By constructing a special cone in C1[0,2π]
and the fixed point theorem, this paper investigates second-order singular semipositone periodic boundary value problems with dependence on the first-order derivative and obtains the existence of multiple positive solutions. Further, an example is given to demonstrate the applications of our main results.

1. Introduction

In this paper, we are concerned with the existence of multiple positive solutions for the second-order singular semipositone periodic boundary value problems (PBVP, for short):
(1.1)u′′(t)+a(t)u(t)=f(t,u(t),u′(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π),
where a∈C[0,2π], the nonlinear term f(t,u,v) may be singular at t=0, t=2π, and u=0, also may be negative for some value of t, u, and v.

In recent years, second-order singular periodic boundary value problems have been studied extensively because they can be used to model many systems in celestial mechanics such as the N-body problem (see [1–11] and references therein). By applying the Krasnosel'skii's fixed point theorem, Jiang [5] proves the existence of one positive solution for the second-order PBVP
(1.2)u′′(t)+m2u=f(t,u),t∈[0,2π],u(0)=u(2π),u′(0)=u′(2π),
where m∈(0,1/2) is a constant and f∈C([0,2π]×[0,+∞),[0,+∞)). Zhang and Wang [6] used the same fixed point theorem to prove the existence of multiple positive solutions for PBVP (1.2) when f(t,u) is nonnegative and singular at u=0, not singular at t=0, t=2π. Lin et al. [7] only obtained the existence of one positive solution to PBVP (1.1) when f(t,u,v)=f(t,u),f is semipositone and singular only at u=0.All the above works were done under the assumption that the first-order derivative u′ is not involved explicitly in the nonlinear term f.

Motivated by the works of [5–7], the present paper investigates the existence of multiple positive solutions to PBVP (1.1). PBVP (1.1) has two special features. The first one is that the nonlinearity f may depend on the first-order derivative of the unknown function u, and the second one is that the nonlinearity f(t,u,v) is semipositone and singular at t=0, t=2π, and u=0. We first construct a special cone different from that in [5–7] and then deduce the existence of multiple positive solutions by employing the fixed point theorem on a cone. Our results improve and generalize some related results obtained in [5–7].

A map u∈C1[0,2π]∩C2(0,2π) is said to be a positive solution to PBVP(1.1) if and only if u satisfies PBVP (1.1) and u(t)>0 for t∈[0,2π].

The contents of this paper are distributed as follows. In Section 2, we introduce some lemmas and construct a special cone, which will be used in Section 3. We state and prove the existence of at least two positive solutions to PBVP (1.1) in Section 3. Finally, an example is worked out to demonstrate our main results.

2. Some Preliminaries and Lemmas

Define the set functions
(2.1)Λ={a∈C[0,2π]:a≻0,t∈[0,2π],(∫02πapdt)1/p≤K(2q)forsomep≥1},
where q is the conjugate exponent of p,
(2.2)K(q)={1q(2π)2/q(22+q)1-2/q(Γ(1/q)Γ(1/2+1/q))2,1≤q<∞,2π,q=∞,
where Γ is the Gamma function.

Given a∈Λ, let G(t,s) be the Green function for the equation
(2.3)u′′+a(t)u(t)=0,t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π).

Now, the following Lemma follows immediately from the paper [7].

Lemma 2.1.

G(t,s) has the following properties:

G(t,s) is continuous in t and s for all t,s∈[0,2π];

G(t,s)>0 for all (t,s)∈[0,2π]×[0,2π],G(0,s)=G(2π,s) and ∂G/∂t|(0,s)=∂G/∂t|(2π,s);

denote l1=min0≤t,s≤2πG(t,s) and l2=max0≤t,s≤2πG(t,s), then l2>l1>0;

there exist functions h,H∈C2[0,2π] such that
(2.4)G(t,s)={(α+1)H(t)h(s)+(β-1)h(t)H(s)+cH(t)H(s)+dh(t)h(s),0≤s≤t≤2π,αH(t)h(s)+βh(t)H(s)+cH(t)H(s)+dh(t)h(s),0≤t≤s≤2π,

where α,β,c,d are constants, H,h are independent solutions of the linear differential equation u′′+a(t)u(t)=0, and H′(t)h(t)-h′(t)H(t)=1;

Gt′(t,s) is bounded on [0,2π]×[0,2π].

Denote l3=max0≤t,s≤2π|Gt′(t,s)|, then l3>0.

Remark 2.2.

