S. L. Singh and S. N. Mishra

Volume 2010, Article ID 898109,14pages doi:10.1155/2010/898109

Research Article

Coincidence Theorems for Certain Classes of Hybrid Contractions

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 27 August 2009; Accepted 9 October 2009

Academic Editor: Mohamed A. Khamsi

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 hybrid contractions consisting of a pair of single-valued and multivalued maps on an arbitrary nonempty set with values in a metric space are proved. In addition, the existence of a common solution for certain class of functional equations arising in dynamic programming, under much weaker conditions are discussed. The results obtained here in generalize many well known results.

1. Introduction

Nadler’s multivalued contraction theorem 1 see also Covitz and Nadler, Jr. 2 was subsequently generalized among others by Reich 3 and ´Ciri´c 4. For a fundamental development of fixed point theory for multivalued maps, one may refer to Rus 5.

Hybrid contractive conditions, that is, contractive conditions involving single-valued and multivalued maps are the further addition to metric fixed point theory and its applications.

For a comprehensive survey of fundamental development of hybrid contractions and historical remarks, refer to Singh and Mishra 6 see also Naimpally et al.7 and Singh and Mishra8.

Recently Suzuki 9, Theorem 2 obtained a forceful generalization of the classical Banach contraction theorem in a remarkable way. Its further outcomes by Kikkawa and Suzuki 10, 11, Mot¸ and Petrus¸el 12 and Dhompongsa and Yingtaweesittikul 13, are important contributions to metric fixed point theory. Indeed, 10, Theorem 2 see Theorem 2.1 below presents an extension of 9, Theorem 2 and a generalization of the multivalued contraction theorem due to Nadler, Jr.1. In this paper we obtain a coincidence theorem Theorem 3.1 for a pair of single-valued and multivalued maps on an arbitrary

nonempty set with values in a metric space and derive fixed point theorems which generalize Theorem 2.1and certain results of Reich 3, Zamfirescu 14, Mot¸ and Petrus¸el12, and others. Further, using a corollary ofTheorem 3.1, we obtain another fixed point theorem for multivalued maps. We also deduce the existence of a common solution for Suzuki-Zamfirescu type class of functional equations under much weaker contractive conditions than those in Bellman15, Bellman and Lee16, Bhakta and Mitra17, Baskaran and Subrahmanyam 18, and Pathak et al.19.

2. Suzuki-Zamfirescu Hybrid Contraction

For the sake of brevity, we follow the following notations, whereinP andT are maps to be defined specifically in a particular context whilex,andyare the elements of specific domains:

M P;x, y

d x, y

,dx, P x d y, P y

2 ,d

x, P y d

y, P x 2

,

M

P;Tx, Ty

d

Tx, Ty

,dTx, P x d

Ty, P y

2 ,d

Tx, P y d

Ty, P x 2

,

m P;x, y

d x, y

, dx, P x, d y, P y

,d x, P y

d y, P x 2

.

2.1

Consistent with Nadler, Jr.20, page 620,Y will denote an arbitrary nonempty set, X, da metric space, and CLX resp. CBX the collection of nonempty closedresp., closed and boundedsubsets ofX.ForA, B∈CLXand >0,

N, A {x∈X:dx, a< for somea∈A}, E_A,B{ >0 :A⊆N, B, B⊆N, A},

HA, B

⎧⎨

⎩

infE_A,B, ifE_A,B/φ

∞, ifEA,Bφ.

2.2

The hyperspaceCLX, His called the generalized Hausdorffmetric space induced by the metricdonX.

For any subsetsA, BofX,dA, Bdenotes the ordinary distance between the subsets AandB,while

ρA, B sup{da, b:a∈A, b∈B},

BNX A:φ /A⊆X and the diameter ofAis finite

. 2.3

As usual, we writedx, B resp.,ρx, BfordA, B resp.,ρA, BwhenA{x}.

In all that followsηis a strictly decreasing function from0,1onto1/2,1defined by

ηr 1

1r. 2.4

Recently Kikkawa and Suzuki10obtained the following generalization of Nadler, Jr.

1.

Theorem 2.1. LetX, dbe a complete metric space andP :X → CBX.Assume that there exists r∈0,1such that

KSCηrdx, P x≤dx, yimpliesHP x, P y≤rdx, y for allx, y∈X.ThenPhas a fixed point.

For the sake of brevity and proper reference, the assumption (KSC) will be called Kikkawa- Suzuki multivalued contraction.

