Volume 2010, Article ID 452905,18pages doi:10.1155/2010/452905
Research Article
Remarks on Recent Fixed Point Theorems
S. L. Singh and S. N. Mishra
Department of Mathematics, School of Mathematical & Computational Sciences, Walter Sisulu University, Nelson Mandela Drive, Mthatha 5117, South Africa
Correspondence should be addressed to S. N. Mishra,[email protected] Received 10 October 2009; Revised 11 February 2010; Accepted 27 April 2010 Academic Editor: Tomonari Suzuki
Copyrightq2010 S. L. Singh and S. N. Mishra. 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.
Coincidence and fixed point theorems for a new class of contractive, nonexpansive and hybrid contractions are proved. Applications regarding the existence of common solutions of certain functional equations are also discussed.
1. Introduction
The following remarkable generalization of the classical Banach contraction theorem, due to Suzuki 1, has led to some important contribution in metric fixed point theorysee, e.g., 1–8.
Theorem 1.1. LetX, dbe a complete metric space andS:X → X.Define a nonincreasing function θfrom0,1onto1/2,1by
θr
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
1 if 0≤r≤
√5−1 2 , 1−r
r2 if
√5−1
2 ≤r≤ 1 2, 1
1r if 1
2 ≤r <1.
1.1
Assume that there existsr∈0,1such that θrdx, Sx≤d
x, y
impliesd
Sx, Sy
≤rd x, y
SC for allx, y ∈X.ThenShas a unique fixed point. A map satisfying conditionSCis called Suzuki contraction and the above theorem as the Suzuki contraction theorem (see [9]).
FollowingTheorem 1.1, Edelstein’s theorem for contractive maps has been generalized in7 cf. Theorem 2.1. Fixed point theorems for nonexpansive maps due to Browder10, 11and G ¨ohde12have been generalized in6 cf.Theorem 3.1below.Theorem 1.1and Nadler’s multivalued contraction theorem have been generalized by Kikkawa and Suzuki 2 cf.Theorem 4.1below. Further,Theorem 4.1has been generalized by Mot¸ and Petrus¸el 3, Dhompongsa and Yingtaweesittikul 4, Singh and Mishra9and others. Combining the ideas of Suzuki6,7, Goebel13and Naimpally et al.14, first we generalize Theorems 2.1and3.1to a wider class of maps on an arbitrary nonempty set with values in a metric resp. Banachspace. Using the notion of IT-commuting maps due to Itoh and Takahashi 15, we obtain generalizations of multivalued fixed point theorems due to Reich16, Iseki 17, Kikkawa and Suzuki2, Mot¸ and Petrus¸el3, Dhompongsa and Yingtaweesittikul4 and others to the case of Suzuki generalized hybrid contractioncf.Theorem 4.1. Various examples presented inSection 5demonstrate the generality of the assumptions used in our results. An experimental approach regarding the sequence of Jungck iterates18for the new class of contractive and nonexpansive maps is also discussed, which leads to a conjecture.
Finally, we deduce the existence of a common solution for the Suzuki class of functional equations under much weaker conditions than those in19–21.
2. Contractive Maps
The following result of Suzuki 7 generalizes the well-known fixed point theorem of Edelstein22.
Theorem 2.1. LetX, dbe a compact metric space andS:X → X.Assume that 1
2dx, Sx< d x, y
impliesd Sx, Sy
< d
x, y 2.1
forx, y∈X.ThenShas a unique fixed point.
Throughout this paperY will denote an arbitrary nonempty set. As a generalization of the results of Goebel13, Edelstein22and Naimpally et al.14, Corollary 3, we extend Theorem 2.1for a pair of Suzuki contractive mapsS, T:Y → Xcf.2.2and2.3, wherein X, dis a metric space.
Theorem 2.2. LetS, T :Y → X be such thatSY ⊆TYandTYis a compact subspace ofX.
Assume that forx, y∈Y,
1
2dTx, Sx< d
Tx, Ty 2.2
implies
d
Sx, Sy
< d
Tx, Ty
, 2.3
andTxTyimpliesSxSy. ThenSandThave a coincidence, that is, there existsz∈Ysuch that Sz Tz.Further, ifY X,thenSandT have a unique common fixed point provided thatSandT commute atz.
Proof. DefineF : TY → TYbyFa ST−1afor eacha∈ TY.To see thatF is well defined, observe bySY⊆TYthat forx∈T−1a,
FaSx, Fa⊆TY. 2.4
Takex, y∈T−1asuch thatbSx, cSy.Then, sinceTxTy,we havebc.ThereforeFis well-defined.
