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Some new fixed point theorems for set-valued contractions in complete metric spaces

Fixed Point Theory and Applications20112011:72

https://doi.org/10.1186/1687-1812-2011-72

Received: 27 July 2011

Accepted: 31 October 2011

Published: 31 October 2011

Abstract

In this article, we obtain some new fixed point theorems for set-valued contractions in complete metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

MSC: 47H10, 54C60, 54H25, 55M20.

Keywords

fixed point theoremset-valued contraction

1 Introduction and preliminaries

Let (X, d) be a metric space, D a subset of X and f : DX be a map. We say f is contractive if there exists α [0, 1) such that for all x, y D,
d ( f x , f y ) α d ( x , y ) .
The well-known Banach's fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping f : XX is called a quasi-contraction if there exists k < 1 such that
d ( f x , f y ) k max { d ( x , y ) , d ( x , f x ) , d ( y , f y ) , d ( x , f y ) , d ( y , f x ) }

for any x, y X. In 1974, C'iric' [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

Throughout we denote the family of all nonempty closed and bounded subsets of X by CB(X). The existence of fixed points for various multi-valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler [3] extended the famous Banach Contraction Principle from single-valued mapping to multi-valued mapping and proved the below fixed point theorem for multi-valued contraction.

Theorem 1[3]Let (X, d) be a complete metric space and T : XCB(X). Assume that there exists c [0, 1) such that
H ( T x , T y ) c d ( x , y ) f o r a l l x , y X ,

where H denotes the Hausdorff metric on CB(X) induced by d, that is, H(A, B) = max{supxAD(x, B), supyBD(y, A)}, for all A, B CB(X) and D(x, B) = infzBd(x, z). Then, T has a fixed point in X.

In 1989, Mizoguchi-Takahashi [4] proved the following fixed point theorem.

Theorem 2[4]Let (X, d) be a complete metric space and T : XCB(X). Assume that
H ( T x , T y ) ξ ( d ( x , y ) ) d ( x , y )

for all x, y X, where ξ : [0, ∞) → [0, 1) satisfies lim sup s t + ξ ( s ) < 1 for all t [0, ∞). Then, T has a fixed point in X.

In the recent, Amini-Harandi [5] gave the following fixed point theorem for set-valued quasi-contraction maps in metric spaces.

Theorem 3[5]Let (X, d) be a complete metric space. Let T : XCB(X) be a k-set-valued quasi-contraction with k < 1 2 , that is,
( T x , T y ) k max { ( x , y ), D ( x , T x ), D ( y , T y ), D ( x , T y ) ), D ( y , T x ) }

for any x, y X. Then, T has a fixed point in X.

2 Fixed point theorem (I)

In this section, we assume that the function ψ : +5+ satisfies the following conditions:

(C1) ψ is a strictly increasing, continuous function in each coordinate, and

(C2) for all t +, ψ(t, t, t, 0, 2t) < t, ψ(t, t, t, 2t, 0) < t, ψ(0, 0, t, t, 0) < t and ψ(t, 0, 0, t, t) < t.

Definition 1 Let (X, d) be a metric space. The set-valued map T : XX is said to be a set-valued ψ-contraction, if
( T x , T y ) ψ ( d ( x , y ), D ( x , T x ), D ( y , T y ), D ( x , T y ) ), D ( y , T x ) )

for all x, y X.

We now state the main fixed point theorem for a set-valued ψ-contraction in metric spaces, as follows:

Theorem 4 Let (X, d) be a complete metric space. Let T : XCB(X) be a set-valued ψ-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B CB(X), a A and γ > 0 with H ( A , B ) < γ , there exists b B such that d(a, b) < γ. Since T : XCB(X) is a set-valued ψ-contraction, we have
( T x , T y ) ψ ( d ( x , y ), D ( x , T x ), D ( y , T y ), D ( x , T y ) ), D ( y , T x ) )
for all x, y X. Suppose that x0 X and that x1 X. Then, by induction and by the above observation, we can find a sequence {x n } in X such that xn+1 Tx n and for each n ,
d ( x n + 1 , x n ) ψ ( d ( x n , x n - 1 ) , D ( x n , T x n ) , D ( x n - 1 , T x n - 1 ) , D ( x n , T x n - 1 ) , D ( x n - 1 , T x n ) ) ψ ( d ( x n , x n - 1 ) , d ( x n , x n + 1 ) , d ( x n - 1 , x n ) , d ( x n , x n ) , d ( x n - 1 , x n + 1 ) ) ψ ( d ( x n , x n - 1 ) , d ( x n , x n + 1 ) , d ( x n - 1 , x n ) , 0 , d ( x n - 1 , x n ) + d ( x n , x n + 1 ) ) ,
and hence, we can deduce that for each n ,
d ( x n + 1 , x n ) d ( x n , x n - 1 ) .
Let we denote c m = d(xm+1, x m ). Then, c m is a non-increasing sequence and bounded below. Thus, it must converges to some c ≥ 0. If c > 0, then by the above inequalities, we have
c c n + 1 ψ ( c n , c n , c n , 0 , 2 c n ) .
Passing to the limit, as n → ∞, we have
c c ψ ( c , c , c , 0 , 2 c ) < c ,