Definition 2.2. MapsP:Y → CLXandT:Y → Xare said to be Suzuki-Zamfirescu hybrid contraction if and only if there existsr ∈0,1such that

S-ZηrdTx, P x≤dTx, TyimpliesHP x, P y≤r·maxMP;Tx, Ty for allx, y∈Y.

A mapP:X → CLXsatisfying CGHP x, P y≤r·maxmP;x, y

for all x, y ∈ X, where 0 ≤ r < 1, is called ´Ciri´c-generalized contraction. Indeed, ´Ciri´c4 showed that a ´Ciri´c generalized contraction has a fixed point in aP-orbitally complete metric spaceX.

It may be mentioned that in a comprehensive comparison of 25 contractive conditions for a single-valued map in a metric space, Rhoades21has shown that the conditionsCG andZare, respectively, the conditions21and19whenPis a single-valued map, where

ZHP x, P y≤r·maxMP;x, yfor allx, y∈X.

Obiviously,ZimpliesCG. Further, Zamfirescu’s condition14is equivalent toZ whenPis single-valuedsee Rhoades21, pages 259 and 266.

The following example indicates the importance of the conditionS-Z.

Example 2.3. LetX {1,2,3}be endowed with the usual metric and letP andT be defined by

P x

⎧⎨

⎩

2,3 ifx /3, 3 ifx3,

Tx

⎧⎨

⎩

1 ifx /1, 3 ifx1.

2.5

ThenPdoes not satisfy the conditionKSC. Indeed, forx2, y3,

ηrd2, P2 0≤d2,3, 2.6

and this does not imply

1HP2, P3≤d2,3 r. 2.7

Further, as easily seen,P does not satisfyCGforx 2, y 3. However, it can be verified that the pairP andT satisfies the assumptionS-Z. Notice thatP does not satisfy the conditionS-ZwhenY XandTis the identity map.

We will need the following definitions as well.

Definition 2.4see4. An orbit forP : X → CLX atx0 ∈ X is a sequence{xn : xn ∈ P x_n−1}, n 1, 2, . . . .A spaceX is calledP-orbitally complete if and only if every Cauchy sequence of the form{xni :x_ni ∈P x_ni−1}, i1, 2, . . . converges inX.

Definition 2.5. LetP : Y → CLX and T : Y → X.If for a point x0 ∈ Y, there exists a sequence{xn}inYsuch thatTx_n1 ∈P x_n, n0,1,2, . . . ,then

OTx0 {Txn :n1,2, . . .} 2.8

is the orbit for P, T at x₀. We will use O_Tx0 as a set and a sequence as the situation demands. Further, a spaceXisP, T-orbitally complete if and only if every Cauchy sequence of the form{Txni :Txni ∈P xni−1}converges inX.

As regards the existence of a sequence {Txn} in the metric space X, the sufficient condition is that PY ⊆ TY. However, in the absence of this requirement, for some x0∈Y,a sequence{Txn}may be constructed some times. For instance, in the above example, the range of P is not contained in the range of T, but we have the sequence {Txn} for x₀2, x₁ x₂ · · ·1.So we have the following definition.

Definition 2.6. If for a pointx0 ∈Y,there exists a sequence{xn}inY such that the sequence O_Tx0converges inX,thenXis calledP, T-orbitally complete with respect tox₀or simply P, T, x0-orbitally complete.

We remark that Definitions2.5and2.6are essentially due to Rhoades et al.22when Y X. InDefinition 2.6, if Y X and T is the identity map on X, the P, T, x0-orbital completeness will be denoted simply byP, x0-orbitally complete.

Definition 2.723, see also8. MapsP :X → CLXandT :X → Xare IT-commuting at z∈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. Notice that if P is also single- valued, then their IT-commutativity and commutativity are the same.

3. Coincidence and Fixed Point Theorems

Theorem 3.1. Assume that the pair of maps P : Y → CLXand T : Y → X is a Suzuki- Zamfirescu hybrid contraction such thatPY⊆ TY.If there exists anu₀ ∈Y such thatTYis P, T, u0-orbitally complete, thenPandT have a coincidence point; that is, there existsz ∈Y such thatTz∈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. Without any loss of generality, we may taker > 0 andT a nonconstant map. Letq r^−1/2.Pick u₀ ∈ Y.We construct two sequences{un} ⊆ Y and {yn Tu_n} ⊆ TYin the following manner. SincePY ⊆ TY,we take an element u1 ∈ Y such that Tu1 ∈ P u0. Similarly, we chooseTu2∈P u1such that

dTu1, Tu2≤qHP u0, P u1. 3.1

IfTu₁ Tu₂,thenTu₁∈P u₁and we are done asu₁is a coincidence point ofT andP.