Now, fora /banda, b∈TY, T−1a∩T−1bφ.So, for distincta, b∈TY, we suppose 1/2da, Fa< da, b. Then forx∈T−1aandy∈T−1b,we have
1
2dTx, Sx 1
2da, Fa< da, b d
Tx, Ty
. 2.5
This inequality implies that dSx, Sy < dTx, Ty. SodFa, Fb < da, b. Therefore, by Theorem 2.1,Fhas a unique fixed pointw.Then for anyz∈T−1w, SzFw w Tz.So,z is a coincidence point ofSandT.IfSandTare commuting atz,thenSzTz⇒SSzSTz TSz TTzand SwTw.IfSz /SSz,then1/2dTz, Sz 0 < dTz, TTz dTz, TSz, and this implies thatdw, Sw dSz, SSz< dTz, TSz dw, Sw,a contradiction. So,w is a common fixed point ofSandT.
We conclude the proof by showing the unicity of the common fixed point. Suppose thatv /wis another common fixed point ofSandT.Since1/2dTw, Sw 0< dTw, Tv, we havedw, v dSw, Sv< dTw, Tv w, v,a contradiction. Hencevw.
3. Nonexpansive Maps
A self-mapS of a metric spaceX is nonexpansive ifdSx, Sy ≤ dx, y for allx, y ∈ X.
The theory of nonexpansive maps is exciting and plays a vital role in nonlinear analysis and applicationssee, e.g.,10–12,23–28. Recently, Suzuki6obtained the following theorem that generalizes the results of Browder10,11and G ¨ohde12.
Theorem 3.1. Let C be a convex subset of a Banach spaceEandS:C → C.Assume that 1
2 x−Sx ≤x−y impliesSx−Sy≤x−y 3.1 for allx, y∈C. Assume further that one of the following holds:
iCis compact;
iiCis weakly compact andEhas the Opial property;
iiiCis weakly compact andEis uniformly convex in every direction (UCED).
ThenShas a fixed point.
For definitions and details of the Opial property25, uniform convexity and UCED, one may refer to Goebel 23, Goebel and Kirk 24, Prus26, Suzuki 6 and Takahashi 27,28.
Now we present the following extension of Theorem 3.1 for a pair of Suzuki nonexpansive mapscf.3.2.
Theorem 3.2. LetEbe a Banach space andS, T : Y → Esuch thatSY ⊆ TYandTYis a convex subset ofE.Assume that forx, y∈Y,
1
2 Tx−Sx ≤Tx−TyimpliesSx−Sy≤Tx−Ty, 3.2 andTxTyimpliesSxSy. Assume further that either of the following holds:
iTYis compact;
iiTYis weakly compact andEhas the Opial property;
iiiTYis weakly compact andEis UCED.
ThenSandT have a coincidence.
Proof. As in the proof ofTheorem 2.1, lettingFaST−1afora∈TY,it suffices to show thatF:TY → TYhas a fixed point.
Leta, b∈TYsuch that1/2 a−Fa ≤ a−b .Then forx∈T−1aandy∈T−1b, we have1/2 Tx−Sx 1/2 a−Fa ≤ a−b Tx−Ty .By3.2, we obtain Sx−Sy ≤ Tx−Ty ,and thus Fa−Fb ≤ a−b .So, byTheorem 3.1,Fhas a fixed point.
4. Multivalued Contractions
In all that follows, let CBX resp. CLXdenote the family of all nonempty closed bounded resp. closedsubsets ofX. LetHdenote the Hausdorffmetric induced by the metricdof the metric spaceX.For any subsetsA, BofX,dA, Bdenotes the gap between the subsetsAand B,whileρA, B sup{da, b:a∈A, b∈B}and
BNX A:φ /A⊆X and the diameter ofAis finite
. 4.1
The following result of Kikkawa and Suzuki2is a generalization of Nadler29.
Theorem 4.1. Let X, d be a complete metric space and P : X → CBX. Define a strictly decreasing functionηfrom0,1onto1/2,1by
ηr 1
1r. 4.2
Assume that there existsr∈0,1such that ηrdx, P x≤d
x, y
impliesH
P x, P y
≤rd x, y
KSMC
for allx, y∈X.ThenPhas a fixed point, that is, there existsz∈Xsuch thatz∈P z.
Theorem 4.1has further been generalized by Mot¸ and Petrus¸el3, Dhompongsa and Yingtaweesittikul4and Singh and Mishra9.
Fora, b, c, e, f∈0,1,letβandγbe defined by β 1−b−c
1a , γ 1−c−e
1a . 4.3
For a metric spaceX,we considerP:Y → CLXandT :Y → Xsatisfying γdTx, P x≤d
Tx, Ty
impliesH
P x, P y
≤M x, y
4.4
for allx, y∈Y,where M
x, y ad
Tx, Ty
bdTx, P x cd
Ty, P y ed
Tx, P y fd
Ty, P x
4.5 andabcef <1.
