Contractive Mappings in Partially Ordered Partial Metric Spaces

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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,

¹

Hassen Aydi,

²

and Stojan Radenovi ´c

³

1Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan

2Institut Sup´erieur d’Informatique et de Technologies de Communication de Hammam Sousse, Universit´e de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia

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

Correspondence should be addressed to Stojan Radenovi´c,[email protected] 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 14. The aim of this paper is to study the necessary conditions for existence of fixed point of mapping satisfyingT-Hardy-Rogers conditions in the framework of partially ordered partial metric spaces. Our results extend and strengthen various known results8,24. In the sequel, the letters R, R, and N will denote the set of real numbers, the set of nonnegative real

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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, completeness, 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 lim_{n→ ∞}x_n vorx_n → v;ii{xn}is called a Cauchy sequence if there exists and is finitelim_{n, m→ ∞}pxn, x_m.

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 lim_{n, m→ ∞}pxn, x_m. Ifp is a partial metric onX, then the functionp^s : X×X → R given byp^sx, 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, p^s;

b X, p is complete if and only if the metric space X, p^s is complete. Furthermore, limn→ ∞p^sxn, x 0 if and only if

px, x lim

n→ ∞pxn, x lim

n, m→ ∞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 thatx_n → xand y_n → y impliespxn, y_n → 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 → ∞.

2 see7However, ifpxn, x → px, x 0 thenpxn, y → px, yfor ally∈X.

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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, τp^s → X2, τp^sare continuous.

Remark 1.7. It is worth to notice that the notions p-continuous and p^s-continuous of any function in the context of partial metric spaces are incomparable, in general. Indeed, if X 0,∞,px, y max{x, y},p^sx, y |x−y|,f0 1, and fx x² for allx > 0 andgx |sinx|, thenf is ap-continuous andp^s-discontinuous at pointx 0; whileg is a p-discontinuous andp^s-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, p^simplies that{yn}is convergent inX, p^s,

iisubsequentially convergent if for any sequence {yn} in X such that {Tyn} is convergent inX, p^simplies that{yn}has a convergent subsequence inX, p^s. 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 exista_i ≥ 0,i 1, . . . ,5 with a₁a₂a₃a₄a₅<1 such that for allx, y∈X

p

Tfx, Tfy

≤a1p

Tx, Ty a2p

Tx, Tfx a3p

Ty, Tfy a₄p

Tx, Tfy a₅p

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

A pointx∈X is called a fixed point of mappingf :X → Xifx fx. The set of all fixed points of the mappingfis denoted byF_f.

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2. Fixed Point Results

In this section, we obtain fixed point results for a mapping satisfying aT-Hardy-Rogers contractive 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→ ∞p^sz, xn 0 it followsxn zfor alln∈N;

thenF_f/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}inXwithx_nfⁿx₀and sox_n1fx_nforn∈N. Sincex_n−1, x_n∈Δ therefore by replacingxbyx_n−1andybyxnin1.2, we have

pTxn, Tx_n1 p

Tfx_n−1, Tfxn

≤a₁pTx_n−1, Tx_n a₂p

Tx_n−1, Tfx_n−1 a₃p

Tx_n, Tfx_n a₄p

Tx_n−1, Tfxn a₅p

Txn, Tfx_n−1

a1pTxn−1, Txn a2pTxn−1, Txn a3pTxn, Tx_n1 a₄pTx_n−1, Tx_n1 a₅pTxn, Tx_n

≤a1a₂pTxn−1, Txn a₃pTxn, Tx_n1 a₄

pTx_n−1, Tx_n pTx_n, Tx_n1−pTx_n, Tx_n

a₅pTx_n, Tx_n a₁a₂a₄pTx_n−1, Tx_n a3a₄pTxn, Tx_n1

a5−a4pTxn, Txn,

2.2

that is,

1−a₃−a₄pTxn, Tx_n1≤a1a₂a₄pTxn−1, Txn a5−a₄pTxn, Txn. 2.3 Similarly, replacingxbyx_nandybyx_n−1in1.2, we obtain

