On a Suzuki Type General Fixed Point Theorem with Applications

Volume 2010, Article ID 234717,15pages doi:10.1155/2010/234717

Research Article

On a Suzuki Type General Fixed Point Theorem

with Applications

S. L. Singh,

_{H. K. Pathak,}

_{and S. N. Mishra}

3_{School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, India}

Correspondence should be addressed to S. N. Mishra,[email protected]

Received 29 October 2010; Accepted 2 December 2010

Academic Editor: A. T. M. Lau

Copyrightq2010 S. L. Singh 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 main result of this paper is a fixed-point theorem which extends numerous fixed point theorems for contractions on metric spaces and recently developed Suzuki type contractions. Applications to certain functional equations and variational inequalities are also discussed.

1. Introduction

The classical Banach contraction theorem has numerous generalizations, extensions, and applications. In a comprehensive comparison of contractive conditions, Rhoades 1 recognized that ´Ciri´c’s quasicontraction2 see condition C below is the most general condition for a self-mapT of a metric space which ensures the existence of a unique fixed point. Pal and Maiti 3 proposed a set of conditions see PM.1–PM.4 below as an extension of the principle of quasicontractionC, under whichT may have more than one fixed pointseeExample 2.7below. Thus the conditionCis independent of the conditions PM.1–PM.4 see also Rhoades4, page 42.

Consider the following conditions for a mapT from a metric spaceX, dto itself for

x, y∈X:

C dTx, Ty≤kmax{dx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx}, 0< k <1,

PM.1 dx, Tx dy, Ty≤adx, y, 1< a <2,

PM.2 dx, Tx dy, Ty≤bdx, Ty dy, Tx dx, y, 1/2< b <2/3,

PM.3 dx, Tx dy, Ty dTx, Ty≤cdx, Ty dy, Tx, 1< c <3/2,

PM.4 dTx, Ty≤kmax{dx, y, dx, Tx, dy, Ty,1/2dx, Ty, dy, Tx}, 0< k <1.

2. Main Results

Throughout this paper, we denote byNthe set of natural numbers. We suppose that

ηmin

1

a,

1−b

3b ,

2−c

2c−1, 1 1k

, 2.1

wherea, b, c, andkare as in conditionsPM.1–PM.4. Notice that

1 2 <

1

a<1,

1 6 <

1−b

3b <

1 3, 1

4 < 2−c

2c−1 <1, 1 2 <

1 1k <1.

2.2

Evidently,η1k≤1.

An orbitOT, x0ofT :X → Xatx0 ∈Xis a sequence{xn :xn Tnx0, n1,2, . . .}.

A spaceXisT-orbitally complete if and only if every Cauchy sequence contained in the orbit

OT, x0converges inX, for allx0 ∈X.

An orbit of a multivalued mapP :X → 2X_{, the collection of nonempty subsets ofX,}

atx0∈Xis a sequence{xn:xn∈P xn−1, n1,2, . . .}.Xis calledP-orbitally complete if every

Cauchy sequence of the form{xni :xni ∈P xni−1, i 1,2, . . .}converges inX, for allx0 ∈X.

For details, refer to ´Ciri´c2,21.

The following theorem is our main result.

Theorem 2.1. LetT be a self-map of a metric spaceXandX beT-orbitally complete. Assume that there exists anx0∈Xsuch that for any two elementsx, y∈OT, x0,

ηdx, Tx≤dx, y 2.3

implies that at least one of the conditions (PM.1), (PM.2), (PM.3), and (PM.4) is true. Then, the sequence{Tn_{x0}converges inXandzlim}

Proof. Define a sequence {dn} such that dn dxn, xn1, wherexn Tnx0,n ∈ N. Since

ηdxn, Txn≤ dxn, Txnfor anyn∈N, one of the conditionsPM.1–PM.4is true for the

pairxn, xn1. IfPM.1is true, then

dxn, xn1 dxn1, xn2≤adxn, xn1. 2.4

This yields

dn1≤a−1dn. 2.5

Similarly, ifPM.2,PM.3, andPM.4are true, then correspondingly we obtain

dn1≤

2b−1 1−b dn,

dn1 ≤ c−

1 2−cdn, dn1≤kdn.

