . ,n– }, we have
u(T) –u()
T– ≥
u(T) –u(ξi)
T–ξi ,
i.e.,
Tu(ξi) –ξiu(T)≥(T–ξi)u(),
so that
T
n–
i=
αiu(ξi) – n–
i=
αiξiu(T)≥ n–
i=
With the boundary conditionu() =ni=–αiu(ξi), we have
u()≥
n–
i= αiξi
T–ni=–αi(T–ξi)u(T).
This completes the proof.
Define the operatorF:P→Eby
(Fu)(t) =
t
φq
n–
i= βi
T
ξi q(τ)f(τ,u,u
)τ
–ni=–βi
+
T
s
q(τ)fτ,u,uτ
s
+
n–
i= αi –ni=–αi
ξi
φq
n–
i= βi
T
ξi q(τ)f(τ,u,u
)τ
–ni=–βi +
T
s
q(τ)fτ,u,uτ
s
fort∈[,T]T. By the definition ofF, the monotonicity ofφq(u) and the assumptions
(H)-(H), it is easy to see that for eachu∈P,Fu∈PandFu(T) is the maximum value ofFu(t). Moreover, by direct calculation, we get that each fixed point of the operatorFinPis a positive solution of the BVP (.) and (.). It is easy to see thatF:P→Pis completely continuous.
Foru∈P, we define
α(u) = max t∈[,T]T
u(t)=u(T), β(u) = sup t∈[,T]T
u(t)=u(),
ψ(u) = min
t∈[η,T]Tu(t) =u(η).
It is easy to see thatα,β:P→[,∞) are nonnegative continuous convex functionals with
u=max{α(u),β(u)};ψ:P→[,∞) is nonnegative concave functional. We haveψ(u)≤
α(u) foru∈Pand the assumptions (A), (A) and (A) in Lemma . hold. For notational convenience, we denoteλ,LandQby
λ=
η
φq
n–
i= βi
T
ξi q(τ)τ
–ni=–βi +
T
η
q(τ)τ
s,
L=
T
φq
n–
i= βi
T
ξi q(τ)τ
–ni=–βi +
T
s
q(τ)τ
s
+
n–
i= αi –ni=–αi
ξi
φq
n–
i= βi
T
ξi q(τ)τ
–ni=–βi +
T
s
q(τ)τ
s,
Q=φq
n–
i= βi
T
ξi q(τ)τ
–ni=–βi +
T
q(τ)τ
.
Theorem . Assume that(H)-(H)hold,and there exists <r<b< b≤r, <l≤l such thatλb≤min{r/L,l/Q}.If f satisfies the following conditions:
(D) f(t,w,v)≤min{φp(r/L),φp(l/Q)}for(t,w,v)∈[,T]T×[,r]×[–l,l];
(D) f(t,w,v) <min{φp(r/L),φp(l/Q)}for(t,w,v)∈[,T]T×[,r]×[–l,l];
Proof In order to show that Lemma . holds, it is sufficient to show that the conditions in Lemma . are satisfied with respect to operatorF.
We first prove that if the assumption (D) is satisfied, thenF:P¯(α,r;β,l)→ ¯P(α,r;
and assumption (D) implies that
Similarly, ifu∈ ¯P(α,r;β,l), then the assumption (D) implies that
ft,u(t),u(t)<minφp(r/L),φp(l/Q)
fort∈[,T]T.
We can get thatF:P¯(α,r;β,l)→P(α,r;β,l).
So condition (B) of Lemma . is satisfied.
To prove condition (B) of Lemma . holds. We chooseu(t) = bfort∈[,T]T. It is
obvious that
u(t) = b∈ ¯P(α, b;β,l;ψ,b) and ψ(u) = b>b
and, consequently,
u∈ ¯P(α, b;β,l;ψ,b) :ψ(u) >b
=∅.
So, foru∈ ¯P(α, b;β,l;ψ,b), there areb≤u(t)≤band|u(t)| ≤lfort∈[η,T]T.
Thus, from the assumption (D), we have
ft,u(t),u(t)>φp(b/λ) fort∈[η,T]T.
