Volume 2010, Article ID 108343,11pages doi:10.1155/2010/108343
Research Article
Fixed Point Theorems for Nonlinear
Operators with and without Monotonicity in Partially Ordered Banach Spaces
Hui-Sheng Ding,
1Jin Liang,
2and Ti-Jun Xiao
31College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
2Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
3Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Correspondence should be addressed to Ti-Jun Xiao,[email protected]
Received 30 September 2009; Revised 5 December 2009; Accepted 6 December 2009 Academic Editor: Mohamed A. Khamsi
Copyrightq2010 Hui-Sheng Ding et al. 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 establish two fixed point theorems for nonlinear operators on Banach spaces partially ordered by a cone. The first fixed point theorem is concerned with a class of mixed monotone operators.
In the second fixed point theorem, the nonlinear operators are neither monotone nor mixed monotone. We also provide an illustrative example for our second result.
1. Introduction
Fixed point theorems for nonlinear operators on partially ordered Banach spaces have many applications in nonlinear equations and many other subjectscf., e.g.,1–7and references therein; in particular, various kinds of fixed point theorems for mixed monotone operators are proved and appliedsee, e.g.,1,3,5,7and references therein.
Stimulated by7,8, we investigate further, in this paper, the existence of fixed points of nonlinear operators with and without monotonicity in partially ordered Banach spaces.
In Section 2, a fixed point theorem for a class of mixed monotone operators is established. In Section 3, without any monotonicity assumption for a class of nonlinear operators, we obtain a fixed point theorem by using Hilbert’s projection metric.
Let us recall some basic notations about conefor more details, we refer the reader to 2. LetX be a real Banach space. A closed convex setP inX is called a convex cone if the
following conditions are satisfied:
iifx∈P, thenλx∈Pfor anyλ≥0, iiifx∈Pand−x∈P, thenx0.
A conePinduces a partial ordering≤inXby
x≤y iffy−x∈P. 1.1
For any givenu, v∈P,
u, v:{x∈X|u≤x≤v}. 1.2
A conePis called normal if there exists a constantk >0 such that
0≤x≤yimplies thatx ≤ky, 1.3
where · is the norm onX.
Throughout this paper, we denote byNthe set of nonnegative integers,Rthe set of real numbers,Xa real Banach space,Pa convex cone inX,ean element inP\ {θ}θis the zero element ofX, andPethe following set:
Pe
x∈P : ∃α, β >0 such thatαe≤x≤βe
. 1.4
2. Monotonic Operators
Theorem 2.1. Suppose that the operatorA:Pe×Pe×Pe → Pesatisfies the following.
S1A·, y, zis increasing,Ax,·, zis decreasing, andAx, y,·is decreasing.
S2There exist a constantt0∈0,1and a functionφ:0,1×Pe×Pe → 0,∞such that for eachx, y, z∈Peandt∈t0,1,φt, x, y> tand
A
tx, t−1y, z
≥φ t, x, y
A x, y, z
. 2.1
S3There existx0, y0 ∈Pesuch thatx0≤y0,x0≤Ax0, y0, x0,Ay0, x0, y0≤y0and
x,y∈xinf0,y0φ t, x, y
> t, ∀t∈t0,1. 2.2
S4There exists a constantL >0 such that, for allx, y, z1, z2 ∈Pewithz1≥z2,
A x, y, z1
−A x, y, z2
≥ −L·z1−z2. 2.3
ThenAhas a unique fixed pointx∗inx0, y0, that is,Ax∗, x∗, x∗ x∗.
Proof. The proof is divided into 4 steps.
Step 1. Lett1∈t0,1and
ψ t, x, y
φ t1, x, y
t1 t, t∈0,1, x, y∈Pe. 2.4
For eacht ∈ 0,1, there exists a nonnegative integerk such thattk11 ≤ t < tk1, that is,t1 ≤ t/tk1 <1. Now, byS2, we deduce, for allx, y, z∈Pe,
A
tx, t−1y, z
A t
tk1 ·tk1x,tk1
t ·t−k1 y, z
≥φ t
tk1, tk1x, t−k1 y
A
tk1x, t−k1 y, z
≥ t tk1A
tk1x, t−k1 y, z
≥ t t1A
t1x, t−11 y, z
≥ t t1φ
t1, x, y A
x, y, z ψ
t, x, y A
x, y, z .
2.5
Moreover, byS3, we get
x,y∈xinf0,y0ψ t, x, y
infx,y∈x0,y0φ t1, x, y
t1 ·t > t, ∀t∈0,1. 2.6
Hence, in the following proof, one can assume thatt00 inS2andS3without loss.