2.6

Hence, from2.5-2.6,

dn1≤λdn, 2.7

where

λmax

a−1,2b−1

1−b, c−1 2−c, k

. 2.8

Since 0< λ <1, the sequence{xn}is Cauchy. By theT-orbital completeness ofX, the limitz

of the sequence{xn}is inX. Moreover, there existsn0∈Nsuch that

ηdxn, Txn≤dxn, x 2.9

forn≥n0, wherex /z. Therefore, by conditionsPM.1–PM.4, we have one of the following

for_{x /}z:

dxn, Txn dx, Tx≤adxn, x, 2.10

which yields on makingn → ∞,

and similarly

dx, Tx≤ 3b

1−bdx, z, 2.12

dx, Tx≤ 2c−1

2−c dx, z, 2.13 dz, Tx≤kmax{dx, z, dx, Tx}, 2.14

that is,

dz, Tx≤kdx, Tx, 2.15

or

dz, Tx≤kdx, z, 2.16

and in this case

dx, Tx≤dx, z dz, Tx≤dx, z kdx, z, 2.17

that is,

1

1kdx, Tx≤dx, z. 2.18

Thus, in view of2.11,2.12,2.13,2.18, and2.15, one of the following is true for_{x /}z:

ηdx, Tx≤dx, z, 2.19

dz, Tx≤kdx, Tx. 2.20

Case 1. Suppose that2.19is true. Then, by the assumption, one ofPM.1–PM.4is true, that is,

dx, Tx dz, Tz≤adx, z,

dx, Tx dz, Tz≤bdx, Tz dz, Tx dx, z,

dx, Tx dz, Tz dTx, Tz≤cdx, Tz dz, Tx,

dTx, Tz≤kmax

dx, z, dx, Tx, dz, Tz,1

2dx, Tz dz, Tx

.

Takingx xnin these inequaliteis and makingn → ∞, we see that one of the following is

true:

dz, Tz≤0, 1−bdz, Tz≤0, 2−cdz, Tz≤0, 1−kdz, Tz≤0.

2.22

All these possibilities lead to the fact thatTzz.

Case 2. Suppose that2.20is true. We show that there exists a subsequence{nj}of{n}such

that

ηdxnj, xnj1

≤dxnj, z

, j ∈N. 2.23

Recall that by2.7,

dxn, xn1≤λdxn−1, xn. 2.24

Suppose that

ηdxn−1, xn> dxn−1, z, ηdxn, xn1> dxn, z. 2.25

Then

dxn−1, xn≤dxn−1, z dxn, z

< ηdxn−1, xn ηdxn, xn1 ≤ηdxn−1, xn ηλdxn−1, xn

η1λdxn−1, xn.

2.26

Since without loss of generality, we may takeλk, we have

dxn−1, xn< η1kdxn−1, xn

≤dxn−1, xn.

2.27

This is a contradiction. Therefore, either

ηdxn−1, xn≤dxn−1, z, or ηdxn, xn1≤dxn, z. 2.28

This implies that either

holds forn∈N. Thus, there exists a subsequence{nj}of{n}such that

ηdxnj, xnj1

≤dxnj, z

, 2.30

that is,

ηdxnj, Txnj

≤dxnj, z

forj ∈N. 2.31

Hence, by the assumption, one of the conditionsPM.1–PM.4is satisfied forx xnj and yz, and makingj → ∞, we obtainzTz.

Remark 2.2. If only the conditionPM.4is satisfied inTheorem 2.1, then the uniqueness of the fixed-pointzfollows easily. Hence, we have the followingsee also10, Corollary 2.1. Corollary 2.3. LetT be a self-map of a metric spaceX andX beT-orbitally complete. Assume that there exists anx0∈Xsuch that for any two elementsx, y∈OT, x0,

1

1kdx, Tx≤d

x, y 2.32

implies the condition (PM.4). ThenThas a unique fixed point.

Remark 2.4. Corollary 2.3generalizes certain theorems from7,9–11and others.

Remark 2.5. It is clear from the proof ofTheorem 2.1that the best value ofηin classPM.1– PM.4is, respectively, 1/2, 1/6, 1/4, and 1/2.

The following result is close in spirit to several generalizations of the Banach con-traction theorem by Edelstein22, Sehgal23, Chatterjea24, Rhoades1, conditions 20 and22, and Suzuki15, Theorem 3.

Theorem 2.6. LetT be a self-map of a metric spaceX. Assume that

ithere exists a pointx0∈Xsuch that the orbitOT, x0has a cluster pointz∈X,

iiT andT2_{are continuous atz,}

iiifor any two distinct elementsx, y∈OT, x0,

1

2dx, Tx< d

x, y 2.33

implies one of the following conditions:

PM.1∗ dx, Tx dy, Ty<2dx, y,

PM.3∗ dx, Tx dy, Ty dTx, Ty<3/2dx, Ty dy, Tx,

PM.4∗ dTx, Ty<max{dx, y, dx, Tx, dy, Ty,1/2dx, Ty, dy, Tx}.

Thenzis a fixed point ofT.