From the definition of the functionalψ, we see that
ψ(Fu) = min
t∈[η,T]TFu(t) =Fu(η)
=
η
φq
n–
i= βi
T
ξi q(τ)f(τ,u,u
)τ
–ni=–βi
+
T
s
q(τ)fτ,u,uτ
s
+
n–
i= αi –ni=–αi
ξi
φq
n–
i= βi
T
ξiq(τ)f(τ,u,u
)τ
–ni=–βi +
T
s
q(τ)fτ,u,uτ
s
≥ η
φq
n–
i= βi
T
ξi q(τ)f(τ,u,u
)τ
–ni=–βi
+
T
η
q(τ)fτ,u,uτ
s
>
η
φq
n–
i= βi
T
ξi q(τ)φp(b/λ)τ
–ni=–βi
+
T
η
q(τ)φp(b/λ)τ
s
= b
λ
η
φq
n–
i= βi
T
ξi q(τ)τ
–ni=–βi +
T
η
q(τ)τ
s=b.
So, we getψ(Fu) >bforu∈ ¯P(α, b;β,l;ψ,b), and condition (B) of Lemma . holds.
Finally, we prove that condition (B) of Lemma . holds. Ifu∈ ¯P(α,r;β,l;ψ,b) andα(Fu) > b, we have
ψ(Fu) = min
t∈[η,T]TFu(t) =Fu(η)≥
η
T t∈max[,T]TFu(t)≥
Hence, condition (B) of Lemma . is satisfied. Then using Lemma . and the assumption
denotes the set of all nonnegative integers.
Takeα=,α=,β=,β=,ξ=,ξ=,T= ,p=q= andq(t)≡,t∈[,T]T.
Consider the following BVP
As a result,f(t,w,v) satisfies
f(t,w,v)≤min
φp
r
˜ L
,φp
l Q
≈. <min
φp
r
L
,φp
l Q
for ≤t≤, ≤w≤,|v| ≤;
f(t,w,v) >φp
b
λ
≈. for ≤t≤, ≤w≤,|v| ≤;
f(t,w,v) <min
φp
r
˜ L
,φp
l Q
≈. <min
φp
r
L
,φp
l Q
for ≤t≤, ≤w≤
,|v| ≤ .
Hence, by Theorem ., we have that the BVP (.) and (.) has at least three positive solutionsu,uandusuch that
max t∈[,]T
u(t)
<
, t∈sup[,T]T u
(t)<
;
< min t∈[,]T
u(t)
≤ max
t∈[,]T
u(t)
≤, sup t∈[,]T
u(t)≤;
min t∈[,]T
u(t)
< ,
<t∈max[,]T
u(t)
< ,
<t∈sup[,]T
u(t)≤.
4 Existence of triple positive solutions to (1.3) and (1.4) The following conditions will be used in this section:
(H) ,T∈T, <ξ<ξ<· · ·<ξm–<ρ(T),ξi∈T,ai,bi∈[,∞)satisfy ≤mi=–ai< and≤
m–
i= bi< ; (H) η=min{t∈T:T/≤t<T}exists;
(H) a(t)∈Cld([,T]T, [,∞))with <ηTa(t)∇t<∞;
(H) f : (,T)T×[,∞)×R→[,∞)is continuous;
(H) a(t)f(t, , )≡,f(t, , )≥on[,T]T.
Let the Banach space
E=Cld[,T]T
=u|[,T]T→R|uis-differentiable on [,T]T,
anduis ld-continuous on [,T] T
be endowed with the norm
u=max sup t∈[,T]T
u(t), sup t∈[,T]T
u(t).
Define
K=u∈E|u(t)≥,u(t)≤, andu(t) is concave on [,T] T.
We note thatu(t) is a solution of (.) and (.) if and only if
Au(t). Moreover, by direct calculation, we get that each fixed point of the operatorAinK
is a positive solution of (.), (.). It is easy to see thatA:K→Kis completely continuous. Foru∈K, we define For notational convenience, we denoteM,NandQby
(D) f(t,w,v) >φp(b/M)for(t,w,v)∈[η,T]T×[b, b]×[–l,l];
Proof In order to show that Lemma . holds, it is sufficient to show that the conditions in Lemma . are satisfied with respect to the operatorA.
