Volume 2010, Article ID 178421,10pages doi:10.1155/2010/178421
Research Article
The Theory of Reich’s Fixed Point Theorem for Multivalued Operators
Tania Laz ˘ar,
1Ghiocel Mot¸,
2Gabriela Petrus¸el,
3and Silviu Szentesi
41Commercial Academy of Satu Mare, Mihai Eminescu Street No. 5, Satu Mare, Romania
2Aurel Vlaicu University of Arad, Elena Dragoi Street, No. 2, 310330 Arad, Romania
3Department of Business, Babes¸-Bolyai University, Cluj-Napoca, Horea Street No. 7, 400174 Cluj-Napoca, Romania
4Aurel Vlaicu University of Arad, Revoult¸iei Bd., No. 77, 310130 Arad, Romania
Correspondence should be addressed to Ghiocel Mot¸,[email protected] Received 12 April 2010; Revised 12 July 2010; Accepted 18 July 2010
Academic Editor: S. Reich
Copyrightq2010 Tania Laz˘ar 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.
The purpose of this paper is to present a theory of Reich’s fixed point theorem for multivalued operators in terms of fixed points, strict fixed points, multivalued weakly Picard operators, multivalued Picard operators, data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well- posedness of the fixed point problem, and the generated fractal operator.
1. Introduction
LetX, dbe a metric space and consider the following family of subsetsPclX :{Y ⊆X | Y is nonempty and closed}.We also consider the followinggeneralizedfunctionals:
D:PX×PX−→R, DA, B:inf{da, b|a∈A, b∈B}, 1.1 Dis called the gap functional betweenAandB. In particular, ifx0∈X,thenDx0, B:D{x0}, B:
ρ:PX×PX−→R∪ {∞}, ρA, B:sup{Da, B|a∈A}, 1.2 ρis called thegeneralizedexcess functional:
H:PX×PX−→R∪ {∞}, HA, B:max
ρA, B, ρB, A
, 1.3
His thegeneralizedPompeiu-Hausdorfffunctional.
It is well known that ifX, dis a complete metric space, then the pairPclX, His a complete generalized metric space.See1,2.
Definition 1.1. IfX, dis a metric space, then a multivalued operatorT :X → PclXis said to be a Reich-type multivalueda, b, c-contraction if and only if there exista, b, c∈Rwith abc <1 such that
H
Tx, T y
≤ad x, y
bDx, Tx cD y, T
y
, for eachx, y∈X. 1.4 Reich proved that any Reich-type multivalued a, b, c-contraction on a complete metric space has at least one fixed pointsee3.
In a recent paper Petrus¸el and Rus introduced the concept of “theory of a metric fixed point theorem” and used this theory for the case of multivalued contractionsee4. For the singlevalued case, see5.
The purpose of this paper is to extend this approach to the case of Reich-type multivalueda, b, c-contraction. We will discuss Reich’s fixed point theorem in terms of
ifixed points and strict fixed points, iimultivalued weakly Picard operators, iiimultivalued Picard operators,
ivdata dependence of the fixed point set,
vsequence of multivalued operators and fixed points, viUlam-Hyers stability of a multivaled fixed point equation, viiwell-posedness of the fixed point problem;
viiifractal operators.
Notice also that the theory of fixed points and strict fixed points for multivalued operators is closely related to some important models in mathematical economics, such as optimal preferences, game theory, and equilibrium of an abstract economy. See6for a nice survey.
2. Notations and Basic Concepts
Throughout this paper, the standard notations and terminologies in nonlinear analysis are usedsee the papers by Kirk and Sims7, Granas and Dugundji8, Hu and Papageorgiou 2, Rus et al.9, Petrus¸el10, and Rus11.
LetXbe a nonempty set. Then we denote.
PX {Y |Y is a subset ofX}, PX
Y ∈ PX|Y is nonempty
. 2.1
LetX, dbe a metric space. ThenδY sup{da, b|a, b∈Y}and PbX {Y ∈PX|δY<∞}, PcpX
Y ∈PX|Y is compact
. 2.2
LetT :X → PXbe a multivalued operator. Then the operatorT :PX → PX, which is defined by
TY:
x∈Y
Tx, forY ∈PX, 2.3
is called the fractal operator generated byT. For a well-written introduction on the theory of fractals see the papers of Barnsley12, Hutchinson13, Yamaguti et al.14.
It is known that ifX, dis a metric space andT : X → PcpX, then the following statements hold:
aifT is upper semicontinuous, thenTY∈PcpX, for everyY ∈PcpX;
bthe continuity ofT implies the continuity ofT:PcpX → PcpX.
