End-Point Results for Multivalued Mappings in Partially Ordered Metric Spaces

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Volume 2012, Article ID 580250,19pages doi:10.1155/2012/580250

Research Article

End-Point Results for Multivalued Mappings in Partially Ordered Metric Spaces

Ismat Beg

¹

and Hemant Kumar Nashine

²

1Department of Mathematics and Statistics, Faculty of Management Studies, University of Central Punjab, Lahore, Pakistan

2Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur 492101, India

Correspondence should be addressed to Ismat Beg,ibeg@lums.edu.pk Received 27 March 2012; Accepted 7 June 2012

Academic Editor: J. Dydak

Copyrightq2012 I. Beg and H. K. Nashine. 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 purpose of this paper is to prove end-point theorems for multivalued mappings satisfying comparatively a more general contractive condition in ordered complete metric spaces. After- wards, we extend the results of previous sections and prove common end-point results for a pair of T-weakly isotone increasing multivalued mappings in the underlying spaces. Finally, we present common end point for a pair ofT-weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

1. Introduction and Preliminaries

Fixed-point theory for multivalued mappings was originally initiated by Von Neumann in the study of game theory. Fixed-point theorem for multivalued mappings is quite useful in control theory and has been frequently used in solving the problem of economics and game theory.

The theory of multivalued nonexpansive mappings is comparatively complicated as compare to the corresponding theory of single-valued nonexpansive mappings. It is therefore natural to expect that the theory of noncontinuous nonself-multivalued mappings would be much more complicated.

The study of fixed-points for multivalued contraction mappings was equally an active topic as single-valued mappings. The development of geometric fixed-point theory for multivalued was initiated with the work of Nadler Jr.1in the year 1969. He used the concept

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of Hausdorﬀmetric to establish the multivalued contraction principle containing the Banach contraction principle as a special case, as following.

Theorem 1.1. LetX, dbe a complete metric space andTis a mapping fromXintoCBXsuch that for allx, y∈ X,

H

Tx,Ty

≤λd x, y

, 1.1

where 0≤λ <1. ThenThas a fixed-point.

Since then, this discipline has been further developed, and many profound concepts and results have been established; for example, the work of Border2, ´Ciri´c3, Corley4, Itoh and Takahashi5, Mizoguchi and Takahashi6, Petrus¸el and Luca 7, Rhoades8, Tarafdar and Yuan9, and references cited therein.

LetX, dbe a metric space. We denote the class of nonempty and bounded subsets of XbyBX. ForA,B ∈BX, functionsDA,B, andδA,Bare defined as follows:

DA,B inf{da, b:a∈ A, b∈ B},

δA,B sup{da, b:a∈ A, b∈ B}. 1.2

If A {a}, then we write DA,B Da,B and δA,B δa,B. Also in addition, if B{b}, thenDA,B da, bandδA,B da, b. Obviously,DA,B≤δA,B.

For allA,B,C ∈BX, the definition ofδA,Byields the following:

δA,B δB,A, δA,B≤δA,C δC,B, δA,B 0 iﬀAB{a},

δA,A diamA.

1.3

A pointx ∈ Xis called a fixed-point of a multivalued mappingT : X → BXif x∈ Tx. If there exists a pointx∈ Xsuch thatTx{x}, thenxis called an end-point ofT 10.

Definition 1.2. LetXbe a nonempty set. ThenX, d,is called an ordered metric space if and only if:

i X, dis a metric space,

ii X,is a partially ordered set.

Definition 1.3. LetX,be a partial ordered set. Thenx, y∈ Xare called comparable ifxy oryxholds.

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Definition 1.4 see11. LetA and Bbe two nonempty subsets of a partially ordered set X,. The relation betweenAandBis denoted and defined as follows:

A ≺1B, if for everya∈ Athere existsb∈ Bsuch thatab. 1.4

Definition 1.5see12. A functionψ:0,∞ → 0,∞is called an altering distance function if the following properties are satisfied:

iψis monotone increasing and continuous, iiψt 0 if and only ift0.