We first prove that if the assumption (D) is satisfied, thenA:P¯(α,r;β,l)→ ¯P(α,r;
and assumption (D) implies that
Similarly ifu∈ ¯P(α,r;β,l), then the assumption (D) implies that ft,u(t),u(t)<minφp(r/N),φp(l/Q)
fort∈[,T]T.
We can get thatA:P¯(α,r;β,l)→P(α,r;β,l).
So condition (B) of Lemma . is satisfied.
To prove condition (B) of Lemma . holds. We chooseu(t) = bfort∈[,T]T. It is
obvious that
u(t) = b∈ ¯P(α, b;β,l;ψ,b) and ψ(u) = b>b,
and, consequently,
u∈ ¯P(α, b;β,l;ψ,b) :ψ(u) >b
=∅.
So, foru∈ ¯P(α, b;β,l;ψ,b), there areb≤u(t)≤band|u(t)| ≤lfort∈[η,T]T.
Thus, from the assumption (D), we have
ft,u(t),u(t)>φp(b/M) fort∈[η,T]T.
From the definition of the functionalψ, we see that
ψ(Au) = min t∈[η,T]TAu(t)
=
m–
i= bi –mi=–bi
T
ξi
φq
m–
i= ai
ξi
a(r)f(r,u,u
)∇r
–mi=–ai
+
s
a(r)fr,u,u∇r
s
≥ m–
i= bi –mi=–bi
T
ξi
φq
m–
i= ai
ξi
a(r)f(r,u,u
)∇r
–mi=–ai
+
η
a(r)fr,u,u∇r
s
>
m–
i= bi –mi=–bi
T
ξi
φq
m–
i= ai
ξi
a(r)φp(b/M)∇r
–mi=–ai
+
η
a(r)φp(b/M)∇r
s
= b
M
m–
i= bi –mi=–bi
T
ξi
φq
m–
i= ai
ξi
a(r)∇r
–mi=–ai +
η
a(r)∇r
s
=b.
So, we getψ(Au) >bforu∈ ¯P(α, b;β,l;ψ,b), and condition (B) of Lemma . holds.
Finally, we prove that condition (B) of Lemma . holds. Ifu∈ ¯P(α,r;β,l;ψ,b) and
α(Au) > b, we have
ψ(Au) = min
t∈[η,T]TAu(t) =Au(T)≥
max
t∈[,T]TAu(t)≥
α(Au) > b>b.
and (.) such that
denotes the set of all nonnegative integers.
As a result,f(t,w,v) satisfies
f(t,w,v)≤minφp(r/N˜),φp(l/Q)
≈. <minφp(r/N),φp(l/Q)
,
when ≤t≤, ≤w≤, –≤v≤;
f(t,w,v) >φp(b/M)≈., when ≤t≤, ≤w≤, –≤v≤;
f(t,w,v) <minφp(r/N˜),φp(l/Q)
≈. <minφp(r/N),φp(l/Q)
,
when ≤t≤, ≤w≤/, –/≤v≤/.
Then all conditions of Theorem . hold. Therefore, the BVP (.) and (.) have at least three positive solutionsu,uandusuch that
max t∈[,]T
u(t)
< /, sup t∈[,T]T
u
(t)< /;
< min t∈[,]T
u(t)
≤ max
t∈[,]T
u(t)
≤, sup t∈[,]T
u(t)≤;
min t∈[,]T
u(t)
< , max t∈[,]T
u(t)
< , sup t∈[,]T
u(t)≤.
5 Existence of double positive solutions to (1.5) and (1.6)
Throughout the section, we assume that the following conditions are satisfied: (H) ,T∈T, <ξ<ξ<· · ·<ξm–<ρ(T),ai∈[, +∞)satisfy <
m–
i= ai< ; (H) q(t)∈Cld([,T]T, [, +∞))with <ξiq(r)∇r<∞;
(H) f : [,T]T×[, +∞)→(–∞, +∞)is continuous;
(H) q(t)f(t,u(t))≡,f(t, )≥on[,T]T.
Lemma . If h∈Cld[,T],then
φu(t)∇+h(t) = , t∈(,T)
T, (.)
