Volume 2008, Article ID 645419,9pages doi:10.1155/2008/645419
Research Article
Well-Posedness and Fractals via Fixed Point Theory
Cristian Chifu and Gabriela Petrus¸el
Department of Business, Faculty of Business, Babes¸-Bolyai University Cluj-Napoca, Horea 7, 400174 Cluj-Napoca, Romania
Correspondence should be addressed to Gabriela Petrus¸el,[email protected] Received 25 August 2008; Accepted 6 October 2008
Recommended by Andrzej Szulkin
The purpose of this paper is to present existence, uniqueness, and data dependence results for the strict fixed points of a multivalued operator of Reich type, as well as, some sufficient conditions for the well-posedness of a fixed point problem for the multivalued operator.
Copyrightq2008 C. Chifu and G. Petrus¸el. 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.
1. Introduction
LetX, dbe a metric space. We will use the following symbolssee also1:
PX {Y ⊂X|Y /∅};
PbX {Y ∈PX|Y is bounded};
PclX {Y ∈PX|Y is closed};
PcpX {Y ∈PX|Y is compact}.
IfT : X→PXis a multivalued operator, then forY ∈ PX,TY
x∈YTxwe will denote the image of the setY throughT.
Throughout the paperFT :{x∈X |x∈Tx}resp.,SFT :{x∈X | {x}Tx}
denotes the fixed point setresp., the strict fixed point setof the multivalued operatorT. We introduce the following generalized functionals.
Theδgeneralized functional
δd:PX×PX−→R∪ {∞}, δdA, B sup
da, b|a∈A, b∈B .
1.1
The gap functional
Dd :PX×PX−→R∪ {∞}, DdA, B inf
da, b|a∈A, b∈B
. 1.2
The excess generalized functional
ρd:PX×PX−→R∪ {∞}, ρdA, B sup
Dda, B|a∈A
. 1.3
The Pompeiu-Hausdorffgeneralized functional
Hd:PX×PX−→R∪ {∞}, HdA, B max
ρdA, B, ρdB, A
. 1.4
The first purpose of this paper is to present existence, uniqueness, and data dependence results for the strict fixed point of a multivalued operator of Reich type. Since, in our approach, the strict fixed point is constructed by iterations, this generates the possibility to give some sufficient conditions for the well-posedness of a fixed point problem for the multivalued operator mentioned below.
Definition 1.1. LetX, dbe a metric space andT :X→PclX. ThenT is called a multivalued δ-contraction of Reich type, if there exista, b, c∈Rwithabc <1 such that
δ
Tx, Ty
≤adx, y bδ
x, Tx cδ
y, Ty
, 1.5
for allx, y∈X.
The notion of well-posed fixed point problem for single valued and multivalued operator was defined and studied by F.S. De Blasi and J. Myjak, S. Reich and A.J. Zaslavski, Rus and Petrus¸el2, Petrus¸el et al.3.
Definition 1.2see Petrus¸el and Rus2and3. ALetX, dbe a metric space,Y ∈PX andT :Y→PclXbe a multivalued operator.
Then the fixed point problem is well posed forTwith respect toDdif a1FT {x∗}i.e.,x∗∈Tx∗;
b1Ifxn∈Y,n∈NandDdxn, Txn→0 asn→ ∞thenxn→x∗asn→ ∞.
B Let X, d be a metric space, Y ∈ PX and T : Y→PclX be a multivalued operator.
Then the fixed problem is well posed forTwith respect toHdif a2 SFT {x∗}i.e.,{x∗}Tx∗;
b2Ifxn∈Y,n∈NandHdTxn→0 asn→ ∞thenxn→x∗asn→ ∞.
The second aim is to study the existence of an attractori.e., the fixed point of the multifractal operator, see 4–7 for an iterated multifunction system consisting of nonself multivalued operators.
2. Main results
We will give first another proofa constructive oneof a result given by Reich8in 1972.
For some similar results, see9,10. In our proof, the strict fixed point will be obtained by iterations.
Theorem 2.1Reich’s theorem. LetX, dbe a complete metric space and letT :X→PbXbe a multivalued operator, for which there exista, b, c∈Rwithabc <1 such that
δ
Tx, Ty
≤adx, y bδ
x, Tx cδ
y, Ty
, ∀x, y∈X. 2.1
ThenT has a unique strict fixed point inX, that is,SFT{x∗}.
Proof. Letq >1 andx0∈Xbe arbitrarily chosen. Then there existsx1∈Tx0such that δ
x0, T x0
≤qd x0, x1
. 2.2
We have δ
x1, T x1
≤δ T
x0
, T x1
≤ad x0, x1
bδ x0, T
x0
cδ x1, T
x1
≤abqd x0, x1
cδ x1, T
x1
.
2.3
It follows that
δ x1, T
x1
≤ abq 1−c d
x0, x1
. 2.4
Forx1∈Tx0,there existsx2∈Tx1such that δ
x1, T x1
≤qd x1, x2
. 2.5
Then
δ x2, T
x2≤δ T
x1
, T x2
≤ad x1, x2
bδ x1, T
x1
cδ x2, T
x2
≤abqd x1, x2
cδ x2, T
x2
.