Using paper [5], we can get G(t,s) when a(t)≡m2 and m∈(0,1/2), obtaining
(2.5)G(t,s)={sinm(t-s)+sinm(2π-t+s)2m(1-cos2mπ),0≤s≤t≤2π,sinm(s-t)+sinm(2π-s+t)2m(1-cos2mπ),0≤t≤s≤2π,Gt′(t,s)={cosm(t-s)-cosm(2π-t+s)2(1-cos2mπ),0≤s≤t≤2π,-cosm(s-t)+cosm(2π-s+t)2(1-cos2mπ),0≤t<s≤2π,l1=sin2mπ2m(1-cos2mπ),l2=sinmπm(1-cos2mπ),l3=12.

Let E={u∈C1[0,2π]:u(0)=u(2π),u′(0)=u′(2π)} with norm ∥u∥=max{∥u∥0,∥u′∥0}, where ∥u∥0=maxt∈[0,2π]|u(t)|. Then (E,∥·∥) is a Banach space. Let σ=:min{l1/l2,l1/l3},L=:l3/l1, from Lemma 2.1, we know that σ,L are both constants and 0<σ<1, L>0.

It is easy to conclude that K is a cone of E and Ωr is an open set of E.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let E be a Banach space and P a cone in E. Suppose Ω1 and Ω2 are bounded open sets of E such that θ∈Ω1⊂Ω1¯⊂Ω2 and suppose that A:P∩(Ω2¯∖Ω1)→P is a completely continuous operator such that

infu∈P∩∂Ω1∥Au∥>0 and u≠λAu for u∈P∩∂Ω1,λ≥1; u≠λAu for u∈P∩∂Ω2,0<λ≤1,or

infu∈P∩∂Ω2∥Au∥>0 and u≠λAu for u∈P∩∂Ω2,λ≥1; u≠λAu for u∈P∩∂Ω1,0<λ≤1.

Then A has a fixed point in P∩(Ω2¯∖Ω1).

For convenience, let us list some conditions for later use.

a(t)∈Λ,f:(0,2π)×(0,+∞)×R→R is continuous and there exists a constant M>0 such that
(2.7)0≤f(t,u,v)+M≤g(t)h(u,v),∀(t,u,v)∈(0,2π)×(0,+∞)×R,

where g∈C((0,2π),R+),h∈C((0,+∞)×R,R+), and 0<∫02πg(t)dt<+∞;

there exist r1>σ-12πMl2 and a(t)∈L[0,2π] with ∫02πa(t)dt>(≥)r1l1-1 such that
(2.8)M+f(t,u,v)≥(>)a(t),∀t∈(0,2π),u∈(0,r1],v∈[-(Lr1+2πMl3),(Lr1+2πMl3)];

there exists R1>r1 such that
(2.9)max{l2,l3}∫02πg(t)dt<R1M0-1,

where M0=:max{h(u,v):u∈[σR1-2πMl2,R1],v∈[-(LR1+2πMl3),(LR1+2πMl3)]};

there exists [α*,β*]⊂(0,2π) such that
(2.10)limu→+∞f(t,u,v)u=+∞uniformly with respect to t∈[α*,β*],v∈R.

3. Main ResultsTheorem 3.1.

Assume that conditions (H_{0})–(H_{3}) are satisfied, then PBVP (1.1) has at least two positive solutions u1,u2∈C1[0,2π]∩C2(0,2π) such that r1<∥u1+Mω∥<R1<∥u2+Mω∥, where ω(t)=:∫02πG(t,s)ds.

Proof.

We consider the following PBVP:
(3.1)u′′(t)+a(t)u(t)=f(t,u(t)-Mω(t),u′(t)-Mω′(t))+M,t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π).

It is easy to see that if u∈C1[0,2π]∩C2(0,2π) and r1<∥u∥<R1 is a positive solution of PBVP (3.1) with u(t)>Mω(t) for t∈[0,2π], then x(t)=u(t)-Mω(t) is a positive solution of PBVP (1.1) and r1<∥x+Mω∥<R1.

As a result, we will only concentrate our study on PBVP (3.1).

Define an operator T:K∖{θ}→E by
(3.2)(Tu)(t)=:∫02πG(t,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds,∀t∈[0,2π],
where G(t,s) is the Green function to problem (2.3).

(1) We first show that T:K∩(ΩR¯∖Ωr1)→K is completely continuous for any R>r1.

