Fixed Point of Hardy Rogers Contractive Mappings in Partially Ordered Partial Metric Spaces

Volume 2012, Article ID 313675,11pages doi:10.1155/2012/313675

Research Article

Fixed Point of

T

-Hardy-Rogers

Contractive Mappings in Partially Ordered

Partial Metric Spaces

Mujahid Abbas,

_{Hassen Aydi,}

_{and Stojan Radenovi ´c}

1_{Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan} 2_{Institut Sup´erieur d’Informatique et de Technologies de Communication de Hammam Sousse,}

Universit´e de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia

3_{Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia}

Correspondence should be addressed to Stojan Radenovi´c,sradenovic@mas.bg.ac.rs

Received 20 March 2012; Accepted 29 August 2012

Academic Editor: Brigitte Forster-Heinlein

Copyrightq2012 Mujahid Abbas 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.

We prove some fixed point theorems for a T-Hardy-Rogers contraction in the setting of partially ordered partial metric spaces. We apply our results to study periodic point problems for such mappings. We also provide examples to illustrate the results presented herein.

1. Introduction and Preliminaries

The notion of a partial metric space was introduced by Matthews in 1. In partial metric spaces, the distance of a point in the self may not be zero. After the definition of a partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle, more suitable in this context1. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions e.g.,2–21, 22. Existence of fixed points in partially ordered metric spaces has been initiated in 2004 by Ran and Reurings

23. Subsequently, several interesting and valuable results have appeared in this direction

numbers, and the set of nonnegative integer numbers, respectively. The usual order on R

resp., onRwill be indistinctly denoted by≤or by≥.

Consistent with1,8 see25–29the following definitions and results will be needed in the sequel.

Definition 1.1see1. A partial metric on a nonempty setXis a mappingp:X×X → R

such that for allx, y, z∈X,

p1xy⇔px, x px, y py, y,

p2px, x≤px, y,

p3px, y py, x,

p4px, y≤px, z pz, y−pz, z.

A partial metric space is a pairX, psuch thatX is a nonempty set andpis a partial metric onX. Ifpx, y 0, thenp1andp2imply thatxy. But converse does not hold always. A

trivial example of a partial metric space is the pairR, p, wherep:R×R → Ris defined aspx, y max{x, y}. Each partial metricponXgenerates aT0topologyτponXwhich has

as a base the family openp-balls{Bpx, ε:x∈X, ε >0}, whereBpx, ε {y∈X:px, y<

px, x ε}.

On a partial metric space the concepts of convergence, Cauchy sequence, complete-ness, and continuity are defined as follows.

Definition 1.2see1. LetX, pbe a partial metric space and let{xn}be a sequence inX.

Theni{xn}converges to a pointx∈Xif and only ifpx, x limn→ ∞px, xn we may still

write this as limn→ ∞xn vorxn → v;ii{xn}is called a Cauchy sequence if there exists

and is finitelim_{n, m→ ∞}pxn, xm.

Definition 1.3see1. A partial metric spaceX, pis said to be complete if every Cauchy

sequence{xn}inX converges to a pointx∈X, such thatpx, x limn, m→ ∞pxn, xm. Ifp

is a partial metric onX, then the functionps : X×X → R given bypsx, y 2px, y− px, x−py, yis a metric onX.

Lemma 1.4see1,20. LetX, pbe a partial metric space. Then

a{xn}is a Cauchy sequence inX, pif and only if it is a Cauchy sequence in the metric

spaceX, ps;

b X, p is complete if and only if the metric space X, ps is complete. Furthermore,

limn→ ∞psxn, x 0 if and only if

px, x lim

n→ ∞pxn, x n, mlim→ ∞pxn, xm. 1.1

Remark 1.5. 1 see 19 Clearly, a limit of a sequence in a partial metric space does not

need to be unique. Moreover, the functionp·,·does not need to be continuous in the sense thatxn → xand yn → y impliespxn, yn → px, y. For example, if X 0,∞and

px, y max{x, y}forx, y ∈ X, then for{xn} {1}, pxn, x x px, xfor eachx ≥1

and so, for example,xn → 2 andxn → 3 whenn → ∞.