On the other hand, fixed-point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings13, Theorem 2.1who presented its applications to matrix equations. Subsequently, Nieto and Rodr´ıguez-L ´opez14extended the result of Ran and Reurings for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary diﬀerential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed-point theorems in ordered metric spaces. For detail see14–28and references cited therein. Beg and Butt11,17,29worked on set-valued mappings and proved common fixed-point for mapping satisfying implicit relation in partially ordered metric space. Recently, Choudhury and Metiya30proved fixed-point theorems for multivalued mappings in the framework of a partially ordered metric space.

The results of this paper are divided in three sections. In the first section we establish the existence of end-points for a multivalued mapping under a more general contractive condition in partially ordered metric spaces. The consequences of the main theorem are also given. The second section is devoted for common end-point results for a pair of weakly isotone increasing multivalued mappings. In the third section, we present common end- point results for a pair of weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

2. End-Point Theorems for a Multivalued Mapping

In this section, we prove end-point theorems for a multivalued mapping in ordered complete metric space.

Theorem 2.1. LetX, d,be an ordered complete metric space. LetT :X → BXbe such that the following conditions are satisfied:

ithere existsx0∈ Xsuch that{x0} ≺1Tx0, iiforx, y∈ X,xyimpliesTx≺1Ty, iii

ψ δ

Tx,Ty

≤αψ M

x, y

Lmin

Dx,Tx, D y,Ty

, D x,Ty

, D

y,Tx

, 2.1

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for all comparablex, y∈ X, whereL≥0, 0< α <1 andψis an altering distance function and

M x, y

max

d x, y

, Dx,Tx, D y,Ty

,D x,Ty

D y,Tx 2

. 2.2

If the property

{xn} ⊂ Xis a nondecreasing sequence withxn−→zinX, thenxn≺z∀n 2.3

holds, thenThas a end-point.

Proof. By the assumptioni, there existsx1 ∈ Tx0 such that x0 x1. By the assumption ii,Tx0≺1Tx1. Then there existsx2 ∈ Tx1 such thatx1 x2. Continuing this process we construct a monotone increasing sequence{xn}inXsuch thatx_n1∈ Txn, for alln≥0. Thus we have

x0x1x2x3 · · · xn x_n1 · · ·. 2.4

Ifxn0∈ Txn0for somen0, then the proof is finished. So assumexn/xn1for alln≥0.

Using the monotone property ofψand the conditioniii, we have for alln≥0, ψdxn1, xn2≤ψδTxn,Txn1

≤αψ

max dxn, xn1, Dxn,Txn, Dxn1,Txn1, Dxn,Txn1 Dxn1,Txn

2

Lmin{Dxn,Txn, Dxn1,Tx_n1, Dxn,Tx_n1, Dxn1,Txn}

≤αψ

max dxn, xn1, dxn, xn1, dxn1, xn2, dxn, x_n2 dx_n1, x_n1

2

Lmin{dxn, x_n1, dx_n1, x_n2, dxn, x_n2, dx_n1, x_n1}.

2.5

Sincedxn, x_n2/2≤max{dxn, x_n1, dx_n1, x_n2}, it follows that

ψdxn1, xn2≤αψmax{dxn, xn1, dxn1, xn2}. 2.6

Suppose thatdxn, x_n1≤dx_n1, x_n2, for some positive integern.

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Then from2.6, we have

ψdxn1, xn2≤αψdxn1, xn2, 2.7

it implies that dx_n1, xn2 0, or that xn1 xn2, contradicting our assumption that xn/x_n1, for eachn.