So we takeTu1/Tu2. In an analogous manner, chooseTu3∈P u2such that

dTu2, Tu3≤qHP u1,P u2. 3.2

If Tu₂ Tu₃, then Tu₂ ∈ P u₂ and we are done. So we take Tu₂/Tu₃, and continue the process. Inductively, we construct sequences{un}and{Tun}such thatTu_n2 ∈ P u_n1, Tu_n1/Tu_n2and

dTun1, Tu_n2≤qHP un, P u_n1. 3.3

Now we see that

ηrdTun, P u_n≤ηrdTun, Tu_n1≤dTun, Tu_n1. 3.4

Therefore by the conditionS-Z,

d

y_n1, y_n2

≤qHP un, P u_n1

≤qr·max

dTun, Tu_n1,dTun, P un dTun1, P u_n1

2 ,

dTun, P u_n1 dTu_n1, P u_n 2

≤qr·max

⎧⎪

⎪⎨

⎪⎪

⎩ d

y_n, y_n1 ,d

yn, y_n1 d

y_n1, y_n2

2 ,

1 2d

y_n, y_n2

⎫⎪

⎪⎬

⎪⎪

⎭.

3.5

This yields

d

y_n1, y_n2

≤r₁d

y_n, y_n1

, 3.6

wherer₁qr <1.

Therefore the sequence {yn} is Cauchy in TY. Since TY is P, T, u0-orbitally complete, it has a limit inTY.Call itu.Letz∈T⁻¹u.Thenz∈YanduTz.

Now as in10, we show that

dTz, P x≤rdTz, Tx 3.7

for anyTx∈TY− {Tz}.Sincey_n → Tz,there exists a positive integern₀such that dTz, Tun≤ 1

3dTz, Tx ∀n≥n₀. 3.8

Therefore forn≥n₀,

ηrdTun, P u_n≤dTun, P u_n≤dTun, Tu_n1

≤dTun, Tz dTun1,Tz

≤ 2

3dTz, Tx dTz, Tx−1

3dTz, Tx

≤dTz, Tx−dTz, Tun≤dTun, Tx.

3.9

Therefore by the conditionS-Z, d

y_n1, P x

≤HP un, P x

≤r·max

d y_n, Tx

,d yn, P un

dTx, P x

2 ,d

yn, P x

dTx, P un 2

≤r·max

d y_n, Tx

,d

yn, y_n1

dTx, P x

2 ,d

yn, P x d

Tx, y_n1 2

. 3.10

Makingn → ∞, dTz, P x≤r·max

dTz, Tx,1

2dTx, P x,dTz, P x dTx, Tz 2

. 3.11

This yields3.7;Tx/Tz.

Next we show that HP x, P z≤r·max

dTx, Tz,dTx, P x dTz, P z

2 ,dTx, P z dTz, P x 2

3.12

for anyx∈Y.Ifxz,then it holds trivially. So we supposex /zsuch thatTx /Tz.Such a choice is permissible asT is not a constant map.

Therefore using3.7,

dTx, P x≤dTx, Tz dTz, P x

≤dTx, Tz rdTx, Tz. 3.13

Hence

1

1rdTx, P x≤dTx, Tz. 3.14

This implies3.12, and so

d

y_n1, P z

≤HP un, P z

≤r·max

dTun, Tz,dTun, P u_n dTz, P z

2 ,dTun, P z dTz, P un 2

≤r·max

d y_n, Tz

,d

y_n, y_n1

dTz, P z

2 ,d

y_n, P z d

Tz, y_n1 2

. 3.15

Makingn → ∞,

dTz, P z≤rdTz, P z. 3.16

SoTz∈P z,sinceP zis closed.

Further, ifY X, TTzTz,andP, T are IT-commuting atz,that is,TP z⊆P Tz,then Tz∈P z⇒TTz∈TP z⊆P Tz, and this proves thatTzis a fixed point ofP.

We remark that, in general, a pair of continuous commuting maps at their coincidences need not have a common fixed point unlessThas a fixed pointsee, e.g.,6–8.

Corollary 3.2. LetP :X → CLX.Assume that there existsr∈0,1such that ηrdx, P x≤d

x, y

impliesH

P x, P y

≤r·maxM P;x, y

3.17 for allx, y∈X.If there exists au0 ∈Xsuch thatXisP, u0-orbitally complete, thenPhas a fixed point.