Proof. An appropriate blend of the proof of Theorems 2.1 and 2 of Pal and Maiti3works. If only the conditionPM.4∗ is satisfied inTheorem 2.6, then the uniqueness of the fixed-pointzfollows easily.

Example 2.7. LetX {0,1/4,3/4,1}and T0 T1/4 0, T3/4 T1 3/4. Then, the mapT satisfies all the requirements ofTheorem 2.1with a 3/2,b 7/12, andk 4/5. Further,T is not a ´Ciri´c-Suzuki contraction, that is,T does not satify the requirements of10, Corollary 2.1. Evidently,Tis not a quasicontraction.

Example 2.8. LetX 0,1and

Tx

⎧ ⎪ ⎨ ⎪ ⎩

0, if 0≤x < 1

2, 1

2, if 1

2 ≤x≤1.

2.34

Then, one of the conditionsPM.1–PM.4is satisfiede.g.,x 49/100,y 1/2. AsT has two fixed points, it cannot satisfy any of the conditions which guarantee the existence of a unique fixed point.

Example 2.9. LetX{3,5,6,7}and

Tx

⎧ ⎨ ⎩

3, if _{x /}6,

6, if x6. 2.35

Then, the mapT satisfies all the requirements of Theorem 2.6. If inTheorem 2.6, the initial choice isx0 6resp.,x0/6, then{Tnx0}converges to 6resp., 3.

For any subsetsA, BofX,dA, Bdenotes the gap betweenAandB, while

ρA, B sup{dA, B:a∈A, b∈B},

BNX A:_{φ /}A⊆X and diameter ofAis finite. 2.36

As usual, we writedx, B resp.,ρx, BfordA, B resp.,ρA, BwhenA{x}. We useTheorem 2.1to obtain the following result for a multivalued map.

Theorem 2.10. LetP : X → BNXand letX beP-orbitally complete. Assume that there exist a, b, c, k, andηas defined inSection 2such that for anyx, y∈X

implies that at least one of the following conditions is true:

PM.1∗∗ ρx, P x ρy, P y≤adx, y,

PM.2∗∗ ρx, P x ρy, P y≤bdx, P y dy, P x dx, y,

PM.3∗∗ ρx, P x ρy, P y ρP x, P y≤cdx, P y dy, P x,

PM.4∗∗ ρP x, P y≤kmax{dx, y, dx, P x, dy, P y,1/2dx, P y, dy, P x}.

ThenP has a fixed point.

Proof. It may be completed following Reich 25, ´Ciri´c 2, and Singh and Mishra 11. However, a basic skech of the same is given below.

Letδ√k. Define a single-valued mapf :X → Xas follows. For eachx∈X, letfx

be a point ofP xsuch that

dx, fx≥δρx, P x. 2.38

Sincefx∈P x,dx, fx≤ρx, P x. So,2.37gives

ηdx, fx≤dx, y, 2.39

and in view of conditionsPM.1∗∗–PM.4∗∗, this implies that one of the following is true:

dx, fxdy, fy≤adx, y,

dx, fxdy, fy≤bdx, fydy, fxdx, y,

dx, fxdy, fydfx, fy≤cdx, fydy, fx,

dfx, fy ≤ k δmax

δdx, y, δρx, P x, δρy, P y,δ

2

dx, fy, dy, fx

≤√kmax

dx, y, dx, P x, dy, P y,1

2

dx, fydy, fx.

2.40

This meansTheorem 2.1applies as “x, y∈OT, x0” in the statement ofTheorem 2.1may be

replaced by “x, y∈X”. Hence, there exists a pointz∈Xsuch thatzfz, andz∈P z.

3. Applications

3.1. Application to Dynamic Programming

In this section, we assume thatUandV are Banach spaces,W⊆UandD⊆V. LetRdenote the field of reals,τ :W×D → W,f :W×D → RandG:W×D×R → R. The subspaces

WandDare considered as the state and decision spaces, respectively. Then, the problem of dynamic programming reduces to the problem of solving the functional equation

p:sup

y∈D

In multistage processes, some functional equations arise in a natural waycf. Bellman26 and Bellman and Lee27. The intent of this section is to study the existence of the solution of the functional equation3.1arising in dynamic programming.

LetBWdenote the set of all bounded real-valued functions onW. For an arbitrary

h∈W, definehsup_x∈W|hx|. Then,BW, · is a Banach space. Assume thatθk

1/1k, 0< k <1 and the following conditions hold: DP.1 G, fare bounded.