Step 2. Fixx, y∈Pe. Then, there existsα∈0,1such thatx, y∈αx0, α−1y0. Let
Ψxyz A x, y, z
Lz
1L , z∈Pe. 2.7
ThenΨxyis an operator fromPetoPe, and byS4,Ψxy is increasing inPe. CombiningS1–
S3, we have A
x, y, αx0
≥A
αx0, α−1y0, x0
≥φ
α, x0, y0 A
x0, y0, x0
≥αx0, 2.8
provided thatα∈0,1. Moreover, it is easy to see that2.8holds whenα1. Similarly, one can show that
A
x, y, α−1y0
≤α−1y0. 2.9
Then, it follows that
Ψxyαx0≥αx0, Ψxy
α−1y0
≤α−1y0. 2.10
Let
xxyn Ψxy
xn−1xy
, ynxy Ψxy
yn−1xy
, n1,2, . . . , x0xyαx0, y0xy α−1y0.
2.11
Then, using arguments similar to those in the proof of7, Theorem 2.1, one can show that Ψxyhas a unique fixed pointx∗xyinαx0, α−1y0, and
xxyn −→xxy∗ , ynxy −→x∗xy n−→ ∞. 2.12
We claim thatx∗xyis the unique fixed point ofΨxyinPe. In fact, letyxy∗ be a fixed point ofΨxy
inPe, andβ∈0, αsuch thatyxy∗ ∈βx0, β−1y0. By the above proof,Ψxy has a unique fixed point inβx0, β−1y0, which means thatx∗xy y∗xy. In addition, it follows from
x∗xy Ψxy
x∗xy
A
x, y, x∗xy Lxxy∗ 1L
2.13
thatxxy∗ Ax, y, x∗xy.
Step 3. ByStep 2, we can define an operatorΦ:Pe×Pe → Peby Φ
x, y
xxy∗ Ψxy
x∗xy
A
x, y, x∗xy
. 2.14
Let x, x ∈ x0, y0 with x ≤ x and α ∈ 0,1 with x, x, y ∈ αx0, α−1y0. Denote by {xnxy},{xnxy}the corresponding sequences in the proof ofStep 2. Then
x1xy Ψxy
x0xy
Ψxyαx0
A
x, y, αx0 Lαx0 1L
≤ A
x, y, αx0
Lαx0
1L Ψxyαx0 x1xy.
2.15
Next, by induction andΨxy being increasing, one can show thatxxyn ≤xnxyfor alln∈N. So x∗xy lim
n→ ∞xnxy ≤ lim
n→ ∞xnxyx∗xy, 2.16
that is,Φx, y≤Φx, y. Thus,Φ·, yis increasing. By a similar method, one can prove that Φx,·is decreasing. On the other hand, byS3, forx, y∈Peandt∈0,1,
Φ
tx, t−1y A
tx, t−1y,Φ
tx, t−1y
≥A
tx, t−1y,Φ x, y
≥φ t, x, y
A x, y,Φ
x, y φ
t, x, y Φ
x, y .
2.17
Letu0x0, v0 y0, and
un Φun−1, vn−1, vn Φvn−1, un−1, for n1,2, . . . . 2.18 By choosingα1 inStep 1, we getx∗x0y0∈x0, y0. Then
u1 Φ x0, y0
xx∗0y0 ≥x0 u0, v1 Φ y0, x0
x∗y0x0≤y0v0. 2.19
AsΦ·, yis increasing andΦx,·is decreasing, it follows immediately that
u0≤u1≤ · · · ≤un≤ · · · ≤vn≤ · · · ≤v0. 2.20
Next, by making some needed modifications in the proof of3, Theorem 2.11, one can show thatΦhas a fixed pointx∗∈x0, y0. Suppose thaty∗∈x0, y0is a fixed point ofΦ. It follows from the definition ofunandvnthatun≤y∗ ≤vnfor alln∈N. Then, by the normality ofΦ, we gety∗x∗. Sox∗is the unique fixed point ofΦinx0, y0.
Step 4. By Steps2and3, we get
x∗ Φx∗, x∗ Ax∗, x∗,Φx∗, x∗ Ax∗, x∗, x∗. 2.21
Letx∈x0, y0such thatxAx, x, x. Then it follows fromStep 2thatΦx, x x, that is, xis a fixed point ofΦinx0, y0. Thus, byStep 3,xx∗, which means thatx∗is the unique fixed point ofAinx0, y0.
Remark 2.2. Compared with7, Remark 2.4, the nonlinear operatorAinTheorem 2.1is more general, and soTheorem 2.1may have a wider range of applications.
3. Nonmonotonic Case
First, let us recall some definitions and basic results about Hilbert’s projection metric for more details, see6.
Definition 3.1. Elementsxandybelonging toP not both zeroare said to be linked if there existλ, μ >0 such that
λx≤y≤μx. 3.1
This defines an equivalence relation onP and dividesP into disjoint subsets which we call constituents ofP.