φu()= , u(T) = m–
i=
aiu(ξi) (.)
has the unique solution
u(t) = –
t
φ–
s
h(r)∇r
s
+
–mi=–ai
– m–
i= ai
ξi
φ–
s
h(r)∇r
s
+
T
φ–
s
h(r)∇r
s
. (.)
Lemma . Assume that(H)holds,for h∈Cld[,T]and h≥,then the unique solution u of(.)and(.)satisfies
Lemma . Assume that(H)holds,if h∈Cld[,T]and h≥,then the unique solution u of(.)and(.)satisfies
inf
t∈[,T]u(t)≥γu, where
γ =
m–
i= ai(T–ξi) T–mi=–aiξi
, u= max t∈[,T]
u(t).
LetE=Cld([,T],R) be the set of all ld-continuous functions from [,T] toR, and let
the norm onCld([,T]) be the maximum. Then theCld([,T],R) is a Banach space. We
define three cones by
P=u:u∈E,u(t)≥,∀t∈[,T]T,
P=u:u∈E,u(t) is nonnegative, concave and decreasing on [,T]T
and
K= u:u∈E,u(t) is nonnegative and decreasing on [,T]T, inf
t∈[,T]Tu(t)≥
γu
,
whereγ is the same as in Lemma .. It is easy to see that the BVP (.) and (.) has a solutionu=u(t) if and only ifusolves the equation
u(t) = –
t
φ–
s
q(r)fr,u(r)∇r
s
+
–mi=–ai
– m–
i= ai
ξi
φ–
s
q(r)fr,u(r)∇r
s
+
T
φ–
s
q(r)fr,u(r)∇r
s
.
We define the operatorsA:P→E,T:P→EandS:K→Eas follows:
(Au)(t) = –
t
φ–
s
q(r)fr,u(r)∇r
s
+
–mi=–ai
– m–
i= ai
ξi
φ–
s
q(r)fr,u(r)∇r
s
+
T
φ–
s
q(r)fr,u(r)∇r
s
,
(Tu)(t) =
–
t
φ–
s
q(r)fr,u(r)∇r
s
+
–mi=–ai
– m–
i= ai
ξi
φ–
s
q(r)fr,u(r)∇r
+
T
φ–
s
q(r)fr,u(r)∇r
s +
,
(Su)(t) = –
t
φ–
s
q(r)f+r,u(r)∇r
s
+
–mi=–ai
– m–
i= ai
ξi
φ–
s
q(r)f+r,u(r)∇r
s
+
T
φ–
s
q(r)f+r,u(r)∇r
s
, (.)
where
(C)+=max{C, }, f+t,u(t)=maxft,u(t), , t∈[,T]T.
Lemma . S:K→K is completely continuous.
Proof It is obvious thatKis a cone inE. Byf+≥, we get
s
q(r)f+r,u(r)∇r≥.
Consequently,
φ–
s
q(r)f+r,u(r)∇r
≥.
This together with (.) imply that (Su)(t)≥ andSuis decreasing on [,T]T. From
(Su)(t) = –φ–
t
q(r)f+r,u(r)∇r
≥,
we see that (Su)is decreasing on [,T] TK.
(i) Iftis a left-scattered point, we have
(Su)∇=(Su)
(ρ(t)) – (Su)(t)
ρ(t) –t ≤.
(ii) Iftis a left-dense point, we have
(Su)∇=lim
s→t
(Su)(t) – (Su)(s)
t–s ≤.
By (i) and (ii), we have (Su)∇ ≤,t∈[,T]
TK∩TK. Hence, Lemma . implies thatS: K→K.
We now prove thatSis completely continuous.
(a) From the continuity off andq(t)∈Cld([,T]T, [, +∞)), we find thatSis
(b) LetDbe a bounded subset ofK, and letM> be the constant such thatu ≤M
foru∈D. The continuity off+guarantees that there is aL> such that
s
q(r)f+r,u(r)∇r≤ T
q(r)f+r,u(r)∇r≤φ(L).