Therefore,dxn1, xn2 < dxn, xn1, for alln≥ 0 and{dxn, xn1}is a monotone decreasing sequence of nonnegative real numbers. Hence there exists anr ≥0 such that

dxn, xn1−→r asn−→ ∞. 2.8 Taking the limit asn → ∞ in2.6and using the continuity ofψ, we have ψr ≤ αψr, which is a contradiction unlessr 0. Hence

nlim→ ∞dxn, xn1 0. 2.9

Next we show that{xn}is a Cauchy sequence. If otherwise, there exists an > 0 for which we can find two sequences of positive integers{mk}and{nk}such that for all positive integersk,nk> mk> kanddx_mk, x_nk≥.

Assuming thatnkis the smallest such positive integer, we getnk> mk> k, d

xmk, xnk

≥, d

xmk, xnk−1

< . 2.10

Now,

≤d

xmk, xnk

≤d

xmk, xnk−1 d

xnk−1, xnk

, 2.11

that is,

≤d

xmk, xnk

< d

xnk−1, xnk

. 2.12

Taking the limit ask → ∞in the above inequality and using2.9, we have

k→ ∞lim d

xmk, xnk

. 2.13

Again, d

xmk, xnk

≤d

xmk, xmk1 d

xmk1, xnk1 d

xnk1, xnk , d

x_mk1, x_nk1

≤d

x_mk1, x_mk d

x_mk, x_nk d

x_nk, x_nk1

. 2.14

Taking the limit ask → ∞in the above inequalities and using2.9and2.13, we have

klim→ ∞d

xmk1, xnk1

. 2.15

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Again,

d

xmk, xnk

≤d

xmk, xnk1 d

xnk1, xnk , d

xmk, xnk1

≤d

xmk, xnk d

xnk, xnk1

. 2.16

Lettingk → ∞in the above inequalities and using2.9and2.13, we have

klim→ ∞d

xmk, xnk1

. 2.17

Similarly, we have that

klim→ ∞d

x_nk, x_mk1

. 2.18

For each positive integerk,x_mkandx_nkare comparable. Then using the monotone property ofψand the conditioniii, we have

ψ d

x_mk1, x_nk1

≤ψ δ

Tx_mk,Tx_nk

≤αψ

max

d

x_mk, x_nk , D

x_mk,Tx_mk , D

x_nk,Tx_nk , D

xmk,Txnk D

xnk,Txmk 2

Lmin D

xmk,Tx_mk , D

xnk,Tx_nk , D

xmk,Tx_nk , D

xnk,Txmk

≤αψ

max

d

x_mk, x_nk , d

x_mk, x_mk1 , d

x_nk, x_nk1 , d

xmk, xnk1 d

xnk, xmk1 2

Lmin d

x_mk, x_mk1 , d

x_nk, x_nk1 , d

x_mk, x_nk1 , d

xnk, xmk1 .

2.19 Letting k → ∞in above inequality, using 2.9,2.13, 2.15, 2.17, and 2.18 and the continuity ofψ, we have

ψ≤αψ, 2.20

which is a contradiction by virtue of a property ofψ.

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Hence{xn}is a Cauchy sequence. From the completeness ofX, there exists az ∈ X such that

xn−→z asn−→ ∞. 2.21

By the assumption2.3,xnz, for alln.

Then by the monotone property ofψand the conditioniii, we have ψδxn1,Tz≤ψδTxn,Tz

≤αψ

max dxn, z, Dxn,Txn, Dz,Tz,Dxn,Tz Dz,Txn 2

Lmin{Dxn,Txn, Dz,Tz, Dxn,Tz, Dz,Txn}

≤αψ

max dxn, z, dxn, xn1, Dz, Tz,Dxn, Tz dz, xn1 2

Lmin{dxn, xn1, Dz,Tz, Dxn,Tz, dz, xn1}.

2.22 Taking the limit asn → ∞in the above inequality, using2.9and2.21and the continuity ofψ, we have

ψδz,Tz≤αψDz,Tz≤αψδz,Tz, 2.23

which implies thatδz,Tz 0, or that{z}Tz. Moreover,zis a end-point ofT.

