JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 295209 10.1155/2012/295209 295209 Research Article Multiple Positive Solutions for Singular Semipositone Periodic Boundary Value Problems with Derivative Dependence Lu Huiqin Liu Yansheng 1 School of Mathematical Sciences Shandong Normal University Shandong Jinan 250014 China sdnu.edu.cn 2012 24 7 2012 2012 29 03 2012 19 05 2012 2012 Copyright © 2012 Huiqin Lu. 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.

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 aC[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  and references therein). By applying the Krasnosel'skii's fixed point theorem, Jiang  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 fC([0,2π]×[0,+),[0,+)). Zhang and Wang  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.  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 , 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  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 .

A map uC1[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={aC[0,2π]:a0,t[0,2π],(02πapdt)1/pK(2q)forsomep1}, 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))2,1q<,2π,q=, where Γ is the Gamma function.

Given aΛ, 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 .

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=min0t,s2πG(t,s) and l2=max0t,s2πG(t,s), then l2>l1>0;

there exist functions h,HC2[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),0st2π,αH(t)h(s)+βh(t)H(s)+cH(t)H(s)+dh(t)h(s),0ts2π,

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=max0t,s2π|Gt(t,s)|, then l3>0.

Remark 2.2.

Using paper , 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),0st2π,sinm(s-t)+sinm(2π-s+t)2m(1-cos2mπ),0ts2π,Gt(t,s)={cosm(t-s)-cosm(2π-t+s)2(1-cos2mπ),0st2π,-cosm(s-t)+cosm(2π-s+t)2(1-cos2mπ),0t<s2π,l1=sin2mπ2m(1-cos2mπ),l2=sinmπm(1-cos2mπ),l3=12.

Let E={uC1[0,2π]:u(0)=u(2π),u(0)=u(2π)} with norm u=max{u0,u0}, where u0=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.

Define (2.6)K={uE:    u(t)σu,|u(t)|Lu,t[0,2π]},Ωr={uE:    u<r},r>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 θΩ1Ω1¯Ω2 and suppose that A:P(Ω2¯Ω1)P is a completely continuous operator such that

infuPΩ1Au>0 and uλAu for uPΩ1,  λ1; uλAu for uPΩ2,0<λ1,or

infuPΩ2Au>0 and uλAu for uPΩ2,  λ1; uλAu for uPΩ1,0<λ1.

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,+)×RR is continuous and there exists a constant M>0 such that (2.7)0f(t,u,v)+Mg(t)h(u,v),(t,u,v)(0,2π)×(0,+)×R,

where gC((0,2π),R+),  hC((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*],vR.

3. Main Results Theorem 3.1.

Assume that conditions (H0)–(H3) are satisfied, then PBVP (1.1) has at least two positive solutions u1,u2C1[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 uC1[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 uK(ΩR¯Ωr1), from (H1), 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]dsl1l2l202πG(t,s)[f(s,u(s)-Mω(s),u(s)-Mω(s))+M]dsl1l2maxτ[0,2π]02πG(τ,s)[f(s,u(s)-Mω(s),u(s)-Mω(s))+M]ds=l1l2Tu0σTu0,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]dsl3l1l102π[f(s,u(s)-Mω(s),u(s)-Mω(s))+M]dsl3l102πG(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)σTu, |(Tu)(t)|LTu,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 (H1), 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))|dt0,n+. So, T:K(ΩR¯Ωr1)K is continuous.

For any bounded DK(ΩR¯Ωr1), From Lemma 2.1 and (H1), for any uD, we have (3.8)Tumax{l2,l3}02π[f(s,u(s)-Mω(s),u(s)-Mω(s))+M]dsmax{l2,l3}02πg(s)h(u(s)-Mω(s),u(s)-Mω(s))dsmax{l2,l3}M102πg(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)|l3M102πg(s)ds,      t[0,2π],uD, 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),0st2π,αH(t)h(s)+βh(t)H(s)+cH(t)H(s)+dh(t)h(s),0t<s2π, where α,β,c,d are constants, h,HC2[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 0st2π and 0t<s2π. So, for any t1,  t2[0,2π],t1<t2, we have (3.11)0t1|Gt(t1,s)-Gt(t2,s)|g(s)ds0,ast1t2-,ort2t1+,t1t2|Gt(t1,s)-Gt(t2,s)|g(s)ds2l3t1t2g(s)ds0,ast1t2-,ort2t1+,t22π|Gt(t1,s)-Gt(t2,s)|g(s)  ds0,ast1t2-,ort2t1+.

