Báo cáo hóa học: "Research Article Fixed Point Results for Generalized Contractive Multimaps in Metric Space" docx

Khamsi The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditi

Volume 2009, Article ID 432130, 16 pages

doi:10.1155/2009/432130

Research Article

Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces

Abdul Latif1 and Afrah A N Abdou2

1 Department of Mathematics, King Abdulaziz University, P.O Box 80203, Jeddah 21589, Saudi Arabia

2 Girls College of Education, King Abdulaziz University, P.O Box 14884, Jeddah 21434, Saudi Arabia

Correspondence should be addressed to Abdul Latif,latifmath@yahoo.com

Received 17 May 2009; Accepted 10 August 2009

Recommended by Mohamed A Khamsi

The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions Consequently, our results either generalize or improve a number of fixed point results including the corresponding recent fixed point results of Ciric2008, Latif-Albar 2008, Klim-Wardowski

2007, and Feng-Liu 2006 Examples are also given

Copyrightq 2009 A Latif and A A N Abdou 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

1 Introduction and Preliminaries

LetX, d be a metric space, 2 X a collection of nonempty subsets of X, CBX a collection

of nonempty closed bounded subsets of X, ClX a collection of nonempty closed subsets of

X, K X a collection of nonempty compact subsets of X and H the Hausdorff metric induced

by d Then for any A, B ∈ CBX,

H A, B  max

 sup

x ∈A d x, B, sup

y ∈B d

y, A

where dx, B  inf y ∈B d x, y.

An element x ∈ X is called a fixed point of a multivalued map T : X → 2 X if x ∈ Tx.

We denote FixT  {x ∈ X : x ∈ Tx} A sequence {xn } in X is called an orbit of T at x0∈ X

if x n ∈ Tx n−1 for all n ≥ 1.

A map f : X → R is called lower semicontinuous if for any sequence {x n } ⊂ X with

x n → x ∈ X it implies that fx ≤ lim inf n→ ∞f x n

Using the concept of Hausdorff metric, Nadler 1 established the following fixed point result for multivalued contraction maps, known as Nadler’s contraction principle which in turn is a generalization of the well-known Banach contraction principle

Theorem 1.1 see 1 Let X, d be a complete metric space and let T : X → CBX be a

contraction map Then Fix T / ∅.

Using the concept of the Hausdorff metric, many authors have generalized Nadler’s contraction principle in many directions But, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorff metric Recently, Feng and Liu

2 extended Nadler’s fixed point theorem without using the concept of Hausdorff metric They proved the following result

Theorem 1.2 Let X, d be a complete metric space and let T : X → ClX be a map such that for

any fixed constants h, b ∈ 0, 1, h < b, and for each x ∈ X there is y ∈ Tx satisfying the following

conditions:

bd

x, y

≤ dx, Tx,

d

y, T

y

≤ hdx, y

.

1.2

Then Fix T / ∅ provided a real-valued function g on X, gx  dx, Tx is lower semicontinuous.

Recently, Klim and Wardowski3 generalizedTheorem 1.2and proved the following two results

Theorem 1.3 Let X, d be a complete metric space and let T : X → ClX Assume that the

following conditions hold:

i there exist a number b ∈ 0, 1 and a function k : 0, ∞ → 0, b such that for each

t ∈ 0, ∞,

lim sup

r → t k r < b, 1.3

ii for any x ∈ X there is y ∈ Tx satisfying

bd

x, y

≤ dx, Tx,

d

y, T

y

≤ kd

x, y

d

x, y

Then Fix T / ∅ provided a real-valued function g on X, gx  dx, Tx is lower semicontinuous.

Theorem 1.4 Let X, d be a complete metric space and let T : X → KX Assume that the

following conditions hold:

i there exists a function k : 0, ∞ → 0, 1 such that for each t ∈ 0, ∞,

lim sup

r → t k r < 1, 1.5

ii for any x ∈ X there is y ∈ Tx satisfying

d

x, y

 dx, Tx,

d

y, T

y

≤ kd

x, y

d

x, y

Then Fix T / ∅ provided a real-valued function g on X, gx  dx, Tx is lower semicontinuous.

