R E S E A R C H
Open Access
Multiple positive solutions to some
second-order integral boundary value
problems with singularity on space variable
Qiuyan Zhong
1and Xingqiu Zhang
2,3* *Correspondence:2School of Medical Information
Engineering, Jining Medical College, Rizhao, Shandong 276826, P.R. China
3School of Mathematics, Liaocheng
University, Liaocheng, Shandong 252059, P.R. China
Full list of author information is available at the end of the article
Abstract
This article deals with integral boundary value problems of the second-order differential equations
u(t) +a(t)u(t) +b(t)u(t) +f(t,u(t)) = 0, t∈J+,
u(0) =01g(s)u(s) ds, u(1) =01h(s)u(s) ds,
wherea∈C(J),b∈C(J,R–),f∈C(J+×R+,R+) andg,h∈L1(J) are nonnegative. The result of the existence of two positive solutions is established by virtue of fixed point index theory on cones. Especially, the nonlinearityfpermits the singularity on the space variable.
MSC: 34B15; 34B18
Keywords: integral boundary value; cone; two positive solutions; singularity
1 Introduction
The aim of this article is to study the existence of two positive solutions to the follow-ing nonlinear second-order differential equation involvfollow-ing integral boundary value con-ditions:
u(t) +a(t)u(t) +b(t)u(t) +f(t,u(t)) = , t∈J+,
u() =g(s)u(s) ds, u() =h(s)u(s) ds, ()
wherea∈C(J),b∈C(J,R–),f∈C(J+×R+,R+) andg,h∈L(J) are nonnegative,J= [, ],
J+= (, ),R+= [, +∞),R+= (, +∞),R–= (–∞, ). Singularities of the nonlinearityf are
related to botht= , andu= .
Recently, there has been a considerable increase in the investigation of nonlocal bound-ary value problems; see [–] for integer order and [–] for fractional differential equations. Based on a specially constructed cone, the existence as well as nonexistence results on positive solutions for the following second-order integral BVPs are obtained in
an abstract space in Fenget al.[]:
⎧ ⎪ ⎨ ⎪ ⎩
u(t) +f(t,u(t)) =θ, t∈J+,
u() =g(t)u(t) dt, u() =θ, or u() = , u() =g(t)u(t) dt,
heref ∈C([, ]×P,P),θrepresents the zero element ofE, andgis nonnegative and in-tegrable. Also, in an abstract space, by the fixed point theorem of strict set contraction operators, Zhanget al.[] obtained the existence results of solutions for some second-order integral boundary value problems with the impulsive effect.
For general differential operator, when the nonlinearity f is continuous, Feng and Ge [] studied multiple positive solutions for the following singularm-point boundary value problems:
Lu=λw(t)f(t,u), t∈J+,
u() = , u() = mi=–αiu(ξi),
hereλ> ,ξi∈(, ),αi∈R+(i= , , . . . ,m– ) are known constants andLrepresents the
linear operator
Lu:= –u–au+bu,
wherea∈C(J) andb∈C(J,R+),f ∈C(J×R+,R+),w∈C(J+,R+). As is well known, two,
three and multi-point BVPs may be looked upon as a special case of integral boundary value problems. For integral conditions, with some so-called first eigenvalue of the re-lated linear operator, Liuet al.[] formulated the existence results for BVP (). The whole discussion relied on the fixed point index theorems. With the impulsive effect, under dif-ferent combinations of super-linear and sub-linear condition on nonlinear term and the impulses, some results of existence of multiple positive solutions as well as nonexistence results for BVP () are obtained in Haoet al.[].
We attempt in this article to study the existence of two positive solutions of BVP (). The interesting points focus on two aspects. First, singularities of the nonlinearityf are related not only to the time but also to the space variables. Second, compared with [], the method and conditions used to get result of multiple positive solutions are quite different from that used in []. The integral of the nonlinearity on some special bounded set is considered in this paper. The tools used to obtain the main result are fixed point index theorems on cones. Obviously, the result obtained in this paper can be analogously given for the Riemann-Stieltjes integral case after some minor modifications.
2 Preliminaries and several lemmas
Letψandψbe the unique solution of the BVP
ψ(t) +a(t)ψ(t) +b(t)ψ(t) = ,
and
ψ(t) +a(t)ψ(t) +b(t)ψ(t) = , ψ() = , ψ() = ,
respectively. By [], we know thatψ,ψare strictly increasing and strictly decreasing on
J, respectively.
We adopt the following assumptions throughout this article.