We remark thatβ, γ ∈ 1/2,1.As regards the generality of condition4.4, we offer the following remarks whenY XandTis the identity map onX.
Remarks 4.2. iThe Kikkawa-Suzuki multivalued contractionKSMCis4.4witharand bcef0.
iiGeneralizing theKSMC, Mot¸ and Petrusel3have studied4.4withef 0 andγβ.
iiiDhompongsa and Yingtaweesittikul4have discussed4.4when M
x, y
γ·max d x, y
, dx, P x, d y, P y
4.6 withγθrand some additional requirement.
iv Condition 4.4 includes a few important conditions for single-valued and multivalued maps due to Reich16, 30, Hardy and Rogers 31, and Iseki17 see also condition16in Rhoades32.
By virtue of the symmetry inxandyin the expressionMx, y, it is appropriate to consider4.4whenbcandefas follows:
βdTx, P x≤d
Tx, Ty
impliesH
P x, P y
≤m x, y
KSG
for allx, y∈Y,where m
x, y ad
Tx, Ty b
dTx, P x d
Ty, P y c
d
Tx, P y d
Ty, P x
, 4.7
anda2b2c <1.
In all that follows, we consider the nontrivial case 0< a2b2c.
The condition KSGwill be called Kikkawa-Suzuki generalized hybrid contraction for the mapsP andT.Following Itoh and Takahashi15 see also Singh and Mishra33, mapsP :X → CLXandT :X → Xare IT-commuting atz∈XifTP z⊆P Tz.We remark that IT-commuting maps are more general than commuting maps, weakly commuting maps and weakly compatible maps at a point. For details, one may refer to33.
Theorem 4.3. LetXbe a metric space. Assume that the pair of mapsP :Y → CLXandT :Y → Xis Kikkawa-Suzuki generalized hybrid contraction such thatPY⊆TY,andPYorTYis a complete subspace ofX.ThenP andT have a coincidence point, that is, there existsz∈Y such that Tz ∈ P z.Further, ifY X,thenP and T have a common fixed point provided thatP and T are IT-commuting atzandTzis a fixed point ofT.
Proof. Letq a2b2c−1/2.Pick x0 ∈ Y. Following Singh and Kulshrestha 34and Rhoades et al.35, we construct two sequences {xn} ⊆ Y and {yn Txn} ⊆ TY in the following manner. SincePY ⊆ TY,we choose an element x1 ∈ Y such thatTx1 ∈ P x0. Analogously, chooseTx2∈P x1such that
dTx1, Tx2≤qHP x0, P x1. 4.8
In general, we have sequences{xn}and{Txn}such thatTxn1∈P xn, n0,1, . . . , q >
1 and
dTxn1, Txn2≤qHP xn, P xn1, n0,1, . . . . 4.9
Sinceβ < 1,we see thatβdTxn, P xn ≤ dTxn, Txn1.Therefore by the assumption KSG,
d
yn1, yn2
≤qHP xn, P xn1
≤q ad
yn, yn1 bd
yn, P xn bd
yn1, P xn1 c d
yn, P xn1 d
yn1,P xn
≤q
abd
yn, yn1 bd
yn1, yn2 c d
yn, yn1 d
yn1, yn2 , 4.10
yielding
d
yn1, yn2
≤λd
yn, yn1
, 4.11
whereλqabc/1−qbc<1.So the sequence{yn}is Cauchy. IfTYis complete, then it has a limit inTY.IfPYis complete, then the limit is still inTYasPY⊆TY. Call the limitw.Letz∈T−1w.Thenz∈Y andTzw.Now as in2, we show that
dTz, P x≤ abc
β1adTz, Tx 4.12
for anyTx∈TY− {Tz}. Sinceyn → Tz,there exists a positive integern0such that dTz, Txn≤ 1
3dTz, Tx ∀n≥n0. 4.13
Therefore for anyn≥n0,
βdTxn, P xn≤dTxn, Txn1
≤dTxn, Tz dTxn1,Tz
≤ 2
3dTz, Tx dTz, Tx−1
3dTz, Tx
≤dTz, Tx−dTz, Txn≤dTxn, Tx.