The set of all nonempty invariant subsets ofT is denoted byIT, that is,
IT:{Y ∈PX|TY⊂Y}. 2.4
A sequence of successive approximations of T starting from x ∈ X is a sequence xnn∈Nof elements inXwithx0x, xn1∈Txn, forn∈N.
IfT :Y ⊆ X → PX,thenFT :{x∈Y |x∈Tx}denotes the fixed point set ofT andSFT :{x∈Y | {x}Tx}denotes the strict fixed point set ofT. By
GraphT: x, y
∈Y×X:y∈Tx
2.5 we denote the graph of the multivalued operatorT.
IfT :X → PX, thenT0 :1X, T1:T, . . . , Tn1T◦Tn, n∈N,denote the iterate operators ofT.
Definition 2.1 see 15. Let X, d be a metric space. Then, T : X → PX is called a multivalued weakly Picard operator briefly MWP operator if for each x ∈ X and each y∈Txthere exists a sequencexnn∈NinXsuch that
ix0xandx1y;
iixn1 ∈Txnfor alln∈N;
iiithe sequencexnn∈Nis convergent and its limit is a fixed point ofT.
For the following concepts see the papers by Rus et al.15, Petrus¸el10, Petrus¸el and Rus16, and Rus et al.9.
Definition 2.2. LetX, dbe a metric space, and letT :X → PXbe an MWP operator. The multivalued operatorT∞: GraphT → PFTis defined by the formulaT∞x, y {z∈FT | there exists a sequence of successive approximations ofTstarting fromx, ythat converges toz}.
Definition 2.3. LetX, dbe a metric space andT :X → PXan MWP operator. ThenT is said to be ac-multivalued weakly Picard operatorbrieflyc-MWP operatorif and only if there exists a selectiont∞ofT∞such thatdx, t∞x, y≤cdx, yfor allx, y∈GraphT.
We recall now the notion of multivalued Picard operator.
Definition 2.4. LetX, dbe a metric space andT : X → PX. By definition, T is called a multivalued Picard operatorbriefly MP operatorif and only if
i SFT FT {x∗};
iiTnx → {xH ∗}asn → ∞, for eachx∈X.
In10other results on MWP operators are presented. For related concepts and results see, for example,1,17–23.
3. A Theory of Reich’s Fixed Point Principle
We recall the fixed point theorem for a single-valued Reich-type operator, which is needed for the proof of our first main result.
Theorem 3.1see3. LetX, dbe a complete metric space, and letf :X → Xbe a Reich-type single-valueda, b, c-contraction, that is, there exista, b, c∈Rwithabc <1 such that
d
fx, f y
≤ad x, y
bd
x, fx cd
y, f y
, for eachx, y∈X. 3.1
Thenfis a Picard operator, that is, we have:
iFf {x∗};
iifor eachx∈Xthe sequencefnxn∈Nconverges inX, dtox∗.
Our main result concerning Reich’s fixed point theorem is the following.
Theorem 3.2. LetX, d be a complete metric space, and let T : X → PclXbe a Reich-type multivalueda, b, c-contraction. Letα: ab/1−c. Then one has the following
iFT/∅;
iiT is a 1/1−α-multivalued weakly Picard operator;
iiiletS:X → PclXbe a Reich-type multivalueda, b, c-contraction andη >0 such that HSx, Tx≤ηfor eachx∈X, thenHFS, FT≤η/1−α;
ivletTn:X → PclX(n∈N) be a sequence of Reich-type multivalueda, b, c-contraction, such thatTnx →H Txuniformly asn → ∞. Then,FTn
→H FTasn → ∞.
If, moreoverTx∈PcpXfor eachx∈X, then one additionally has:
v(Ulam-Hyers stability of the inclusion x ∈ Tx) Let > 0 and x ∈ X be such that Dx, Tx≤,then there existsx∗∈FTsuch thatdx, x∗≤/1−α;
viT :PcpX, H → PcpX, H,TY:
x∈YTxis a set-to-seta, b, c-contraction and (thus)FT{A∗T};
viiTnx →H A∗Tasn → ∞, for eachx∈X;
viiiFT ⊂A∗T andFTare compact;
ixA∗T
n∈N\{0}Tnxfor eachx∈FT.