(H) a∈C(J),b∈C(J,R–);
(H) g,h:J+→R+are integrable, andk> ,k> ,k> , where
k= –
ψ(s)g(s) ds, k=
ψ(s)g(s) ds,
k=
ψ(s)h(s) ds, k= –
ψ(s)h(s) ds,
k=kk–kk;
(H) f ∈C(J+×R+,R+);
(H) there exist three functionsa,b∈C(J+,R+),g∈C(R+,R+)satisfying
f(t,u)≤a(t)g(u) +b(t), ∀t∈J+,u∈R+,
where, in addition,
a∗r=
H(t)a(t)gr(t) dt< +∞, b∗=
H(t)b(t) dt< +∞,
and
gr(t) =max
g(u) :γ(t)r≤u≤r, ∀r> ,
here,γ(t)is defined in (),H(t)is defined in ();
(H) there exists a functionc∈C(J+,R+)satisfying
f(t,u)
c(t)u →+∞ asu→+∞
uniformly fort∈J+, and in addition,
c∗=
c(t) dt< +∞;
(H) there exists a functiond∈C(J+,R+)satisfying
f(t,u)
d(t) →+∞ asu→
uniformly fort∈J+, and in addition,
d∗=
d(t) dt< +∞.
Lemma ([]) Assume that(H)and(H)hold.Then,for any y∈C(J+)∩L(J),the BVP
u(t) +a(t)u(t) +b(t)u(t) +y(t) = , t∈J+,
u() =g(s)u(s) ds, u() =h(s)u(s) ds, ()
has a unique solution u that can be expressed in the form
u(t) =
H(t,s)y(s) ds, t∈J,
where
H(t,s) =G(t,s)p(s) +ψ(t)k+ψ(t)k k
G(τ,s)p(s)g(τ) dτ
+ψ(t)k+ψ(t)k k
G(τ,s)p(s)h(τ) dτ, ()
p(t) =exp
t
a(s) ds
,
G(t,s) =
ρ
ψ(t)ψ(s), ≤t≤s≤,
ψ(s)ψ(t), ≤s≤t≤, ρ=ψ
(). ()
Moreover,u(t)≥on J provided y≥.
By Remark . in [], we have
Lemma []Suppose that(H)and(H)hold,then,for any t,s∈J,we have
≤G(t,s)≤G(s,s), ≤H(t,s)≤H(s), ()
H(t,s)≥γ(t)H(s), ()
whereγ(t) =min{ψ(t),ψ(t)},t∈J and
H(s) =G(s,s)p(s) +k+k k
G(τ,s)p(s)g(τ) dτ+k+k k
G(τ,s)p(s)h(τ) dτ. ()
LetE=C(J) be the standard Banach space with the maximum norm andPbe the typical cone of nonnegative continuous functions in the form
P=u∈E:u(t)≥γ(t)u,t∈J.
First, we give an operatorT:P\ {} →C(J) as follows:
(Tu)(t) =
H(t,s)fs,u(s)ds, t∈J. ()
Lemma If conditions(H)and(H)are satisfied,then,for any n>m> ,T:Pmn→P is
a completely continuous operator.
Proof For any givenu∈Pmn, we havem≤ u ≤n. From the construction ofP, we have
γ(t)m≤u(t)≤n, ∀t∈J. ()
Clearly, for anyn>m> , condition (H) means that
a∗mn=
H(t)a(t)gmn(t) dt< +∞, ()
where
gmn(t) =max
g(u) :γ(t)m≤u≤n. ()
It follows from (), (H), (H) and Lemma that
ft,u(t)≤a(t)gmn(t) +b(t), ∀t∈J+,u∈Pmn, ()
and
(Tu)(t) =
H(t,s)fs,u(s)ds
≤
H(s)fs,u(s)ds
≤
H(s)a(s)gmn(s) +b(s)
ds
=a∗mn+b∗, ∀t∈J, ()
which shows thatTmakes sense. According to Lemma , we have for anyt∈J
(Tu)(t) =
H(t,s)fs,u(s)ds
≤
H(s)fs,u(s)ds.
Hence,
Tu ≤
At the same time, by Lemma and (), we get
(Tu)(t) =
H(t,s)fs,u(s)ds
≥γ(t)
H(s)fs,u(s)ds
≥γ(t)Tu, ∀t∈J. ()
This indicates thatTmapsPmnintoP.
Next, we shall show the complete continuity of the operatorT. Letun,u¯∈Pmn, with
un–u¯ → (n→ ∞); thenlimn→∞un(t) =u¯(t),t∈J. Let
(Tu)(t) =f
t,u(t), for anyt∈J+,u∈Pmn,
(Tu)(t) =
H(t,s)u(s) ds, for anyt∈J+,u∈L(J).