Takingψan identity function inTheorem 2.1, we have the following result.

Corollary 2.2. LetX, d,be an ordered complete metric space. LetT :X → BXbe such that the following conditions are satisfied:

δ

Tx,Ty

≤αM x, y

Lmin

Dx,Tx, D y,Ty

, D x,Ty

, D

y,Tx

, 2.24

for all comparablex, y∈ X, whereL≥0, 0< α <1 and M

x, y max

d

x, y

, Dx,Tx, D y,Ty

,D x,Ty

D y,Tx 2

. 2.25

If the property

{xn} ⊂ Xis a nondecreasing sequence withxn−→zinX, thenxnz∀n 2.26 holds, thenThas a end-point.

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The following corollary is a special case of Theorem 2.1when T is a single-valued mapping.

Corollary 2.3. LetX, d,be an ordered complete metric space. LetT :X → Xbe such that the following conditions are satisfied:

ithere existsx0∈ Xsuch thatx0 Tx0, iiforx, y∈ X,xyimpliesTx Ty, iii

ψ d

Tx,Ty

≤αψ M

x, y

Lmin

dx,Tx, d y,Ty

, d x,Ty

, d

y,Tx

, 2.27

M x, y

max

d x, y

, dx,Tx, d y,Ty

,d x,Ty

d y,Tx 2

. 2.28

If the property2.3holds, thenThas a fixed-point.

In the following theorem we replace condition2.3of the above corollary by requiring Tto be continuous.

Theorem 2.4. LetX, d,be an ordered complete metric space. LetT: X → Xbe a continuous mapping such that the following conditions are satisfied:

ψ d

Tx,Ty

≤αψ M

x, y

Lmin

dx,Tx, d y,Ty

, d x,Ty

, d

y,Tx

, 2.29

M x, y

max

d x, y

, dx,Tx, d y,Ty

,d x,Ty

d y,Tx 2

. 2.30

ThenThas a end-point.

Proof. If we assumeT as a multivalued mapping in whichTxis a singleton set for every x∈ X. Then we consider the same sequence{xn}as in the proof ofTheorem 2.1. Follows the line of proof ofTheorem 2.1, we have that{xn}is a Cauchy sequence and

n→ ∞lim xnz. 2.31

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Then, the continuity ofTimplies that z lim

n→ ∞x_n1 lim

n→ ∞TxnTz 2.32

and this proves thatzis a end-point ofT.

3. Common End-Point Theorems for a Pair of Multivalued Mappings

In this section, we prove common end-point theorems for a pair of T-weakly isotone increasing multivalued mappings.

To complete the result, we need notion ofT-weakly isotone increasing for multivalued mappings given by Vetro31, Definition 4.2.

Definition 3.1. LetX,be a partially ordered set andS,T :X → BXbe two maps. The mappingS is said to beT-weakly isotone increasing ifSx2Ty2Sz for all any x ∈ X, y∈ Sxandz∈ Ty.

Note that, in particular, for single-valued mappingsT,S : X → X, mapping S is said to beT-weakly isotone increasing if31, Definition 2.2if for eachx∈ Xwe haveSx TSx STSx.

Theorem 3.2. LetX, d,be an ordered complete metric space. LetT,S:X → BXbe such that ψ

δ

Tx,Sy

≤αψ M

x, y

Lmin

Dx,Tx, D y,Sy

, D x,Sy

, D

y,Tx

, 3.1

M x, y

max

d x, y

, Dx,Tx, D y,Sy

,D x,Sy

D y,Tx 2

. 3.2

Also suppose thatSisT-weakly isotone increasing and there exists anx0∈ Xsuch that{x0} ≺2Sx0. If the property

{xn} ⊂ Xis a nondecreasing sequence withxn−→zinX, thenxnz∀n 3.3

holds, thenSandThave a common end-point.