which is a contradiction. Hence, c = 0.

We next claim that the following result holds:

for each γ > 0, there is n0(γ) such that for all m > n > n0(γ),
d ( x m , x n ) < γ . ( * )
We shall prove (*) by contradiction. Suppose that (*)is false. Then, there exists some γ > 0 such that for all k , there exist m k , n k with m k > n k k satisfying:
  1. (1)

    m k is even and n k is odd;

     
  2. (2)

    d ( x m k , x n k ) γ ;

     
  3. (3)

    m k is the smallest even number such that the conditions (1), (2) hold.

     
Since c m 0, by (2), we have lim k d ( x m k , x n k ) = γ and
γ d ( x m k , x n k ) ( T x m k 1 , T x n k 1 ) ψ ( d ( x m k 1 , x n k 1 ), d ( x m k 1 , x m k ), d ( x n k 1 , x n k ), d ( x m k 1 , x n k ), d ( x n k 1 , x m k ) ) ψ ( c m k 1 + d ( x m k , x n k ) + c n k 1 , c m k 1 , c n k 1 , c m k 1 + d ( x m k , x n k ), d ( x m k , x n k ) + c n k 1 ) ).
Letting k → ∞. Then, we get
γ ψ ( γ , 0 , 0 , γ , γ ) < γ ,

a contradiction. It follows from (*) that the sequence {x n } must be a Cauchy sequence.

Similarly, we also conclude that for each n ,
d ( x n , x n + 1 ) ψ ( d ( x n - 1 , x n ) , D ( x n - 1 , T x n - 1 ) , D ( x n , T x n ) , D ( x n - 1 , T x n ) , D ( x n , T x n - 1 ) ) ψ ( d ( x n - 1 , x n ) , d ( x n - 1 , x n ) , d ( x n , x n + 1 ) , d ( x n - 1 , x n + 1 ) , d ( x n , x n ) ) ψ ( d ( x n - 1 , x n ) , d ( x n , x n + 1 ) , d ( x n - 1 , x n ) , d ( x n - 1 , x n ) + d ( x n , x n + 1 ) , 0 ) ,
and hence, we have that for each n ,
d ( x n , x n + 1 ) d ( x n - 1 , x n ) .
Let we denote b m = d(x m , xm+1). Then, b m is a non-increasing sequence and bounded below. Thus, it must converges to some b ≥ 0. If b > 0, then by the above inequalities, we have
b b n + 1 ψ ( b n , b n , b n , 2 b n , 0 ) .
Passing to the limit, as n → ∞, we have
b b ψ ( b , b , b , 2 b , 0 ) < b ,

which is a contradiction. Hence, b = 0. By the above argument, we also conclude that {x n } is a Cauchy sequence.

Since X is complete, there exists μ X such that limn→∞x n = μ. Therefore,
D ( μ , T μ ) = lim n D ( x n + 1 , T μ ) lim n H ( T x n , T μ ) lim n ψ ( d ( x n , μ ) , D ( x n , T x n ) , D ( μ , T μ ) , D ( x n , T μ ) , D ( μ , T x n ) ) lim n ψ ( d ( x n , μ ) , d ( x n , x n + 1 ) , D ( μ , T μ ) , D ( x n , T μ ) , d ( μ , x n + 1 ) ) ψ ( 0 , 0 , D ( μ , T μ ) , D ( μ , T μ ) , 0 ) < D ( μ , T μ ) ,

and hence, D(μ, ) = 0, that is, μ , since is closed.

3 Fixed point theorem (II)

In 1972, Chatterjea [6] introduced the following definition.

