Fixed Points for Discontinuous Monotone Operators

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Volume 2010, Article ID 926209,11pages doi:10.1155/2010/926209

Research Article

Fixed Points for Discontinuous Monotone Operators

Yujun Cui

¹

and Xingqiu Zhang

²

1Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266510, China

2School of Mathematics, Liaocheng University, Liaocheng 252059, China

Correspondence should be addressed to Yujun Cui,[email protected] Received 24 September 2009; Accepted 21 November 2009

Academic Editor: Tomas Dominguez Benavides

Copyrightq2010 Y. Cui and X. Zhang. 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.

We obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space. Moreover, the maximal and minimal fixed points can be achieved by monotone iterative method under some conditions. As an example of the application of our results, we show the existence of extremal solutions to a class of discontinuous initial value problems.

1. Introduction

LetXbe a Banach space. A nonempty convex closed setP ⊂Xis said to be a cone if it satisfies the following two conditions:ix∈P,λ≥0 impliesλx∈P;iix∈P,−x∈Pimpliesxθ, whereθdenotes the zero element. The conePdefines an ordering inEgiven byx≤yif and only ify−x∈P. LetD u0, v0be an ordering interval inX, andA:D → Xan increasing operator such thatu0 ≤Au0,Av0 ≤v0. It is a common knowledge that fixed point theorems on increasing operators are used widely in nonlinear diﬀerential equations and other fields in mathematics1–7.

But in most well-known documents, it is assumed generally that increasing operators possess stronger continuity and compactness. Recently, there have been some papers that considered the existence of fixed points of discontinuous operators. For example, Krasnosel’skii and Lusnikov 8 and Chen 9 discussed the fixed point problems for discontinuous monotonically compact operator. They called an operator A to be a monotonically compact operator if x1 ≤ · · · ≤ xn ≤ · · · ≤ w x1 ≥ · · · ≥ xn ≥ · · · ≥ w implies thatAxnconverges to somex^∗∈Xin norm and thatx^∗sup{Axn}xinf{Axn}.

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A monotonically compact operator is referred to as an MMC-operator. A is said to be h- monotone ifx < y impliesAx < Ay−αx, yh, whereh ∈ P,h /θ, andαx, y > 0. They proved the following theorem.

Theorem 1.1see8. LetA:E → Ebe anh-monotone MMC-operator withu < Au≤Av < v.

ThenAhas at least one fixed pointx^∗∈u, vpossessing the property ofh-continuity.

Motivated by the results of3,8,9, in this paper we study the existence of the minimal and maximal fixed points of a discontinuous operator A, which is expressed as the form CB. We do not assume any continuity on A. It is only required thatCorBis an MMC- operator andBD orADpossesses the quasiseparability, which are satisfied naturally in some spaces. As an example for application, we applied our theorem to study first order discontinuous nonlinear diﬀerential equation to conclude our paper.

We give the following definitions.

Definition 1.2see3. LetY be an Hausdorﬀtopological space with an ordering structure.

Y is called an ordered topological space if for any two sequences{xn}and{yn}inY,xn ≤ ynn1,2, . . .andxn → x,yn → yn → ∞implyx≤y.

Definition 1.3see3. LetYbe an ordered topological space,Sis said to be a quasi-separable set inY if for any totally ordered setMinS, there exists a countable set{yn} ⊂Msuch that {yn}is dense inMi.e., for anyy∈M, there exists{ynj} ⊂ {yn}such thatynj → yn → ∞.

Obviously, the separability implies the quasi-separability.

Definition 1.4see3. LetX, Ybe two ordered topological spaces. An operatorA:X → Yis said to be a monotonically compact operator ifx1≤ · · · ≤xn≤ · · · ≤wx1≥ · · · ≥xn≥ · · · ≥w implies thatAxnconverges to somey^∗∈Yin norm and thaty^∗sup{Axn}y^∗inf{Axn}.

Remark 1.5. The definition of the MMC-operator is slightly diﬀerent from that of8,9.