Proof. It comes fromTheorem 3.1whenY XandTis the identity map onX.

The following two results are the extensions of Suzuki9, Theorem 2.Corollary 3.3 also generalizes the results of Kikkawa and Suzuki10, Theorem 3and Jungck24.

Corollary 3.3. Letf, T :Y → Xbe such thatfY⊆TYand TYis anf, T-orbitally complete subspace ofX.Assume that there existsr∈0,1such that

ηrd Tx, fx

≤d

Tx, Ty

3.18

implies

d

fx, fy

≤r·maxM

f;Tx, Ty

3.19

for allx, y∈Y.ThenfandThave a coincidence point; that is, there existsz∈Y such thatfzTz.

Further, ifY X andf andT commute atz,then f andT have a unique common fixed point.

Proof. SetP x{fx}for everyx∈Y.Then it comes fromTheorem 3.1that there existsz∈Y such thatfzTz.Further, ifY Xandf,andTcommute atz,thenffzfTzTfz. Also, ηrdTz, fz 0≤dTz, Tfz,and this implies

d

fz, ffz

≤r·maxM

f;Tz, Tfz rd

fz, ffz

. 3.20

This yields thatfzis a common fixed point offandT.The uniqueness of the common fixed point follows easily.

Corollary 3.4. Let f : X → X be such thatX isf-orbitally complete. Assume that there exists r∈0,1such that

ηrd x, fx

≤d x, y

implies d fx, fy

≤r·maxM f;x, y

3.21

for allx, y∈X.Thenfhas a unique fixed point.

Proof. It comes fromCorollary 3.2thatf has a fixed point. The uniqueness of the fixed point follows easily.

Theorem 3.5. LetP :Y → BNXandT : Y → X be such thatPY⊆ TYand letTYbe P, T-orbitally complete. Assume that there existsr ∈0,1such that

ηrρTx, P x≤d

Tx, Ty

3.22 implies

ρ

P x, P y

≤r·max

d

Tx, Ty

,ρTx, P x ρ

Ty, P y

2 ,d

Tx, P y d

Ty, P x 2

3.23 for allx, y∈Y.Then there existsz∈Y such thatTz∈P z.

Proof. Chooseλ ∈0,1.Define a single-valued mapf :Y → X as follows. For eachx∈Y, letfxbe a point ofP x,which satisfies

d Tx, fx

≥r^λρTx, P x. 3.24

Sincefx∈P x, dTx, fx≤ρTx, P x.So3.22gives ηrd

Tx, fx

≤ηrρTx, P x≤d

Tx, Ty

, 3.25

and this implies3.23. Therefore d

fx, fy

≤ρ

P x, P y

≤r·r^−λ·max

r^λd

Tx, Ty

,r^λρTx, P x r^λρ

Ty, P y

2 ,

r^λd

Tx, P y r^λd

Ty, P x 2

≤r^1−λ·max

d

Tx, Ty ,d

Tx, fx d

Ty, fy

2 ,d

Tx, fy d

Ty, fx 2

. 3.26

This means thatCorollary 3.3applies as fY ∪ fx∈P x

⊆PY⊆TY. 3.27

HencefandThave a coincidence atz∈Y.ClearlyfzTzimpliesTz∈P z.

Now we have the following.

Theorem 3.6. LetP : X → BNXand let X beP-orbitally complete. Assume that there exists r∈0,1such thatηrρx, P x≤dx, yimplies

ρ

P x, P y

≤r·max

d x, y

,ρx, P x ρ y, P y

2 ,d

x, P y d

y, P x 2

3.28

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

Proof. Forλ∈0,1, define a single-valued mapf:X → Xas follows. For eachx∈X,letfx be a point ofP xsuch that

d x, fx

≥r^λρx, P x. 3.29

Now following the proof technique of Theorem 3.5 and using Corollary 3.4, we conclude thatfhas a unique fixed pointz∈X.Clearlyzfzimplies thatz∈P z.

Now we close this section with the following.

Question 1. Can we replace Assumption3.17inCorollary 3.2by the following:

ηrdx, P x≤d x, y

3.30

implies

H

P x, P y

≤r·max

d x, y

, dx, P x, d y, P y

,1 2

d x, P y

d

y, P x

3.31

for allx, y∈X?

4. Applications

Throughout this section, we assume thatUandV are Banach spaces,W⊆U,andD⊆V.Let Rdenote the field of reals,τ :W×D → W, g, g:W×D → R,andG, F :W×D×R → R.