Definition 2 Let (X, d) be a metric space. A mapping f : XX is said to be a C -contraction if there exists α ( 0 , 1 2 ) such that for all x, y X, the following inequality holds:
d ( f x , f y ) α ( d ( x , f y ) + d ( y , f x ) ) .

Choudhury [7] introduced a generalization of C -contraction, as follows:

Definition 3 Let (X, d) be a metric space. A mapping f : XX is said to be a weakly C -contraction if for all x, y X,
d ( f x , f y ) 1 2 ( d ( x , f y ) + d ( y , f x ) - ϕ ( d ( x , f y ) , d ( y , f x ) ) ) ,

where ϕ : +2+is a continuous function such that ψ(x, y) = 0 if and only if x = y = 0.

In [6, 7], the authors proved some fixed point results for the C -contractions. In this section, we present some fixed point results for the weakly ψ- C -contraction in complete metric spaces.

Definition 4 Let (X, d) be a metric space. The set-valued map T : XX is said to be a set-valued weakly ψ- C -contraction, if for all x, y X
H ( T x , T y ) ψ ( [ D ( x , T y ) + D ( y , T x ) - ϕ ( D ( x , T y ) , D ( y , T x ) ) ] ) ,

where

(1) ψ : ++is a strictly increasing, continuous function with ψ ( t ) 1 2 t for all t > 0 and ψ(0) = 0;

(2) ϕ : +2+is a strictly decreasing, continuous function in each coordinate, such that ϕ(x, y) = 0 if and only if x = y = 0 and ϕ(x, y) ≤ x + y for all x, y +.

Theorem 5 Let (X, d) be a complete metric space. Let T : XCB(X) be a set-valued weakly C -contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B CB(X), a A and γ > 0 with H ( A , B ) < γ , there exists b B such that d(a, b) < γ. Since T : XCB(X) be a set-valued weakly ψ- C -contraction, we have that
H ( T x , T y ) ψ ( [ D ( x , T y ) + D ( y , T x ) - ϕ ( D ( x , T y ) , D ( y , T x ) ) ] )
for all x, y X. Suppose that x0 X and that x1 X. Then, by induction and by the above observation, we can find a sequence {x n } in X such that xn+1 Tx n and for each n ,
d ( x n + 1 , x n ) H ( T x n , T x n - 1 ) ψ ( [ D ( x n , T x n - 1 ) + D ( x n - 1 , T x n ) - ϕ ( D ( x n , T x n - 1 ) , D ( x n - 1 , T x n ) ) ] ) ψ ( [ d ( x n , x n ) + d ( x n - 1 , x n + 1 ) - ϕ ( d ( x n , x n ) , d ( x n - 1 , x n + 1 ) ) ] ) = ψ ( [ 0 + d ( x n - 1 , x n + 1 ) - ϕ ( 0 , d ( x n - 1 , x n + 1 ) ) ] ) ψ ( [ d ( x n - 1 , x n ) + d ( x n , x n + 1 ) ] ) 1 2 [ d ( x n - 1 , x n ) + d ( x n , x n + 1 ) ] ,
and hence, we deduce that for each n ,
d ( x n + 1 , x n ) d ( x n , x n - 1 ) .
Thus, {d(xn+1, x n )} is non-increasing sequence and bounded below and hence it is convergent. Let limn→∞d(xn+1, x n ) = ξ. Letting n → ∞ in (**), we have
ξ = lim n d ( x n + 1 , x n ) lim n ψ ( [ d ( x n - 1 , x n + 1 ) ] ) lim n 1 2 [ d ( x n - 1 , x n + 1 ) ] lim n 1 2 [ d ( x n - 1 , x n ) + d ( x n , x n + 1 ) ] 1 2 [ ξ + ξ ] = ξ ,
that is,
lim n d ( x n - 1 , x n + 1 ) = 2 ξ .
By the continuity of ψ and ϕ, letting n → ∞ in (**), we have
ξ ψ ( 2 ξ - ϕ ( 0 , 2 ξ ) ) ξ - 1 2 ϕ ( 0 , 2 ξ ) ξ .

Hence, we have ϕ(0, 2ξ) = 0, that is, ξ = 0. Thus, limn→∞d(xn+1, x n ) = 0.

We next claim that the following result holds:

for each γ > 0, there is n0(γ) such that for all m > n > n0(γ),
d ( x m , x n ) < γ . ( * * * )
We shall prove (***) by contradiction. Suppose that (***) is false. Then, there exists some γ > 0 such that for all k , there exist m k , n k with m k > n k k satisfying:
  1. (1)

    m k is even and n k is odd;

     
  2. (2)

    d ( x m k , x n k ) γ ;

     
  3. (3)

    m k is the smallest even number such that the conditions (1), (2) hold.