1−a2−a5pTxn, Tx_n1≤a1a3a5pTxn−1, Txn a4−a5pTxn, Txn. 2.4

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Summing2.3and2.4, we obtainpTxn, Tx_n1 ≤ δpTxn−1, Txn, whereδ 2a1a2 a₃a₄a₅/2−a₂−a₃−a₄−a₅. Obviously 0≤δ <1. Therefore, for alln≥1,

pTxn, Tx_n1≤δpTx_n−1, Tx_n≤ · · · ≤δⁿpTx0, Tx₁. 2.5

Now, for anym∈Nwithm > n, we have

pTxn, Tx_m≤pTxn, Tx_n1 pTx_n1, Tx_n2 · · ·pTx_m−1, Tx_m

≤

δⁿδⁿ¹· · ·δ^m−1 pTx0, Tx₁≤ δⁿ

1−δpTx0, Tx₁,

2.6

which implies thatpTxn, Txm → 0 asn, m → ∞. Hence{Txn}is a Cauchy sequence in X, pand inX, p^s. SinceX, pis complete, therefore fromLemma 1.4,X, p^sis a complete metric space. Hence{Txn}converges to somev∈Xwith respect to the metricp^s, that is,

nlim→ ∞p^sTxn, v 0, 2.7

or equivalently,

pv, v lim

n→ ∞pTxn, v lim

n, m→ ∞pTxn, Txm 0. 2.8

Suppose thatT is subsequentially convergent, therefore convergence of{Txn}inX, p^simplies that{xn}has a convergent subsequence{xni}inX, p^s. So

i→ ∞limp^sx_n_i, u 0, 2.9

for some u ∈ X. As T is continuous, so 2.9 and Definition 1.6. imply that limi→ ∞p^s Txni, Tu 0. From 2.7 and by the uniqueness of the limit in metric spaceX, p^s, we obtainTuv. Consequently,

0pTu, Tu lim

i→ ∞pTxni, Tu lim

i, j→ ∞p

Txni, Txnj . 2.10

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

SinceTx_n_i → 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 havex_n_i uandxni, u∈Δ. Now,

p

Tfu, Tu

≤p

Tfu, Tfx_n_i p

Tfx_n_i, Tu

−p

Tfx_n_i, Tfx_n_i

≤a1pTxni, Tu a2p

Txni, Tfxni

a3p

Tu, Tfu a4p

Txni, Tfu a₅p

Tu, Tfx_n_i p

Tfx_n_i, Tu

−p

Tfx_n_i, Tfx_n_i

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a1pTxni, Tu a2pTxni, Txni1 a3p

Tu, Tfu a4p

Txni, Tfu a₅pTu, Txni1 pTxni1, Tu−pTxni1, Tx_n_i₁

≤a₁pTxni, Tu a₂pTxni, Txni1 a₃p

Tu, Tfu a4

pTxni, Tu p

Tu, Tfu

−pTu, Tu

a₅pTu, Txni1

pTxni1, Tu −pTxni1, Txni1.

2.11

On taking limit asi → ∞and applyingRemark 1.5.2we get p

Tu, Tfu

≤a₁pTu, Tu a₂pTu, Tu a₃p

Tu, Tfu a₄

pTu, Tu p

Tu, Tfu

−pTu, Tu a₅pTu, Tu pTu, Tu−pTu, Tu

a1a2a5pTu, Tu a3a4p

Tu, Tfu

≤a1a₂a₃a₄a₅p

Tu, Tfu

< p

Tu, 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 thatF_f ×F_f ⊂ Δ. Let w be a fixed point of f. As F_f ×F_f ⊂ Δ, therefore u, w∈Δ. From1.2, we have

pTu, Tw p

Tfu, Tfw

≤a₁pTu, Tw a₂p

Tu, Tfu a₃p

Tw, Tfw a₄p

Tu, Tfw a₅p

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

≤a1a₂a₃a₃a₄a₅pTu, Tw

< pTu, Tw,

2.13 and hencepTu, Tw 0, which further implies thatuwasTis injective.