Definition 1.6see30. Suppose thatX, pis a partial metric space. Denoteτpits topology. We say T : X, p → X, p is continuous if both T : X, τp → X, τp and T :

X, τps _{→ X}

2, τpsare continuous.

Remark 1.7. It is worth to notice that the notions p-continuous and ps-continuous of any

function in the context of partial metric spaces are incomparable, in general. Indeed, if X 0,∞,px, y max{x, y},psx, y |x−y|,f0 1, and fx x2 _{for allx > 0}

andgx |sinx|, thenf is ap-continuous andps-discontinuous at pointx 0; whileg is a p-discontinuous andps-continuous atxπ.

According to31, we state the following definition.

Definition 1.8. LetX, pbe a partial metric space. A mappingT:X → Xis said to be

isequentially convergent if for any sequence{yn}inXsuch that{Tyn}is convergent inX, psimplies that{yn}is convergent inX, ps,

iisubsequentially convergent if for any sequence {yn} in X such that {Tyn} is convergent inX, psimplies that{yn}has a convergent subsequence inX, ps.

Consistent with24,31we define aT-Hardy-Rogers contraction in the framework of partial metric spaces.

Definition 1.9. Let X, p be a partial metric space andT, f : X → X be two mappings. A

mappingfis said to be aT-Hardy-Rogers contraction if there existai ≥ 0,i 1, . . . ,5 with

a1a2a3a4a5<1 such that for allx, y∈X

pTfx, Tfy≤a1p

Tx, Tya2p

Tx, Tfxa3p

Ty, Tfy

a4p

Tx, Tfya5p

Ty, Tfx. 1.2

Puttinga1 a4 a5 0 anda2 a3/0,resp.,a1 a2 a3 0 anda4 a5/0in the

previous definition, then the inequality1.2is said a T-Kannanresp.,T-Chatterjeatype contraction. Also, ifa4 a50 anda1, a2, a3/0,1.2is said theT-Reich type contraction.

Definition 1.10. LetX be a nonempty set. ThenX, p,is called a partially ordered partial

metric space if and only ifipis a partial metric onXandiiis a partial order onX. LetX, pbe a partial metric space endowed with a partial orderand letf :X → X be a given mapping. We define setsΔ,Δ1⊂X×Xby

Δ x, y∈X×X:xy oryx,

Δ1

x, x∈X×X:xfxorfxx. 1.3

2. Fixed Point Results

In this section, we obtain fixed point results for a mapping satisfying aT-Hardy-Rogers con-tractive condition defined on a partially ordered partial metric space which is complete.

We start with the following result.

Theorem 2.1. LetX,, pbe an partially ordered partial metric space which is complete. Let T :

X → Xbe a continuous, injective mapping andf :X → Xa nondecreasingT-Hardy-Rogers

con-traction for allx, y ∈ Δ. If there exists x0 ∈ X with x0 fx0, and one of the following two

conditions is satisfied

afis a continuous self-map onX;

bfor any nondecreasing sequence{xn}inX,with limn→ ∞psz, xn 0 it followsxn

zfor alln∈N;

thenFf/φprovided thatT is subsequentially or sequentially convergent. Moreover,f has a unique

fixed point ifFf×Ff ⊂Δ.