DP.2Assume that for everyx, y∈W×D,h, q∈BWandt∈W,

ηk|ht−Kht| ≤ht−qt 3.2

implies _G

x, y, ht−Gx, y, qt

≤kmaxht−qt,|ht−Kht|,qt−Kqt,1

2ht−Kqtqt−Kht

,

3.3

whereKis defined as follows:

Khx sup

y∈D

fx, yGx, y, hτx, y, x∈W, h∈BW. 3.4

Theorem 3.1. Assume that the conditions (DP.1) and (DP.2) are satisfied. Then, the functional equation3.1has a unique bounded solution.

Proof. We note thatBW, dis a complete metric space, wheredis the metric induced by the supremum norm onBW. ByDP.1,Kis a self-map ofBW.

Pickx∈W andh1, h2 ∈BW. Letμbe an arbitrary positive number. We can choose

y1, y2∈Dsuch that

Khj < f

x, yj

Gx, yj, hj

xj

μ, 3.5

wherexjτx, yj,j1,2.

Further, we have

Kh1x≥f

x, y2

Gx, y2, h1x2

, 3.6

Kh2x≥f

x, y1

Gx, y1, h2x1

. 3.7

Therefore,3.2becomes

Set

Mk:kmax

dh1, h2, dh1, Kh1, dh2, Kh2,1

2dh1, Kh2 dh2, Kh1

. 3.9

From3.5,3.7, and3.8, we have

Kh1x−Kh2x< G

x, y1, h1x1

−Gx, y1, h2x1

μ

≤Gx, y1, h1x1

−Gx, y1, h2x1μ

≤kmax

|h1x1−h2x1|,|h1x1−Kh1x1|,|h2x1−Kh2x1|,

1

2|h1x1−Kh2x1||h2x1−Kh1x1|

μ

≤Mk μ.

3.10

Similarly, from3.5,3.6, and3.8, we get

Kh2x−Kh1x≤Mk μ. 3.11

From3.10and3.11, we have

|Kh1x−Kh2x| ≤Mk μ. 3.12

Since the inequality3.12is true for anyx∈W, andμ >0 is arbitrary, we find from3.8that

θkdh1, Kh1≤dh1, h2 3.13

implies

dKh1, Kh2≤Mk. 3.14

SoCorollary 2.3 applies, whereinK corresponds to the mapT. Therefore,K has a unique fixed-pointh∗, that is,h∗xis the unique bounded solution of the functional equation3.1.

3.2. Application to Variational Inequalities

very suitable for implementation on parallel computers with single-instruction, multiple-data architecture, particularly on massively parallel processors.

The variational inequality problem is to find a functionusuch that

maxLu−f, u−φ0 onΩ,

u0 on∂Ω,

3.15

whereΩis a nonemptyq-starshaped open bounded subset ofRN_{for someq∈Ωwith smooth}

boundary such that 0∈Ω,Lis an elliptic operator defined onΩby

L−aijx ∂

2

∂xi∂xj bix

∂

∂xi cxIN,

3.16

where summation with respect to repeated indices is implied,cx ≥ 0,aijxis a strictly

positive definite matrix, uniformly inx, forx∈Ω, fandφare smooth functions defined inΩ andφsatisfies the condition:φx≥0,x∈∂Ω.

The corresponding problem of stochastic optimal control can be described as follows:

L−cIis the generator of a diffusion process inRN_{,cis a discount factor,fis the continuous}

cost, andφ represents the cost incurred by stopping the process. The boundary condition “u0 on∂Ω” expresses the fact that stopping takes place either prior or at the time that the diffusion process exists fromΩ.

A problem related to 3.15 is the two-obstacle variational inequality. Given two smooth functions φ and μ defined on Ω such that φ ≤ μ in Ω, φ ≤ 0 ≤ μ on ∂Ω, the corresponding variational inequality is as follows:

maxminLu−f, u−φ, u−μ0 on Ω.

u0 on∂Ω.

3.17

Note that the problem3.17arises in stochastic game theory.

LetAbe anN×Nmatrix corresponding to the finite difference discretizations of the operatorL. We make the following assumptions about the matrixA:

Aii1,

j, j /i

Aij>−1, Aij<0 fori /j. _3.18

These assumptions are related to the definition of “M-matrices”, arising from the finite difference discretization of continuous elliptic operators having the property3.18under the appropriate conditions andQdenotes the set of all discretized vectors inΩ see31,32. Note that the matrixAis anM-matrix if and only if every off-diagonal entry ofAis nonpositive.

LetBIN−A. Then, the corresponding properties for theB-matrices are

Bii0,

j, j /i

LetbmaxijBijandA∗anN×Nmatrix such thatA∗_ii1−bandA_ij∗ −bfori /j. Then,

we haveB∗IN−A∗.