For any u∈K∩(ΩR¯∖Ωr1), from (H_{1}), we have u(t)-Mω(t)≥σr1-2πMl2>0. So, by Lemma 2.1 and (3.2),
(3.3)(Tu)(0)=(Tu)(2π),(Tu)′(0)=(Tu)′(2π),(3.4)(Tu)(t)=∫02πG(t,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≥l1l2l2∫02πG(t,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≥l1l2maxτ∈[0,2π]∫02πG(τ,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds=l1l2‖Tu‖0≥σ‖Tu‖0,∀t∈[0,2π],(3.5)|(Tu)′(t)|=|∫02πGt′(t,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds|≤∫02π|Gt′(t,s)|[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≤l3l1l1∫02π[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≤l3l1∫02πG(τ,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds=l3l1(Tu)(τ),∀t,τ∈[0,2π].

From (3.5), we have (Tu)(t)≥(l1/l3)maxτ∈[0,2π]|(Tu)′(τ)|≥σ∥(Tu)′∥0. Therefore, (Tu)(t)≥σ∥Tu∥, |(Tu)′(t)|≤L∥Tu∥,for allt∈[0,2π],that is,T:K∩(ΩR¯∖Ωr1)→K.

Assume that un,u*∈K∩(ΩR¯∖Ωr1) with ∥un-u*∥→0,n→+∞. Thus, from (H_{1}), we have
(3.6)limn→+∞f(t,un(t)-Mω(t),un′(t)-Mω′(t))=f(t,u*(t)-Mω(t),u*′(t)-Mω′(t)),t∈(0,2π),|f(t,un(t)-Mω(t),un′(t)-Mω′(t))|≤M+M1g(t),t∈(0,2π),[M+M1g(t)]∈L[0,2π],
where M1=:max{h(u,v):u∈[σr1-2πMl2,R],v∈[-(LR+2πMl3),(LR+2πMl3)]}.

Lemma 2.1 and Lebesgue-dominated convergence theorem guarantee that
(3.7)‖Tun-Tu*‖≤max{l2,l3}∫02π|f(t,un(t)-Mω(t),un′(t)-Mω′(t))-f(t,u*(t)-Mω(t),u*′(t)-Mω′(t))|dt⟶0,n⟶+∞.
So, T:K∩(ΩR¯∖Ωr1)→K is continuous.

For any bounded D⊂K∩(ΩR¯∖Ωr1), From Lemma 2.1 and (H_{1}), for any u∈D, we have
(3.8)‖Tu‖≤max{l2,l3}∫02π[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≤max{l2,l3}∫02πg(s)h(u(s)-Mω(s),u′(s)-Mω′(s))ds≤max{l2,l3}M1∫02πg(s)ds,
which means the functions belonging to {(TD)(t)} and the functions belonging to {(TD)′(t)} are uniformly bounded on [0,2π]. Notice that
(3.9)|(Tu)′(t)|≤l3M1∫02πg(s)ds,t∈[0,2π],u∈D,
which implies that the functions belonging to {(TD)(t)} are equicontinuous on [0,2π]. From Lemma 2.1, we have
(3.10)Gt′(t,s)={(α+1)H′(t)h(s)+(β-1)h′(t)H(s)+cH′(t)H(s)+dh′(t)h(s),0≤s≤t≤2π,αH′(t)h(s)+βh′(t)H(s)+cH′(t)H(s)+dh′(t)h(s),0≤t<s≤2π,
where α,β,c,d are constants, h,H∈C2[0,2π] are independent solutions of the linear differential equation u′′+a(t)u(t)=0, and H′(t)h(t)-h′(t)H(t)=1.

It is easy to see that Gt′(t,s) is continuous in t and s for 0≤s≤t≤2π and 0≤t<s≤2π. So, for any t1,t2∈[0,2π],t1<t2, we have
(3.11)∫0t1|Gt′(t1,s)-Gt′(t2,s)|g(s)ds⟶0,ast1⟶t2-,ort2⟶t1+,∫t1t2|Gt′(t1,s)-Gt′(t2,s)|g(s)ds≤2l3∫t1t2g(s)ds⟶0,ast1⟶t2-,ort2⟶t1+,∫t22π|Gt′(t1,s)-Gt′(t2,s)|g(s)ds⟶0,ast1⟶t2-,ort2⟶t1+.

Therefore,
(3.12)|(Tu)′(t1)-(Tu)′(t2)|=|∫02π[Gt′(t1,s)-Gt′(t2,s)][f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds|≤M1∫02π|Gt′(t1,s)-Gt′(t2,s)|g(s)ds=M1{∫0t1|Gt′(t1,s)-Gt′(t2,s)|g(s)ds+∫t1t2|Gt′(t1,s)-Gt′(t2,s)|g(s)ds+∫t22π|Gt′(t1,s)-Gt′(t2,s)|g(s)ds}⟶0,ast1⟶t2-ort2⟶t1+.
Thus, the functions belonging to {TD′(t)} are equicontinuous on [0,2π]. By Arzela-Ascoli theorem, TD is relatively compact in C1[0,2π].