Proof. Define a sequence{xn} ⊂ Xand prove that the limit point of that sequence is a unique common end-point forTandS. For a givenx0∈ Xand nonnegative integernlet

x0x, x2n1∈ Sx2n, x2n2∈ Tx2n1 forn≥0. 3.4 Ifxn0 ∈ Sxn0orxn0 ∈ Txn0for somen0, then the proof is finished. So assumexn/xn1for all n.

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Since{x0} 2Sx0,x1 ∈ Sx0can be chosen so thatx0x1. SinceSisT-weakly isotone increasing, it is Sx02Tx1; in particular, x2 ∈ Tx1 can be chosen so that x1 x2. Now, Tx12Sx2sincex2∈ Tx1; in particular,x3∈ Sx2can be chosen so thatx2x3.

Continuing this process we construct a monotone increasing sequence{xn}inXsuch that

x0x1x2x3 · · · xn xn1 · · ·. 3.5

Ifxn0 ∈ Sxn0orxn0 ∈ Txn0for somen0, then the proof is finished. So assumexn/xn1for all n.

Suppose thatnis an odd number. Substitutingxxnandyxn1in3.1and using properties of functionψ, we have for alln≥0,

ψdx_n1, x_n2≤ψδTxn,Sx_n1

≤αψ

max dxn, x_n1, Dxn,Txn, Dx_n1,Sx_n1, Dxn,Sxn1 Dxn1,Txn

2

Lmin{Dxn,Txn, Dxn1,Sxn1, Dxn,Sxn1, Dxn1,Txn}

≤αψ

max dxn, xn1, dxn, xn1, dxn1, xn2, dxn, x_n2 dx_n1, x_n1

2

Lmin{dxn, x_n1, dx_n1, x_n2, dxn, x_n2, dx_n1, x_n1}.

3.6

Sincedxn, xn2/2≤max{dxn, xn1, dxn1, xn2}, it follows that

ψdxn1, xn2≤αψmax{dxn, xn1, dxn1, xn2}. 3.7

Suppose thatdxn, xn1≤dxn1, xn2, for some positive integern.

Then from3.7, we have

ψdxn1, xn2≤αψdxn1, xn2, 3.8

it implies that dxn1, xn2 0, or that xn1 xn2, contradicting our assumption that xn/xn1, for eachnand so we have

dxn1, xn2< dxn, xn1. 3.9

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In the similar fashion, we can also show inequalities 3.9 when n is an even number.

Therefore, the sequence{dxn, x_n1}is a monotone decreasing sequence of nonnegative real numbers. Hence there exists anr ≥0 such that

dxn, xn1−→r asn−→ ∞. 3.10 Taking the limit asn → ∞ in3.7and using the continuity ofψ, we have ψr ≤ αψr, which is a contradiction unlessr 0. Hence

nlim→ ∞dxn, x_n1 0. 3.11

Next we show that{xn}is a Cauchy sequence. If otherwise, there exists an > 0 for which we can find two sequences of positive integers{mk}and{nk}such that for all positive integersk,nk> mk> kanddx_mk, x_nk≥.

Assuming thatnkis the smallest such positive integer, we getnk> mk> k, d

x_mk, x_nk

≥, d

x_mk, x_nk−1

< . 3.12

Now,

≤d

xmk, xnk

≤d

xmk, xnk−1 d

xnk−1, xnk

, 3.13

that is,

≤d

x_mk, x_nk

< d

x_nk−1, x_nk

. 3.14

Taking the limit ask → ∞in the above inequality and using3.11, we have

klim→ ∞ d

xmk, xnk

. 3.15

Again, d

xmk, xnk

≤d

xmk, xmk1 d

xmk1, xnk1 d

xnk1, xnk , d

xmk1, xnk1

≤d

xmk1, xmk d

xmk, xnk d

xnk, xnk1

. 3.16

Taking the limit ask → ∞in the above inequalities and using3.11and3.15, we have

klim→ ∞d

xmk1, xnk1

. 3.17

Again,

d

x_mk, x_nk

≤d

x_mk, x_nk1 d

x_nk1, x_nk , d

xmk, xnk1

≤d

xmk, xnk d

xnk, xnk1

. 3.18

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Lettingk → ∞in the above inequalities and using2.9and3.15, we have

k→ ∞lim d

x_mk, x_nk1

. 3.19

Similarly, we have that

k→ ∞lim d

x_nk, x_mk1

. 3.20

For each positive integerk,xmkandxnkare comparable. Then using the monotone property ofψand the condition3.1, we have

