On Fixed Point Theorems for Multivalued Contractions

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Volume 2010, Article ID 870980,18pages doi:10.1155/2010/870980

Research Article

On Fixed Point Theorems for Multivalued

Contractions

Zeqing Liu,

1

_{Wei Sun,}

1

_{Shin Min Kang,}

2

_{and Jeong Sheok Ume}

3

1_{Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China}

2_{Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea} 3_{Department of Applied Mathematics, Changwon National University,}

Changwon 641-773, Republic of Korea

Correspondence should be addressed to Shin Min Kang,[email protected]

Received 13 August 2010; Accepted 15 November 2010

Academic Editor: N. J. Huang

Copyrightq2010 Zeqing Liu 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.

Three concepts of multivalued contractions in complete metric spaces are introduced, and the conditions guaranteeing the existence of fixed points for the multivalued contractions are established. The results obtained in this paper extend genuinely a few fixed point theorems due to ´Ciri´c2009Feng and Liu2006and Klim and Wardowski2007. Five examples are given to explain our results.

1. Introduction and Preliminaries

Let X, d be a metric space, and let CLX, CBX, and CX denote the families of all nonempty closed, all nonempty closed and bounded, and all nonempty compact subsets of

X, respectively. For anyU, V ∈CLXandx∈X, letdx, U inf{dx, y:y∈U}and

HU, V

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

max

sup

u∈U

du, V,sup

v∈V

dv, U

, if the maximum exists,

∞, otherwise.

1.1

Such a mappingHis called a generalized Hausdorﬀmetric inCLXinduced byd. Throughout this paper, we assume thatÊ,Ê

_{, and}

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The existence of fixed points for various multivalued contractive mappings had been studied by many authors under diﬀerent conditions. For details, we refer the reader to1–7 and the references therein. In 1969, Nadler Jr7extended the famous Banach Contraction Principle from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.

Theorem 1.1see7. LetX, dbe a complete metric space, and letT be a mapping fromX into

CBX. Assume that there existsc∈0,1such that

HTx, Ty ≤cdx, y , ∀x, y∈X. 1.2

Then,T has a fixed point.

In 1989, Mizoguchi and Takahashi6generalized the Nadler fixed point theorem and got a fixed point theorem for the multivalued contraction as follows.

CBX. Assume that there exists a mapϕ:0,∞ → 0,1such that

lim sup

r→t

ϕr<1, ∀t∈Ê _,

HTx, Ty ≤ϕdx, y dx, y , ∀x, y∈X.

1.3

Then,T has a fixed point.

In 2006, Feng and Liu3obtained a new extension of the Nadler fixed point theorem and proved the following fixed point theorem.

Theorem 1.3see3. LetX, dbe a complete metric space, and letT be a multivalued mapping fromX intoCLX. If there exist constantsb, c ∈ 0,1, c < b, such that for anyx ∈ X there is

y∈Txsatisfying

bdx, y ≤fx, fy ≤cdx, y , 1.4

thenThas a fixed point inXprovided a functionfx dx, Tx,x∈Xis lower semicontinuous.

In 2007, Klim and Wardowski5improved the result of Feng and Liu and proved the following results.

Theorem 1.4see5. LetX, dbe a complete metric space, and letT be a multivalued mapping fromXintoCLX. Assume that

athe mappingf:X → Ê

_{, defined by}_fx_{dx, T}_x_,_x_∈_X_{, is lower semi-continuous;}

bthere existb∈0,1andϕ:Ê

_→ ₀_{, b}_satisfying

lim sup

r→t ϕr< b, t∈

Ê

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and for anyx∈Xthere isy∈Txsatisfying

bdx, y ≤dx, Tx, fy ≤ϕdx, y dx, y . 1.6

Then,T has a fixed point inX.

