On a Suzuki Type General Fixed Point Theorem with Applications

(1)

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,

^{1, 2}

H. K. Pathak,

^{1, 3}

and S. N. Mishra

¹

1Department of Mathematics, Walter Sisulu University, Mthatha 5117, South Africa

221 Govind Nagar, Rishikesh 249201, India

3School 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.

On the other hand, Suzuki5recently obtained a remarkable generalization of the Banach contraction theorem which itself has been extended and generalized on various settingssee, e.g,6–15. With a view of extending Suzuki’s contraction theorem5and its several generalizations, we combine the ideas of Pal and Maiti3, Suzuki5, and Popescu 10to obtain a very general fixed-point theorem. Subsequently, we use our results to solve certain functional equations and variational inequalities under diﬀerent conditions than those considered in Bhakta and Mitra16, Baskaran and Subrahmanyam17, Pathak et al.18,19, Singh and Mishra11,12, and Pathak et al.20, and references thereof.

(2)

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 → 2^X, 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≤d x, 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{Tⁿx0}converges inXandzlim_n_{→ ∞}Tⁿx0is a fixed point ofT.

(3)

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

d_n1≤ 2b−1 1−b dn, dn1 ≤ c−1

2−cdn, dn1≤kdn.

2.6

Hence, from2.5-2.6,

d_n1≤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 forx /z:

dxn, Txn dx, Tx≤adxn, x, 2.10

which yields on makingn → ∞,

dx, Tx≤adx, z, 2.11

(4)

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 forx /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

.

2.21

(5)

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

ηd

xnj, xnj1

≤d xnj, z

, j ∈N. 2.23

Recall that by2.7,

dxn, x_n1≤λdx_n−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

≤ηdx_n−1, xn ηλdx_n−1, xn η1λdxn−1, xn.

2.26

Since without loss of generality, we may takeλk, we have dx_n−1, xn< η1kdx_n−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

ηdx2n−1, x2n≤dx2n−1, z, or ηdx2n, x2n1≤dx2n, z 2.29

(6)

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

xnj, xnj1

≤d xnj, z

, 2.30

that is,

ηd

xnj, Txnj

≤d xnj, 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 contraction 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 andT²are 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.2^∗ dx, Tx dy, Ty<2/3dx, Ty dy, Tx dx, y,

(7)

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

ηρx, P x≤d x, y

2.37

(8)

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

d x, fx

≥δρx, P x. 2.38

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

x, fx

≤d x, y

, 2.39

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

d x, fx

d y, fy

≤ad x, y

, d

x, fx d

y, fy

≤b d

x, fy d

y, fx d

x, y , d

x, fx d

y, fy d

fx, fy

≤c d

x, fy d

y, fx , d

fx, fy

≤ k δmax

δd

x, y

, δρx, P x, δρ y, P y

,δ 2

d x, fy

, d

y, fx

≤√ kmax

d

x, y

, dx, P x, d y, P y

,1 2

d x, fy

d

y, 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

f x, y

G x, y, p

τ

x, y

, x∈W. 3.1

(9)

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

−G

x, y, qt

≤kmaxht−qt,|ht−Kht|,qt−Kqt,1

2ht−Kqtqt−Kht , 3.3

whereKis defined as follows:

Khx sup

y∈D

f x, y

G x, y, h

τ

x, 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

G x, yj, hj

xj

μ, 3.5

wherexjτx, yj,j1,2.

Further, we have

Kh1x≥f x, y2

G

x, y2, h1x2

, 3.6

Kh2x≥f x, y1

G

x, y1, h2x1

. 3.7

Therefore,3.2becomes

θk|h1x−Kh1x| ≤ |h1x−h2x|. 3.8

(10)

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

−G

x, y1, h2x1 μ

≤G

x, y1, h1x1

−G

x, 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

As another application of Corollary 2.3, we show the existence of solutions of variational inequalities as in the work of Belbas and Mayergoyz28. Variational inequalities arise in optimal stochastic control 29 as well as in other problems in mathematical physics, for examples, deformation of elastic bodies stretched over solid obstacles, elastoplastic torsion, and so forth,30. The iterative method for solutions of discrete variational inequalities is

(11)

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 max

Lu−f, u−φ

0 onΩ,

u0 on∂Ω, 3.15

whereΩis a nonemptyq-starshaped open bounded subset ofR^Nfor 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 inR^N,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:

max min

Lu−f, u−φ

, u−μ

0 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

Bij<1, Bij>0 for i /j. 3.19

(12)

Letbmaxi

jBijandA^∗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:

max min

Ax−A^∗dx, Tx−f, x−A^∗dx, Tx−φ

, x−A^∗dx, Tx−μ

0, 3.20

whereTis an operator fromR^Ninto itself implicitly defined by

Txmin max

BxA1−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≤d x, 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. LetTy_i 1−B_ij^∗dyi, Tyi μifor anyy∈Qand anyi, j 1,2, . . . , N. Now, for anyx∈Q, sinceTx_i ≤1−B^∗_ijdxi, Txi μi, we have

Ty

imax

Bijyj 1−B^∗_ij

d yi, Tyi

fi, 1−B_ij^∗

d yi, Tyi

φi

, 3.24

(13)

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

Txmax

BxA1−B^∗dx, Tx fi,1−B^∗dx, Tx φ

. 3.25

Therefore, we have

Ty

i

Ty

i. 3.26

Now, ifTx_iBijxjAij1−B_ij^∗dxi, Txi fi, then since Ty

i≥BijyjAij

1−B_ij^∗

d yi, Tyi

fi, 3.27

by using3.18, we have Tx_i−

Ty

i≤Bijxi−yi 1−B_ij^∗

max

dxi, Txi, d yi, Tyi

≤Bijxi−yi 1−B_ij^∗

×max

dxi, Txi, d yi, Tyi

,1 2

d xi, Tyi

d yi, Txi

.