Proof. Asfis nondecreasing, therefore by given assumption, we have

x1fx0f2x0 · · · fnx0fn1x0 · · · 2.1

Define a sequence{xn}inXwithxnfnx0and soxn1fxnforn∈N. Sincexn−1, xn∈Δ

therefore by replacingxbyxn−1andybyxnin1.2, we have

pTxn, Txn1 p

Tfxn−1, Tfxn

≤a1pTxn−1, Txn a2p

Tx_n−1, Tfx_n−1a3p

Txn, Tfxn

a4p

Txn−1, Tfxna5p

Txn, Tfxn−1

a1pTxn−1, Txn a2pTxn−1, Txn a3pTxn, Txn1

a4pTxn−1, Txn1 a5pTxn, Txn

≤a1a2pTxn−1, Txn a3pTxn, Txn1

a4

pTx_n−1, Txn pTxn, Txn1−pTxn, Txna5pTxn, Txn

a1a2a4pTxn−1, Txn a3a4pTxn, Txn1

a5−a4pTxn, Txn,

2.2

that is,

1−a3−a4pTxn, Txn1≤a1a2a4pTxn−1, Txn a5−a4pTxn, Txn. 2.3

Similarly, replacingxbyxnandybyxn−1in1.2, we obtain

Summing2.3and2.4, we obtainpTxn, Txn1 ≤ δpTxn−1, Txn, whereδ 2a1a2

a3a4a5/2−a2−a3−a4−a5. Obviously 0≤δ <1. Therefore, for alln≥1,

pTxn, Txn1≤δpTxn−1, Txn≤ · · · ≤δnpTx0, Tx1. 2.5

Now, for anym∈Nwithm > n, we have

pTxn, Txm≤pTxn, Txn1 pTxn1, Txn2 · · ·pTxm−1, Txm

≤δn_δn1_{· · ·δ}m−1 _pTx

0, Tx1≤

δn

1−δpTx0, Tx1,

2.6

which implies thatpTxn, Txm → 0 asn, m → ∞. Hence{Txn}is a Cauchy sequence in

X, pand inX, ps. SinceX, pis complete, therefore fromLemma 1.4,X, psis a complete metric space. Hence{Txn}converges to somev∈Xwith respect to the metricps, that is,

lim

n→ ∞p

s_Tx

n, v 0, 2.7

or equivalently,

pv, v lim

n→ ∞pTxn, v n, m→ ∞lim pTxn, Txm 0. 2.8

Suppose thatT is subsequentially convergent, therefore convergence of{Txn}inX, ps im-plies that{xn}has a convergent subsequence{xni}inX, ps. So

lim

i→ ∞p

s_x

ni, u 0, 2.9

for some u ∈ X. As T is continuous, so 2.9 and Definition 1.6. imply that limi→ ∞ps

Txni, Tu 0. From 2.7 and by the uniqueness of the limit in metric spaceX, ps, we

obtainTuv. Consequently,

0pTu, Tu lim

i→ ∞pTxni, Tu i, jlim→ ∞p

Txni, Txnj . 2.10

1◦Iff is a continuous self-map onX, thenfxni → fuandTfxni → Tfuasi → ∞.

SinceTxni → Tuasi → ∞, we obtain thatTfuTu. AsT is injective, so we have

fuu.

2◦Iffis not continuous then by given assumption we havexnufor alln∈N. Thus

for a subsequence{xni}of{xn}we havexni uandxni, u∈Δ. Now,

pTfu, Tu≤pTfu, TfxnipTfxni, Tu−pTfxni, Tfxni

≤a1pTxni, Tu a2p

Txni, Tfxnia3p

Tu, Tfua4p

Txni, Tfu

a5p

a1pTxni, Tu a2pTxni, Txni1 a3p

Tu, Tfua4p

Txni, Tfu

a5pTu, Txni1 pTxni1, Tu−pTxni1, Txni1

≤a1pTxni, Tu a2pTxni, Txni1 a3p

Tu, Tfu

a4

pTxni, Tu pTu, Tfu−pTu, Tua5pTu, Txni1

pTxni1, Tu −pTxni1, Txni1.

2.11

On taking limit asi → ∞and applyingRemark 1.5.2we get

pTu, Tfu≤a1pTu, Tu a2pTu, Tu a3p

Tu, Tfu

a4

pTu, Tu pTu, Tfu−pTu, Tu

a5pTu, Tu pTu, Tu−pTu, Tu

a1a2a5pTu, Tu a3a4p

Tu, Tfu

≤a1a2a3a4a5p

Tu, Tfu

< pTu, Tfu,

2.12

which implies thatpTu, Tfu 0, and so Tu Tfu. Now injectivity of T givesu fu. Following similar arguments to those given above, the result holds whenT is sequentially convergent.