4.14
Hence by the assumptionKSG, d
yn1, P x
≤HP xn, P x≤mxn, x
≤ad yn, Tx
b d
yn, yn1
dTx, P x c
d yn,P x
d
Tx, yn1
. 4.15 Makingn → ∞,we have
dTz, P x≤
abc 1−b−c
dTz, Tx. 4.16
This yields4.12,Tx /Tz.Next we show that
HP x, P z≤mx, z 4.17
for anyx∈Y.Ifxz,then it holds trivially. So we takex /zsuch thatTx /Tz.We can do so since, without any loss of generality, we take the mapTnonconstant. By4.12,
dTx, P x≤dTx, Tz dTz, P x
≤dTx, Tz
abc 1−b−c
dTz, Tx. 4.18
HenceβdTx, P x≤dTx, Tz.This implies4.17. Therefore d
yn1, P z
≤HP xn, P z≤mxn, z
≤ad yn, Tz
b d
yn, yn1
dTz, P z c
d yn, P z
d
Tz, yn1
. 4.19 Makingn → ∞,this yields1−b−cdTz, P z≤0,andTz∈P z.
Further, ifY X, TTzTz, andP andTare IT-commuting atz,thenTz∈P zimplies thatTTz∈TP z⊆P Tz.This proves thatTzis a fixed point ofP.
We remark that the assumption thatT has a fixed point in Theorem 4.3is essential.
Indeed, in general, a pair of continuous commuting maps on the space need not have a common fixed pointsee, e.g.,14,33.
Corollary 4.4. LetXbe a complete metric space andP :X → CLX.Assume there exista, b, c∈ 0,1such that
βdx, P x≤d x, y
impliesH
P x, P y
≤N P;x, y
4.20
for allx, y∈X,where N
P;x, y ad
x, y b
dx, P x d y, P y
c d
x, P y d
y, P x
4.21
anda2b2c <1.
Proof. It comes fromTheorem 4.1whenY XandT is the identity map.
The following two results are the extensions of Suzuki contraction theorem.
Corollary 4.5also generalizes the results of Kikkawa and Suzuki2, Theorem 2, Jungck18 and Dhompongsa and Yingtaweesittikul4, Theorem 3.4v.
Corollary 4.5. Letg, T : Y → X be such that gY ⊆ TYand gY or TYis a complete subspace ofX.Assume that there exista, b, c∈0,1such that
βd
Tx, gx
≤d
Tx, Ty
4.22
implies
d gx, gy
≤ad
Tx, Ty b
d Tx, gx
d
Ty, gy c
d
Tx, gy d
Ty, gx
4.23 for allx, y ∈X,wherea2b2c < 1.Theng andT have a coincidence pointz ∈ Y.Further, if Y Xandg, T commute atz,thengandThave a unique common fixed point.
Proof. SetP x{gx}for everyx∈Y.Then it comes fromTheorem 4.1that there existsz∈Y such thatgz Tz.Further, ifY X andg, T commute atz,then ggz gTz Tgz.Also, βdTz, gz 0≤dTz, Tgzand this implies
d
gz, ggz
≤ad
Tz, Tgz b
d Tz, gz
d
Tgz, ggz c
d
Tz, ggz d
Tgz, gz a2cd
gz, ggz .
4.24 This proves thatgzis a common fixed point ofgandT.The uniqueness of the common fixed point follows easily.
Corollary 4.6. LetXbe a complete metric space andg :X → X.Assume that there exista, b, c∈ 0,1such thatβdx, gx≤dx, yimpliesdgx, gy≤Ng;x, yfor allx, y∈X,wherea2b 2c <1.Thenghas a unique fixed point.
Proof. It follows fromCorollary 4.5ifY XandTis the identity map onX.
Theorem 4.7. LetX be a metric space, and letP : Y → BNXand T : Y → X be such that PY ⊆ TYandTYis a complete subspace ofX.Assume that there exista, b, c ∈0,1such that
βρTx, P x≤d
Tx, Ty
4.25
implies
ρ
P x, P y
≤ad
Tx, Ty b
ρTx, P x ρ
Ty, P y c
d
Tx, P y d
Ty, P x
4.26
for allx, y∈Y,wherea2b2c <1. Then the mapsTandPhave a coincidence.
Proof. It may be completed following Reich 30 and ´Ciri´c 36 and using Corollary 4.5.
However, for the sake of completeness, we give an outline of the same. Letta2b2c.For p∈0,1; define a single-valued mapg :Y → Xas follows. For each x∈Y,letgxbe a point ofP xsuch that dTx, gx ≥ tpρTx, P x.Notice thatfY
{fx ∈ P x} ⊆ PY ⊆ TY. Sincegx∈P x, dTx, gx≤ρTx, P x.So4.25givesβdTx, gx≤βρTx, P x≤dTx, Ty, and this implies condition4.26. Therefore
d gx, gy
≤ρ
P x, P y
≤t−p atpd
Tx, Ty btp
ρTx, P x ρ
Ty, P y ctp
d
Tx, P y d
Ty, P x
≤t−p ad
Tx, Ty b
d Tx, gx
d
Ty, gy c
d
Tx, gy d
Ty, gx .