Proof. iLetx0∈Xandx1∈Tx0be arbitrarily chosen. Then, for each arbitraryq >1 there existsx2∈Tx1such thatdx1, x2≤qHTx0, Tx1. Hence
dx1, x2≤qadx0, x1 bDx0, Tx0 cDx1, Tx1
≤qadx0, x1 bdx0, x1 cdx1, x2. 3.2 Thus
dx1, x2≤ qab
1−qc dx0, x1. 3.3
Denoteβ:qab/1−qc.By an inductive procedure, we obtain a sequence of successive approximations forT starting fromx0, x1 ∈ GraphTsuch that, for eachn ∈N, we have dxn, xn1≤βndx0, x1.Then
d
xn, xnp
≤βn1−βp
1−β dx0, x1, for eachn, p∈N\ {0}. 3.4 If we choose 1< q <1/abc, then by3.4we get that the sequencexnn∈Nis Cauchy and hence convergent inX, dto somex∗∈X
Notice that, by Dx∗, Tx∗ ≤ dx∗, xn1 Dxn1, Tx∗ ≤ dxn1, x∗ HTxn, Tx∗ ≤ dxn1, x∗ adxn, x∗ bDxn, Txn cDx∗,Tx∗ ≤ dxn1, x∗ adxn, x∗ bdxn, xn1 cDx∗, Tx∗, we obtain that
Dx∗, Tx∗≤ 1
1−cdxn1, x∗ adxn, x∗ bdxn, xn1−→ as n−→∞. 3.5 Hencex∗∈FT.
iiLetp → ∞in3.4. Then we get that dxn, x∗≤βn 1
1−βdx0, x1 for eachn∈N\ {0}. 3.6 Forn1,we get
dx1, x∗≤ β
1−βdx0, x1. 3.7
Then
dx0, x∗≤dx0, x1 dx1, x∗≤ 1
1−βdx0, x1. 3.8
Letq1 in3.8, then
dx0, x∗≤ 1
1−αdx0, x1. 3.9
HenceTis a 1/1−α-multivalued weakly Picard operator.
iiiLetx0∈FSbe arbitrarily chosen. Then, byii, we have that
dx0, t∞x0, x1≤ 1
1−αdx0, x1, for eachx1∈Tx0. 3.10 Letq >1 be an arbitrary. Then, there existsx1∈Tx0such that
dx0, t∞x0, x1≤ 1
1−αqHSx0, Tx0≤ qη
1−α. 3.11
In a similar way, we can prove that for eachy0 ∈FTthere existsy1 ∈Sy0such that d
y0, s∞ y0, y1
≤ qη
1−α. 3.12
Thus,3.11and3.12together imply thatHFS, FT≤qη/1−αfor everyq >1. Letq1 and we get the desired conclusion.
ivfollows immediately fromiii.
vLet >0 andx∈Xbe such thatDx, Tx≤. Then, sinceTxis compact, there existsy∈Txsuch thatdx, y≤. From the proof ofi, we have that
d x, t∞
x, y
≤ 1 1−αd
x, y
. 3.13
Sincex∗:t∞x, y∈FT, we get thatdx, x∗≤/1−α.
viWe will prove for anyA, B∈PcpXthat
HTA, TB≤aHA, B bHA, TA cHB, TB. 3.14 For this purpose, letA, B ∈ PcpXand let u ∈ TA. Then, there existsx ∈ A such that u∈Tx. Since the setsA, Bare compact, there existsy∈Bsuch that
d x, y
≤HA, B. 3.15
From3.15we get thatDu, TB≤Du, Ty≤HTx, Ty≤adx, y bDx, Tx cDy, Ty≤adx, y bρA, Tx cρB, Ty≤aHA, B bρA, TA cρB, TB≤ aHA, B bHA, TA cHB, TB. Hence
ρTA, TB≤aHA, B bHA, TA cHB, TB. 3.16 In a similar way we obtain that
ρTB, TA≤aHA, B bHA, TA cHB, TB. 3.17
Thus,3.16and3.17together imply that
HTA, TB≤aHA, B bHA, TA cHB, TB. 3.18 Hence, T is a Reich-type single-valued a, b, c-contraction on the complete metric space PcpX, H. FromTheorem 3.1we obtain that
aFT{A∗T}and
bTnA →H A∗Tasn → ∞, for eachA∈PcpX.
viiFromvi-bwe get thatTn{x} Tn{x} →H A∗Tasn → ∞, for eachx∈X.
viii-ix Letx ∈ FT be an arbitrary. Thenx ∈ Tx ⊂ T2x ⊂ · · · ⊂ Tnx ⊂ · · ·. Hencex ∈ Tnx, for eachn ∈ N∗. Moreover, limn→∞Tnx
n∈N∗Tnx. Fromvii, we immediately get thatA∗T
n∈N∗Tnx. Hencex∈
n∈N∗Tnx A∗T. The proof is complete.
A second result for Reich-type multivalueda, b, c-contractions formulates as follows.