Suppose thatFf ×Ff ⊂ Δ. Let w be a fixed point of f. As Ff ×Ff ⊂ Δ, therefore

u, w∈Δ. From1.2, we have

pTu, Tw pTfu, Tfw

≤a1pTu, Tw a2p

Tu, Tfua3p

Tw, Tfwa4p

Tu, Tfwa5p

Tw, Tfu

a1pTu, Tw a2pTu, Tu a3pTw, Tw a4pTu, Tw a5pTw, Tu

≤a1a2a3a3a4a5pTu, Tw

< pTu, Tw,

2.13

Example 2.2. LetX 0,1be endowed with usual order and letp be the complete partial metric onXdefined bypx, y max{x, y}for allx, y∈X. LetT, f :X → Xbe defined by Tx4x/5 andfxx/4. Note thatΔ X×X. For anyx, y∈Δ, we have

pTfx, Tfy max

_x 5, y 5 1 5max

x, y≤ 12

35max x, y ≤ 2 7max 4x 5 , 4y 5 1 7max 4x 5 , x 5 1 7max 4y 5 , y 5 1 7max 4x 5 , y 5 1 7max _4y 5 , x 5

a1p

Tx, Tya2p

Tx, Tfxa3p

Ty, Tfy

a4p

Tx, Tfya5p

Ty, Tfx.

2.14

Therefore,f is aT-Hardly-Rogers contraction witha12/7, a2 a3a4 a5 1/7.

Obvi-ously,T is continuous and sequentially convergent. Thus, all the conditions ofTheorem 2.1

are satisfied. Moreover, 0 is the unique fixed point off.

Example 2.3. LetX 0,∞be endowed with usual order and letpbe a partial metric onX

defined bypx, y max{x, y}for allx, y∈X. DefineT, f :X → XbyTxx2 _{and fx}

x/3. Note thatΔ X×X. For anyx, y∈Δ, we have

pTfx, Tfy max

x2 9 , y2 9 1 9max

x2_{, y}2_≤ 1

6max

x2_{, y}2

a1p

Tx, Ty≤a1p

Tx, Tya2p

Tx, Tfxa3p

Ty, Tfy

a4p

Tx, Tfya5p

Ty, Tfx.

2.15

Therefore,fis aT-Hardly-Rogers contraction witha1 a2 a3 a4 a5 1/6. Also,T is

continuous and sequentially convergent. Thus all the conditions ofTheorem 2.1are satisfied. Moreover, 0 is the unique fixed point off.

TakingTx xin1.2andTheorem 2.1, we get the Hardy-Rogers type32 and so the Kannan, Chatterjea, and Reichfixed point theorem on partially ordered partial metric spaces.

Corollary 2.4. LetX,, pbe a partially ordered partial metric space which is complete. Letf:X →

Xbe a nondecreasing mapping such that for allx, y∈Δ, we have

pfx, fy≤a1p

x, ya2p

x, fxa3p

y, fya4p

x, fya5p

y, fx, 2.16

whereai≥0,i1, . . . ,5 witha1a2a3a4a5 <1. If there existsx0∈Xwithx0fx0, and

one of the following two conditions is satisfied.

afis a continuous self map onX;

bfor any nondecreasing sequence{xn}inX,with limn→ ∞psz, xn 0 it followsxn

Remark 2.5. Corollary 2.4corresponds to Theorem 2 of Altun et al.8in partially ordered partial metric spaces. For particular choices of the coefficientsaii1,...,5 inTheorem 2.1, we

obtain theT-Kannan,T-Chatterjea, andT-Reich type fixed point theorems. Also,Theorem 2.1

is an extension of Theorem 2.1 of Filipovi´c et al.24from the cone metric spaces to partial metric spaces.

3. Periodic Point Results

Letf :X → X. If the mapfsatisfiesFf Ffn for eachn∈N, then it is said to have the

pro-pertyP, for more details see33.

Definition 3.1. LetX,be a partially ordered set. A mappingf is called1a dominating

map onXifxfxfor eachxinXand2a dominated map onXiffxxfor eachxinX.