3.28

IfTx_i 1−B_ij^∗·dxi, Txi φi, then since Ty

i≥ 1−B_ij^∗

·d yi, Tyi

φi, 3.29

we have Tx_i−

Ty

i≤ 1−B^∗_ij

max

dxi, Txi, d yi, Tyi

≤ 1−B^∗_ij

max

dxi, Txi, d yi, Tyi

,1 2

d xi, Tyi

d yi, Txi

.

3.30

Hence, from3.18–3.20, we have Tx_i−

Ty

i≤bx−y 1−bmax

dx, Tx, d y, Ty

,1 2

d x, Ty

d

y, Tx . 3.31

Sincexandyare arbitrarily chosen, we have Ty

i−Tx_i≤bx−y 1−bmax

dx, Tx, d y, Ty

,1 2

d x, Ty

d

y, Tx . 3.32

(14)

Therefore, from3.31and3.32, it follows that Tx−Ty≤bx−y 1−bmax

dx, Tx, d y, Ty

,1 2

d x, Ty

d

y, Tx

. 3.33

This yields

Tx−Ty≤kmax

d x, y

, dx, Tx, d y, Ty

,1 2

d x, Ty

d

y, Tx

, 3.34

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

θkdx, Tx≤d x, y

3.35

implies

Tx−Ty≤kmax

d x, y

, dx, Tx, d y, Ty

,1 2

d x, Ty

d

y, Tx

. 3.36

Note thatR^N is complete and Qa closed subset ofR^N, 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.

References

1 B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.

2 L. B. ´Ciri´c, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.

3 T. K. Pal and M. Maiti, “Extensions of fixed point theorems of Rhoades and ´Ciri´c,” Proceedings of the American Mathematical Society, vol. 64, no. 2, pp. 283–286, 1977.

4 B. E. Rhoades, “Extensions of some fixed point theorems of ´Ciri´c, Maiti, and Pal,” Mathematics Seminar Notes. Kobe University, vol. 6, no. 1, pp. 41–46, 1978.

5 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.

6 A. Abkar and M. Eslamian, “Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 457935, 10 pages, 2010.

7 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.

(15)

8 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.

9 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.

10 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.

11 S. L. Singh and S. N. Mishra, “Coincidence theorems for certain classes of hybrid contractions,” Fixed Point Theory and Applications, vol. 2010, Article ID 898109, 14 pages, 2010.

12 S. L. Singh and S. N. Mishra, “Remarks on recent fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, Article ID 452905, 18 pages, 2010.

13 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.

14 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.

15 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.

16 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.

17 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.

18 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.

19 H. K. Pathak and B. Fisher, “Common fixed point theorems with applications in dynamic programming,” Glasnik Matematiˇcki, vol. 3151, no. 2, pp. 321–328, 1996.

20 H. K. Pathak, S. N. Mishra, and A. K. Kalinde, “Some Gregus type common fixed point theorems with applications,” Demonstratio Mathematica, vol. 36, no. 2, pp. 413–426, 2003.

21 L. B. ´Ciri´c, “Generalized contractions and fixed-point theorems,” Publications de l’Institut Math´ematique, vol. 1226, pp. 19–26, 1971.

22 M. Edelstein, “On fixed and periodic points under contractive mappings,” Journal of the London Mathematical Society, vol. 37, pp. 74–79, 1962.

23 V. M. Sehgal, “On fixed and periodic points for a class of mappings,” Journal of the London Mathematical Society, vol. 5, pp. 571–576, 1972.

24 S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de l’Acad´emie Bulgare des Sciences, vol. 25, pp.

727–730, 1972.

25 S. Reich, “Fixed points of contractive functions,” Bollettino della Unione Matematica Italiana, vol. 5, pp.

26–42, 1972.

26 R. Bellman, Methods of Nonliner Analysis. Vol. II, Mathematics in Science and Engineering, Vol. 61-II, Academic Press, New York, NY, USA, 1973.

27 R. Bellman and E. S. Lee, “Functional equations in dynamic programming,” Aequationes Mathematicae, vol. 17, no. 1, pp. 1–18, 1978.

28 S. A. Belbas and I. D. Mayergoyz, “Applications of fixed-point methods to discrete variational and quasivariational inequalities,” Numerische Mathematik, vol. 51, no. 6, pp. 631–654, 1987.

29 A. Bensoussan and J.-L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Méthodes Mathématiques de l’Informatique, no. 6, Dunod, Paris, France, 1978.

30 G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 21, Springer, Berlin, Germany, 1976.

31 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics, Academic Press, New York, NY, USA, 1979.

32 R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliﬀs, NJ, USA, 1982.