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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

p

Tfx, 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, Ty a₂p

Tx, Tfx a₃p

Ty, Tfy a₄p

Tx, Tfy a₅p

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

p

Tfx, Tfy max

x² 9 ,y²

9

1 9max

x², y²

≤ 1 6max

x², y² a₁p

Tx, Ty

≤a₁p

Tx, Ty a₂p

Tx, Tfx a₃p

Ty, Tfy a₄p

Tx, Tfy a₅p

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

p fx, fy

≤a₁p x, y

a₂p x, fx

a₃p y, fy

a₄p x, fy

a₅p y, fx

, 2.16

wherea_i≥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→ ∞p^sz, xn 0 it followsxn zfor alln∈N; thenF_f/φ. Moreover,fhas a unique fixed point ifF_f×F_f ⊂Δ.

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Remark 2.5. Corollary 2.4corresponds to Theorem 2 of Altun et al.8in partially ordered partial metric spaces. For particular choices of the coeﬃcientsai_i1,...,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 fxx^1/3, 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 xⁿ 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

p

Tfx, Tf²x ≤λp

Tx, Tfx

, 3.1

for someλ∈0,1and for allx∈X,x /fx. Thenfhas the propertyPprovided thatF_f is nonempty andfis a dominating map onFfⁿ.

Proof. Letu ∈ F_fⁿ for some n > 1. Now we show thatu fu. Sincef is dominating on F_fⁿ, thereforeu fuwhich further implies thatfⁿ⁻¹u fⁿuasf is nondecreasing. Hence fⁿ⁻¹u, fⁿ⁻¹u∈Δ1. Now by using3.1, we have

p

Tu, Tfu p

Tffⁿ⁻¹u, Tf²fⁿ⁻¹u

≤λp

Tfⁿ⁻¹u, Tfⁿu λp

Tffⁿ⁻²u, Tf²fⁿ⁻²u . 3.2

Repeating the above process, we get p

Tu, Tfu

≤λⁿp

Tu, Tfu

. 3.3

Taking limit asn → ∞, we obtainpTu, Tfu 0 andTuTfu. AsT is injective, soufu, that is,u∈F_f.

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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 f²x and hencex, fx∈Δ. Now from1.2,

p

Tfx, Tf²x p

Tfx, Tffx

≤a₁p

Tx, Tfx a₂p

Tx, Tfx a₃p

Tfx, Tf²x a₄p

Tx, Tf²x a₅p

Tfx, Tfx

≤a₁a₂p

Tx, Tfx a₃p

Tfx, Tf²x a4

p

Tx, Tfx p

Tfx, Tf²x −p

Tfx, Tfx a5p

Tfx, Tfx ,

3.4

that is,

1−a₃−a₄p

Tfx, Tf²x ≤a1a₂a₄p

Tx, Tfx

a₅−a₄p

Tfx, Tfx

. 3.5

Again by using1.2, we have

p

Tf²x, Tfx p

Tffx, Tfx

≤a₁p

Tfx, Tx a₂p

Tfx, Tf²x a₃p

Tx, Tfx a₄p

Tfx, Tfx a₅p

Tx, Tf²x

≤a1a₃p

Tx, Tfx a₂p

Tfx, Tf²x a₄p

Tfx, Tfx a₅

p

Tx, Tfx p

Tfx, Tf²x −p

Tfx, Tfx ,

3.6

which implies that

1−a2−a5p

Tfx, Tf²x ≤a1a3a5p

Tx, Tfx

a4−a5p

Tfx, Tfx

. 3.7

Summing3.5and3.7impliespTfx, Tf²x≤λpTx, Tfx,λ 2a1a2a3a₄a5/2− a₂−a₃−a₄−a₅. Obviously,λ∈0,1. ByTheorem 3.4,fhas the propertyP.

Acknowledgment

S. Radenovi´c is thankful to The Ministry of Science and Technology Development of Serbia.

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Contractive Mappings in Partially Ordered Partial Metric Spaces

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. Introduction and Preliminaries

2. Fixed Point Results

3. Periodic Point Results

Acknowledgment

References

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