Example 3.2. LetX 0,1 be endowed with usual ordering. Let f : X → X defined by

fxx1/3_{, thenx≤x}1/3 _{fxfor allx∈X. Thusfis a dominating map.}

Example 3.3. LetX 0,∞be endowed with usual ordering. Letf : X → X defined by

fx √n_{xforx ∈ 0,1and letfx x}n _{for x ∈ 1,∞, for anyn ∈ N, then for allx ∈ X,}

x≤fxthat isfis the dominating map. Note thatΔ1/φiffis a dominating or a dominated

mapping.

We have the following result.

Theorem 3.4. LetX,, pbe a partially ordered partial metric space which is complete. LetT :X →

Xbe an injective mapping andf:X → Xa nondecreasing such that for allx, x∈Δ1, we have

pTfx, Tf2_{x ≤λpTx, Tfx, 3.1}

for someλ∈0,1and for allx∈X,x /fx. Thenfhas the propertyPprovided thatFf is nonempty

andfis a dominating map onFfn.

Proof. Letu ∈ Ffn for some n > 1. Now we show thatu fu. Sincef is dominating on

Ffn, thereforeu fuwhich further implies thatfn−1u fnuasf is nondecreasing. Hence

fn−1_{u, f}n−1_u∈Δ

1. Now by using3.1, we have

pTu, TfupTffn−1_{u, Tf}2_fn−1_u

≤λpTfn−1_{u, Tf}n_{u λpTff}n−2_{u, Tf}2_fn−2_{u .}

3.2

Repeating the above process, we get

pTu, Tfu≤λn_{pTu, Tfu. 3.3}

Theorem 3.5. LetX,, pbe a partially ordered partial metric space which is complete. letT, f :

X → X be mappings satisfy the condition ofTheorem 2.1. Iff is dominating onX, thenf has the

propertyP.

Proof. From Theorem 2.1, Ff/∅. We will prove that 3.1 is satisfied for all x, x ∈ Δ1.

Indeed,f is a dominating map so thatxfxand alsofis nondecreasing so thatfx f2_x

and hencex, fx∈Δ. Now from1.2,

pTfx, Tf2_{x pTfx, Tffx≤a}

1p

Tx, Tfxa2p

Tx, Tfxa3p

Tfx, Tf2_x

a4p

Tx, Tf2_{x a}

5p

Tfx, Tfx

≤a1a2p

Tx, Tfxa3p

Tfx, Tf2_x

a4

pTx, TfxpTfx, Tf2_{x −pTfx, Tfx a}

5p

Tfx, Tfx,

3.4

that is,

1−a3−a4p

Tfx, Tf2_{x ≤a}

1a2a4p

Tx, Tfx a5−a4p

Tfx, Tfx. 3.5

Again by using1.2, we have

pTf2_{x, Tfx pTffx, Tfx≤a}

1p

Tfx, Txa2p

Tfx, Tf2_{x a}

3p

Tx, Tfx

a4p

Tfx, Tfxa5p

Tx, Tf2_x

≤a1a3p

Tx, Tfxa2p

Tfx, Tf2_{x a}

4p

Tfx, Tfx

a5

pTx, TfxpTfx, Tf2_{x −pTfx, Tfx ,}

3.6

which implies that

1−a2−a5p

Tfx, Tf2_{x ≤a}

1a3a5p

Tx, Tfx a4−a5p

Tfx, Tfx. 3.7

Summing3.5and3.7impliespTfx, Tf2_{x≤λpTx, Tfx,λ 2a}

1a2a3a4a5/2−

a2−a3−a4−a5. Obviously,λ∈0,1. ByTheorem 3.4,fhas the propertyP.

Acknowledgment

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Optimization

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Combinatorics

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International Journal of Hindawi Publishing Corporation

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Journal of

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Function Spaces

Abstract and Applied Analysis

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International Journal of Mathematics and Mathematical Sciences

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The Scientific

World Journal

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Algebra

Discrete Dynamics in Nature and Society

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Decision Sciences

Discrete Mathematics

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Stochastic Analysis