Theorem 3.3. LetX, dbe a complete metric space andT :X → PclXa Reich-type multivalued a, b, c-contraction withSFT/∅. Then, the following assertions hold:
(x)FT SFT{x∗};
(xi) (Well-posedness of the fixed point problem with respect toD[24]) Ifxnn∈Nis a sequence inXsuch thatDxn, Txn → 0 asn → ∞, thenxn →d x∗asn → ∞;
(xii) (Well-posedness of the fixed point problem with respect toH[24]) Ifxnn∈Nis a sequence inXsuch thatHxn, Txn → 0 asn → ∞, thenxn →d x∗as n → ∞.
Proof. x Let x∗ ∈ SFT. Note that SFT {x∗}. Indeed, if y ∈ SFT, then dx∗, y HTx∗, Ty≤adx∗, y bDx∗, Tx∗ cDy, Ty adx∗, y. Thusyx∗.
Let us show now thatFT {x∗}. Suppose thaty∈FT. Then,dx∗, y DTx∗, y≤ HTx∗, Ty ≤ adx∗, y bDx∗, Tx∗ cDy, Ty adx∗, y. Thusy x∗. Hence FT ⊂SFT {x∗}. SinceSFT ⊂FT, we get thatSFT FT {x∗}.
xiLet xnn∈Nbe a sequence inX such thatDxn, Txn → 0 asn → ∞. Then, dxn, x∗ ≤ Dxn, Txn HTxn, Tx∗ ≤ Dxn, Txn adxn, x∗ bDxn, Txn
cDx∗, Tx∗ 1 bDxn, Txn adxn, x∗. Then dxn, x∗ ≤ 1 b/1 − aDxn, Txn → 0 asn → ∞.
xiifollows byxisinceDxn, Txn≤Hxn, Txn → 0 asn → ∞.
A third result for the case ofa, b, c-contraction is the following.
Theorem 3.4. Let X, dbe a complete metric space, and let T : X → PcpX be a Reich-type multivalueda, b, c-contraction such thatTFT FT. Then one has
xiiiTnx →H FTasn → ∞, for eachx∈X;
xivTx FT for eachx∈FT;
xvIfxnn∈N⊂Xis a sequence such thatxn →d x∗∈FTasn → ∞andTisH-continuous, thenTxn →H FTasn → ∞.
Proof. xiiiFrom the fact thatTFT FT andTheorem 3.2viwe have thatFT A∗T. The conclusion follows byTheorem 3.2vii.
xivLetx∈FTbe an arbitrary. Thenx∈Tx, and thusFT ⊂Tx. On the other hand Tx⊂TFT⊂FT. ThusTx FT, for eachx∈FT.
xvLetxnn∈N⊂Xbe a sequence such thatxn →d x∗∈FT asn → ∞. Then, we have Txn →H Tx∗ FT asn → ∞. The proof is complete.
For compact metric spaces we have the following result.
Theorem 3.5. LetX, dbe a compact metric space, and letT : X → PclXbe a H-continuous Reich-type multivalueda, b, c-contraction. Then
(xvi) ifxnn∈Nis such thatDxn, Txn → 0 asn → ∞, then there exists a subsequence xnii∈Nofxnn∈Nsuch thatxni
→d x∗∈FTasi → ∞(generalized well-posedness of the fixed point problem with respect toD[24,25]).
Proof. xviLetxnn∈Nbe a sequence inXsuch thatDxn, Txn → 0 asn → ∞. Letxnii∈N be a subsequence ofxnn∈Nsuch thatxni
→d x∗asi → ∞. Then, there existsyni ∈Txni, such thatyni
→d x∗asi → ∞. ThenDx∗, Tx∗≤dx∗, yni Dyni, Txni HTxni, Tx∗≤ dx∗, yni adx∗, xni bDxni, Txni cDx∗, Tx∗. Hence
Dx∗, Tx∗≤ 1 1−c d
x∗, yni
adx∗, xni bDxni, Txni
−→0 3.19
asn → ∞.Hencex∗∈FT.
Remark 3.6. Forb c 0 we obtain the results given in4. On the other hand, our results unify and generalize some results given in12,13,17,26–34. Notice that, if the operatorTis singlevalued, then we obtain the well-posedness concept introduced in35.
Remark 3.7. An open question is to present a theory of the ´Ciri´c-type multivalued contraction theoremsee36. For some problems for other classes of generalized contractions, see for example,17,21,27,34,37.
Acknowledgments
The second and the forth authors wish to thank National Council of Research of Higher Education in RomaniaCNCSISby “Planul National, PN II2007–2013—Programul IDEI- 1239” for the provided financial support. The authors are grateful for the reviewersfor the careful reading of the paper and for the suggestions which improved the quality of this work.
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