R E S E A R C H
Open Access
Multiple positive solutions of nonlinear
third-order boundary value problems with
integral boundary conditions on time scales
Yongkun Li
*and Lingyan Wang
*Correspondence: [email protected]
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China
Abstract
In this paper, we consider a class of nonlinear third-order boundary value problems with integral boundary conditions on time scales. By applying a generalization of the Leggett-Williams fixed point theorem, we establish the existence of at least three positive solutions. The discussed problem involves both an increasing and positive homomorphism, which generalizes thep-Laplacian operator. As an application, we give an example to illustrate our results.
Keywords: time scales; fixed point theorem; integral boundary conditions; multiple positive solutions; cone
1 Introduction
In recent years, much attention has been paid to boundary value problems with integral boundary conditions due to their various applications in chemical engineering, thermo-elasticity, population dynamics, heat conduction, chemical engineering underground wa-ter flow, thermo-elasticity, and plasma physics. Many results for the boundary value prob-lems with integral boundary conditions have been reported in the literature (see [–] and references therein). For example, in [], Guoet al.considered the following boundary value problem:
u(t) +f(t,u(t),u(t)) = , t∈(, ),
u() = , u() = , u() =g(t)u(t)dt, () wheref : [, ]×R+×R–→R+is a nonnegative function. By applying the fixed point
index theory in a cone and spectral radius of a linear operator, they found that () has at least one positive solution.
On the other hand, the study of dynamic equations on time scales, which has been cre-ated in order to unify the study of differential and difference equations, is an area of math-ematics that has recently received a lot of attention; moreover, many results on this issue have been well documented in the monographs [–].
In addition, boundary value problems with integral boundary conditions on time scales represent a very interesting and important class of problems. The study of boundary value problems on time scales has received a lot attention in the literature (see [–]). For ex-ample, in [], Li and Zhang studied the second-orderp-Laplacian dynamic equation on
time scales,
ϕp
x(t)∇+λft,x(t),x(t)= , t∈(,T)T, ()
with integral boundary condition
x() = , αx(T) –βx() =
T
g(s)x(s)∇s. ()
By using the Legget-Williams fixed point theorem, they obtained some criteria for the existence of at least three positive solutions to problems ()-().
In [], Han and Kang considered the existence of multiple positive solutions for the following third-orderp-Laplacian dynamic equation on time scales:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
(p(u(t)))∇+f(t,u(t)) = , t∈[a,b],
αu(ρ(a)) –βu(ρ(a)) = ,
γu(b) +δu(b) = ,
u(ρ(a)) = .
()
By using fixed point theorems in cones, they obtained the existence criteria of at least two positive solutions to problem ().
However, to the best of our knowledge, there is no paper published on the existence of multiple positive solutions to nonlinear third-order boundary value problems with inte-gral boundary conditions on time scales. This paper attempts to fill this gap in the litera-ture.
Motivated by the above, in this paper, we are concerned with the following third-order boundary value problem with integral boundary conditions on time scaleT:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
(φ(–u(t)))+q(t)f(t,u(t),u(t)) = , t∈[, ] T,
au() –bu() =
g(s,u(s))s,
cu() +du() =
g(s)(u(s))s,
u() = ,
()
whereTis a time scale, and are points inT, [, ]T:= [, ]∩T,φ: R→Ris an increas-ing and positive homeomorphism (see Definition .) withφ() = .
Our main purpose of this paper is to study the existence of multiple positive solutions to (). Our method of this paper is based on some recent fixed point theorems derived by Bai and Ge (see []). As an application, we give an example to illustrate our results.
Throughout this paper, we assume that the following conditions are satisfied:
(C) a,b,c,d∈[, +∞)withρ:=ac+ad+bc> ,
(C) f∈C([, ]T×R+×R+, R+)withf(t, , )= for allt∈[, ]T,
(C) g∈C([, ]T×R+, R+),g∈C([, ]T, R+)andq∈C([, ]T, R+).
(C) ρ–
g(s)(b+as)s> . 2 Preliminaries and statements
Definition . LetEbe a real Banach space. A nonempty closed convex setP⊂Eis called a cone if it satisfies the following two conditions:
(i) x∈P,λ≥impliesλx∈P; (ii) x∈P,–x∈Pimpliesx= .