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|M102π|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,ast1t2-ort2t1+. 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)infuKΩr1Tu>0,uλTu,uKΩr1,λ1.

For any uKΩr1, we have (3.14)0<σr1-2πMl2u(t)-Mω(t)r1,|u(t)-Mω(t)||u(t)|+M|ω(t)|Lr1+2πMl3,t[0,2π]. From (H1) and (3.2), (3.15)(Tu)(t)=02πG(t,s)[f(s,u(s)-Mω(s),u(s)-Mω(s))+M]dsl102πa(s)ds>l1r1l1-1=r1>0.

Suppose that there exist λ01 and u0KΩr1 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]dsl102πa(s)ds>l1r1l1-1=r1. This is in contradiction with u0KΩr1 and (3.13) holds.

(3) Next, we show that (3.17)uλTuuKΩR1,0<λ1.

Suppose this is false, then there exist λ0(0,1] and u0KΩR1 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 (H2), we have (3.19)0<σR1-2πMl2u0(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 (H2), it follows that (3.20)u0(t)l202πg(s)h(u0(s)-Mω(s),u0(s)-Mω(s))dsl2M002πg(s)ds<R1,t[0,2π],|u0(t)|l302πg(s)h(u0(s)-Mω(s),u0(s)-Mω(s))dsl3M002πg(s)ds<R1,t[0,2π]. Thus, u<R1. This is in contradiction with u0KΩR1 and (3.17) holds.

(4) Choose N*=(1+2πMl2)[σl1(β*-α*)]-1+1. From (H3), there exists R2>max{R1,1} such that (3.21)f(t,u,v)N*u,uR2,vR,t[α*,β*].

Now, we show that (3.22)infuKΩRTu>0,  uλTu,uKΩR,λ1, where R=(R2+2πMl2)σ-1.

For any uKΩR, 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]dsl1α*β*[f(s,u(s)-Mω(s),u(s)-Mω(s))]dsl1N*R2(β*-α*)>0.

Suppose that there exist λ01 and u0KΩR 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]dsl1α*β*[f(s,u(s)-Mω(s),u(s)-Mω(s))]dsl1N*R2(β*-α*)>(R2+2πMl2)σ-1=R. This is in contradiction with u0KΩR and (3.22) holds.

Now, (3.13), (3.17), (3.22), and Lemma 2.3 guarantee that T has two fixed points u1K(ΩR1Ωr1¯), u2K(ΩRΩR1¯) with r1<u11<R1<u21<R. Clear, PBVP (3.1) has at least two positive solutions u1,u2C1[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 (H0)), Theorem 3.1 still holds.

Corollary 3.3.

Assume that (H0)–(H2) 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 (H0) and (H3) hold, and

(H4) 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,u2C1[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 0f(t,u,v)+Mg(t)h(u,v) and 02πg(t)dt=π. So (H0) 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+120π1/29/4+14πt(2π-t)-330π+120π1/29/44(π)2-330π+120π+14πt(2π-t)>14πt(2π-t). Thus, (H1) holds.

Taking R1=4, then for u[(9/5)2,4],|v|21/20, we have (4.2)M05722π(29/2+(2120)2+1)<5722π(23+2+1)=65362π. So, M0max{l2,l3}02πg(t)dt=22πM0<65/18<4=R1. That is, (H2) holds.

Let [α*,β*]=[π/2,π], then it is easy to check that (H3) holds.

Thus all the conditions of Theorem 3.1 are satisfied, so PBVP (3.27) has at least two positive solutions u1,  u2C1[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.

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