CX. Assume that

bthere exists a functionϕ:Ê

_→ ₀_,₁_satisfying

lim sup

r→t ϕr<1, ∀t∈

Ê

_, _1.7

dx, y fx, fy ≤ϕdx, y dx, y . 1.8

Then,T has a fixed point inX.

In 2008 and 2009, ´Ciri´c1,2introduced new multivalued nonlinear contractions and established a few nice fixed point theorems for the multivalued nonlinear contractions, one of which is as follows.

CLX. Assume that

bthere exists a functionϕ:Ê

_→ _a,₁_,₀_{< a <}₁_{, satisfying}

lim sup

r→t

ϕr<1, ∀t∈Ê

_; _1.9

ϕfx dx, y ≤fx, fy ≤ϕfx dx, y . 1.10

ThenT has a fixed point inX.

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corresponding fixed point theorems of ´Ciri´c2, Feng and Liu3, and Klim and Wardowski

5and are diﬀerent from those results of Mizoguchi and Takahashi6and Nadler Jr7. Next we recall and introduce the following result in 4 and some notions and terminologies.

Lemma 1.7see4. LetX, dbe a complete metric space andD ∈CLX. Then, for eachx∈X

andk >1there exists an elementa∈Dsuch that

dx, a≤kdx, D. 1.11

In the rest of this paper, for a multivalued mappingT :X → CLX, we put

A

⎧ ⎪ ⎨ ⎪ ⎩

0,diamX, if diamX<∞,

0,∞, if diamX ∞,

B

⎧ ⎪ ⎨ ⎪ ⎩

0,supfX, if supfX<∞,

0,∞, if supfX ∞,

1.12

where diamX sup{dx, y :x, y ∈X}andfx dx, Tx, for allx∈X. The function

f:X → Ê

_{is said to be}_T_{-orbitally lower semicontinuous at}_z_∈_X_if_{_x

n}n≥0⊂Xis an arbitrary orbit ofT with limn→ ∞xnzimpling thatfz≤lim infn→ ∞fxn.

2. Main Results

In this section, we establish three fixed point theorems for three new multivalued contractions in complete metric spaces.

Theorem 2.1. LetTbe a multivalued mapping from a complete metric spaceX, dintoCLXsuch that

for each x∈X there existsy∈Txsatisfying

αfx dx, y ≤fxandfy ≤βfx dx, y , 2.1

whereα:B → 0,1andβ:B → 0,1satisfy that

lim inf

r→0 αr>0, lim sup_r_→_t βr

αr <1, ∀t∈

0,supfX . 2.2

Then,

a1for eachx0∈Xthere exists an orbit{xn}n≥0ofTandz∈Xsuch thatlimn→ ∞xnz;

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Proof. Put

γt βt_αt, ∀t∈0,supfX . 2.3

It follows from2.1that for eachx0∈Xthere existsx1 ∈Tx0satisfying

αfx0 dx0, x1≤fx0, fx1≤β

fx0 dx0, x1, 2.4

which together with2.3yield that

fx1≤β

fx0 fx0

αfx0

γfx0 fx0. 2.5

Continuing this process, we choose easily an orbit{xn}n≥0of Tsatisfying

xn1∈Txn, α

fxn dxn, xn1≤fxn,

fxn1≤β

fxn dxn, xn1, ∀n≥0,

2.6

which imply that

fxn1≤β

fxn dxn, xn1≤β

fxn

αfxn

γfxn fxn, ∀n≥0. 2.7

Now, we claim that

lim

n→ ∞fxn 0. 2.8

Notice that the ranges ofα, β,2.2, and2.3ensure that

0≤γt<1, ∀t∈0,supfX . 2.9

Using 2.7 and 2.9, we conclude that {fxn}n≥0 is a nonnegative and nonincreasing sequence, which means that

lim

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for somea≥0. Suppose thata >0. Taking limits superior asn → ∞in2.7and using2.2,