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 that a mapαis a nonnegative continuous convex functional on a conePof a real Banach spaceEifα:P→[,∞) is continuous and
αtx+ ( –t)y≤tα(x) + ( –t)α(y) for allx,y∈Pandt∈[, ].
Definition . A projectionφ: R→Ris called an increasing and positive homomor-phism if the following conditions are satisfied:
(i) Ifx≤y, thenφ(x)≤φ(y)for allx,y∈R;
(ii) φis a continuous bijection, and its inverse mapping is also continuous; (iii) φ(xy) =φ(x)φ(y)for allx,y∈R+, whereR+= [, +∞).
Letψbe a nonnegative continuous concave functional onP, andαandβbe nonnegative continuous convex functionals onP. For nonnegative real numbersr,a, andl, 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 [].
Lemma .[] Let P be a cone in real Banach spaceE.Assume that constants r,b,d,r,
l,and lsatisfy <r<b<d≤rand <l≤l.If there exist two nonnegative continuous
convex functionalsαandβon P and a nonnegative continuous concave functionalψon P such that:
(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)→ ¯P(α,r;β,l)is a completely continuous operator which satisfies
(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,u,and uinP¯(α,r;β,l)with
u∈P(α,r;β,l), u∈
¯
Pα,r;β,l;ψ,b:ψ(u) >b
,
u∈ ¯P(α,r;β,l)\
¯
P(α,r;β,l;ψ,b)∪ ¯P(α,r;β,l)
.
LetE=C[, ] ={uanduare continuous on [, ]
T}. ThenEis a Banach space with
respect to the norm
u=maxmax
t∈[,]T
u(t), max
t∈[,]T
u(t).
Define
P=u∈E:u(t)≥,u(t)≥,u(t) is concave on [, ]T.
Clearly,Pis a cone.
Lemma . Assume that(C)-(C),andρ–
g(s)(b+as)s= hold.Then u∈Eis a
solution of the boundary value problem()if and only if u(t)is a solution of the following integral equation:
u(t) =
G(t,s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s+d+c( –t)f+ (b+at)f,
where
G(t,s) = ρ
(b+aσ(s))(d+c( –t)), ≤σ(s)≤t≤,
(b+at)(d+c( –σ(s))), ≤t≤s≤, () f:=
ρ
g
s,u(s)s, ()
f:=
ρ–
g(s)(b+as)s
–
g(s)f(s)s
+ ρ
g(s)
d+c( –s)s
g
s,u(s)s
, ()
f(t) :=
G(t,s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s.
Proof Let
u(t) = t
ρ
b+aσ(s)d+c( –t)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
+
t
ρ(b+at)
d+c –σ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
Taking the-derivative of (), we get
u(t) = – t
c
ρ
b+aσ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
+
t
a
ρ
d+c –σ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
–cf +af.
It follows that
u(t) = ρ
–cb+aσ(t)–ad+c –σ(t) ×φ–
t
q(τ)fτ,u(τ),u(τ)τ
= –
ρ(ac+ad+bc)φ
–
t
q(τ)fτ,u(τ),u(τ)τ
= –φ–
t
q(τ)fτ,u(τ),u(τ)τ
, ()
hence,u() = –φ–() = and
φ–u(t)=
t
q(τ)fτ,u(τ),u(τ)τ,
so
φ–u(t)= –q(t)ft,u(t),u(t), that is,
φ–u(t)+q(t)ft,u(t),u(t)= . Furthermore, we have
u() =
b
ρ
d+c –σ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s+ (d+c)f +bf,
u() =
a
ρ
d+c –σ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s–cf +af,
hence
au() –bu() =ρ f
=
g
s,u(s)s. ()
Since
u() =
d
ρ
b+aσ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
u() = –
c
ρ
b+aσ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s–cf+af,
we obtain
cu() +du() =ρf
=
g(ζ)
G(ζ,s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
+d+c( –ζ)f+ (b+aζ)f
ζ. ()
From () and (), we find thatf andf satisfy
⎧ ⎪ ⎨ ⎪ ⎩
f =ρ
g(s,u(s))s,
[ –g(s)(d+c( –s))s]f+ [ρ–
g(s)(b+as)s]f
=g(s)f(s)s,
which implies thatf andf are defined by () and (), respectively. The proof is
com-plete.