2.6
It follows that
δ x2, T
x2
≤ abq 1−c d
x1, x2
≤ abq 1−c δ
x1, T x1
≤
abq 1−c
2
d x0, x1
.
2.7
Inductively, we can construct a sequencexnn∈Nhaving the properties 1 αxn∈Txn−1, n∈N∗;
2 βdxn, xn1≤δxn, Txn≤abq/1−cndx0, x1.
We will prove now that the sequencexnn∈Nis Cauchy.
We successively have
d
xn, xnp
≤d
xn, xn1 d
xn1, xn2
· · ·d
xnp−1, xnp
≤ abq 1−c
n
abq 1−c
n1 · · ·
abq 1−c
np−1 d
x0, x1
.
2.8
Let us denoteα: abq/1−c. Then
d
xn, xnp
≤αn
1α· · ·αp−1 d
x0, x1
αnαp−1 α−1 d
x0, x1
. 2.9
If we choseq <1−a−c/b, thenα <1.
Lettingn→ ∞, sinceαn→0, it follows that
d
xn, xnp
−→0 asn−→ ∞. 2.10
Hencexnn∈Nis Cauchy.
By the completeness of the spaceX, d,we get that there exists x∗ ∈ X such that xn→x∗asn→ ∞.
Next, we will prove thatx∗∈SFT. We have
δ x∗, T
x∗
≤d x∗, xn
δ xn, T
xn
δ T
xn
, T x∗
≤d x∗, xn
δ xn, T
xn
ad xn, x∗
bδ xn, T
xn
cδ x∗, T
x∗ . 2.11
Then
δ x∗, T
x∗
≤ 1a 1−cd
x∗, xn
1b 1−cδ
xn, T xn
2.12
becauseδxn, Txn≤αndx0, x1⇒δx∗, Tx∗ 0⇒Tx∗ {x∗}i.e.,x∗∈SFT. For the last part of our proof, we will show the uniqueness of the strict fixed point.
Suppose that there existx∗, y∗∈SFT. Then d
x∗, y∗ δ
T x∗
, T y∗
≤ad x∗, y∗
bδ x∗, T
x∗ cδ
y∗, T y∗
. 2.13
If x∗ and y∗ are distinct points, then we get that a ≥ 1, which contradicts our hypothesis. Thusx∗y∗. The proof is complete.
Regarding the well-posedness of a fixed point problem, we have the following result.
Theorem 2.2. LetX, dbe a complete metric space and letT :X→PbXbe a multivalued operator.
Suppose there exista, b, c∈Rwithabc <1 such that δ
Tx, Ty
≤adx, y bδ
x, Tx cδ
y, Ty
, ∀x, y∈X. 2.14
Then the fixed point problem is well posed forTwith respect toHd. Proof. By Reich’s theorem, we get thatSFT{x∗}.
Letxn∈X,n∈Nsuch thatHdxn, Txn→0 asn→ ∞. Then Hd
xn, T xn
δd
xn, T xn
. 2.15
We have to show thatxn→x∗asn→ ∞. We successively have d
xn, x∗
≤δd
xn, T xn
δd
T xn
, T x∗
≤δd
xn, T xn
ad xn, x∗
bδd
xn, T xn
cδd
x∗, T x∗ 1bδd
xn, T xn
ad xn, x∗
.
2.16
It follows that d
xn, x∗
≤ 1b 1−aδd
xn, T xn
1b 1−aHd
xn, T xn
−→0, n−→ ∞. 2.17
Hence
xn−→x∗, n−→ ∞. 2.18
With respect to the same multivalued operators, a data dependence result can also be established as follows.
Theorem 2.3. LetX, dbe a complete metric space and letT1, T2 :X→PbXbe two multivalued operators. Suppose that
ithere exista, b, c∈Rwithabc <1 such that
δT1x, T1y≤adx, y bδx, T1x cδy, T1y, ∀x, y∈X 2.19
(denote the unique strict fixed point ofT1byx∗1);
ii SFT2/∅;
iiithere existsη >0 such thatδT1x, T2x≤η, for allx∈X.
Then
δ
x∗1,SFT2
≤ 1cη
1−a . 2.20
Proof. Letx2∗∈SFT2.Thenδx∗2, T2x∗2 0.
We have d
x∗1, x∗2 δ
T1
x∗1 , T2
x∗2
≤δ T1
x∗1 , T1
x∗2 δ
T1
x2∗ , T2
x∗2
≤ad x∗1, x∗2
bδ x1∗, T1
x∗1 cδ
x∗2, T1
x∗2 η ad
x∗1, x∗2 cδ
T2
x∗2 , T1
x∗2
η≤ad x∗1, x∗2
1cη.
2.21
It follows that
d x∗1, x∗2
≤ 1c
1−aη. 2.22
By taking supx∗
2∈SFT2, it follows that δ
x1∗,SFT2
≤ 1c
1−aη. 2.23
LetX, dbe a complete metric space and letF1, . . . , Fm :X→PXbe a finite family of multivalued operators.