4.27 So, takingaat−p,bbt−p,cct−pandβ 1−b−c/1a,we see thatβdTx, gx≤ βdTx, gx≤dTx, Tyimplies
d gx, gy
≤ad
Tx, Ty b
d Tx, gx
d
Ty, gy c
d
Tx, gy d
Ty, gx
, 4.28
wherea2b2cat−p2bt−p2ct−pt1−p<1. Hence, by virtue ofCorollary 4.5,gandT have a coincidence atz∈Y. EvidentlyTzgzimpliesTz∈P z.
Theorem 4.8. Let X be a complete metric space andP : Y → BNX.Assume that there exist a, b, c∈0,1such thatβρx, P x≤dx, yimplies
ρ
P x, P y
≤ad x, y
b
ρx, P x ρ y, P y
c d
x, P y d
y, P x
4.29 for allx, y∈X,wherea2b2c <1.ThenPhas a unique fixed point.
Proof. It may be completed, as above, usingCorollary 4.6.
5. Examples and Discussion
The following example shows that the Suzuki contractive conditioncf.2.2and2.3for a pair of maps is indeed more useful than condition2.1for a map on a metric space. In all the examples of this section, spaces are endowed with the usual metric.
Example 5.1. LetX 0,11/10and letS, T :X → Xbe defined by,
Sx
⎧⎪
⎨
⎪⎩
0 if 0≤x≤ 1 2, 1
2 if 1
2 < x≤ 11 10,
Tx
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
0 ifx0, 1
2 if 0< x≤ 1 2, 11
10 if 1
2 < x≤ 11 10.
5.1
Then assumption 2.1of Theorem 2.1is not satisfied for the mapStake, e.g., x 25/100, y 51/100.However,SandT satisfy all the assumptions ofTheorem 2.2. Notice that Sx Sy whenever Tx Ty for any x, y ∈ Y. Moreover, SX {0,1/2} ⊂ {0, 1/2,11/10} TX. So, Theorem 2.2guarantees the existence of a coincidence point, namely, 0 which is the unique common fixed point ofSandT.
Sequence of Iterates
For mapsSandT studied in Theorems2.2and3.2, a sequence of iterates may be constructed following Jungck18. For anyx0 ∈ Y,choose an x1 ∈ Y such thatTx1 Sx0.We can do this sinceSY ⊆ TY.Now choosex2 ∈ Y such thatTx2 Sx1.Continuing this process, we choosexn1 ∈ Y such that Txn1 Sxn, n 0, 1, 2, . . .. For the sake of brevity and appropriate reference, the sequence{Txn}will be called Jungck sequence of iterates or simply Jungck iterates. Notice that the sequence{Txn}is the usual Picard sequence of iterates when T is the identity map onY X.In the case ofExample 5.1, takex0 11/10.Then{Txn} {1/2,0,0, . . .}which converges to 0.However, in general, under the assumptions of Theorems 2.2or3.2, there may not exist a sequence {Txn}which converges. The following examples illustrate this fact.
Example 5.2. LetX −11,−10∪ {0} ∪10,11,
Sx
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
11x100
x9 , if −11≤x <−10, 0, ifx−10,0,10,
−11x−100
x−9 , if 10< x≤11,
5.2
andTxx, x∈X.
Suzuki7has shown thatSsatisfies the assumption ofTheorem 2.2withY Xand Tthe identity map onX.It is also shown in7that the Picard sequence of iterates of the map Sdoes not converge when the initial choicex0falls inX− {−10,0,10},althoughSsatisfies all the hypotheses ofTheorem 2.1. Thus, under the hypotheses ofTheorem 2.2, Jungck iterates forS, Tneed not converge.
Example 5.3. LetX 3,7, and letS, T :X → Xbe such that
Sx
⎧⎪
⎨
⎪⎩
3 ifx /6, 5 ifx6,
Tx
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
3 ifx3, 5 ifx /3, x /6, 7 ifx6.
5.3
Evidently, S is not nonexpansive. Further, S is also not Suzuki nonexpansive. Indeed, for x6, y5,
1
2|x−Sx| 1
2|6−5| 1
2 ≤1x−y, 5.4
while
Sx−Sy|5−3|2 /≤1x−y. 5.5
Notice thatSX⊆TX,and the assumption3.2, namely,
1
2|Tx−Sx| ≤Tx−Ty⇒Sx−Sy≤Tx−Ty, 5.6 is satisfied for allx, y∈X.AlsoSxSywheneverTxTyfor anyx, y∈X.
The sequence{Txn}constructed beforeExample 5.2may be used to approximate the coincidence values of the mapsSandTunder the hypotheses ofTheorem 3.2. Note that ifzis such thatSzTzw,thenwis the coincidence value ofSandT at their coincidence point.
