Positive Solutions for Impulsive Equations of Third Order in Banach Space

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doi:10.1155/2010/185701

Research Article

Positive Solutions for Impulsive Equations of

Third Order in Banach Space

Jingjing Cai

Department of Mathematics, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Jingjing Cai,[email protected]

Received 4 September 2010; Accepted 30 November 2010

Academic Editor: John Graef

Copyrightq2010 Jingjing Cai. 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.

Using the fixed-point theorem, this paper is devoted to study the multiple and single positive solutions of third-order boundary value problems for impulsive diﬀerential equations in ordered Banach spaces. The arguments are based on a specially constructed cone. At last, an example is given to illustrate the main results.

1. Introduction

The purpose of this paper is to establish the existence of positive solutions for the following third-order three-point boundary value problemsBVP, for shortin Banach spaceE

−xt λf1

t, xt, yt, t∈0,1\ {t1, t2, . . . , tm},

−yt μf2

t, xt, yt, t∈0,1\ {t1, t2, . . . , tm},

Δxtk −I1,kxtk, Δytk −I2,k

ytk

, k1,2, . . . , m,

x0 x0 θ, x1−αxηθ, y0 y0 θ, y1−αyηθ,

1.1

wherefi ∈C0,1×P×P, P,Ii,k∈CP, P,i1,2,k1,2, . . . , m.Δxtk xt_k−xt−_k,

Δytk yt_k−yt−_k, μ >0,λ >0.θis the zero element ofE.

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concern mainly about the existence of positive solutions for the cases in which the spaces are real and the equations have no parameters. And many authors consider nonlinear term have same linearity. In this paper, we consider the existence of solutions when the nonlinear terms have diﬀerent properties, the space is abstract and the equations have two diﬀerent parameters.

In3, Guo et al. studied the following nonlinear three-point boundary value problem:

ut atfut 0,

u0 u0 0, u1 αuη, 1.2

where a ∈ C0,1,0, ∞, f ∈ C0, ∞,0, ∞. The authors obtained at least one positive solutions of BVP1.2by using fixed-point theorem whenfis sublinear or suplinear. In 8, Yao and Feng used the upper and lower solutions method proved some existence results for the following third-order two-point boundary value problem

ut ft, ut 0, 0≤t≤1,

u0 u0 u1 0. 1.3

Inspired by the above work, the aim of this paper is to establish some simple criteria for the existence of nontrivial solutions for BVP1.1under some weaker conditions. The new features of this paper mainly include the following aspects. Firstly, we consider the system

1.1in abstract space while3,8talk about equations in real spaceE R. Secondly, we obtained the positive solutions when the two parameters have diﬀerent ranges. Thirdly,f1 andf2in system1.1may have diﬀerent properties. Fourthly,fi i1,2in system1.1not

only containsx, y but alsot, which is much more complicated. Finally, the main technique used here is the fixed-point theory and a special cone is constructed to study the existence of nontrivial solutions.

We recall some basic facts about ordered Banach spacesE. The coneP inEinduces a partial order on E, that is, x ≤ y if and only if y− x ∈ P, P is said to be normal if there exists a positive constant N such thatθ ≤ x ≤ y impliesx ≤ Ny, without loss of generality, suppose, in present paper, the normal constantN1.α·denotes the measure of noncompactnesscf.10.

Some preliminaries and a number of lemmas to the derivation of the main results are given inSection 2, then the proofs of the theorems are given in Section 3, followed by an example, inSection 4, to demonstrate the validity of our main results.

2. Preliminaries and Lemmas

In this paper we will consider the Banach spaceE,·, denoteJ 0,1and PC2J, E {x| x∈CJ, E,xis continuous att /tkandxis left continuous atttk, the right limitxt_k

exists,k 1,2, . . . , m}. For anyx ∈PC2J, Ewe definex1 sup_t∈Jxtandx, y2

x1 y1forx, y∈PC2J, E×PC2J, E.

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Afi ∈ C0,1×P ×P, P,Ii,k ∈ CP, P,i 1,2,k 1,2, . . . , m. For anyt ∈ 0,1

and r > 0,ft, Pr, Pr {ft, u, v : u, v ∈ Pr} is relatively compact in E, where

Pr {x∈P | x ≤r}.