Define an operatorF:P→Eby
(Fu)(t) =
G(t,s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
+d+c( –t)f+ (b+at)f, ()
whereG,f, andf are defined by (), (), and (), respectively. Lemma . Assume that(C)-(C)hold.Then for u∈P,we have:
(i) (Fu)(t)is concave on[, ]T.
(ii) (Fu)(t)≥fort∈[, ]T.
Proof Foru∈P:
(i) By the definition ofFand similar to the proof of(), we have
(Fu)(t) = –φ–
t
q(τ)fτ,u(τ),u(τ)τ
≤, ()
so(Fu)(t)is nonincreasing. This implies that(Fu)(t)is concave.
(ii) From()and(), we can verify that(Fu)(t)≥fort∈[, ]T.
The proof is complete.
Lemma . Let(C)-(C)hold.Assume that
(C) c–
g(s)s< .
Proof On the contrary, we assume that the inequality (Fu)(t) < holds foru∈Pand
t∈[, ]T. By Lemma ., we see that (Fu)(t) is nonincreasing on [, ]T. Hence
(Fu)()≤(Fu)(t), t∈[, ] T.
Similar to the proof of (), we easily obtain
c(Fu)() +d(Fu)() =
g(s)(Fu)(s)s.
Hence
–c
d(Fu)() +
d
g(s)(Fu)(s)s= (Fu)()≤(Fu)(t) < .
Clearly, we have
(Fu)()
g(s)s=
g(s)(Fu)()s≤
g(s)(Fu)(s)s<c(Fu)(),
that is,
c–
g(s)s
(Fu)() > .
According to Lemma ., we have (Fu)()≥. So,c–g(s)s> . However, this
con-tradicts condition (C). Consequently, (Fu)(t)≥ fort∈[, ]T. The proof is complete.
Lemma . Suppose that(C)-(C)hold.Then F:P→P is a completely continuous
op-erator.
Proof From Lemmas . and . it follows thatF:P→Pis well defined. Next, we show thatFis completely continuous. To this end, we assume thatmis a positive constant and
u∈ ¯Pm={u∈P:u ≤m}. Note that the continuity off(t,u(t),u(t)) andq(t) guarantees
that there is anM> such thatq(t)f(t,u(t),u(t))≤φ(M) for allt∈[, ]
T. Therefore, according to Lemma . and Lemma .(i), we have
max
t∈[,]T
(Fu)(t)= (Fu)()
=
d
ρ
b+aσ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
+df+ (b+a)f
≤
d
ρ(b+a)φ
–
q(τ)fτ,u(τ),u(τ)τ
s+df+ (b+a)f
≤
d
ρ(b+a)φ
–
φ(M)τ
s+df+ (b+a)f
≤ dM
and
max
t∈[,]T
(Fu)(t)= (Fu)()
=
a
ρ
d+c –σ(s)φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
–cf+af
≤
a
ρ(d+c)φ
–
q(τ)fτ,u(τ),u(τ)τ
s+af
≤
a
ρ(d+c)φ
–
φ(M)τ
s+af
≤aM
ρ (d+c) +af,
which imply thatFP¯mis uniformly bounded. In addition, sinceFP¯mis uniformly bounded,
according to the mean value theorem (Theorem . in []), one can easily see that, for
u∈ ¯Pm,t,t∈[, ]T,
(Fu)(t) – (Fu)(t)→, ast–t→. ()
From (), for allu∈ ¯Pm,t∈[, ]T, it follows that
(Fu)(t) = (Fu)() – t
φ–
s
q(τ)fτ,u(τ),u(τ)τ
s.