For example, in the case ofExample 5.3, for anyx0 ∈ X,the sequence {Txn}converges to 3.The following example reveals some strange pattern regarding the convergence of Jungck iterates{Txn}under the assumption3.2.
Example 5.4. LetX 3,7andS, T :X → Xbe defined by
Sx
⎧⎨
⎩
3, ifx /6 5, ifx6,
Tx
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
7, if x3 3, if x /3, /6 5 if x6.
5.7
Notice the following.
1Sis not nonexpansive.
2Sis not Suzuki nonexpansivetakex5, y6.
3SX⊆TX.
4SandT satisfy assumption3.2withY EX.
5SxSywheneverTxTyfor anyx, y∈X.
6For anyz /3, /6, Sz Tz 3.Note that coincidence pointzis different from the coincidence valuew3.
7As regards the Jungck sequence of iterates{Txn},we examine some cases below.
iForx03,considerxn3n/n1, n1,2, . . . . Evidently,Txn → 3.
iiForx06 andxn6, n1, 2, . . . , Txn → 5 andS6T65.
Here it is very interesting to note thatSandTare commuting atx6,which is not a common fixed point ofSandT.
The following example illustrates the validity and superiority of the Kikkawa-Suzuki generalized contraction for a pair of maps.
Example 5.5. LetX 0,∞and for everyx∈X,defineP x 0,3xandTx5x.ThenPdoes not satisfy the assumptionKSMCofTheorem 4.1. Indeed, for anyr ∈0,1andx3 and y1, ηrd3, P3 0≤d3,1andHP3, P1 6> d3,1.Further, asd1, P1 d2, P2 0,the mapP does not satisfy either of the conditions studied by Mot¸ and Petrus¸el3and Dhompongsa and Yingtaweesittikul4 see Remarks4.2ii–iii. However, for everyx, y∈ X, HP x, P y≤adTx, Ty,wherea∈3/5,1,bc0.So,PandTsatisfy the assumption KSGofTheorem 4.3withY X.
The following example shows the usefulness of domain Y different from X in Theorem 4.3.
Example 5.6. LetRbe the set of real numbers,Y Cthe set of complex numbersandX 0,∞. Forx, y ∈R andz x, y ∈ Y,defineP z 0, x2y2andTz 2x2y2.Then PY ⊆ TYand P satisfies the assumptionKSG with a 1/2, b c 0. Evidently Theorem 4.3applies andTz∈P zforz 0,0.
In view of the foregoing discussion regarding the convergence of Jungck iterates of Suzuki class of nonexpansive pair of maps, we present the following.
Conjecture 5.7. Let Cbe a nonempty subset of a Banach spaceE andS, T : C → C satisfying assumption3.2. LetSC⊆TCand letTCbe a compact convex subset ofE.Forx0∈C,define a sequence{Txn}such that
Txn1λSxn 1−λTxn, n0,1,2, . . . , 5.8
whereλ∈1/2,1.Then the sequence{Txn}converges to a coincidence point ofSandT.
We remark that its particular case withY XandTthe identity map is Theorem 2 of Suzuki6.
6. Applications
Throughout this section, we assume thatUandV are Banach spaces,W ⊆UandD⊆V.Let Rdenote the field of reals,τ :W×D → W, f, g :W×D → RandG, F :W×D×R → R.
ConsideringWandDas the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:
p:sup
y∈D f x, y
G x, y, p
τ
x, y
, x∈W, 6.1
q:sup
y∈D g x, y
F x, y, q
τ
x, y
, x∈W. 6.2
In the multistage process, some functional equations arise in a natural way cf., Bellman 19 and Bellman and Lee 20 see also 37–39. In this section, we study the existence of a common solution of the functional equations6.1and6.2arising in dynamic programming.
LetBWdenote the set of all bounded real-valued functions onW.For an arbitrary h∈BW, define h supx∈W|hx|.ThenBW, · is a Banach space. Suppose that the following conditions hold.
DP-1G, F, fandgare bounded.
DP-2aAssume that for everyx, y∈W×D, h, k∈BWandt∈W, 1
2|Kht−Jht| ≤ |Jht−Jkt| 6.3
implies
G
x, y, ht
−G
x, y, kt≤ |Jht−Jkt|, 6.4
whereKandJare defined as follows:
Khx sup
y∈D f x, y
G x, y, h
τ
x, y
, x∈W, h∈BW, ∗
Jhx sup
y∈D g x, y
F x, y, h
τ
x, y
, x∈W, h∈BW. 6.5
DP-2bLetβbe defined as inSection 4. Assume that there exista, b, c ∈ 0,1such that for everyx, y∈W×D, h, k∈BWandt∈W,
β|Kht−Jht| ≤ |Jht−Jkt| 6.6
implies G
x, y, ht
−G
x, y, kt≤a|Jht−Jkt|b|Jht−Kht||Jkt−Kkt|
c|Jht−Kkt||Jkt−Kht|, 6.7
wherea2b2c <1.