Now, we show the existence of iterative solutions of variational inequalities. Consider the following discrete variational inequalities mentioned above:

maxminAx−A∗dx, Tx−f, x−A∗dx, Tx−φ, x−A∗dx, Tx−μ0, 3.20

whereTis an operator fromRN_{into itself implicitly defined by}

TxminmaxBxA1−B∗dx, Tx f,1−B∗dx, Tx φ,1−B∗dx, Tx μ

3.21

for allx∈Qsuch that for allx, y∈Q, the condition

θkdx, Tx≤dx, y, θk 1

1k, wherekmax{b,1−b} 3.22

holds. Suppose that the condition3.22implies thatTis defined inQas in3.21, then3.20 is equivalent to the fixed-point problem

xTx, 3.23

that is,Q∩FT/∅.

Notice that in two-person game, we have to determine the best strategies for each player on the basis of maximin and minimax criterion of optimality. This criterion will be well stated as follows: a player lists his/her worst possible outcomes, and then he/she chooses that strategy which corresponds to the best of these worst outcomes. Here, the problem3.20 exhibits the situation in which two players are trying to control a diffusion process; the first player is trying to maximize a cost functional, and the second player is trying to minimize a similar functional. The first player is called the maximizing player and the second one the minimizing player. Here,f represents the continuous rate of cost for both players,φis the stopping cost for the maximizing player, andμis the stopping cost for the minimizing player. This problem is fixed by inducting an operatorT implicitly defined for allx∈Qas in3.21. Theorem 3.2. Under the assumptions3.18and3.19, a solution for3.23exists.

Proof. LetTyi 1−Bij∗dyi, Tyi μifor anyy∈Qand anyi, j 1,2, . . . , N. Now, for

anyx∈Q, sinceTxi ≤1−B∗ijdxi, Txi μi, we have

Ty_imaxBijyj

1−B∗_ijdyi, Tyi

fi,

1−B_ij∗dyi, Tyi

φi

that is, if the maximizing player succeeds to maximize a cost functional in his/her strategy which corresponds to the best ofNworst outcomes from his/her list, then the game would be one-sided. In this situation, we introduce the one sided operator

TxmaxBxA1−B∗dx, Tx fi,1−B∗dx, Tx φ

. 3.25

Therefore, we have

Ty_iTy_i. 3.26

Now, ifTxiBijxjAij1−Bij∗dxi, Txi fi, then since

Ty_i≥BijyjAij

1−Bij∗

dyi, Tyi

fi, 3.27

by using3.18, we have

Txi−

Ty_i≤Bijxi−yi

1−B_ij∗maxdxi, Txi, d

yi, Tyi

≤Bijxi−yi

1−B_ij∗

×max

dxi, Txi, d

yi, Tyi

,1

2

dxi, Tyi

dyi, Txi

.

3.28

IfTxi 1−Bij∗·dxi, Txi φi, then since

Ty_i≥1−B_ij∗·dyi, Tyi

φi, 3.29

we have

Txi−

Ty_i≤1−B∗ij

maxdxi, Txi, d

yi, Tyi

≤1−B∗_ijmax

dxi, Txi, d

yi, Tyi

,1

2

dxi, Tyi

dyi, Txi

.

3.30

Hence, from3.18–3.20, we have

Txi−

Ty_i≤bx−y 1−bmax

dx, Tx, dy, Ty,1

2

dx, Tydy, Tx.

3.31

Sincexandyare arbitrarily chosen, we have

Ty_i−Txi≤bx−y 1−bmax

dx, Tx, dy, Ty,1

2

dx, Tydy, Tx.

Therefore, from3.31and3.32, it follows that

_{Tx−Ty≤bx−y} 1−bmax

dx, Tx, dy, Ty,1

2

dx, Tydy, Tx. 3.33

This yields

_Tx−Ty≤kmax

dx, y, dx, Tx, dy, Ty,1

2

dx, Tydy, Tx, 3.34

wherek max{b,1−b}. Thus, we see that under the assumptions3.18and3.19, for all

x, y∈Q,

θkdx, Tx≤dx, y 3.35

implies

_{Tx−Ty≤kmaxd}

x, y, dx, Tx, dy, Ty,1

2

dx, Tydy, Tx. 3.36

Note thatRN is complete and Qa closed subset ofRN, it follows thatQis complete. As a consequence,Qis orbitally complete.

Hence, we conclude that all the conditions ofCorollary 2.3are satisfied inQ. Therefore, Corollary 2.3ensures the existence of a solution of3.23.

Acknowledgment

This research is supported by the Directorate of Research Development, Walter Sisulu University.

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