Hence, for allu∈ ¯Pm,t,t∈[, ]T, we have
(Fu)(t
) – (Fu)(t)=
t
φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
– t
φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
= t
t
φ–
s
q(τ)fτ,u(τ),u(τ)τ
s
≤ t
t
φ–φ(M)s
=M|t–t|,
which implies that
(Fu)(t
) – (Fu)(t)→, ast–t→. ()
Therefore, () and () imply thatFuis equicontinuous for allu∈ ¯Pm. By applying the
Arzela-Ascoli theorem on time scales, we can see thatFP¯mis relatively compact. In view
of Lebesgue’s dominated convergence theorem on time scales, it is clear thatFis a con-tinuous operator. Hence,F:P→Pis completely continuous operator. The proof is
3 Main results
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∈P, this means that assumptions (A)-(A) in Lemma . hold.
Suppose thatω∈Twith <ω< . For convenience, we introduce the following nota-tions:
the following three conditions:
(i) f(t,u,v)≤min{φ(rE),φ(l)},g(s,u)s≤min{rE,l},for
(t,u,v)∈[, ]T×[,r]×[–l,l];
(ii) f(t,u,v) >φ(r)for(t,u,v)∈[ω, ]T×[r,ωr]×[–l,l];
(iii) f(t,u,v) <min{φ(rE),φ(l)},g(s,u)s≤min{rE,l},for
(t,u,v)∈[, ]T×[,r]×[–l,l].
Then problem()has at least three nonnegative solutions u,u,u,which satisfy
We first prove that if assumption (i) is satisfied, thenF:P¯(α,r;β,l)→ ¯P(α,r;β,l).
and assumption (i) implies
Secondly, we show that condition (B) of Lemma . holds. We letu(t) = ωr fort∈[, ]T.
ω>r, and consequently
assumption (ii) we get
ft,u(t),u(t)>φ
From the definition of the functionalψwe see that
+ (b+a)r
Finally, we show that
Therefore, condition (B) of Lemma . is satisfied. So, all the conditions of Lemma . are
The proof is complete.
Ifφ(x) =p(x) =|x|p–xfor somep> , where–p =q, then () can be written as a BVP
with ap-Laplace operator: ⎧
Then, by Theorem ., we have the following.
Corollary . Assume that(C)-(C)hold.If there exist constants r,r,r,l,and lwith
the following three conditions:
(i) f(t,u,v)≤min{p(rE),p(l)},
Then problem()has at least three nonnegative solutions u,u,u,which satisfy
where
f(t,u,v) =
t +
u+ (
v
), u≤, t
+
+ ( v )
, u> ,
and
φ(u) =u, g(t,u) =tu+
u
t, g(t) = .
By calculating, we haveρ=, and
G(t,s) =
( + s)( –t), ≤s≤t≤, ( + t)( –s), ≤t≤s≤. Setω=, then we obtain
=
, = ,
,, E= , ,.
Clearly, assumptions (C)-(C) hold andf(t, , )= on [, ]. We chooser =,r= ,
r= , andl=,l= . So <r<r<ωr and <l<l, and it is easy to check that r
≤min{
r E,
l
}. Now, we show that conditions (i)-(iii) are satisfied:
(i) f(t,u,v)≤. < .≈min{φ(rE),φ(l)},
g(s,u(s))s≤
(t+ (
)t)s= . < .≈min{ r E,
l }, for
(t,u,v)∈[, ]×[, ]×[–,];
(ii) f(t,u,v)≥. > .≈φ(r), for(t,u,v)∈[, ]×[, ]×[–, ]; (iii) f(t,u,v)≤. < .≈min{φ(rE),φ(l)},
g(s,u(s))s≤
ts≈. < .≈min{ r E,
l }, for
(t,u,v)∈[, ]×[,]×[–,].
From the above, we see that all the conditions of Theorem . are satisfied. Hence, by Theorem ., BVP () has at least three nonnegative solutionsu,u,usuch that
max
t∈[,]T
u(t)
<
, t∈max[,]T
u(t)< ;
< min
t∈[,]T
u(t)
≤ max
t∈[,]T
u(t)
≤, max
t∈[,]T
u
(t)≤;
min
t∈[ω,]T
u(t)
< ,
<t∈max[,]T
u(t)
< ,
<t∈max[,]T
u
(t)≤.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors wrote, read, and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. This study was supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072.
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