Lemma 2.1. Assume thatαη /1, then for anyy∈C0,1, the following boundary value problem:

−ut yt, t∈0,1\ {t1, t2, . . . , tm},

Δutk −Ikutk, k1,2, . . . , m,

u0 u0 θ, u1−αuηθ

2.1

has a unique solution

ut ₁

0

Gt, sysds

m

k1

Gt, tkIkutk, 2.2

where

Gt, s 1

21−αη ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2ts−s2₁₋_αη _t2_s_α₋₁_, _s_≤_min_{η, t}_,

t2₁₋_αη _t2_s_α₋₁_, _t_≤_s_≤_η,

2ts−s2₁₋_αη _t2_αη₋_s_, _η_≤_s_≤_t,

t21−s, maxη, t≤s.

2.3

Proof. The proof is similar to Lemma 2.2 in3, we omit it.

Lemma 2.2see3. Assume that0 < η < 1and1 < α < 1/η. Then0 ≤Gt, s≤ gsfor any

t, s∈0,1×0,1, wheregs 1 α/1−αηs1−s,s∈0,1.

Lemma 2.3see3. Let0 < η < 1and 1 < α < 1/η, then for anyt, s ∈ η/α, η×0,1,

Gt, s≥σgs, where

0< σ η

2

2α2₁ _αmin{α−1,1}<1. 2.4

In the paper, we define coneKas follows:

Kx∈PC2J, E|xt≥θ, xt≥σxs, t∈η α, η

, s∈0,1. 2.5

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By2.11–2.13and Lebesgue-dominated convergence theorem

_T₁_x_n_{, y}_n₋_T₁_x₀_{, y}₀

1−→0 n−→ ∞. 2.14

SoT1is continuous. Similarly,T2is continuous. It follows thatTis continuous.

Next we proveT is compact. LetV {xn, yn} ⊂K×Kbe bounded,V1 {xn}and

V2 {yn}. Letxn, yn2 ≤rfor somer >0, thenxn1 ≤r,yn1 ≤ r. It is easy to see that

{T1xn, ynt}is equicontinuous. By conditionAwe have

αT1Vt α

1

0

Gt, sf1

s, xns, yns

ds

m

k1

Gt, tkI1,kxntk:xn∈V1, yn ∈V2

≤2

1

0

αGt, sf1s, V1s, V2s

m

k1

αGt, tkI1,kV1tk

0

2.15

which implies thatαT1V 0. So, αTV 0, it follows thatT is compact. The lemma is proved.

In this paper, denote

f_iβ lim sup

x y→β

max

t∈0,1

ft, x, y

x y , fi,βxlim infy→βt∈minη/α,η

ft, x, y

x y ,

ψfi

β _{lim sup}

x y→β

max

t∈0,1

ψft, x, y

x y ,

ψfi

β lim inf

x y→βt∈minη/α,η

ψft, x, y

x y .

Ii,βk lim inf x →β

Ii,kx

x , I

β

ik lim sup x →β

Ii,kx

x , k1,2, . . . , m.

2.16

whereβ0 orβ ∞,ψ∈P∗ {ψ∈E∗:ψx≥θ, ∀x∈P}andψ1.P∗is a dual cone ofP.

We list the assumptions:

H1 ψf10> m1,ψf2∞> m2, wherem1, m2 ∈0, ∞;

H2f_i0 < m3,ψf1_∞> m4, Ii,0k 0,i1,2, wherem3, m4>0 andm3m4;

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Sinceψf10 > m1, there existε1 > 0 and 0 < R1 < 1 such thatψf1t, u, v ≥ m1

ε1u vfor 0≤ u v ≤R1andt∈η/α, η. LetΩ2 {x, y∈K×K:x, y2< R1}. Then for anyx, y∈∂Ω2, byH1and the definition ofψ, we obtain

_T₁_{x, y}

1≥ψ

T1

x, yη≥λ η

η/α

Gη, sψf1

t, xs, ysds

≥m1 ε1λσ

_η

η/α

Gη, sx₁ y₁ds

R1m1 ε1λσ

η

η/α

Gη, sds.

3.6

By3.6andH

Tx, y₂≥T1x, y₁≥R1m1 ε1λσ

_η

η/α

Gη, sds

> R1x, y₂,

x, y∈∂Ω2,

3.7

Similarly, by ψf2∞ > m2, there existε2 > 0 and R2 > 1 such thatψf2t, u, v ≥

m2 ε2u vfort∈η/α, ηandu, v∈P with 0≤ u v ≤R2. LetΩ3{x, y∈

K×K:x, y2< R2}. Then for anyx, y∈∂Ω3,

T2x, y₁≥R2μm2 ε2σ

_η

η/α

Gη, sds. 3.8

So we have by3.8andH

Tx, y₂≥T2x, y₁> R2 x, y₂, for any

x, y∈∂Ω3. 3.9

By3.5,3.7,3.9andLemma 2.4we get that BVP1.1has at least two positive solutions withx1, y12<1<x2, y22.