DP-2cJh1Jh2impliesKh1Kh2.
DP-3For anyh∈BW,there existsk∈BWsuch that
Khx Jkx, x∈W. 6.8
DP-4There existsh∈BWsuch that
Jhx Khx impliesJKhx KJhx. 6.9
Theorem 6.1. Assume that conditions (DP-1), (DP-2a), (DP-2c) and (DP-3) are satisfied. If JBWis a compact convex subspace ofBW,then the functional equations6.1and6.2have a conicidence bounded solution.
Proof. Letdbe the metric induced by the supremum norm onBW. ThenBWis a complete metric space. By DP-1, J and K are self-maps of BW. Condition DP-3 implies that
KBW⊆JBW.
Letλbe an arbitrary positive number andh1, h2 ∈BW.Letx∈Wbe arbitrary and choosey1, y2∈Dsuch that
Khj< f x, yj
G
x, yj, hj xj
λ, 6.10
wherexjτx, yj, j 1,2.
Further,
Kh1x≥f x, y2
G
x, y2, h1x2
, 6.11
Kh2x≥f x, y1
G
x, y1, h2x1
. 6.12
Therefore, the first inequality inDP-2abecomes 1
2|Kh1x−Jh1x| ≤ |Jh1x−Jh2x|, 6.13 and this together with6.10and6.12implies
Kh1x−Kh2x< G
x, y1, h1x1
−G
x, y1, h2x1 λ
≤G
x, y1, h1x1
−G
x, y1, h2x1λ, 6.14 that is
Kh1x−Kh2x≤ |Jh1x−Jh2x|λ. 6.15
Similarly,6.10,6.11and6.13imply
Kh2x−Kh1x≤ |Jh1x−Jh2x|λ. 6.16
So, from6.15and6.16,we have
|Kh1x−Kh2x| ≤ |Jh1x−Jh2x|λ. 6.17
Sincex∈Wandλ >0 is arbitrary, we find from6.13that 1
2dKh1, Jh1≤dJh1, Jh2 6.18
implies
dKh1, Kh2≤dJh1, Jh2. 6.19
Hence taking also the notice ofDP-2c, we see thatTheorem 3.2iapplies, whereinK andJcorrespond, respectively, to the mapsSandTSo,KandJhave a coincidence pointh∗, that is,h∗xis a bounded coincidence solution of the functional equations6.1and6.2.
Corollary 6.2. Suppose that the following conditions hold:
iGandfare bounded.
iifor everyx, y∈W×D, h, k∈BWandt∈W, 1
2|ht−Kht| ≤ |ht−kt| 6.20
implies
G
x, y, ht
−G
x, y, kt≤ |ht−kt|, 6.21
whereKis defined by∗. Then the functional equation6.1has a bounded solution inW provided thatBWis compact.
Proof. It comes from Theorem 6.1 when g 0, τx, y x and Fx, y, t t as the assumptionsDP-2candDP-3become redundant in this context.
We remark thatTheorem 6.1does not guarantee the existence of a common solution even if we add to it the commutativity requirementDP-4. Further, a solution guaranteed byCorollary 6.2 need not be unique. These observations add importance to the following formulation regarding the existence of a unique common bounded solution.
Theorem 6.3. Assume that conditions (DP-1), (DP-2b), (DP-3), and (DP-4) are satisfied. If KBW or JBW is a closed convex subspace of BW, then the functional equations 6.1 and6.2have a unique common bounded solution.
Proof. Recall thatBW, dis a complete metric space. The self-mapsJ andKofBWare commuting at their coincidence points byDP-4. Proceeding as in the proof ofTheorem 6.1, we see thatKandJcorrespond, respectively, to the mapsgandT ofCorollary 4.5. HenceK andJhave a unique bounded common solutionh∗xof the functional equations6.1and 6.2.
Acknowledgments
The authors thank the referees and Professor Tomonari Suzuki for their perspicacious comments and suggestions regarding this work. This research is supported by the Directorate of Research Development, Walter Sisulu University.
References
1 T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,”
Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
2 M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constants in complete metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2942–
2949, 2008.
3 G. Mot¸ and A. Petrus¸el, “Fixed point theory for a new type of contractive multivalued operators,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3371–3377, 2009.
4 S. Dhompongsa and H. Yingtaweesittikul, “Fixed points for multivalued mappings and the metric completeness,” Fixed Point Theory and Applications, vol. 2009, Article ID 972395, 15 pages, 2009.