Note thatTheorem 1.3 generalizes Nadler’s contraction principle andTheorem 1.2 Most recently, Ciric 4 obtained some interesting fixed point results which extend and generalize the cited results Namely,4, Theorem 5 generalizes 5, Theorem 5, 4, Theorem 6 generalizes 4, Theorems 1.2, 1.3, and 3, theorem 7 generalizesTheorem 1.4

In6, Kada et al introduced the concept of w-distance on a metric space as follows:

A function ω : X × X → 0, ∞ is called w-distance on X if it satisfies the following for each x, y, z ∈ X:

w1 ωx, z ≤ ωx, y  ωy, z;

w2 a map ωx, · : X → 0, ∞ is lower semicontinuous; that is, if a sequence {y n} in

X with yn → y ∈ X, then ωx, y ≤ lim inf n→ ∞ω x, y n;

w3 for any  > 0, there exists δ > 0 such that ωz, x ≤ δ and ωz, y ≤ δ imply

d x, y ≤ .

Note that, in general for x, y ∈ X, ωx, y / ωy, x and not either of the implications

ω

be a normed space Then the functions ω1, ω2: Y ×Y → 0, ∞ defined by ω1

ω2 6 Many other examples and properties

of the w-distance can be found in6,7

The following lemma is crucial for the proofs of our results

Lemma 1.5 see 8 Let K be a closed subset of X and ω be a w-distance on X Suppose that there

exists u ∈ X such that ωu, u  0 Then ωu, K  0 ⇔ u ∈ K, where ωu, K  inf y ∈K ω u, y.

Most recently, the authors of this paper generalized Latif and Albar9, Theorem 1.3

as follows

Theorem 1.6 see 10 Let X, d be a complete metric space with a w-distance ω Let T : X →

Cl X be a multivalued map satisfying that for any constant b ∈ 0, 1 and for each x ∈ X there is

y ∈ J x

b such that

ω

y, T

y

≤ kω

x, y

ω

x, y

where J x

b  {y ∈ Tx : bωx, y ≤ ωx, Tx} and k is a function from 0, ∞ to 0, b with

lim supr → tk r < b, for every t ∈ 0, ∞ Suppose that a real-valued function g on X defined by

g x  ωx, Tx is lower semicontinuous Then there exists v o ∈ X such that gv o   0 Further,

if ω v o , v o   0, then v0∈ FixT.

The aim of this paper is to present some more general results on the existence of fixed points for multivalued maps satisfying certain conditions Our results unify and generalize

the corresponding results of Mizoguchi and Takahashi5, Klim and Wardowski 3, Latif and Abdou10, Ciric 4, Feng and Liu 2, Latif and Albar 9 and several others

2 The Results

First we prove a theorem which is a generalization of Ciric4, Theorem 5 and due to Klim and Wardowski3, Theorem 1.4

Theorem 2.1 Let X, d be a complete metric space with a w-distance ω Let T : X → ClX be a

multivalued map Assume that the following conditions hold:

i there exists a function ϕ : 0, ∞ → 0, 1 such that for each t ∈ 0, ∞

lim sup

r → t ϕ r < 1 2.1

ii for any x ∈ X, there exists y ∈ Tx satisfying

ω

x, y

≤2− ϕω

x, y

ω x, Tx,

ω

y, T

y

≤ ϕω

x, y

ω

x, y 2.2

iii the map f : X → R, defined by fx  ωx, Tx is lower semicontinuous.

Then there exists v0∈ X such that fv0  0 Further if ωv0, v0  0, then v0 ∈ Tv0.