2. Main Results

Theorem 2.1. LetX be an ordered topological space, andD u0, v0an order interval inX. Let A:D → Xbe an operator. Assume that

ithere exist ordered topological spaceY, increasing operatorC : D → Y, and increasing operatorB:Cu0, Cv0 {y∈Y |Cu0≤y≤Cv0} → Xsuch thatABC;

iiADis quasiseparable andCis an MMC-operator;

iiiu0≤Au0,Av0≤v0.

ThenAhas at least one fixed point inD.

Proof. It follows from the monotonicity ofAand conditioniiithatA:D → D. SetR{x∈ AD|x≤Ax}. SinceAu0∈R,Ris nonempty. Suppose thatMis a totally ordered set inR.

We now show thatMhas an upper bound inR.

SinceM⊂AD, by conditioniithere exists a countable subset{xi}ofMsuch that {xi}is dense inM. Consider the sequence

z1x1, zimax{zi−1, xi}, i1,2, . . . . 2.1

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SinceMis a totally ordered set,zimakes sense and

z1≤z2≤ · · · ≤zi≤ · · · . 2.2

By conditionii,M⊂D u0, v0andDefinition 1.4, there existsy^∗∈Y such that

Czi−→y^∗sup{Czi}, i−→ ∞, 2.3

Cu0 ≤y^∗≤Cv0, 2.4

and henceBy^∗make sense.

Set

x^∗By^∗. 2.5

Using2.1and2.2, we have

xi≤AxiBCxi≤BCzi≤By^∗x^∗. 2.6

Since{xi} is dense inM, for anyx ∈ Mthere exists a subsequence{xij}of{xi}such that xij → xj → ∞. By2.6andDefinition 1.2, we get

x≤x^∗, ∀x∈M. 2.7

Hencex≤Ax≤Ax^∗, thereforeAx^∗is an upper bound ofM.

Now we showAx^∗∈R. By virtue of2.4and conditioniii

u0≤Au0BCu0 ≤By^∗x^∗≤BCv0≤v0. 2.8

Thusx^∗ ∈ u0, v0 Dand henceAx^∗ ∈D. By2.7and conditionii, we getzi ≤ x^∗and henceCzi≤Cx^∗. By2.3andDefinition 1.2, we gety^∗≤Cx^∗and

x^∗By^∗≤BCx^∗Ax^∗. 2.9

HenceAx^∗≤AAx^∗, and thereforeAx^∗∈R.

This shows thatAx^∗is an upper bound ofMinR. It follows from Zorn’s lemma that Rhas maximal elementx. Thusx≤Ax. And soAx≤AAx, which implies thatAx∈Rand x≤Ax. Asxis a maximal element ofR,xAx; that is,xis a fixed point ofA.

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ii Cu0, Cv0is quasiseparable andBis an MMC-operator;

iiiu0≤Au0,Av0≤v0.

ThenAhas at least one fixed point inD.

Proof. Lety1Cu0,y2Cv0. By the conditionsiandiii, we have

y1Cu0≤CAu0CBCu0CBy1, CBy2CBCv0CAv0≤Cv0y2. 2.10

SinceCBis increasing, for anyy∈y1, y2, we get

y1≤CBy1 ≤CBy≤CBy2≤y2, 2.11

that is,CB : y1, y2 → y1, y2; therefore the quasiseparability ofCu0, Cv0implies that CBy1, y2is quasiseparable. ApplyingTheorem 2.1, the operatorCBhas at least one fixed pointy^∗iny1, y2, that is,

y^∗CBy^∗, y^∗∈ y1, y2

. 2.12

Setx^∗By^∗. SinceBis increasing, by2.12, we have

u0≤Au0 BCu0≤By^∗x^∗≤Bcv0Av0≤v0, x^∗By^∗B

CBy^∗ BC

By^∗

Ax^∗; 2.13

that is,x^∗is a fixed point of the operatorAinu0, v0.

Theorem 2.3. If the conditions inTheorem 2.1are satisfied, thenAhas the minimal fixed pointu^∗ and the maximal fixed pointv^∗inD; that is,u^∗andv^∗are fixed points ofA, and for any fixed pointx ofAinD, one hasu^∗≤x≤v^∗.

