A Common End Point Theorem for Set-Valued Generalized -Weak Contraction

Volume 2010, Article ID 509658,8pages doi:10.1155/2010/509658

Research Article

A Common End Point Theorem for Set-Valued

Generalized

ψ, ϕ

-Weak Contraction

Mujahid Abbas

_{and Dragan D}

−

_ori´c

1_{Department of Mathematics, Centre for Advanced Studies in Mathematics, Lahore University of}

Management Sciences, 54792 Lahore, Pakistan

2_{Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca 154, 11000 Beograd, Serbia}

Correspondence should be addressed to Dragan D−ori´c,[email protected]

Received 21 August 2010; Accepted 18 October 2010

Academic Editor: Satit Saejung

Copyrightq2010 M. Abbas and D. D−ori´c. 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 introduce the class of generalizedψ, ϕ-weak contractive set-valued mappings on a metric space. We establish that such mappings have a unique common end point under certain weak conditions. The theorem obtained generalizes several recent results on single-valued as well as certain set-valued mappings.

1. Introduction and Preliminaries

mappings, it is essential for mappings to satisfy certain contractive conditions which involve Hausdorffmetric.

The aim of this paper is to obtain the common end point, a special case of fixed point, of two multivlaued mappings without appeal to continuity of any map involved therein. It is also noted that our results do not require any commutativity condition to prove an existence of common end point of two mappings. These results extend, unify, and improve the earlier comparable results of a number of authors.

LetX, dbe a metric space, and letBXbe the class of all nonempty bounded subsets ofX. We define the functionsδ:BX×BX → RandD:BX×BX → Ras follows:

δA, B sup{da, b :a∈A, b∈B},

DA, B inf{da, b :a∈A, b∈B},

1.1

where R denotes the set of all positive real numbers. For δ{a}, B and δ{a},{b}, we writeδa, Bandda, b, respectively. Clearly,δA, B δB, A. We appeal to the fact that

δA, B 0 if and only ifAB{x}forA, B∈BXand

0≤δA, B≤δA, B δA, B, 1.2

forA, B, C∈BX. A pointx∈Xis called a fixed point ofT ifx∈Tx. If there exists a point

x∈Xsuch thatTx{x}, thenxis termed as an end point of the mappingT.

2. Main Results

In this section, we established an end point theorem which is a generalization of fixed point theorem for generalizedψ, ϕ-weak contractions. The idea is in line with Theorem 2.1 in6 and theorem 1 in5.

Definition 2.1. Two set-valued mappingsT, S:X → BXare said to satisfy the property of generalizedψ, φ-weak contractionif the inequality

ψδSx, Ty≤ψMx, y−ϕMx, y, 2.1

where

Mx, ymax

dx, y, δx, Sx, δy, Ty,1

2

Dx, TyDy, Sx 2.2

holds for allx, y∈Xand for given functionsψ, ϕ:R → R.

aψis a continuous monotone nondecreasing function withψt 0if and only ift0,

bϕis a lower semicontinuous function withϕt 0if and only ift0

then there exists the unique pointu∈Xsuch that{u}TuSu.

Proof. We construct the convergent sequence{xn}inXand prove that the limit point of that

sequence is a unique common fixed point forT andS. For a givenx0 ∈X and nonnegative

integernlet

x2n1 ∈Sx2nA2n, x2n2∈Tx2n1A2n1, 2.3

and let

anδAn, An1, cndxn, xn1. 2.4

The sequencesanandcnare convergent. Suppose thatnis an odd number. Substituting

xxn1andyxnin2.1and using properties of functionsψandϕ, we obtain

ψδAn1, An ψδSxn1, Txn

≤ψMxn1, xn−ϕMxn1, xn

≤ψMxn1, xn,

2.5

which implies that

δAn1, An≤Mxn1, xn. 2.6

Now from2.2and from triangle inequality forδ, we have

Mxn1, xn

max

dxn1, xn, δxn1, Sn1, δxn, Tn,

1

2Dxn1, Tn Dxn, Sn1

≤max

δAn, An−1, δAn, An1, δAn−1, An,1

2Dxn1, An δAn−1, An1

max

δAn, An−1, δAn, An1,1

2δAn−1, An1

≤max

δAn, An−1, δAn, An1,1

2δAn−1, An δAn, An1

max{δAn−1, An, δAn, An1}.

IfδAn, An1> δAn−1, An, then

Mxn, xn1≤δAn1, An. 2.8

From2.6and2.8it follows that

Mxn, xn1 δAn1, An> δAn−1, An≥0. 2.9

It furthermore implies that

ψδAn, An1≤ψMxn, xn1−ϕMxn, xn1

< ψMxn1, xn

ψδAn, An1

2.10

which is a contradiction. So, we have

δAn, An1≤Mxn, xn1≤δAn−1, An. 2.11

Similarly, we can obtain inequalities2.11also in the case whennis an even number. Therefore, the sequence{an}defined in2.4is monotone nonincreasing and bounded. Let

an → awhenn → ∞. From2.11, we have

lim

n→ ∞δAn, An1 nlim→ ∞Mxn, xn1 a≥0. 2.12

Lettingn → ∞in inequality

ψδA2n, A2n1≤ψMx2n, x2n1−ϕMx2n, x2n1, 2.13

we obtain

ψa≤ψa−ϕa, 2.14

which is a contradiction unlessa0. Hence,

lim

n→ ∞annlim→ ∞δAn, An1 0. 2.15

From2.15and2.3, it follows that

lim

The sequence{xn}is a Cauchy sequence. First, we prove that for eachε > 0 there exists