2.3,2.9, and2.10, we get that

alim sup

n→ ∞

fxn1≤lim sup

n→ ∞

γfxn fxn

≤lim sup

n→ ∞

γfxn lim sup n→ ∞

fxn alim sup n→ ∞

γfxn < a,

2.11

which is a contradiction. Thus,a0; that is,2.8holds. Next, we show that{xn}n≥0is a Cauchy sequence. Let

blim sup

n→ ∞ γ

fxn , clim inf n→ ∞ α

fxn . 2.12

It follows from2.2,2.3,2.9, and2.12that

0≤b <1, c >0. 2.13

Letp ∈ 0, candq ∈ b,1. In light of2.12and2.13, we deduce that there exists some

n0∈Æsuch that

γfxn < q, α

fxn > p, ∀n≥n0, 2.14

which together with2.6and2.7yield that

fxn1≤qfxn, dxn, xn1≤

fxn

p , ∀n≥n0, 2.15

which imply that

fxn1≤qn1−n0fxn0, dxn, xn1≤

fxn0

p q

n−n0_, ∀_n≥_n

0, 2.16

which give that

dxn, xm≤ m−1

kn

dxk, xk1≤

fxn0

p

m−1

kn

qk−n0≤ fxn0

p1−q q

n1−n0_, ∀_{m > n}≥_n

0, 2.17

which implies that{xn}n≥0is a Cauchy sequence becauseq <1. It follows from completeness ofX, dthat there is somez∈Xsuch that limn→ ∞xnz.

Suppose thatfisT-orbitally lower semi-continuous inz. It follows from2.8that

0≤dz, Tz fz≤lim inf

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which means thatz∈TzbecauseTzis closed.

Suppose thatzis a fixed point ofT inX. For any orbit{yn}_n_≥₀ofTwithyn1 ∈Tyn

and limn→ ∞ynz, we have

fz dz, Tz 0≤lim inf

n→ ∞ d

yn, T

yn

lim inf

n→ ∞ f

yn , 2.19

that is,fisT-orbitally lower semi-continuous inz. This completes the proof.

fy ≤ϕfx dx, y , ∀x, y ∈X×Tx, 2.20

whereϕ:B → 0,1satisfies that

lim inf

r→0 ϕr>0, lim sup_r_→_t ϕr<1, ∀t∈

0,supfX . 2.21

Then,

a2zis a fixed point ofTinXif and only if the functionfisT-orbitally lower semi-continuous atz.

Proof. It follows from Lemma1.7that for eachx∈Xthere existsy∈Txwith

dx, y ≤ fx ϕfx

, 2.22

which together with2.20and2.21ensures that2.1and2.2hold withα√ϕandβϕ. Thus, Theorem2.2follows from Theorem2.1. This completes the proof.

for eachx∈X there existsy∈Txsatisfying

αdx, y dx, y ≤fx andfy ≤βdx, y dx, y , 2.23

whereα:A → 0,1andβ:A → 0,1satisfy that

lim inf

r→t αr>0, lim sup

r→t βr

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and one ofαandβis nondecreasing. Then,

a2zis a fixed point ofTinXif and only if the functionfisT-orbitally lower semi-continuous atz.

Proof. Put

γt βt_αt, ∀t∈0,diamX. 2.25

It follows from the ranges ofα, β,2.24, and2.25that

0≤γt<1, ∀t∈0,diamX. 2.26

As in the proof of Theorem2.1, we select an orbit{xn}n≥0ofT satisfying

xn1 ∈Txn, αdxn, xn1dxn, xn1≤fxn,

fxn1≤βdxn, xn1dxn, xn1, ∀n≥0,

2.27

which imply that

fxn1≤βdxn, xn1_αdxfxn

n, xn1 γdxn, xn1fxn, ∀n≥0, 2.28

dxn1, xn2≤

fxn1

αdxn1, xn2 ≤

βdxn, xn1

αdxn1, xn2dxn, xn1, ∀n≥

0. 2.29

Using 2.26 and 2.28, we conclude easily that {fxn}_n_≥₀ is a nonnegative and

nonincreasing sequence. Consequently there exists someθ≥0 satisfying

lim

n→ ∞fxn θ. 2.30

Now we claim that

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Suppose to the contrary, that is, there exists some n0 ≥ 0 satisfying dxn01, xn02 >