Proof let x0∈ X be any initial point Then there exists x1 ∈ Tx0 such that

ω x0, x1 ≤2− ϕωx0, x1ω x0, T x0,

ω x1, T x1 ≤ ϕωx0, x1ωx0, x1. 2.3

From2.3 we get

ω x1, T x1 ≤ ϕωx0, x12− ϕωx0, x1ω x0, T x0. 2.4

Define a function ψ : 0, ∞ → 0, ∞ by

ψ t  ϕt2− ϕt 1 −1− ϕt2

Using the facts that for each t ∈ 0, ∞, ϕt < 1 and lim r → tsup ϕr < 1, we have

ψ t < 1 , 2.6 lim sup

r → t ψ r < 1 ∀t ∈ 0, ∞ 2.7

From2.4 and 2.5, we have

ω x1, T x1 ≤ ψωx0, x1ωx0, T x0. 2.8

Similarly, for x1 ∈ X, there exists x2 ∈ Tx1 such that

ω x1, x2 ≤2− ϕωx1, x2ω x1, T x1,

ω x2, T x2 ≤ ϕωx1, x2ωx1, x2. 2.9

Thus

ω x2, T x2 ≤ ψωx1, x2ωx1, T x1. 2.10 Continuing this process we can get an orbit{x n } of T in X satisfying the following:

ω x n , x n1 ≤2− ϕωx n , x n1ω x n , T x n , 2.11

ω x n1, T x n1 ≤ ψωx n , x n1ωx n , T x n , 2.12

for each integer n ≥ 0 Since ψt < 1 for each t ∈ 0, ∞ and from 2.12, we have for all n ≥ 0

ω x n1, T x n1 < ωx n , T x n . 2.13

Thus the sequence of nonnegative real numbers {ωx n , T x n} is decreasing and bounded

below, thus convergent Therefore, there is some δ≥ 0 such that

lim

n→ ∞ω x n, T x n   δ. 2.14

From2.11, as ϕt < 1 for all t ≥ 0, we get

ω x n , T x n  ≤ ωx n , x n1 < 2ωx n , T x n , 2.15

Thus, we conclude that the sequence of nonnegative reals {ωx n, xn1} is bounded

Therefore, there is some θ≥ 0 such that

lim inf

n→ ∞ ω x n , x n1  θ. 2.16

Note that ωx n , x n1 ≥ ωx n , T x n  for each n ≥ 0, so we have θ ≥ δ Now we will show that

θ  δ Suppose that δ  0 Then we get

lim

Now consider δ > 0 Suppose to the contrary, that θ > δ Then θ − δ > 0 and so from 2.14 and2.16 there is a positive integer n0such that

ω x n , T x n  < δ  θ − δ

4 ∀n ≥ n0, 2.18

θ−θ − δ

Then from2.19, 2.11 and 2.18, we get

θ−θ − δ

4 < ω x n , x n1

≤2− ϕωx n , x n1ω x n , T x n

<

2− ϕωx n , x n1δθ − δ

4



.

2.20

Thus for all n ≥ n0,



2− ϕωx n , x n1> 3θ  δ

3δ  θ , 2.21

that is,

11− ϕωx n, xn1> 12θ − δ

3δ  θ , 2.22

and we get

−1− ϕωx n, xn12

<−

2θ − δ

3δ  θ

2

Thus for all n ≥ n0,

ψ ωx n , x n1  1 −1− ϕωx n , x n12

< 1−

2θ − δ

3δ  θ

2

.

2.24

Thus, from2.12 and 2.24, we get

ω x n1, T x n1 ≤ hωx n, T x n  ∀n ≥ n0, 2.25

where h  1 − 2θ − δ/3δ  θ2 Clearly h < 1 as θ > δ From2.18 and 2.25, we have for

any k≥ 1

ω x n0k , T x n0k  ≤ h k ω x n0, T x n0. 2.26

Since δ > 0 and h < 1, there is a positive integer k0such that h k0ω x n0, T x n0 < δ Now, since

δ ≤ ωx n , T x n  for each n ≥ 0, by 2.26 we have

δ ≤ ωx n0k0, T x n0k0 ≤ h k0ω x n0, T x n0 < δ. 2.27

a contradiction Hence, our assumption θ > δ is wrong Thus δ  θ Now we will show that

