Remarks on Recent Fixed Point Theorems

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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 if0≤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≤dx, y impliesdSx, Sy≤rdx, y SC

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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 impliesdSx, Sy< dx, 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

dSx, Sy< dTx, 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

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Proof. DefineF : TY → TYbyFa ST−1_a_{for each}_a_∈ _T_Y_._{To see that}_F _{is well}

defined, observe bySY⊆TYthat forx∈T−1_a_,

FaSx, Fa⊆TY. 2.4

Takex, y∈T−1_a_{such that}_b_{Sx, c}_Sy._{Then, since}_Tx_Ty,_{we have}_b_c._Therefore_F_is

well-defined.

Now, fora /banda, b∈TY, T−1_a_∩_T−1_b_φ._{So, for distinct}_{a, b}_∈_T_Y_{, we suppose} 1/2da, Fa< da, b. Then forx∈T−1_a_and_y_∈_T−1_b,_{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−1_{w, Sz}_Fw _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.

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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−1_a_and_y_∈_T−1_b_{, 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≤dx, y impliesHP x, P y≤rdx, y KSMC

for allx, y∈X.ThenPhas a fixed point, that is, there existsz∈Xsuch thatz∈P z.

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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≤dTx, Ty impliesHP x, P y≤Mx, y 4.4

for allx, y∈Y,where

Mx, yadTx, TybdTx, P x cdTy, P yedTx, P yfdTy, 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

Mx, yγ·max dx, y, dx, P x, dy, 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≤dTx, Ty impliesHP x, P y≤mx, y KSG

for allx, y∈Y,where

mx, yadTx, TybdTx, P x dTy, P ycdTx, P ydTy, P x, 4.7

anda2b2c <1.

In all that follows, we consider the nontrivial case 0< a2b2c.

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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,

dyn1, yn2≤qHP xn, P xn1

≤qadyn, yn1bdyn, P xn

bdyn1, P xn1

c dyn, P xn1dyn1,P xn

≤qabdyn, yn1bdyn1, yn2c dyn, yn1dyn1, yn2, 4.10

yielding

dyn1, yn2≤λdyn, 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

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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,

dyn1, P x

≤HP xn, P x≤mxn, x ≤adyn, Tx

bdyn, yn1dTx, P xcdyn,P x

dTx, 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

dyn1, P z≤HP xn, P z≤mxn, z ≤adyn, Tz

bdyn, yn1dTz, P zcdyn, P z

dTz, 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.

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Corollary 4.4. LetXbe a complete metric space andP :X → CLX.Assume there exista, b, c∈

0,1such that

βdx, P x≤dx, y impliesHP x, P y≤NP;x, y 4.20

for allx, y∈X,where

NP;x, yadx, ybdx, P x dy, P ycdx, P ydy, 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

βdTx, gx≤dTx, Ty 4.22

implies

dgx, gy≤adTx, TybdTx, gxdTy, gycdTx, gydTy, 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

dgz, ggz≤adTz, TgzbdTz, gzdTgz, ggzcdTz, ggzdTgz, gz

a2cdgz, 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.

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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≤dTx, Ty 4.25

implies

ρP x, P y≤adTx, TybρTx, P x ρTy, P ycdTx, P ydTy, 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 ≥ tp_ρ_{Tx, P x}_._{Notice that}_f_Y _{_fx _∈ _{P x}_{} ⊆} _P_Y _⊆ _T_Y_. Sincegx∈P x, dTx, gx≤ρTx, P x.So4.25givesβdTx, gx≤βρTx, P x≤dTx, Ty, and this implies condition4.26. Therefore

dgx, gy≤ρP x, P y

≤t−p atpdTx, TybtpρTx, P x ρTy, P yctpdTx, P ydTy, P x

≤t−p adTx, TybdTx, gxdTy, gycdTx, gydTy, gx.

4.27

So, takingaat−p,bbt−p,cct−pandβ 1−b−c/1a,we see thatβdTx, gx≤ βdTx, gx≤dTx, Tyimplies

dgx, gy≤adTx, TybdTx, gxdTy, gycdTx, gydTy, gx, 4.28

wherea2b2cat−p₂_bt−p₂_ct−p_t1−p_<_{1. Hence, by virtue of}_{Corollary 4.5,}_g_and_T 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

ρP x, P y≤adx, ybρx, P x ρy, P ycdx, P ydy, P x 4.29

for allx, y∈X,wherea2b2c <1.ThenPhas a unique fixed point.

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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

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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 _≤_/₁_x₋_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.

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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, x2_y2_and_Tz ₂_x2_y2_._Then

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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

fx, yGx, y, pτx, y, x∈W, _6.1

q:sup y∈D

gx, yFx, y, qτx, 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 sup_x_∈_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

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whereKandJare defined as follows:

Khx sup y∈D

fx, yGx, y, hτx, y, x∈W, h∈BW, _∗

Jhx sup y∈D

gx, yFx, y, hτx, 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

Gx, y, ht−Gx, 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

Gx, yj, hj

xj

λ, 6.10

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Further,

Kh1x≥f

x, y2

Gx, y2, h1x2

, 6.11

Kh2x≥f

x, y1

Gx, 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

−Gx, y1, h2x1

λ

≤Gx, y1, h1x1

−Gx, 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∗,

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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

Gx, y, ht−Gx, 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.

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