dxn0, xn01. Note that one ofαandβis nondecreasing. In view of2.25,2.26, and2.29,

we get that

dxn0, xn01< dxn01, xn02≤

βdxn0, xn01

αdxn01, xn02

dxn0, xn01

≤maxγdxn0, xn01, γdxn01, xn02

dxn0, xn01

< dxn0, xn01,

2.32

which is a contradiction. Thus,2.31holds. Therefore, there exists someη≥0 such that

lim

n→ ∞dxn, xn1 η

_. _2.33

Next, we show thatθ 0. Suppose thatθ > 0. Taking limits superior asn → ∞in

2.28and using2.24,2.25,2.30, and2.33, we get that

θlim sup

n→ ∞ fxn1≤lim supn→ ∞ γdxn, xn1lim supn→ ∞ fxn

θlim sup

n→ ∞ γdxn, xn1< θ,

2.34

which is impossible. Thus,θ0. Let

blim sup

n→ ∞ γdxn, xn1, clim infn→ ∞ αdxn, xn1. 2.35

It follows from2.24,2.25,2.33, and2.35that

0≤b <1, c >0. 2.36

Letp ∈ 0, candq ∈ b,1. By means of2.35and 2.36, we infer that there exists some

n0∈Nsuch that

γdxn, xn1< q, αdxn, xn1> p, ∀n≥n0, 2.37

which together with2.27and2.28yield that

fxn1≤qfxn, dxn, xn1≤ fxn

p , ∀n≥n0. 2.38

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3. Remarks and Examples

In this section, we construct five examples to illustrate the superiority and applications of the results presented in this paper.

Remark 3.1. In case αt b and βt c for all t ∈ 0,supfX, where b and c are constants in0,1withc < b, then Theorem2.1reduces to a result, which is a generalization of Theorem 1.3. The following example reveals that Theorem2.1 extends both essentially Theorem1.3and is diﬀerent from Theorems1.1and1.2.

Example 3.2. LetX 0,1be endowed with the Euclidean metricd| · |, and letT : X → CLXbe defined by

Tx ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x2 3

, x∈

0,23

48

∪

23 48,1

, 1 12, 131 432

, x 23

48.

3.1

It is easy to see that

fx dx, Tx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x−x

2

3 , x∈

0,23

48

∪

23 48,1

,

19

108, x

23 48

3.2

isT-orbitally lower semi-continuous inXandB 0,2/3. Defineα:B → 0,1andβ:B → 0,1by

αt ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 8, t∈

0, 1

13

,

3t

2, t∈

1 13, 2 3 ,

βt max

1 9, 13t 9

, t∈

0,2

3 . 3.3 Obviously, lim inf

r→0 αr 1

8 >0. 3.4

Fort∈0,1/13, we have

lim sup

r→t βr

αr lim supr→t 1 9·

8

1

8

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Fort∈1/13,2/3, we infer that

lim sup

r→t βr

αr lim supr→t 13r

9 · 2 3r

26

27 <1. 3.6

Forx∈0,23/48∪23/48,1, there existsyx2_/₃_∈_T_x_satisfying

αfx dx, y α

x−x

2

3

x−x

2

3

≤x−x

2

3 fx,

fy dy, Ty

x

2

3 −

x4

27

1 3

xx

2

3

x−x

2

3

≤max

1 9,

13 9

x− x

2

3

dx, y

βfx dx, y .

3.7

Forx23/48, there existsy1/12∈Txsatisfying

αfx dx, y 19

72· 19 48 <

19

108fx,

fy dy, Ty d

1 12,

1 432

35

432< 4693 46656β

fx dx, y .