Letxandybe linked. Define M
x, y inf
μ >0 :y≤μx , d
x, y ln
max M
x, y , M
y, x
. 3.2
Then, the following holds.
Theorem 3.2. d·,·defines a complete metric on each constituent ofP. Proof. See6.
We will also need the following result.
Theorem 3.3. [9] LetMbe a complete metric space and suppose thatf:M → Msatisfies d
fx, f y
≤Ψ d
x, y
, ∀x, y∈M, 3.3
whereΨ:0,∞ → 0,∞is upper semicontinuous from the right and satisfiesΨt< tfor all t >0. Thenfhas a unique fixed point inM.
Theorem 3.3 is a generalization of the classical Banach’s contraction mapping principle. There are many generalizations of the classical Banach’s contraction mapping principle see, e.g., 10, 11 and references therein, and these generalizations play an important role in research work about fixed points of nonlinear operators in partially ordered Banach spaces; see, for example,1and the proof of the following theorem.
Now, we are ready to present our fixed point theorem, in which no monotone condition is assumed on the nonlinear operator.
Theorem 3.4. LetTbe an operator fromPetoPe. Assume that there exist a constantε∈0,1and a functionφ:ε,1 → 0,∞such thatφλ> λfor allλ∈ε,1, and
Ty≥φλTx, 3.4
for allx, y∈Peandλ∈ε,1satisfyingλx≤y≤λ−1x. ThenThas a unique fixed point inPe.
Proof. We divided the proof into 2 steps.
Step 1. Letλ∈0,1,x, y∈Pe, andλx≤y≤λ−1x. Then, there existsk∈Nsuch that
ε≤ λ
εk <1. 3.5
In view of
λ εk ·
εkx
λx≤y≤λ−1x≤ε2kλ−1x λ
εk −1
· εkx
, 3.6
by the assumptions, we have
Ty≥φ λ
εk
·T εkx
≥ λ εk ·T
εkx
. 3.7
Similar to the above proof, sinceε·εk−1xεkx≤ε−1·εk−1x, one can deduce
Ty≥ λ εk ·T
εkx
≥ λ
εk ·φε·T εk−1x
≥ λ εk−1 ·T
εk−1x
. 3.8
Continuing by this way, one can get
Ty≥ λ
ε ·Tεx≥ φε
ε λ·Tx. 3.9
Let
ψλ φε
ε λ, λ∈0,1. 3.10
Thenψis continuous,ψλ> λfor allλ∈0,1, and
Ty≥ψλTx, 3.11
for allx, y∈Peandλ∈0,1satisfyingλx≤y≤λ−1x.
Step 2. Next, letx, y∈Pewithx /yand
λ 1
max M
x, y , M
y, x. 3.12
Thenλ∈0,1,λx≤y≤λ−1x, anddx, y lnλ−1.Moreover, byStep 1, we have
Ty≥ψλTx. 3.13
On the other hand, sinceλy≤x≤λ−1y, we also have
Tx≥ψλTy. 3.14
Thus, we get
ψλTx≤Ty≤ Tx
ψλ. 3.15
Now, by the definition ofd·,·, we have
d
Tx, Ty
≤ln 1
ψλ
. 3.16
Let
Ψt
⎧⎨
⎩
−ln ψ
e−t
, t∈0,∞,
0, t0. 3.17
Then,Ψis a continuous function from0,∞to0,∞, and d
Tx, Ty
≤Ψ d
x, y
. 3.18
Moreover, sinceψλ> λfor allλ∈0,1, we get Ψt ln 1
ψe−t <ln 1
e−t t, t >0. 3.19
On the other hand, Pe is obviously a constituent of P, and thus Pe, d is complete by Theorem 3.2. Now,Theorem 3.3yields thatThas a unique fixed point inPe.
Corollary 3.5. Assume thatA : Pe ×Pe → Pe is a mixed monotone operator, that is,A·, y is increasing andAx,·is decreasing. Moreover, there exist a constantε ∈ 0,1and a functionφ : ε,1 → 0,∞such thatφλ> λfor allλ∈ε,1, and
A
λx, λ−1y
≥φλA x, y
, 3.20
for allx, y∈Peandλ∈ε,1. ThenAhas a unique fixed point inPe.
Proof. LetTzAz, z, z∈Pe. Then, sinceAis a mixed monotone operator, we have TyA
y, y
≥A
λx, λ−1x
≥φλAx, x φλTx, 3.21
for all x, y ∈ Pe and λ ∈ ε,1 satisfying λx ≤ y ≤ λ−1x. Then, Theorem 3.4yields the conclusion.