θ  0 Since θ  δ ≤ ωx n, T x n  ≤ ωx n, xn1, then from 2.16 we can read as

lim inf

n→ ∞ ω x n , x n1  θ, 2.28

so, there exists a subsequence{ωx n k , x n k1} of {ωx n , x n1} such that

lim

k→ ∞ω x n k , xn k1  θ  2.29

Now from2.7 we have

lim sup

ω x nk ,x nk1  → θ

ψ ωx n k , x n k1 < 1, 2.30

and from2.12, we have

ω x n k , T x n k1 ≤ ψωx n k , x n k1ωx n k , T x n k 2.31

Taking the limit as k → ∞ and using 2.14, we get

δ lim sup

k→ ∞ ω x n k1 , T x n k1

≤

 lim sup

k→ ∞ ψ ωx n k1 , x n k1

 lim sup

k→ ∞ ω x n k , T x n k



⎛

⎝ lim sup

ω x nk ,x nk1  → θ

ψ ωx n k , x n k1

⎞

⎠δ.

2.32

If we suppose that δ > 0, then from last inequality, we have

lim sup

ω x ,x  → θ

ψ ωx n k , x n k1 ≥ 1, 2.33

which contradicts with2.30 Thus δ  0 Then from 2.14 and 2.15, we have

lim

n→ ∞ω x n , T x n   0, 2.34

and thus

lim

Now, let

ω x nk ,x nk1 → 0 sup ψωx n k , xn k1. 2.36

Then by2.7, α < 1 Let q be such that α < q < 1 Then there is some n0∈ N such that

ψ ωx n , x n1 < q ∀n ≥ n0. 2.37 Thus it follows from2.12,

ω x n1, T x n1 ≤ qωx n , T x n  ∀n ≥ n0. 2.38

By induction we get

ω x n1, T x n1 ≤ q n 1−n0ω x n0, T x n0 ∀n ≥ n0. 2.39 Now, using2.15 and 2.39, we have

ω x n , x n1 ≤ 2q n −n0ω x n0, T x n0 ∀n ≥ n0. 2.40 Now, we show that{x n } is a Cauchy sequence, for all m > n ≥ n0, we get

ω x n , x m ≤m−1

k n

ω x k , x k1

≤ 2m−1

k n

q k −n0ω x n0, T x n0

≤ 2



q n −n0

1− q



ω x n0, T x n0.

2.41

Hence we conclude, as q < 1, that {x n } is Cauchy sequence Due to the completeness of X, there exists some v0∈ X such that lim n→ ∞xn  v0 Since f is lower semicontinuous and from

2.34, we have

0≤ fv0 ≤ lim inf

n→ ∞ f x n   ωx n, T x n   0, 2.42

and thus, fv0  ωv0, T v0  0 Since ωv0, v0  0, and Tv0 is closed, it follows from

Lemma 1.5that v0 ∈ Tv0.

We also have the following interesting result by replacing the hypothesis iii of

Theorem 2.1with another natural condition

Theorem 2.2 Suppose that all the hypotheses of Theorem 2.1 except (iii) hold Assume that

inf{ωx, v  ωx, Tx : x ∈ X} > 0, 2.43

for every v ∈ X with v /∈ Tv Then FixT / ∅.

T x n−1 Due to the completeness of X, there exists v0 ∈ X such that lim n→ ∞x n  v0 Since

ω x, · is lower semicontinuous and x m → v0 ∈ X, it follows for all n ≥ n0

ω x n, v0 ≤ lim

m→ ∞inf ωx n, xm ≤

2q n −n0

1− q



ω x n0, T x n0,

ω x n, T x n  ≤ ωx n, xn1 ≤ 2q n −n0ω x n0, T x n0.

2.44

Assume that v0/ ∈ Tv0 Then, we have

0 < inf{ωx, v0  ωx, Tx : x ∈ X}

≤ inf{ωx n, v0  ωx n, T x n  : n ≥ n0}

≤ inf

2q n −n0

1− q



ω x n0, T x n0  2q n −n0ω x n0, T x n0 : n ≥ n0



 2



2− q



1− qq n0 ω x n0, T x n0 infq n : n ≥ n0



 0,

2.45

which is impossible and hence v0∈ FixT.