3.8

That is, the conditions of Theorem2.1are fulfilled. It follows from Theorem2.1thatT has a fixed point inX. However, we cannot invoke each of Theorems 1.1–1.3to show that the mappingThas a fixed point inX.

In fact, for anyb∈0,1, we consider two possible cases as follows.

Case 1. Let b∈0,4/9. Takex0 1 andy0 ∈Tx0 {1/3}. Notice thatc∈0,1andc < b. It is clear that

dy0, T

y0

8

27

4 9 ·

2 3/≤

2c

3 cd

x0, y0 ; 3.9

Case 2. Let b∈ 4/9,1. Putx0 23/48 andy0 1/12 ∈Tx0 {1/12,131/432}. It follows that

bdx0, y0 19b 48 /≤

4 9 ·

19

48

19

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Setx0 23/48 andy0131/432∈Tx0 {1/12,131/432}. Note thatc∈0,1andc < b. It is easy to verify that

dy0, T

y0 131 432− 1 3 · 1312 4322 152615 559872/≤

19c

108cd

x0, y0 . 3.11

That is, the conditions of Theorem1.3do not hold. Putx01/2 andy023/48. Clearly

HTx0, T

y0

95

432/≤

c

48 cd

x0, y0 , ∀c∈0,1,

HTx0, T

y0

95

432/≤

ϕdx0, y0

48 ϕ

dx0, y0

3.12

for anyϕ:0,∞ → 0,1with lim sup_r_→_tϕr<1, for all t∈Ê

_{. That is, the conditions} of Theorems1.1and1.2do not hold.

Remark 3.3. Ifαt ϕtandβt ϕtfor allt∈0,supfX, then Theorem2.1changes into a result, which is an extension of Theorem1.6. The following example demonstrates that Theorem2.1generalizes substantially Theorem1.6.

Example 3.4. LetX 0,1be endowed with the Euclidean metricd|·|. LetT :X → CLX

be defined by

Tx ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x2 2

, x∈

0,7

8

∪

7 8,1

, 1 2, 2 5

, x 7

8.

3.13

fx dx, Tx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x−x

2

2 , x∈

0,7

8

∪

7 8,1

,

3

8, x

7 8

3.14

isT-orbitally lower semi-continuous inX andB 0,supfX 0,1/2. Defineα: B → 0,1andβ:B → 0,1by

αt 15

19, βt

3 4, t∈

0,1

2

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Obviously,

lim inf

r→0 αr 15 19 >0,

lim sup

r→t βr

αr lim supr→t 3 4 ·

19

15

57

60<1, t∈

0,1

2

.

3.16

Forx∈0,7/8∪7/8,1, there existsyx2_/₂_∈_T_x_{_x2_/₂_}_satisfying

αfx dx, y α

x−x

2

x−x

2

≤x−x

2

2 fx,

fy dy, Ty x

2 2 − x4 8 1 2

xx

2

x−x

2 2 ≤ 3 4·

x−x

2

βfx dx, y .

3.17

Forx7/8, there existsy2/5∈Txsatisfying

αfx dx, y 15

19· 19

40

3

8 fx,

fy dy, Ty d

2 5, 2 25 8 25 < 57 160 3 4 · 19 40β

fx dx, y .

3.18

That is, the conditions of Theorem2.1are fulfilled. It follows from Theorem2.1thatT has a fixed point inX. However, Theorem1.6is inapplicable in ensuring the existence of fixed points for the mappingT inX because there does not existϕ : Ê

_→ _a,₁_and_a_∈ ₀_,₁ satisfying the assumptions of Theorem1.6. In fact, for anyϕ:Ê

_→ _a,₁_and_a_∈₀_,₁_{, we} consider the following two possible cases.