Thus,
Su=Su()
=
–mi=–ai
– m–
i= ai
ξi
φ–
s
q(r)f+r,u(r)∇r
s
+
T
φ–
s
q(r)f+r,u(r)∇r
s
≤ T
φ–( s
q(r)f+(r,u(r))∇r)s
–mi=–ai
≤ T
φ–( T
q(r)f+(r,u(r))∇r)s
–mi=–ai
≤ LT
–mi=–ai .
(c) Lett,t∈[,T]T, and letu∈D. Then we have
(Su)(t) – (Su)(t)
=–
t
φ–
s
q(r)f+r,u(r)∇r
s+
t
φ–
s
q(r)f+r,u(r)∇r
s
=
t
t
φ–
s
q(r)f+r,u(r)∇r
s≤L|t–t|.
From (a)-(c) together with the Arzela-Ascoli theorem on time scales, we can find that
S:K→Kis completely continuous. This proof is complete.
The following fixed point index theory will play an important role in the proof our re-sults.
Lemma .([, ]) Let E be a Banach space,and let K be a cone in E.For r> ,define Kr={u∈K:u<r}.Assume that S:K¯r→K is a completely continuous operator such
that Tu=u for u∈∂Kr={u∈K:u=r}.
(a) IfTu ≤ ufor allu∈∂Kr,theni(T,Kr,K) = . (b) IfTu ≥ ufor allu∈∂Kr,theni(T,Kr,K) = .
For notational convenience, we introduce the following notations. Let
R=φ–
T
q(r)∇r
T–mi=–aiξi –mi=–ai
,
Q=
–mi=–ai
φ–
ξi
q(r)∇r
T–
m–
i= aiφ–
ξi
q(r)∇r
ξi
Fort∈[,T]T, we haveγb≤u(t)≤b. By condition (iii), we find foru(t)∈∂Pb
s
q(r)f+r,u(r)∇r>φ
b Q
s
q(r)∇r
so that
φ–
s
q(r)f+r,u(r)∇r
> b
Qφ –
s
q(r)∇r
.
Moreover, we get
Su= (Su)()
=
–mi=–ai
T
φ–
s
q(r)f+r,u(r)∇r
s
– m–
i= ai
ξi
φ–
s
q(r)f+r,u(r)∇r
s
> b Q
T
φ –(s
q(r)∇r)s–
b Q
m–
i= ai
ξi
φ –(s
q(r)∇r)s
–mi=–ai
> b Qφ–(
ξi
q(r)∇r)T–
b Q
m–
i= aiφ–(
ξi
q(r)∇r)ξi
–mi=–ai
=b=u.
It follows from Lemma . thati(S,Pc,P) = ,i(S,Pb,P) = .
Thus,i(S,Pb\ ¯Pc,P) = –. Hence,Shas a fixed point inPb\ ¯Pcsuch thatc<u ≤b.
In the end, we show thatuis also a fixed point ofAinPb\ ¯Pc. We only have to verify thatAu=Su,∀u(t)∈(Pb\ ¯Pc)∩ {u:Su=u}, and we getu() =u>c. By Lemma .,
we find that
inf
t∈[,T]Tu(t)≥
γu>γc>d.
Hence,b≥u(t)≥dfort∈[,T]T. Using condition (i), we understand thatf+(t,u(t)) = f(t,u(t)). This implies thatAu=Su=u. It means thatuis a positive solution with
b≥ u>c. This completes the proof.
Example . Let T={ – ( )N} ∪ {
, }, whereNdenotes the set of all nonnegative
integers. Takea=,ξ=,T= andq(t)≡,t∈[,T]T. Consider the following BVP
on time scales
φu(t)∇+ft,u(t)= , t∈[,T]T, (.)
φu()= , u(T) =
u
where
By calculating, we find
γ =
So all the conditions of Theorem . hold. Thus, by Theorem ., problems (.) and (.) have at least two positive solutions.
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
The author carried out the proof of the theorem and approved the final manuscript.
Acknowledgements
The author is very grateful to Professor Ravi P Agarwal and Donal O’Regan for their important comments and suggestions in improving this manuscript. The project is supported by Abdullah Gul University Foundation of Turkey (Project No. 5).
Received: 12 March 2013 Accepted: 31 July 2013 Published: 7 August 2013 References
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doi:10.1186/1687-1847-2013-238