Hence, T:K∩(ΩR¯∖Ωr1)→K is completely continuous for any R>r1.

(2) We now show that
(3.13)infu∈K∩∂Ωr1‖Tu‖>0,u≠λTu,∀u∈K∩∂Ωr1,λ≥1.

For any u∈K∩∂Ωr1, we have
(3.14)0<σr1-2πMl2≤u(t)-Mω(t)≤r1,|u′(t)-Mω′(t)|≤|u′(t)|+M|ω′(t)|≤Lr1+2πMl3,∀t∈[0,2π].
From (H_{1}) and (3.2),
(3.15)(Tu)(t)=∫02πG(t,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≥l1∫02πa(s)ds>l1r1l1-1=r1>0.

Suppose that there exist λ0≥1 and u0∈K∩∂Ωr1 such that u0=λ0Tu0, that is, for t∈[0,2π],
(3.16)u0(t)≥(Tu0)(t)=∫02πG(t,s)[f(s,u0(s)-Mω(s),u0′(s)-Mω′(s))+M]ds≥l1∫02πa(s)ds>l1r1l1-1=r1.
This is in contradiction with u0∈K∩∂Ωr1 and (3.13) holds.

(3) Next, we show that
(3.17)u≠λTu∀u∈K∩∂ΩR1,0<λ≤1.

Suppose this is false, then there exist λ0∈(0,1] and u0∈K∩∂ΩR1 with u0=λ0Tu0, that is, for t∈[0,2π], we have
(3.18)u0(t)≤(Tu0)(t)=∫02πG(t,s)[f(s,u0(s)-Mω(s),u0′(s)-Mω′(s))+M]ds,|u0′(t)|=λ0|(Tu0)′(t)|≤∫02π|Gt′(t,s)|[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds.

From (H_{2}), we have
(3.19)0<σR1-2πMl2≤u0(t)-Mω(t)≤R1,|u0′(t)-Mω′(t)|≤|u0′(t)|+M|ω′(t)|≤LR1+2πMl3,∀t∈[0,2π].
Therefore, by (3.18), (3.19), and (H_{2}), it follows that
(3.20)u0(t)≤l2∫02πg(s)h(u0(s)-Mω(s),u0′(s)-Mω′(s))ds≤l2M0∫02πg(s)ds<R1,∀t∈[0,2π],|u0′(t)|≤l3∫02πg(s)h(u0(s)-Mω(s),u0′(s)-Mω′(s))ds≤l3M0∫02πg(s)ds<R1,∀t∈[0,2π].
Thus, ∥u∥<R1. This is in contradiction with u0∈K∩∂ΩR1 and (3.17) holds.

(4) Choose N*=(1+2πMl2)[σl1(β*-α*)]-1+1. From (H_{3}), there exists R2>max{R1,1} such that
(3.21)f(t,u,v)≥N*u,∀u≥R2,v∈R,t∈[α*,β*].

Now, we show that
(3.22)infu∈K∩∂ΩR‖Tu‖>0,u≠λTu,∀u∈K∩∂ΩR,λ≥1,
where R=(R2+2πMl2)σ-1.

For any u∈K∩∂ΩR, we have
(3.23)u(t)-Mω(t)≥σR-2πMl2=R2,∀t∈[0,2π].
This and (3.21) together with (3.2) imply
(3.24)(Tu)(t)=∫02πG(t,s)[f(s,u(s)-Mω(s),u′(s)-Mω′(s))+M]ds≥l1∫α*β*[f(s,u(s)-Mω(s),u′(s)-Mω′(s))]ds≥l1N*R2(β*-α*)>0.

Suppose that there exist λ0≥1 and u0∈K∩∂ΩR such that u0=λ0Tu0,then, for t∈[α*,β*], we have
(3.25)u0(t)≥(Tu0)(t)=∫02πG(t,s)[f(s,u0(s)-Mω(s),u0′(s)-Mω′(s))+M]ds≥l1∫α*β*[f(s,u(s)-Mω(s),u′(s)-Mω′(s))]ds≥l1N*R2(β*-α*)>(R2+2πMl2)σ-1=R.
This is in contradiction with u0∈K∩∂ΩR and (3.22) holds.