Case 1. Leta∈ 225/361,1. Putx0 7/8. Note thatϕfx0 ⊆ a,1. Ify0 2/5 ∈ Tx0

{1/2,2/5}, we see that

ϕfx0 d

x0, y0 19 40 ϕ 3 8 / ≤3

8 fx0. 3.19

Ify01/2∈Tx0{1/2,2/5}, then we infer that

dy0, T

y0 1 2− 1 8 3

8/≤ 3 8ϕ 3 8

ϕfx0 d

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Case 2. Leta ∈ 0,225/361. Put x0 7/8. Note that ϕfx0 ⊆ a,1. Suppose that

ϕfx0∈a,225/361. Lety01/2∈Tx0 {1/2,2/5}. It is clear that

dy0, T

y0 1 2− 1 8 3

8 d

x0, y0 /≤ϕ

fx0 d

x0, y0 . 3.21

Takey02/5∈Tx0 {1/2,2/5}. Obviously,

dy0, T

y0 2 5 − 2 25 8

25/≤ 19 40ϕ 3 8

ϕfx0 d

x0, y0 . 3.22

Suppose that ϕfx0 ∈ 225/361,1. As in the proof of Case1 stated first, we conclude similarly the conclusion of Case1stated after then. Therefore, the assumptions of Theorem1.6 do not hold.

Remark 3.5. The following example is an application of Theorem2.2.

Example 3.6. LetX 0,1be endowed with the Euclidean metricd|·|. LetT :X → CLX

be defined by

Tx ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x2 3

, x∈

0,23

48

∪

23 48,1

, 1 12, 1 24, 7 48

, x 23

48.

3.23

fx ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

dx, Tx x−x

2

3 , x∈

0,23

48

∪

23 48,1

,

1

3, x

23 48

3.24

isT-orbitally lower semi-continuous inXandB 0,2/3. Defineϕ:B → 0,1by

ϕt max

1 9, 13t 9

, t∈

0,2

3

. 3.25

Clearly,

lim inf

r→0 ϕr lim inf_r_→₀ max

1 9, 13t 9 1

9 >0,

lim sup

r→t ϕr<1, t∈

0,2

3

.

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Forx, y∈0,1\ {23/48}×Tx, we infer that

fy dy, Ty

x

2

3 −

x4 27

13

xx

2

3

x−x

2

3

≤max

1 9,

13 9

x−x

2

3

x−x

2

3

ϕfx dx, y .

3.27

Forx, y 23/48,1/12∈X×Tx, we conclude that

fy dy, Ty 1

12− 1 3·

1 122

35

432< 247

1296

13 27·

19

48 ϕ

fx dx, y . 3.28

Forx, y 23/48,1/24∈X×Tx, we obtain that

fy dy, Ty 1

24− 1 3 ·

1 242

71

1728 < 273

1296

13 27·

21

48 ϕ

fx dx, y . 3.29

Forx, y 23/48,7/48∈X×Tx, we get that

fy dy, Ty

487 −

1 3 ·

72 482

6912959 < 13

81

13 27·

1 3 ϕ

fx dx, y . 3.30

Therefore, all assumptions of Theorem2.2are satisfied. It follows from Theorem2.2thatT

has a fixed point inX.

Remark 3.7. Ifαt 1 andβt ϕtfor allt∈0,diamX, then Theorem2.3comes down to a result, which extends Theorem1.5. The following example shows that Theorem2.3is a genuine generalization of Theorem1.5.

Example 3.8. LetX Ê

_{be endowed with the Euclidean metric}_d _{| · |}_{. Define}_T _: _X _→

CLX,α:Ê

_→ ₀_,₁_{, and}_β_: Ê

_→ ₀_,₁_by

Tx

x

3

∪3x,∞, x∈X,

αt 1, βt 4

9, t∈Ê _,

3.31

respectively. Clearly,A 0,∞and

fx dx, Tx x− x

3

2x

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is continuous inX. Note that for eachx∈Xthere existsyx/3∈Txsuch that

αdx, y dx, y 1·

x−x

3

2x

3 fx,

fy dy, Ty x

3 − x 9 2x 9 ≤ 4 9· 2x

3 β

dx, y dx, y .

