Volume 2010, Article ID 926209,11pages doi:10.1155/2010/926209
Research Article
Fixed Points for Discontinuous
Monotone Operators
Yujun Cui
1and Xingqiu Zhang
21Department 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
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
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.
Theorem 2.2. LetX be an ordered topological space, andD u0, v0an order interval inX. Let
A:D → Xbe an operator. Assume that
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∗BCBy∗BCBy∗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
FixAx∈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
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
wi−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
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.
Theorem 2.5. 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;
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
withlimn→ ∞Anu0u∗,andlimn→ ∞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
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
LpJ, R
xt:J → R|xtis measurable and
J
|xt|pdt <∞
. 3.1
ThenLpJ, Ris a Banach space by the normx p
J|xt| p
dt1/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∈LPJ, R, ft, v0t∈LPJ, R. 3.3
H4There existsM≥0 such thatft, x Mxis nondecreasing for a.a.t∈J.
Consider the differential equation
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
unt ft, un−1t−Mtunt−un−1t, un0 x0,
vnt 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 t0Musds. 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
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:LpJ, 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 ∈ LpJ, R. Hence C maps u0, v0 into LpJ, R and C is an increasing
operator. Set
CJ, R X, LpJ, 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
lim
n→ ∞ft, hnt Mhnt ft, h
∗t Mh∗t, for a.a.t∈J. 3.15
For anyϕt∈LqJ, R p−1q−11, by2.29, we have
0 ≤ft, hnt Mhnt−ft, u0t Mu0t
≤ft, v0t Mv0t−ft, u0t Mu0t,
and hence
ft, hnt Mhnt≤Ht, 3.17
whereHt |ft, v0t Mv0t|2|ft, u0t Mu0t|. ByH3,Ht∈LpJ, R; thus
ϕtft, hnt Mhnt≤ϕtHt, 3.18
whereϕtHt∈L1J, R. Applying the Lebesgue dominated convergence theorem, we have
lim
n→ ∞ Jϕt
ft, hnt Mhntdt
J
ϕtft, h∗t Mh∗tdt. 3.19
This implies thatChn −−−→w Ch∗inLpJ, R; that is,Cis a demicontinuous operator. Since the
cone inLpJ, 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 Mu0sds
−u0t
x0
t
0
fs, u0s Mu0sds−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
References
1 D. J. Guo and V. Lakshmikantham,Nonlinear Problems in Abstract Cones, vol. 5 ofNotes and Reports in Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1988.
2 H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620–709, 1976.
3 J. Sun and Z. Zhao, “Fixed point theorems of increasing operators and applications to nonlinear integro-differential equations with discontinuous terms,” Journal of Mathematical Analysis and Applications, vol. 175, no. 1, pp. 33–45, 1993.
4 S. Heikkil¨a and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, vol. 181 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1994.
5 V. Lakshmikantham and S. Leela,Differential and Integral Inequalities, Academic Press, New York, NY, USA, 1969.
6 C. Klin-eam and S. Suantai, “Strong convergence of monotone hybrid method for maximal monotone operators and hemirelatively nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2009, Article ID 261932, 14 pages, 2009.
7 S. Plubtieng and W. Sriprad, “An extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces,”Fixed Point Theory and Applications, vol. 2009, Article ID 591874, 16 pages, 2009.
8 M. A. Krasnosel’skii and A. B. Lusnikov, “Regular fixed points and stable invariant sets of monotone operators,”Applied Functional Analysis, vol. 30, no. 3, pp. 174–183, 1996.