ψ d

xmk1, xnk2

≤ψ δ

Txmk,Sxnk1

≤αψ

max

d

xmk, xnk1 , D

xmk,Txmk , D

xnk1,Sxnk1 , D

xmk,Sx_nk1 D

xnk1,Tx_mk 2

Lmin D

xmk,Txmk , D

xnk1,Sxnk1 , D

xmk,Sxnk1 , D

xnk1,Txmk αψ

max

d

xmk, xnk1 , d

xmk, xmk1 , d

xnk1, xnk1 , d

x_mk, x_nk2 d

x_nk1, x_mk1 2

Lmin d

xmk, xmk1 , d

xnk1, xnk2 , d

xmk, xnk2 , d

xnk1, xmk1 .

3.21 Lettingk → ∞in above inequality, using3.11,3.15,3.17,3.19, and3.20and using the continuity ofψ, we have

ψ≤αψ, 3.22

which is a contradiction by virtue of a property ofψ.

Hence{xn}is a Cauchy sequence. From the completeness ofX, there exists az ∈ X such that

xn−→z asn−→ ∞. 3.23

By the assumption3.3,xnz, for alln.

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Then by the monotone property ofψand the condition3.1, we have ψδxn1,Sz≤ψδTxn,Sz

≤αψ

max dxn, z, Dxn,Txn, Dz,Sz,Dxn,Sz Dz,Txn 2

Lmin{Dxn,Txn, Dz,Sz, Dxn,Sz, Dz,Txn}

≤αψ

max dxn, z, dxn, xn1, Dz,Sz,Dxn,Sz dz, xn1 2

Lmin{dxn, xn1, Dz,Sz, Dxn,Sz, dz, x_n1}.

3.24 Taking the limit asn → ∞in the above inequality, using3.11and3.23and the continuity ofψ, we have

ψδz,Sz≤αψDz,Sz≤αψδz,Sz, 3.25

it implies thatδz,Sz 0, or that{z} Sz. Similarly{z} Tz. Moreover,zis a common end-point ofTandS.

PuttingSTinTheorem 3.2, we immediately obtain the following result.

Corollary 3.3. LetX, d,be an ordered complete metric space. LetT:X → BXbe such that ψ

δ

Tx,Ty

≤αψ M

x, y

Lmin

Dx,Tx, D y,Ty

, D x,Ty

, D

y,Tx

, 3.26

M x, y

max

d x, y

, Dx,Tx, D y,Ty

,D x,Ty

D y,Tx 2

. 3.27

Also suppose that Tx1TTxfor allx ∈ Xand there isx0 ∈ Xsuch that {x0} ≺1Tx0. If the property

{xn} ⊂ X is a nondecreasing sequence withxn −→zin X, thenxn≺z∀n 3.28

holds, thenThas a end-point.

InTheorem 3.2, ifT,Sare single valued mappings, then we have the following result.

Theorem 3.4. LetX, d,be an ordered complete metric space. LetT,S:X → Xbe such that ψ

d

Tx,Sy

≤αψ M

x, y

Lmin

dx,Tx, d y,Sy

, d x,Sy

, d

y,Tx

, 3.29

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for all comparablex, y∈ X, whereL≥0, 0< α <1 andψis an altering distance function and M

x, y max

d

x, y

, dx,Tx, d y,Sy

,d x,Sy

d y,Tx 2

. 3.30

Also suppose thatSandTare weakly isotone increasing. If

S is continuous 3.31

or

Tis continuous 3.32

or

{xn} ⊂ X is a nondecreasing sequence withxn −→zin X, thenxnz∀n 3.33 holds, thenSandThave a common end-point.