3.33

It is easy to verify that the assumptions of Theorem 2.3 are satisfied. Consequently, Theorem2.3guarantees thatThas a fixed point inX. ButTdoes not satisfy the conditions of Theorem1.5becauseTxis not compact for allx∈X.

Remark 3.9. In caseαt bandβt ϕtfor allt∈0,diamX, then Theorem2.3reduces to a result, which extends Theorem 1.4. The following example reveals that Theorem 2.3 generalizes properly Theorem1.4.

Example 3.10. LetX, dandTbe as in example 3.1. Clearly,

fx dx, Tx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x−x

2

3 , x∈

0,23

48

∪

23 48,1

,

19

108, x

23 48

3.34

isT-orbitally lower semi-continuous inX andA 0,1. Defineα:A → 0,1andβ:A → 0,1by

αt ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 10, t∈

0, 3

22

,

t, t∈

3 22,1

,

βt max

1 11, 2t 3

, t∈0,1.

3.35

It is easy to verify thatαis nondecreasing and

0<lim inf

r→t αr

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 10, t∈

0, 3

22

,

t, t∈

3 22,1

.

3.36

Fort∈0,3/22, we have

lim sup

r→t βr

αr lim supr→t 1 11·

10

1

10

(17)

Fort∈3/22,1, we infer that

lim sup

r→t

βr

αr lim supr→t

2r

3 · 1

r

2

3 <1. 3.38

Forx∈0,23/48∪23/48,1, there existsyx2_/₃_∈_T_x_satisfying

αdx, y dx, y α

x− x

2

3

x−x

2

3

≤x−x

2

3 fx,

fy dy, Ty

x 2 3 − x4 27 1 3

xx

2

3

x−x

2 3 ≤max 1 11, 2 3

x− x

2

3

dx, y

βdx, y dx, y .

3.39

Forx23/48, there existsy1/12∈Tx, satisfying

αdx, y dx, y 19

48· 19 48 <

19

108fx,

fy dy, Ty d

1 12, 1 432 35 432 < 361 3456 19 72· 19 48 β

dx, y dx, y .

3.40

That is, the conditions of Theorem2.3are fulfilled. It follows from Theorem2.3thatT has a fixed point inX. However, we cannot use Theorem1.4to show that the mappingThas a fixed point inXsince there does not existb∈0,1andϕ:Ê

_→ ₀_{, b}_{satisfying the assumptions} in Theorem1.4. In fact, for anyb∈0,1andϕ:Ê

_→ ₀_{, b}_{, we consider two possible cases} as follows.

Case 1. Let b∈0,4/9. Takex0 1 andy0 ∈Tx0 {1/3}. Note thatϕdx0, y0< b. It is clear that

dy0, T

y0 8 27 4 9 · 2 3/≤ϕ

dx0, y0

2 3 ϕ

dx0, y0

dx0, y0 . 3.41

Case 2. Let b∈ 4/9,1. Putx0 23/48 andy0 1/12 ∈Tx0 {1/12,131/432}. It follows that

bdx0, y0 19b 48 /≤

4 9 ·

19

48

19

(18)

Letx0 23/48 andy0 131/432∈Tx0 {1/12,131/432}. Note thatϕdx0, y0 < b. It is easy to verify that

dy0, T

y0

131

432− 1 3 ·

1312 4322

152615 559872/≤ϕ

dx0, y0

19

108ϕ

dx0, y0

dx0, y0 . 3.43

Therefore, the assumptions of Theorem1.4are not satisfied.

Acknowledgment

This work was supported by the Korea Research Foundation KRF Grant funded by the Korea governmentMEST 2009-0073655.

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