5 T. Suzuki, “Some remarks on recent generalization of the Banach contraction principle,” in Proceedings of the 8th International Conference on Fixed Point Theory and Its Applications, pp. 751–761, 2007.
6 T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1088–1095, 2008.
7 T. Suzuki, “A new type of fixed point theorem in metric spaces,” Nonlinear Analysis: Theory, Methods
& Applications, vol. 71, no. 11, pp. 5313–5317, 2009.
8 O. Popescu, “Two fixed point theorems for generalized contractions with constants in complete metric space,” Central European Journal of Mathematics, vol. 7, no. 3, pp. 529–538, 2009.
9 S. L. Singh and S. N. Mishra, “Fixed point theorems for single-valued and multivalued maps,”
communicated.
10 F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.
11 F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.
12 D. G ¨ohde, “Zum Prinzip der kontraktiven Abbildung,” Mathematische Nachrichten, vol. 30, pp. 251–
258, 1965.
13 K. Goebel, “A coincidence theorem,” Bulletin de l’Acad´emie Polonaise des Sciences. S´erie des Sciences Math´ematiques, vol. 16, pp. 733–735, 1968.
14 S. A. Naimpally, S. L. Singh, and J. H. M. Whitfield, “Coincidence theorems for hybrid contractions,”
Mathematische Nachrichten, vol. 127, pp. 177–180, 1986.
15 S. Itoh and W. Takahashi, “Single-valued mappings, multivalued mappings and fixed-point theorems,” Journal of Mathematical Analysis and Applications, vol. 59, no. 3, pp. 514–521, 1977.
16 S. Reich, “Fixed points of contractive functions,” Bollettino della Unione Matematica Italiana, vol. 5, pp.
26–42, 1972.
17 K. Iseki, “Multi-valued contraction mappings in complete metric spaces,” Rendiconti del Seminario Matematico della Universit`a di Padova, vol. 53, pp. 15–19, 1975.
18 G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no.
4, pp. 261–263, 1976.
19 R. Bellman, Methods of Nonliner Analysis. Vol. II, Academic Press, New York, NY, USA, 1973.
20 R. Bellman and E. S. Lee, “Functional equations in dynamic programming,” Aequationes Mathematicae, vol. 17, no. 1, pp. 1–18, 1978.
21 P. C. Bhakta and S. Mitra, “Some existence theorems for functional equations arising in dynamic programming,” Journal of Mathematical Analysis and Applications, vol. 98, no. 2, pp. 348–362, 1984.
22 M. Edelstein, “On fixed and periodic points under contractive mappings,” Journal of the London Mathematical Society. Second Series, vol. 37, pp. 74–79, 1962.
23 K. Goebel, Concise Course on Fixed Point Theorems, Yokohama Publishers, Yokohama, Japan, 2002.
24 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
25 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
26 S. Prus, “Geometrical background of metric fixed point theory,” in Handbook of Metric Fixed Point Theory, pp. 93–132, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
27 W. Takahashi, “Recent results in fixed point theory,” Southeast Asian Bulletin of Mathematics, vol. 4, no.
2, pp. 59–85, 1980.
28 W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000.
29 S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–
488, 1969.
30 S. Reich, “Kannan’s fixed point theorem,” Bollettino della Unione Matematica Italiana, vol. 4, pp. 1–11, 1971.
31 G. E. Hardy and T. D. Rogers, “A generalization of a fixed point theorem of Reich,” Canadian Mathematical Bulletin, vol. 16, pp. 201–206, 1973.
32 B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.
33 S. L. Singh and S. N. Mishra, “Coincidences and fixed points of nonself hybrid contractions,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 486–497, 2001.
34 S. L. Singh and C. Kulshrestha, “Coincidence theorems in metric spaces,” Indian Journal of Physics, Natural Sciences, no. 2, pp. 19–22, 1982.
35 B. E. Rhoades, S. L. Singh, and C. Kulshrestha, “Coincidence theorems for some multivalued mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 7, no. 3, pp. 429–434, 1984.
36 Lj. B. ´Ciri´c, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.
37 R. Baskaran and P. V. Subrahmanyam, “A note on the solution of a class of functional equations,”
Applicable Analysis, vol. 22, no. 3-4, pp. 235–241, 1986.
38 H. K. Pathak, Y. J. Cho, S. M. Kang, and B. S. Lee, “Fixed point theorems for compatible mappings of typePand applications to dynamic programming,” Le Matematiche, vol. 50, no. 1, pp. 15–33, 1995.
39 S. L. Singh and S. N. Mishra, “On a Ljubomir ´Ciri´c fixed point theorem for nonexpansive type maps with applications,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 531–542, 2002.