Proof. If we assumeTandSas a multivalued mapping in whichTxandSxare a singleton set for everyx∈ X. Then we consider the same sequence{xn}as in the proof ofTheorem 3.4.

Follows the line of proof ofTheorem 3.4, we have that{xn}is a Cauchy sequence and

n→ ∞lim xnz. 3.34

Then, ifTis continuous, we have z lim

n→ ∞x_n1 lim

n→ ∞TxnTz 3.35

and this proves thatz is a end-point of T and soz is a end-point of S. Similarly, ifS is continuous, we have the result. Thus it is immediate to conclude thatTandShave a common end-point.

4. Common End-Point Theorems for a Pair of Multivalued Mappings Satisfying Weakly Contractive Condition

In this section, we prove common end-point theorems for a pair of weakly isotone increasing multivalued mappings under weakly contractive condition.

To complete the result, we need notion of weakly contractive condition given by Rhoades32.

Definition 4.1Weakly Contractive Mapping. LetXbe a metric space. A mappingT:X → Xis called weakly contractive if and only if

d

Tx,Ty

≤d x, y

−ϕ d

x, y

, ∀x, y∈ X, 4.1 whereϕis an altering distance function.

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Theorem 4.2. LetX, d,be an ordered complete metric space. LetT,S:X → BXbe such that

ψ δ

Tx,Sy

≤ψ

max

d x, y

, Dx,Tx, D y,Sy

, D x,Sy

D y,Tx 2

−φ max

d x, y

, δ

y,Sy ,

4.2

for all comparablex, y∈ X, whereψ, φ:0,∞ → 0, ∞are an altering distance functions.

Also suppose thatS isT-weakly isotone increasing and there exists anx0 ∈ X such that {x0} ≺2Sx0. If the property

{xn} ⊂ Xis a nondecreasing sequence withxn−→zinX, thenxnz∀n 4.3 holds, thenSandThave a common end-point.

Proof. Define a sequence{xn} ⊂ Xand prove that the limit point of that sequence is a unique common end-point forTandS. For a givenx0∈ Xand nonnegative integernlet

x0 x, x2n1 ∈ Sx2n, x2n2∈ Tx2n1 forn≥0. 4.4 Since {x0} 2Sx0, x1 ∈ Sx0 can be chosen so that x0 x1. Since S is T-weakly isotone increasing, it is Sx02Tx1; in particular, x2 ∈ Tx1 can be chosen so that x1 x2. Now, Tx12Sx2sincex2∈ Tx1; in particular,x3∈ Sx2can be chosen so thatx2x3.

Continuing this process, we conclude that{xn}can be an increasing sequence inX:

x1x2 · · · xnx_n1 · · ·. 4.5 If there exists a positive integerNsuch thatxNxN1, thenxNis a common end-point ofT andS. Hence we will assume thatxn/xn1, for alln≥0.

Suppose thatnis an odd number. Substitutingxxnandyx_n1in2.6and using properties of functionsψandφ, we have for alln≥0,

ψdxn1, xn2≤ψδTxn,Sx_n1

≤ψ

max dxn, xn1, Dxn,Txn, Dxn1,Sxn1, Dxn,Sx_n1 Dx_n1,Txn

2

−φmax{dxn, x_n1, δx_n1,Sx_n1}

≤ψ

max dxn, x_n1, dxn, x_n1, dx_n1, x_n2, dxn, xn2 dxn1, xn1

2

−φmax{dxn, xn1, dxn1, xn2}.

4.6

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Sincedxn, xn2/2≤max{dxn, xn1, dxn1, xn2}, it follows that

ψdxn1, xn2≤ψmax{dxn, xn1, dxn1, xn2}−φmax{dxn, xn1, dxn1, xn2}.