Proof. Set

FixA

x∈Dxis a fixed point of A

. 2.14

ByTheorem 2.1, FixA /∅. Set

S{u, v|u, vis an order interval inX, u, v∈AD, u≤Au, Av≤v, FixA⊂u, v}.

2.15 SinceAis increasing, for anyx∈FixA, we have

u0≤Au0≤Axx≤Av0≤v0, 2.16

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and hence

Au0≤A²u0≤Axx≤A²v0≤Av0, 2.17 thereforeAu0, Av0∈ S, and thusS /∅. An order ofSis defined by the inclusion relation, that is, for anyI1 ∈ S,I2 ∈ S, and ifI1 ⊂ I2, then we defineI1 ≤ I2. We show thatShas a minimal element. Let{uα, vα | α ∈ T}be a totally subset of SandM {uα | α ∈ T}.

Obviously, M is a totally ordered set inX. Since ADis quasiseparable, it follows from M⊂ADthat there exists a countable subset{yi}ofMsuch that{yi}is dense inM. Let

w1y1, wimax w_i−1, yi

, i2,3, . . . . 2.18

SinceMis a totally ordered set,wimakes sense and

w1 ≤w2≤ · · · ≤wi ≤ · · ·. 2.19

Then there existsw∈Y such that

Cwi−→wsup{Cwi}. 2.20

Using the same method as inTheorem 2.1, we can prove thatw makes sense, Au where uBwis an upper bound ofM, and

Au≤AAu. 2.21

Since FixA⊂ uα, vα for allα∈T, for anyx ∈FixA, we haveuα ≤x, for allα∈T. Since wi∈M,wi ≤x. By2.20,w≤Cx, and henceuBw≤BCxAxx, for allx∈FixA, and therefore

Au≤Axx, ∀x∈FixA. 2.22

ConsiderN {vα |α∈T}. Similarly, we can prove that there existsv ∈Dsuch that Avis a lower bound ofNand

AAv≤Av, Av≥x, ∀x∈FixA. 2.23

By2.22and2.23,Au≤Av. SetI Au, Av. By virtue of2.21,2.22, and2.23,I∈S.

It is easy to see thatIis a lower bound of{uα, vα|α∈T}inS. It follows from Zorn’s lemma thatShas a minimal element.

Letu^∗, v^∗be a minimal element ofS. Therefore,u^∗ ≤ Au^∗,Av^∗ ≤ v^∗, and FixA ⊂ u^∗, v^∗. Obviously, u^∗ is a fixed point ofA. In fact, on the contrary,u^∗/Au^∗andu^∗ ≤ Au^∗. Hence

Au^∗≤AAu^∗, Au^∗≤Axx, ∀x∈FixA. 2.24

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SinceAis an increasing operator, this implies that FixA ⊂ Au^∗, v^∗ andu^∗, v^∗includes properlyAu^∗, v^∗. This contradicts thatu^∗, v^∗is the minimal element ofS. Similarly,v^∗is a fixed point ofA. Since FixA ⊂ u^∗, v^∗,u^∗ is the minimal fixed point ofAand v^∗ is the maximal fixed point ofA.

Theorem 2.4. If the conditions inTheorem 2.2are satisfied, thenAhas the minimal fixed pointu^∗ and the maximal fixed pointv^∗inD; that is,u^∗andv^∗are fixed points ofA, and for any fixed pointx ofAinD, one hasu^∗≤x≤v^∗.

Proof. It is similar to the proof ofTheorem 2.4; so we omit it.

iiBis an continuous operator;

iiiCis a demicontinuous MMC-operator;

ivu0≤Au0,Av0≤v0.