The systemF F1, . . . , Fmis said to be an iterated multifunction system.
The operator
TF :PX−→PX, TFY m
i1
FiY, Y ∈PX 2.24
is called the multifractal operator generated by the iterated multifunction systemF F1, . . . , Fm.
Remark 2.4. iIfFi : X→PcpXare multivaluedαi-contractions for eachi ∈ {1,2, . . . , m}, then the multifractal operatorTFis anα-contraction too, whereα:max{αi|i∈ {1, . . . , m}}
Nadler Jr.7.
iiIfFi :X→PcpXare multivaluedϕi-contractionssee4for eachi∈ {1,2, . . . , m}, then the multifractal operatorTFis anϕ-contraction too, see Andres and Fiˇser4for the definitions and the result.
iiiIfF F1, . . . , Fmis an iterated multifunction system, such thatFi:X→PcpXis upper semicontinuous for eachi∈ {1, . . . , m}, then the multifractal operator
TF :PcpX−→PcpX, TFY m
i1
FiY 2.25
is well defined. A fixed pointY∗ ∈ PcpXofTF is called an attractor of the iterated multi- function systemF.
The following result is well known, see, for example, Granas and Dugundji11.
Lemma 2.5. LetX, dbe a complete metric space,x0∈X,r >0 and
B:B x0, r
x∈X|d x, x0
≤r
. 2.26
Letf:B→Xbe anα-contraction.
Ifdx0, fx0≤1−αr, thenfhas a unique fixed point inB.
Our next result concerns with the existence of an attractor for an iterated multifunction system.
Theorem 2.6. LetX, dbe a complete metric space,x0 ∈Xandr >0. LetFi :Bx 0, r→PcpX, i∈ {1, . . . , m}a finite family of multivalued operators.
Suppose that
iFiis anαi-contraction, for eachi∈ {1, . . . , m};
iiδx0, Fix0≤1−max{αi|i∈ {1, . . . , m}}r, for alli∈ {1, . . . , m}.
Then there existsY∗ ∈ B{x 0}, r ⊂ PcpXa unique attractor of the iterated multifunction systemF F1, . . . , Fm.
Proof. SinceFi:Bx 0, r→PcpXis anαi-contraction, for eachi∈ {1, . . . , m}it follows thatFi
is upper semicontinuous, for eachi∈ {1, . . . , m}. ByRemark 2.4iii, we get that the operator TF :B{x 0}, r⊂PcpX→PcpX,TFY m
i1FiY,Y ∈B{x 0}, ris well defined.
Any fixed point Y∗ ∈ B{x 0}, r ⊂ PcpX of TF is an attractor of the iterated multifunction systemF F1, . . . , Fm.
Notice first that, ifY ∈ B{x 0}, r ⊂ PcpX, H, thenH{x0}, Y≤ r, which implies thatdx0, y≤r, for ally∈Y. Thusy∈Bx 0, r, for ally∈Y.
We will show thatTFsatisfies the following two conditions:
iTF is anα-contraction, withα:max{αi|i∈ {1, . . . , m}}, that is, H TF
Y1
,TF
Y2
≤αH Y1, Y2
, ∀Y1, Y2∈B x0
, r
⊂PcpX; 2.27
iiH{x0},TF{x0}≤1−αr.
Indeed, we have
iLet Y1, Y2 ∈ B{x 0}, r ⊂ PcpXs¸iu ∈ TFY1. By the definition of TF, it follows that there existsj ∈ {1, . . . , m}and there existsy1 ∈Y1such thatu∈Fjy1. Since Y1, Y2∈PcpX, there existsy2∈Y2such thatdy1, y2≤HY1, Y2.
Since, for arbitraryε >0 and eachA, B∈PcpXwithHA, B≤ε,we have that for all a∈Athere existsb∈Bsuch thatda, b≤ε, by the following relations
H Fj
y1
, Fj
y2
≤αjd y1, y2
≤αjH Y1, Y2
, 2.28
we obtain that foru∈ Fjy1⊂ TFY1, there existsv ∈Fjy2 ⊂ TFY2such thatdu, v ≤ αjHY1, Y2≤αHY1, Y2.
By the above relation and by the similar onewhere the roles ofTFY1andTFY2are reversed, the first conclusion follows.
iiWe have to show that δ
x0
,TF
x0
≤1−αr 2.29
or equivalently for allu∈TF{x0},we havedx0, u≤1−αr. Sinceu∈TF{x0} it follows that there existsj∈ {1, . . . , m}such thatu∈Fjx0. Then
d x0, u
≤δ x0, Fj
x0
≤1−αr. 2.30
By Lemma 2.5, applied to TF, we get that there exists Y∗ ∈ B{x 0}, r ⊂ PcpX a unique fixed point for TF, that is, a unique attractor of the iterated multifunction system F F1, . . . , Fm. The proof is complete.
Remark 2.7. An interesting extension of the above results could be the case of a set endowed with two metrics, see12for other details.
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