Corollary 3.2. Assume that (A) and the following condition hold, then the conclusion ofTheorem 3.1

also holds.

ψf1

0 > m1, ψ

f2

∞> m2, wherem1, m2∈0, ∞. 3.10

Theorem 3.3. Assume that (A) and (H2) hold, then BVP1.1has at least one positive solution when

λ∈a2, a1andμ∈0, a1.

Proof. ByLemma 2.6, we see that T : K×K → K×Kis completely continuous. ByH2, there existsr1>0,ε3>0,ε >0 such that fori1,2,

_f_i

t, xt, yt≤m3−ε3

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By the definition ofT1we get

_T₁_{x, y}

1≥ψ

T1

x, yη≥λ _η

η/α

Gη, sψf1

t, xs, ysds

≥λm4 ε4

_η

η/α

Gη, sxs ysds

≥λm4 ε4σ

_η

η/α

Gη, sxu yuds.

3.18

So

_T₁_{x, y}

1≥λm4 ε4σ

η

η/α

Gη, sx₁ y₁dsR1λm4 ε4σ

η

η/α

Gη, sds.

3.19

Hence

_T_{x, y}

2≥T1x, y1≥R1λm4 ε4σ

η

η/α

Gη, sds≥R1x, y₂. 3.20

Therefore

_T_{x, y}

2≥x, y2, ∀

x, y∈∂Ω5. 3.21

By 3.16,3.21and Lemma 2.4, it is easily seen that T has a fixed-point x∗, y∗ ∈ Ω5\

Ω4.

Corollary 3.4. Let (A) and the following conditions hold, then BVP1.1has at least one positive solution whileμ∈a2, a1andλ∈0, a1.

f_i0< m3,

ψf2

∞> m4, Ii,0k 0, i1,2. 3.22

Theorem 3.5. Let (A) and (H3) hold, then BVP1.1has at least one positive solution while λ ∈

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Proof. Sinceψf10> m5, we chooseR3>0,ε5 >0 such thatψfit, u, v≥m5 ε5u v for 0≤ u v ≤R3andt∈η/α, η. LetΩ6 {x, y∈K×K :x, y2 < R3}. Then for anyx, y∈∂Ω6,

_T₁_{x, y}

1≥ψ

T1

x, yη≥λ _η

η/α

Gη, sψf1

t, xs, ysds

≥λm5 ε5σ

_η

η/α

Gη, sx₁ y₁ds

R2λm5 ε5σ

_η

η/α

Gη, sds.

3.23

So

_T_{x, y}

2T1x, y1 T2x, y1≥T1x, y1

≥R3λm5 ε5σ

η

η/α

Gη, sds

≥R3,

3.24

which implies

_T_{x, y}

2≥x, y2, ∀

x, y∈∂Ω6. 3.25

On the other hand, byf_i∞ < m6andI_i∞k 0 i1,2, there existM >0,ε6>0,ε >0 such thatm6−ε6>0 and

fi

t, xt, yt≤m6−ε6

xt yt, Ii,kxtk ≤εxtk,

for anyx₁ y₁x, y₂≥M, t∈0,1,

3.26

whereεsatisfies

ε

m

k1

gtk≤ 1

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LetR4 max{M,2R3} andΩ7 {x, y | x, y ∈ K×K : x, y2 < R4}. Then for any

Corollary 3.6. Assume that (A) and the following conditions hold, then BVP1.1has at least one positive solution whileμ∈a3, a4andλ∈0, a4.

ψf2

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4. An Example

In this section, we construct an example to demonstrate the application of our main results obtained inSection 3. Consider the following third-order boundary value problem:

−x_nt λt2e−txnt ynt

Conclusion 1. BVP4.1has at least one positive solution.

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Above all, the conditions ofTheorem 3.3are satisfied. Then for anyλ ∈ a2, ∞and μ ∈

0, a1, BVP4.1has at least one positive solution.

Acknowledgments

The author thanks Professor Liu and Professor Lou for many useful discussions and helpful suggestions. The work was partially supported by NSFC10971155and Innovation program of Shanghai Municipal Education Commission09ZZ33.

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