4.7 Suppose thatdxn, xn1 ≤ dxn1, xn2, for some positive integern. Then from4.7, we have

ψdxn1, xn2≤ψdxn1, xn2−φdxn1, xn2, 4.8 that is, φdx_n1, x_n2 ≤ 0, which implies that dx_n1, x_n2 0, or that x_n1 x_n2, contradicting our assumption thatxn/xn1. So we have

dx_n1, x_n2< dxn, x_n1. 4.9

In the similar fashion, we can also show inequalities 4.9 when n is an even number.

Therefore, for alln≥0 and{dxn, xn1}is a monotone decreasing sequence of nonnegative real numbers. Hence there exists anr≥0 such that

nlim→ ∞dxn, xn1 r. 4.10

In view of the above facts, from4.7we have for alln≥0,

ψdx_n1, x_n2≤ψdxn, x_n1−φdxn, x_n1. 4.11

Taking the limit asn → ∞in the above inequality, using4.10and the continuities ofφand ψ, we have

ψr≤ψr−φr, 4.12

which is a contradiction unlessr 0. Hence

nlim→ ∞dxn, x_n1 0. 4.13

Next we show that{xn}is a Cauchy sequence. If{xn}is not a Cauchy sequence, then using an argument similar to that given inTheorem 3.2, we can find two sequences of positive integers {mk}and{nk}for which

klim→ ∞d

x_mk, x_nk

, lim

k→ ∞d

x_mk1, x_nk1 ,

klim→ ∞d

xmk, xnk1

, lim

k→ ∞d

xnk, xmk1 .

4.14

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For each positive integerk,xmkandxnkare comparable. Then using the monotone property ofψand4.2, we have

ψ d

xmk1, xnk2

≤ψ δ

Tx_mk,Sx_nk1

≤ψ

max

d

xmk, xnk1 , D

xmk,Tx_mk , D

xnk1,Sx_nk1 , D

x_mk,Sx_nk1 D

x_nk1,Tx_mk 2

−φ max

d

xmk, xnk1 , δ

xnk1,Sxnk1 , ψ

max

d

xmk, xnk1 , d

xmk, xmk1 , d

xnk1, xnk2 , d

x_mk, x_nk2 d

x_nk1, Tx_mk1 2

−φ max

d

xmk, xnk1 , d

xnk1, xnk2 .

4.15

Lettingk → ∞in the above inequality, using4.14and the continuities ofψandφ, we have

ψ≤ψ−φ, 4.16

which is a contradiction by virtue of a property ofφ. Hence{xn}is a Cauchy sequence. From the completeness ofX, there exists az∈ Xsuch that

xn−→z asn−→ ∞. 4.17

By the condition4.3,xn z, for alln. Then by the monotone property ofψ and4.2, we have

ψδxn1,Sz≤ψδTxn,Sz

≤ψ

max dxn, z, Dxn,Txn, Dz,Sz,Dxn,Sz Dz,Txn 2

−φmax{dxn, z, δz,Sz}

≤ψ

max dxn, z, dxn, xn1, Dz,Sz,Dxn,Sz dz, xn1 2

−φmax{dxn, z, δz,Sz}.

4.18

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Taking the limit asn → ∞in the above inequality, using4.13,4.17and the continuities of ψandφ, we have

ψδz,Sz≤ψDz,Sz−φδz,Sz, 4.19

which implies that

ψδz,Sz≤ψδz,Sz−φδz,Sz, 4.20

which is a contradiction unlessδz,Sz 0, or that{z} Sz; that is,zis a end-point ofS.

Similarly{z}Tz. Moreover,zis a common end-point ofTandS.

Similar corollaries can be derived fromTheorem 4.2.

Acknowledgment

The present version of the paper owes much to the precise and kind remarks of the learned referees.

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