ThenAhas both the minimal fixed pointu^∗and the maximal fixed pointv^∗inu0, v0, andu^∗andv^∗ can be obtained via monotone iterates:

u0≤Au0≤ · · · ≤Aⁿu0≤ · · · ≤Aⁿv0≤ · · · ≤Av0≤v0 2.25

with lim_n_{→ ∞}Aⁿu0u^∗,and lim_n_{→ ∞}Aⁿv0v^∗. Proof. We define the sequences

unAⁿu0, vnAⁿv0, n1,2, . . . 2.26

and conclude from the monotonicity of operatorAand the conditionivthat

u0≤u1≤ · · · ≤un≤ · · ·vn≤ · · · ≤v1≤v0. 2.27

Let

ynCun, n1,2, . . . . 2.28

SinceCis increasing,y0 ≤y1≤ · · · ≤yn≤ · · · ≤Cv0by2.27. By the conditioniii, we get yn−→y^∗sup

yn

, n−→ ∞. 2.29

By2.29andDefinition 1.2, we have

y^∗∈Cu0, Cv0, 2.30

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and henceBy^∗makes sense. Setu^∗By^∗, thenu^∗∈u0, v0. SinceBis continuous,

unAunBCunByn−→By^∗u^∗. 2.31

By the conditioniii,Cun w

−−−→Cu^∗, that is,yn w

−−−→Cu^∗. Note thatyn → y^∗; we havey^∗Cu^∗; henceu^∗ By^∗ BCu^∗ Au^∗; that is,u^∗is a fixed point ofA. Similarly, there existsv^∗ ∈D such thatvn → v^∗andv^∗is a fixed point of A. By the routine standard proof, it is easy to prove thatu^∗is the minimal fixed point ofAandv^∗is the maximal fixed point ofAinD.

3. Applications

As some simple applications ofTheorem 2.5, we consider the existence of extremal solutions for a class of discontinuous scalar diﬀerential equations.

In the following,R stands for the set of real numbers andJ 0, aa compact real interval. LetCJ, Rbe the class of continuous functions onJ.CJ, Ris a normed linear space with the maximum norm and partially ordered by the coneK {x∈CJ, R:xt≥0}.Kis a normal cone inCJ, R.

For any 1≤p <∞, set

L^pJ, R

xt:J → R|xtis measurable and

J

|xt|^pdt <∞

. 3.1

ThenL^pJ, Ris a Banach space by the normx_p

J|xt|^pdt^1/p.

A functionf:J×R → Ris said to be a Carath´eodory function iffx, yis measurable as a function ofxfor each fixedyand continuous as a function ofyfor a.a.almost allx∈J.

We list for convenience the following assumptions.

H1u0, v0∈ACJ, R,u0≤v0,

u₀t≤ft, u0t, v₀t≥ft, v0t for a.a. t∈J. 3.2

H2f :J×R → Ris a Carath´eodory function.

H3There existsp >1 such that

ft, u0t∈L^PJ, R, ft, v0t∈L^PJ, R. 3.3

H4There existsM≥0 such thatft, x Mxis nondecreasing for a.a.t∈J.

Consider the diﬀerential equation

xft, x, x0 x0, 3.4

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where f : J ×R → R. It is a common knowledge that the initial value problem 3.4 is equivalent to the equation

xt x0 ^t

0

fs, xsds 3.5

ifft, xis continuous. Therefore, whenft, xis not continuous, we define the solution of the integral equation3.5as the solution of the equation3.4.

Theorem 3.1. Under the hypotheses (H1)–(H4), the IVP3.4has the minimal solutionu^∗and max- imal solutionv^∗ inu0, v0. Moreover, there exist monotone iteration sequences{unt},{vnt} ⊂ u0, v0such that

unt−→u^∗t, vnt−→v^∗t asn−→ ∞uniformly ont∈J, 3.6

where{unt}and{vnt}satisfy

u_nt ft, un−1t−Mtunt−un−1t, un0 x0, v_nt ft, vn−1t−Mtvnt−vn−1t, vn0 x0, u0≤u1≤ · · · ≤un ≤ · · · ≤u^∗≤v^∗≤ · · · ≤vn≤ · · · ≤v1≤v0.