Remark 3.6. Corollary 3.5is an improvement of1, Corollary 3.2in the sense that thereφis lower semicontinuous on0,1, and the corresponding conditions need to hold on the whole interval0,1.
4. An Example
In this section, we give an example to illustrateTheorem 3.4. Let us consider the following nonlinear delay integral equation:
xt t
t−τfs, xsds, 4.1
which is a classical model for the spread of some infectious diseasecf.12. In fact,4.1has been of great interest for many authorssee, e.g.,3,8and references therein.
In the rest of this paper, letτ 1 and
ft, x
⎧⎪
⎪⎨
⎪⎪
⎩
1sin2tsin2πt√
x, t∈−∞,∞, 0≤x≤1, 1sin2tsin2πt
√x , t∈−∞,∞, x≥1.
4.2
Next, let us investigate the existence of positive almost periodic solution to 4.1. For the reader’s convenience, we recall some definitions and basic results about almost periodic functionsfor more details, see13.
Definition 4.1. A continuous functionf : R → Ris called almost periodic if for eachε > 0 there existslε > 0 such that every intervalI of lengthlεcontains a numberτ with the property that
ftτ−ft< ε ∀t∈R. 4.3
Denote byAPRthe set of all such functions.
Lemma 4.2. Assume thatf,g∈APR. Then the following hold.
aThe rangeRf {ft:t∈R}is precompact inR, and sofis bounded.
bFf∈APRprovided thatFis continuous onRf.
cfg,f·g ∈APR. Moreover,f/g∈APRprovided that inft∈R|gt|>0.
dEquipped with the sup norm
fsup
t∈R
ft, 4.4 APRturns out to be a Banach space.
Now, letP {x∈APR:xt≥0,∀t∈R}, ande∈Pis defined byet≡1. It is not difficult to verify thatPis a normal cone inAPR, and
Pe {x∈APR:∃ε >0 such thatxt> ε, ∀t∈R}. 4.5
Define a nonlinear operatorT onPeby
Txt t
t−1fs, xsds, x∈Pe, t∈R. 4.6
ByLemma 4.2and3, Corollary 3.3, it is not difficult to verify thatT is an operator fromPe toPe. In addition, in view of4.2, one can verify that
Ty t
t
t−1f s, ys
ds≥ λ
t
t−1fs, xsds
λTxt, t∈R, 4.7
that is,Ty≥√
λTxfor allx, y ∈Peandλ∈0,1withλx≤y≤λ−1x. Then, byTheorem 3.4, T has a unique fixed point in Pe, that is, 4.1has a unique almost periodic solution with positive infimum.
Acknowledgments
The authors are very grateful to the referees for valuable suggestions and comments. In addition, Hui-Sheng Ding acknowledges support from the NSF of China10826066, the NSF of Jiangxi Province of China2008GQS0057, and the Youth Foundation of Jiangxi Provincial Education DepartmentGJJ09456; Jin Liang and Ti-Jun Xiao acknowledge support from the NSF of China10771202, the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics08DZ2271900, and the Specialized Research Fund for the Doctoral Program of Higher Education of China2007035805.
References
1 Y. Z. Chen, “Thompson’s metric and mixed monotone operators,” Journal of Mathematical Analysis and Applications, vol. 177, no. 1, pp. 31–37, 1993.
2 K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
3 H.-S. Ding, T.-J. Xiao, and J. Liang, “Existence of positive almost automorphic solutions to nonlinear delay integral equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2216–
2231, 2009.
4 J. A. Gatica and W. A. Kirk, “A fixed point theorem fork-set-contractions defined in a cone,” Pacific Journal of Mathematics, vol. 53, pp. 131–136, 1974.
5 K. Li, J. Liang, and T.-J. Xiao, “New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation,” Journal of Mathematical Analysis and Applications, vol.
328, no. 2, pp. 753–766, 2007.
6 A. C. Thompson, “On certain contraction mappings in a partially ordered vector space,” Proceedings of the American Mathematical Society, vol. 14, pp. 438–443, 1963.
7 Z. Zhang and K. Wang, “On fixed point theorems of mixed monotone operators and applications,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3279–3284, 2009.
8 E. Ait Dads, P. Cieutat, and L. Lhachimi, “Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 721–739, 2009.
9 D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458–464, 1969.
10 W. A. Kirk, “Fixed points of asymptotic contractions,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 645–650, 2003.
11 W. A. Kirk and H.-K. Xu, “Asymptotic pointwise contractions,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 69, no. 12, pp. 4706–4712, 2008.
12 K. L. Cooke and J. L. Kaplan, “A periodicity threshold theorem for epidemics and population growth,” Mathematical Biosciences, vol. 31, no. 1-2, pp. 87–104, 1976.
13 A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1974.