Now, (3.13), (3.17), (3.22), and Lemma 2.3 guarantee that T has two fixed points u1∈K∩(ΩR1∖Ωr1¯), u2∈K∩(ΩR∖ΩR1¯) with r1<∥u1∥1<R1<∥u2∥1<R. Clear, PBVP (3.1) has at least two positive solutions u1,u2∈C1[0,2π]∩C2(0,2π).

Remark 3.2.

From the proof of Theorem 3.1, when f(t,u,v) is nonnegative (i.e., M=0 in (H_{0})), Theorem 3.1 still holds.

Corollary 3.3.

Assume that (H_{0})–(H_{2}) hold, then PBVP (1.1) has at least one positive solution u(t) such that r1<∥u+Mω∥<R1, where ω(t)=:∫02πG(t,s)ds.

Corollary 3.4.

Assume that (H_{0}) and (H_{3}) hold, and

(H_{4}) there exist R1>σ-12πMl2 such that
(3.26)max{l2,l3}∫02πg(t)dt<R1M0-1,
where M0=:max{h(u,v):u∈[σR1-2πMl2,R1]andv∈[-(LR1+2πMl3),(LR1+2πMl3)]}. Then PBVP (1.1) has at least one positive solution u(t) such that ∥u+Mω∥>R1, where ω(t)=:∫02πG(t,s)ds.

Example 3.5.

Consider the following second-order singular semipositone PBVP:
(3.27)u′′+116u=u9/4+(u′)2+18πut(2π-t)-330πcost12,t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π).

4. Conclusion

PBVP (3.27) has at least two positive solutions u1,u2∈C1[0,2π]∩C2(0,2π) and u1(t),u2(t)>0 for t∈[0,2π].

To see this, we will apply Theorem 3.1 with m=1/4, f(t,u,v)=((u9/4+v2+1)/8πut(2π-t))-(3/30π)cos(t/12), g(t)=1/t(2π-t), h(u,v)=(u9/4+v2+1)/8πu, M=1/20π.

From Remark 2.2, it is easy to see that l1=2, l2=22, l3=1/2, σ=2/2, and L=1/4.

By simple computation, we easily get 0≤f(t,u,v)+M≤g(t)h(u,v) and ∫02πg(t)dt=π. So (H_{0}) holds.

Taking r1=1/2, a(t)=1/4πt(2π-t), then σ-12πMl2=2·2π·(1/20π)·22=2/5<r1, ∫02πa(t)dt=1/4=r1l1-1 and for any t∈(0,2π), u∈(0,1/2], v∈[-7/40,7/40],
(4.1)u9/4+v2+18πut(2π-t)-330πcost12+120π≥1/29/4+14πt(2π-t)-330π+120π≥1/29/44(π)2-330π+120π+14πt(2π-t)>14πt(2π-t).
Thus, (H_{1}) holds.

Taking R1=4, then for u∈[(9/5)2,4],|v|≤21/20, we have
(4.2)M0≤5722π(29/2+(2120)2+1)<5722π(23+2+1)=65362π.
So, M0max{l2,l3}∫02πg(t)dt=22πM0<65/18<4=R1. That is, (H_{2}) holds.

Let [α*,β*]=[π/2,π], then it is easy to check that (H_{3}) holds.

Thus all the conditions of Theorem 3.1 are satisfied, so PBVP (3.27) has at least two positive solutions u1,u2∈C1[0,2π]∩C2(0,2π) and u1(t),u2(t)>0 for t∈[0,2π].

Acknowledgments

Research supported by the Project of Shandong Province Higher Educational Science and Technology Program (J09LA08) and Reward Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2011SF022), China.

GordonW. B.Conservative dynamical systems involving strong forcesMajerP.TerraciniS.Periodic solutions to some problems of n-body typeZhangS.Multiple closed orbits of fixed energy for n-body-type problems with gravitational potentialsJiangD.ChuJ.O'ReganD.AgarwalR. P.Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forcesJiangD.On the existence of positive solutions to second order periodic BVPSZhangZ.WangJ.On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equationsLinX.LiX.JiangD.Positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forcesSunJ.LiuY.Multiple positive solutions of singular third-order periodic boundary value problemHaoX.LiuL.WuY.Existence and multiplicity results for nonlinear periodic boundary value problemsLiuB.LiuL.WuY.Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problemMaR.XuJ.HanX.Global structure of positive solutions for superlinear second-order periodic boundary value problemsGuoD. J.LakshmikanthamV.