Now, we present an improved version of Ciric 4, Theorem 6 and which also generalizes due to Latif and Abdou10, Theorem 1.6 and due to Klim and Wardowski 3, Theorem 1.3

Theorem 2.3 Let X, d be a complete metric space with a w-distance ω Let T : X → ClX, be a

multivalued map Assume that the following condition hold:

i there exist functions ϕ : 0, ∞ → 0, 1 and μ : 0, ∞ → b, 1, with b > 0, μ

nondecreasing such that

ϕ t < μt, lim sup

r → t ϕ r < lim sup

r → t μ r, 2.46

ii for any x ∈ X, there exists y ∈ Tx satisfying the following conditions:

μ

ω

x, y

ω

x, y

≤ ωx, Tx,

ω

y, T

y

≤ ϕω

x, y

ω

x, y

iii the map f : X → R, defined by fx  ωx, Tx is lower semicontinuous.

Then there exists v0∈ X such that fv0  0 Further if ωv0, v0  0, then v0 ∈ Tv0.

Proof Let x0be an arbitrary, then there exists x1∈ Tx0 such that

μ ωx0, x1ωx0, x1 ≤ ωx0, T x0,

ω x1, T x1 ≤ ϕωx0, x1ωx0, x1. 2.48

From2.48 we have

ω x1, T x1 ≤ ϕ ωx0, x1

μ ωx0, x1ω x0, T x0. 2.49

Define a function ψ : 0, ∞ → 0, ∞ by

ψ t  ϕ t

μ t ∀t ∈ 0, ∞. 2.50

Since ϕt < μt, we have

ψ t < 1, 2.51 lim sup

r → t ψ r < 1 ∀t ∈ 0, ∞. 2.52

Thus from2.49

ω x1, T x1 ≤ ψωx0, x1ωx0, T x0. 2.53

Similarly, there exists x2 ∈ Tx1 such that

μ ωx1, x2ωx1, x2 ≤ ωx1, T x1,

ω x2, T x2 ≤ ϕωx1, x2ωx1, x2. 2.54 Then by definition of ψ, we get

ω x2, T x2 ≤ ψωx1, x2ωx1, T x1. 2.55

Continuing this process, we get an orbit{x n } of T at x0such that

μ ωx n , x n1ωx n , x n1 ≤ ωx n , T x n , 2.56

ω x n1, T x n1 ≤ ϕωx n , x n1ωx n , x n1. 2.57 Thus

ω x n1, T x n1 ≤ ψωx n , x n1ωx n , T x n . 2.58

Since ψt < 1 for all t ∈ 0, ∞, we get

ω x n1, T x n1 < ωx n , T x n . 2.59

Thus the sequence of nonnegative real numbers{ωx n, T x n} is decreasing and bounded below, thus convergent Now, we want to show that the sequence {ωx n , x n1} is also

decreasing Suppose to the contrary, that ωx n , x n1 ≤ ωx n1, x n2, then as μt is

nondecreasing, we have

μ ωx n , x n1 ≤ μωx n1, x n2, 2.60 Now using2.56, 2.57 and 2.60 with n  n  1, we get

ω x n1, xn2 ≤ ϕ ωx n, xn1

μ ωx n1, x n2ω x n, xn1

≤ ϕ ωx n , x n1

μ ωx n, xn1ω x n , x n1

< ψ ωx n , x n1ωx n , x n1

< ω x n , x n1,

2.61

a contradiction Thus the sequences{ωx n , x n1} is decreasing Now let

lim sup

n→ ∞ ψ ωx n , x n1  α. 2.62

Thus by2.52, α < 1 Then for any q ∈ α, 1, there exists n0 ∈ N such that

So, from2.58, for all n ≥ n0, we get

ω x n1, T x n1 < qωx n, T x n . 2.64