By (H),
ft,un(t)
→ft,u¯(t) (n→+∞), for anyt∈J+. ()
Similar to (), forun,u¯∈Pmn, one has
ft,un(t)
≤a(t)gmn(t) +b(t), f
t,u¯(t)≤a(t)gmn(t) +b(t), ∀t∈J+.
Then one gets
ft,un(t)
–ft,u¯(t)≤a(t)gmn(t) + b(t)
=σ(t)∈L(J). ()
The Lebesgue dominated convergence theorem together with () and () generates
lim n→∞
(Tun)(t) –
Tu¯(t)dt= .
That is to sayT :Pmn→L(J) is continuous. Furthermore, the complete continuity of
the operatorT:L(J)→C(J) can easily be verified by the Arzela-Ascoli theorem and
a standard discussion. Hence, by the property of compound operators we see thatT =
T◦T:Pmn→C(J) is completely continuous.
Lemma ([]) Let E be a Banach space,P⊂E a cone in E.For r> ,define Pr={u∈P:
u ≤r}.Assume that T:Pr→P is a compact map such that Tu=u for u∈∂Pr={u∈P:
u=r}.
(i) Ifu ≤ Tu,∀u∈∂Pr,then
i(T,Pr,P) = .
(ii) Ifu ≥ Tu,∀u∈∂Pr,then
3 Main results
Theorem Let conditions(H)-(H)be satisfied.Furthermore,there exists a constantr>
satisfying
a∗r+b∗<r, ()
here,a∗r andb∗are given in(H).Then the BVP()admits at least two positive solutions
x∗and x∗∗such that <x∗<r<x∗∗.
Proof From Lemma , we know that for anyn>m> , the operatorT mapsPmnintoP
and is completely continuous. Next, we are in a position to show thatT has two distinct positive fixed pointsx∗,x∗∗such that <x∗<r<x∗∗.
From (H) we know there exists a constantr> satisfying
f(t,u) > γ H ,s
c(s) ds
–
c(t)u, ∀t∈J+,u≥r. ()
Let <γ
=mint∈[,]{ψ(t),ψ(t)}< . Choose
r>max
r γ
,r
. ()
Foru∈∂Pr, considering the definition of coneP, we have
u(t)≥γ(t)r≥γ
r>r, ∀t∈ , . ()
So, we get from (), (), () that
(Tu) = H ,s
fs,u(s)ds
> γ H ,s
c(s) ds
– H ,s
c(s)u(s) ds
≥ γ H ,s
c(s) ds
– H ,s
c(s) ds·γ
r=r, ()
i.e.,Tu>u,u∈∂Pr. Therefore, by Lemma ,
i(T,Pr,P) = . ()
By condition (H), there exists a constantr> satisfying
f(t,u) >
H ,s
d(s) ds
–
d(t)r, ∀t∈J+, <u<r. ()
Choose
Foru∈∂Pr, we have
r>r=u ≥γ
r> , ∀t∈J+.
So, we get from () and () that
(Tu)
=
H
,s
fs,u(s)ds
>
H
,s
d(s) ds
–
H
,s
d(s)rds
≥
H
,s
d(s) ds
–
H
,s
d(s) ds·r=r>r, ()
i.e.,Tu>u,u∈∂Pr. Therefore, by Lemma ,
i(T,Pr,P) = . ()
In a similar manner, foru∈∂Pr, by (H), Lemma and (), we get
(Tu)(t) =
H(t,s)fs,u(s)ds
≤
H(s)fs,u(s)ds
≤
H(s)a(s)gr(s) +b(s)
ds
≤a∗r+b∗<r, ∀t∈J, ()
i.e.,Tu<u,u∈∂Pr. Therefore, by Lemma , we get
i(T,Pr,P) = . ()
Now, the additivity of the fixed point index together with (), (), () implies that
i(T,Pr\P˚r,P) = –
and
i(T,Pr\P˚r,P) = .
Hence,Thas two fixed pointsx∗andx∗∗which belong toPr\P˚randPr\P˚r, respectively,
such that <r<x∗<r<x∗∗ ≤r.
4 An example
Example Consider the following second-order singular integral BVPs:
u(t) +u(t) – u(t) + √(–t)(u
+
√u) +√t(–t)= , <t< ,
Conclusion BVP () admits at least two positive solutionsx∗ andx∗∗ satisfying <
x∗< <x∗∗.
Proof Clearly, BVP () has the form (), in whicha(t)≡,b(t)≡–,f(t,u) = √(–t)×
(u+ √u) +
√t(–t),g(s) =s,h(s) =s
. Obviously,f(t,u) permits singularities att= ,
andu= .
Letψandψsatisfy
ψ(t) +ψ(t) – ψ(t) = , ψ() = , ψ() = ,
ψ(t) +ψ(t) – ψ(t) = , ψ() = , ψ() = .