Viewing W and D as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:

p:sup

y∈D g x, y

G x, y, p

τ

x, y

, x∈W, 4.1

q:sup

y∈D g x, y

F x, y, q

τ

x, y

, x∈W. 4.2

In the multistage process, some functional equations arise in a natural waycf. Bellman 15and Bellman and Lee16; see also17–19,25. In this section, we study the existence of the common solution of the functional equations4.1,4.2arising in dynamic programming.

LetBWdenote the set of all bounded real-valued functions onW.For an arbitrary h∈BW, definehsup_x∈W|hx|.ThenBW, · is a Banach space. Suppose that the following conditions hold:

DP-1G, F, gandgare bounded.

DP-2Letηbe defined as in the previous section. There existsr ∈0,1such that for every x, y∈W×D, h, k∈BWandt∈W,

ηr|Kht−Jht| ≤ |Jht−Jkt| 4.3

implies G

x, y, ht

−G

x, y, kt

≤r·max

|Jht−Jkt|,|Jht−Kht||Jkt−Kkt|

2 ,

|Jht−Kkt||Jkt−Kht|

2

,

4.4

whereKandJare defined as follows:

Khx sup

y∈D g 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. 4.5

DP-3For anyh∈BW,there existsk∈BWsuch that

Khx Jkx, x∈W. 4.6

DP-4There existsh∈BWsuch that

Jhx Khx impliesJKhx KJhx. 4.7

Theorem 4.1. Assume that the conditions (DP-1)–(DP-4) are satisfied. IfJBWis a closed convex subspace of BW,then the functional equations 4.1and 4.2 have a unique common bounded solution.

Proof. Notice thatBW, dis a complete metric space, wheredis the metric induced by the supremum norm onBW.ByDP-1, J andKare self-maps ofBW.The conditionDP- 3implies thatKBW⊆ JBW.It follows fromDP-4thatJ andKcommute at their coincidence points.

Letλ be an arbitrary positive number and h₁, h₂ ∈ BW. Pickx ∈ W and choose y1, y2∈Dsuch that

Kh_j< g x, y_j

G

x, y_j, h_j x_j

λ, 4.8

wherexjτx, yj, j 1,2.

Further,

Kh₁x≥g x, y₂

G

x, y₂, h₁x2

, 4.9

Kh₂x≥g x, y₁

G

x, y₁, h₂x1

. 4.10

Therefore, the first inequality inDP-2becomes

ηr|Kh1x−Jh1x| ≤ |Jh1x−Jh2x|, 4.11

and this together with4.8and4.10implies Kh1x−Kh2x< G

x, y1, h1x1

−G

x, y1, h2x1 λ

≤G

x, y₁, h₁x1

−G

x, y₁, h₂x1λ

≤r·maxMK;Jh₁, Jh₂ λ.

4.12

Similarly,4.8,4.9, and4.11imply

Kh2x−Kh1x≤r·maxMK;Jh1, Jh2 λ. 4.13 So, from4.12and4.13, we have

|Kh1x−Kh2x| ≤r·maxMK;Jh1, Jh2 λ. 4.14 Since the above inequality is true for anyx∈W,andλ >0 is arbitrary, we find from 4.17that

ηrdKh1, Jh1≤dJh1, Jh2 4.15

implies

dKh1, Kh2≤r·maxMK;Jh1, Jh2. 4.16 ThereforeCorollary 3.3applies, whereinKandJcorrespond, respectively, to the maps fandT,Therefore,KandJhave a unique common fixed pointh^∗,that is,h^∗xis the unique bounded common solution of the functional equations4.1and4.2.

Corollary 4.2. Suppose that the following conditions hold.

iGandgare bounded.

iiForηdefined earlier (cf. (DP-2) above), there existsr ∈0,1such that for everyx, y∈ W×D, h, k∈BWandt∈W,

ηr|ht−Kht| ≤ |ht−kt| 4.17

implies G

x, y, ht

−G

x, y, kt≤r·maxMK;ht, kt, 4.18 whereKis defined by∗. Then the functional equation4.1possesses a unique bounded solution inW.

Proof. It comes fromTheorem 4.1whenqp, FG,andggas the conditionsDP-3and DP-4become redundant in the present context.

Acknowledgments

The authors thank the referees and Professor M. A. Khamsi for their appreciation and suggestions regarding this work. This research is supported by the Directorate of Research Development, Walter Sisulu University.

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