n0εsuch that

m, n≥n0⇒δA2m, A2n< ε. 2.17

Suppose opposite that 2.17does not hold then there exists ε > 0 for which we can find nonnegative integer sequences{mk}and{nk}, such thatnkis the smallest element of the sequence{nk}for which

nk> mk> k, δA2mk, A2nk

≥ε. 2.18

This means that

δA2mk, A2nk−2

< ε. 2.19

From2.19and triangle inequality forδ, we have

ε≤δA2mk, A2nk

≤δA2mk, A2nk−2

δA2nk−2, A2nk−1

δA2nk−1, A2nk

< εδA2nk−2, A2nk−1

δA2nk−1, A2nk.

2.20

Lettingk → ∞and using2.15, we can conclude that

lim

k→ ∞δ

A2mk, A2nkε. 2.21

Moreover, from _δ

A2mk, A2nk1

−δA2mk, A2nk≤δA2nk, A2nk1

,

_δ

A2mk−1, A2nk−δA2mk, A2nk≤δA2mk, A2mk−1

,

2.22

using2.15and2.21, we get

lim

k→ ∞δ

A2mk−1, A2nk lim

k→ ∞δ

A2mk, A2nk1

ε, 2.23

and from

_δ

A2mk−1, A2nk1

−δA2mk−1, A2nk≤δ

A2nk, A2nk1

, 2.24

using2.15and2.23, we get

lim

k→ ∞δ

A2mk−1, A2nk1

Also, from the definition ofM2.2and from2.15,2.23, and2.25, we have

lim

k→ ∞M

x2mk, x2nk1

ε. 2.26

Puttingxx2mk,yx2nk1in2.1, we have

ψδA2mk, A2nk1

ψδSx2mk, Tx2nk1

≤ψMx2mk, x2nk1

−ϕMx2mk, x2nk1

.

2.27

Lettingk → ∞and using2.23,2.26, we get

ψε≤ψε−ϕε, 2.28

which is a contradiction withε >0.

Therefore, conclusion 2.17 is true. From the construction of the sequence {xn}, it

follows that the same conclusion holds for{xn}. Thus, for eachε >0 there existsn0εsuch

that

m, n≥n0⇒dx2m, x2n< ε. 2.29

From2.4and2.29, we conclude that{xn}is a Cauchy sequence.

In complete metric spaceX, there existsusuch thatxn → uasn → ∞.

The pointuis end point of S. As the limit pointuis independent of the choice ofxn∈An,

we also get

lim

n→ ∞δSx2n, u nlim→ ∞δTx2n1, u 0. 2.30

From

Mu, x2n1 max

du, x2n1, δu, Su, δx2n1, Tx2n1,

1

2Du, Tx2n1 Dx2n1, Su

,

2.31

we haveMu, x2n1 → δu, Suasn → ∞. Since

lettingn → ∞and using2.30, we obtain

ψδSu, u≤ψδu, Su−ϕδu, Su, 2.33

which impliesψδu, Su 0. Hence,δu, Su 0 orSu{u}.

The pointuis also end point forT. It is easy to see thatMu, u δu, Tu. Using thatu

is fixed point forS, we have

ψδu, Tu ψδSu, Tu

≤ψMu, u−ϕMu, u

ψδu, Tu−ϕδu, Tu,

2.34

and using an argument similar to the above, we conclude thatδu, Tu 0 or{u}Tu. The pointuis a unique end point forSandT. If there exists another fixed pointv ∈ X, thenMu, v du, vand from

ψdu, v ψδSu, Tv

≤ψMu, v−ϕMu, v

ψdu, v−ϕdu, v,

2.35

we conclude thatuv. The proof is completed.

The Theorem2.2established that set-valued mappingsSandTunder weak condition 2.1have the unique common end pointu. Now, we give an example to support our result.

Example 2.3. ConsiderX {1,2,3,4,5}as a subspace of real line with usual metric,dx, y

|y−x|. LetS, T:X → BXbe defined as

Sx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

{4,5} forx∈ {1,2}

{4} forx∈ {3,4}

{3,4} forx5

, Tx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

{3,4} for x∈ {1,2}

{4} for x∈ {3,4}

{3} forx5

. 2.36

and takeψ, ϕ:0,∞ → 0,∞asψt 2tand ϕt t/2.

From Tables1and2, it is easy to verify that mappingsSandTsatisfy condition2.1. Therefore,SandT satisfy the property of generalizedψ, φ−weak contraction. Note thatSand T have unique common end point.S4 T4 {4}. Also, note that forψt t

condition2.1, which became analog to condition2.1in5, does not hold. For example,

Table1

δSx, Ty 1 2 3 4 5

1 2 2 1 1 2

2 2 2 1 1 2

3 1 1 0 0 1

4 1 1 0 0 1

5 1 1 1 1 1

Table2

Mx, y 1 2 3 4 5

1 4 4 4 4 4

2 3 3 3 3 3

3 3 2 1 1 2

4 3 2 1 0 2

5 4 3 2 2 2

Remark 2.4. The Theorem2.2generalizes recent results on single-valued weak contractions given in3,5,6. The example above shows that functionψ in2.1gives an improvement over condition2.1in5.

References

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