By a simple computation, we get
ψ(t) =
e–e–
et–e–t, ψ(t) =
e–e–
e–t–et–,
ρ=
e–e–, p(t) =e t,
k= –
– e
–+ e–e
–
e–e– = ., k= e
–+
e–e– = .,
k=
– e
–+ e+ e
–
e–e– = ., k= –
e+ e
––
e–e– = .,
k=kk–kk= . > ,
G(t,s) = (e–e–)
(et–e–t)(e–s–es–), ≤t≤s≤,
(es–e–s)(e–t–et–), ≤s≤t≤.
It is not difficult to see that <G(t,s) < sand
H(s) =G(s,s)p(s) +k+k k
G(τ,s)p(s)g(τ) dτ+k+k k
G(τ,s)p(s)h(τ) dτ< s.
For any given r> , we can see that (H) is valid fora(t) = √(–t), g(u) =u+ √u, b(t) =
√t(–t), and it follows from ≤
(e–e–)t( –t)≤γ(t)≤ that
a∗r=
H(t)a(t)gr(t) dt
<
t
√ ( –t)
r+ e–e
–
(e–e–)t( –t)r
dt< +∞, ()
b∗=H(t)b(t) dt< .. Clearly, (H) and (H) hold forc(t) =d(t) =√(–t), and c∗=d∗= .. Taker= ; we have by ()
a∗r+b∗<
t
√ ( –t)
+ e–e
–
(e–e–)t( –t)
dt+ .
= . + . + . = . < =r.
Acknowledgements
The project is supported financially by a Project of Shandong Province Higher Educational Science and Technology Program (J15LI16), NSFC cultivation project of Jining Medical University, the Natural Science Foundation of Shandong Province of China (ZR2015AL002), the National Natural Science Foundation of China (11571296, 11571197, 11371221, 11071141) and the Foundations for Jining Medical College Natural Science (JY2015KJ019, JY2015BS07, JYQ14KJ06).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
QZ and XZ obtained the results in an equally joint work. All the authors read and approved the final manuscript.
Author details
1Center for Information Technology, Jining Medical College, Jining, Shandong 272067, P.R. China.2School of Medical
Information Engineering, Jining Medical College, Rizhao, Shandong 276826, P.R. China.3School of Mathematics,
Liaocheng University, Liaocheng, Shandong 252059, P.R. China.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 8 December 2016 Accepted: 7 June 2017 References
1. Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math.222, 351-363 (2008)
2. Zhang, X, Feng, M, Ge, W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math.233, 1915-1926 (2010)
3. Feng, M, Ge, W: Positive solutions for a class ofm-point singular boundary value problems. Math. Comput. Model.46, 375-383 (2007)
4. Liu, L, Hao, X, Wu, Y: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model.57, 836-847 (2013)
5. Hao, X, Liu, L, Wu, Y: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul.16, 101-111 (2011)
6. Zhang, X, Feng, M, Ge, W: Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl.353, 311-319 (2009)
7. Kong, L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal.72, 2628-2638 (2010)
8. Ma, R, Wang, H: Positive solutions of nonlinear three-point boundary-value problems. J. Math. Anal. Appl.279, 216-227 (2003)
9. Cui, Y, Zou, Y: An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions. Appl. Math. Comput.256, 438-444 (2015)
10. Kang, P, Wei, Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Anal.70, 444-451 (2009)
11. Jiang, J, Liu, L, Wu, Y: Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. Appl. Math. Comput.215, 1573-1582 (2009)
12. Zhang, X: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett.39, 22-27 (2015)
13. Wang, G: Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval. Appl. Math. Lett.47, 1-7 (2015)
14. Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst.4, 134-141 (2010)
15. Yuan, C: Two positive solutions for (n– 1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul.17, 930-942 (2012) 16. Cabada, A, Hamdi, Z: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math.
Comput.228, 251-257 (2014)
17. Zhang, X, Wang, L, Sun, Q: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter. Appl. Math. Comput.226, 708-718 (2014)
18. Zhao, K, Liang, J: Solvability of triple-point integral boundary value problems for a class of impulsive fractional differential equations. Adv. Differ. Equ.2017, 50 (2017)
19. Zhao, K, Liu, J: Multiple monotone positive solutions of integral BVPs for a higher-order fractional differential equation with monotone homomorphism. Adv. Differ. Equ.2016, 20 (2016)
20. Zhao, K: Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments. Adv. Differ. Equ.2015, 382 (2015)
21. Zhao, K: Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions. Bound. Value Probl.2015, 181 (2015)
22. Zhao, K: Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays. Dyn. Syst.30(2), 208-223 (2015)