3.7

Proof. For anyh∈CJ, R, we consider the linear integral equation:

xt ht−Txt, 3.8

where Txt ^Δ _t

0Musds. Obviously, T : CJ, R → CJ, R is a linear completely continuous operator. By direct computation, the operator equationxTxθhas only zero solution; then by Fredholm theorem, for anyh∈ CJ, R, the operator equation3.8has a unique solution inCJ, R. We definition the mappingN:CJ, R → CJ, Rby

Nhuh, 3.9

where uh is the unique solution of 3.8 corresponding to h. Obviously N is a linear continuous operator; now we show that N is increasing. Suppose that h1, h2 ∈ CJ, R,

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h1≤h2. Setmt Nh2t−Nh1t. By the definition of the operatorNwe get mt Nh2t−Nh1t

h2t−M

t 0

Nh2sds−

h1t− ^t

0

Nh1sds

h2t−h1t−M

t 0

Nh2sds−Nh1sds

≥ −M ^t

0

msds.

3.10

This integral inequality impliesmt ≥ 0for allt ∈J; that is,Nis an increasing operator.

Set

Qvx0 ^t

0

vsds. 3.11

Obviously,Q:L^pJ, R → CJ, Ris an increasing continuous operator. Set

Cxt ft, xt Mxt, x∈CJ, R. 3.12

ByH2,Cmaps element ofCJ, Rinto measurable functions. For anyu∈u0, v0, byH3 andH4we get

Cu0≤Cu≤Cv0. 3.13

This implies Cu ∈ L^pJ, R. Hence C maps u0, v0 into L^pJ, R and C is an increasing operator. Set

CJ, R X, L^pJ, R Y, BNQ, ABC, D u0, v0. 3.14

By above discussions we know thatC : D → Y and B : Y → X are all increasing. Thus conditionsiandiiinTheorem 2.5are satisfied.

Lethn, h^∗∈Dsuch thathn → h^∗inCJ, R; byH2we have

nlim→ ∞ft, hnt Mhnt ft, h^∗t Mh^∗t, for a.a.t∈J. 3.15

For anyϕt∈L^qJ, R p⁻¹q⁻¹1, by2.29, we have 0 ≤ft, hnt Mhnt−

ft, u0t Mu0t

≤ft, v0t Mv0t−

ft, u0t Mu0t

, 3.16

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and hence

ft, hnt Mhnt≤Ht, 3.17

whereHt |ft, v0t Mv0t|2|ft, u0t Mu0t|. ByH3,Ht∈L^pJ, R; thus

ϕtft, hnt Mhnt≤ϕtHt, 3.18 whereϕtHt∈L¹J, R. Applying the Lebesgue dominated convergence theorem, we have

nlim→ ∞ J

ϕt

ft, hnt Mhnt dt

J

ϕt

ft, h^∗t Mh^∗t

dt. 3.19

This implies thatChn −−−→w Ch^∗inL^pJ, R; that is,Cis a demicontinuous operator. Since the cone inL^pJ, Ris regular, it is easy to see thatCis an MMC-operator. Thus conditioniiiin Theorem 2.5is satisfied.

We now show that conditionivinTheorem 2.5is fulfilled. ByH1and3.5, and noting the definition of operatorN, we get

Au0t−u0t NQCu0t−u0t N

x0 ^t

0

fs, u0s Mu0s ds

−u0t

x0 ^t

0

fs, u0s Mu0s ds−M

t 0

Au0sds−u0t

≥ −M ^t

0

Au0s−u0sds.

3.20

This implies thatAu0t−u0t ≥0,for allt ∈J, that is,u0 ≤ Au0. Similarly we can show thatAv0≤v0.

Since all conditions inTheorem 2.5are satisfied, byTheorem 2.5,Ahas the maximal fixed point and the minimal fixed point inD. Observing that fixed point ofAis equivalent to solutions of3.5, and3.5is equivalent to3.4, the conclusions ofTheorem 3.1hold.

Remark 3.2. In the proof of Theorem 3.1, we obtain the uniformly convergence of the monotone sequences without the compactness condition.

Acknowledgment

The project supported by the National Science Foundation of China10971179.

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