     
Since d(xn+1, x n ) 0, by (2), we have lim k d ( x m k , x n k ) = γ and
γ d ( x m k , x n k ) H ( T x m k - 1 , T x n k - 1 ) ψ ( [ D ( x m k - 1 , T x n k - 1 ) + D ( x n k - 1 , T x m k - 1 ) - ϕ ( D ( x m k - 1 , T x n k - 1 ) , D ( x n k - 1 , T x m k - 1 ) ) ] ) ψ ( [ d ( x m k - 1 , x n k ) + d ( x n k - 1 , x m k ) - ϕ ( d ( x m k - 1 , x n k ) , d ( x n k - 1 , T x m k ) ) ] ) .
Since
d ( x m k - 1 , x n k ) + d ( x n k - 1 , x m k ) d ( x m k - 1 , x m k ) + d ( x m k , x n k ) + d ( x n k , x m k ) + d ( x n k - 1 , x n k ) ,
letting k → ∞, then we get
γ ψ ( 2 γ - ϕ ( γ , γ ) ) γ ,

and hence, ϕ(γ, γ)) = 0. By the definition of ϕ, we get γ = 0, a contradiction. This proves that the sequence {x n } must be a Cauchy sequence.

Since X is complete, there exists z X such that limn→∞x n = z. Therefore,
D ( z , T z ) = lim n D ( x n + 1 , T z ) lim n H ( T x n , T z ) lim n ψ ( [ D ( x n , T z ) + D ( z , T x n ) - ϕ ( D ( x n , T z ) , D ( z , T x n ) ) ] ) lim n ψ ( [ D ( x n , T z ) + d ( z , x n + 1 ) - ϕ ( D ( x n , T z ) , d ( z , x n + 1 ) ) ] ) 1 2 D ( z , T z )

and hence, D(z, Tz) = 0, that is, z Tz, since Tz is closed.

4 Fixed point theorem (III)

In this section, we recall the notion of the Meir-Keeler type function (see [8]). A function φ : ++ is said to be a Meir-Keeler type function, if for each η > 0, there exists δ > 0 such that for t + with ηt < η + δ, we have φ(t) < η. We now introduce the new notions of the weaker Meir-Keeler type function φ : ++ in a metric space and the φ-function using the weaker Meir-Keeler type function, as follow:

Definition 5 Let (X, d) be a metric space. We call φ : ++a weaker Meir-Keeler type function, if for each η > 0, there exists δ > 0 such that for x, y X with ηd(x, y) < δ + η, there exists n0 such that φ n 0 ( d ( x , y ) ) < γ η .

Definition 6 Let (X, d) be a metric space. A weaker Meir-Keeler type function φ ; ++is called a φ-function, if the following conditions hold:

1) φ(0) = 0, 0 < φ(t) < t for all t > 0;

2 ) φ is a strictly increasing function;

3) for each t +, {φ n (t)}nis decreasing;

4) for each t n + \ { 0 } , if limn→∞t n = γ > 0, then limn→∞φ(t n ) < γ;

5) for each t n +, if limn→∞t n = 0, then limn→∞φ(t n ) = 0.

Definition 7 Let (X, d) be a metric space. The set-valued map T : XX is said to be a set-valued weaker Meir-Keeler type φ-contraction, if
H ( T x , T y ) φ 1 2 [ D ( x , T y ) + D ( y , T x ) ]

for all x, y X.

We now state the main fixed point theorem for a set-valued weaker Meir-Keeler type ψ-contraction in metric spaces, as follows:

Theorem 6 Let (X, d) be a complete metric space. Let T : CB(X) be a set-valued weaker Meir-Keeler type ψ-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B CB(X), a A and γ > 0 with H ( A , B ) < γ , there exists b B such that d(a, b) < γ. Since T : XCB(X) be a set-valued ψ-contraction, we have that
H ( T x , T y ) φ 1 2 [ D ( x , T y ) + D ( y , T x ) ]
for all x, y X. Suppose that x0 X and that x1 X. Then, by induction and by the above observation, we can find a sequence {x n } in X such that xn+1 Tx n and for each n ,
d ( x n + 1 , x n ) φ 1 2 [ D ( x n , T x n - 1 ) + D ( x n - 1 , T x n ) ] φ 1 2 [ d ( x n , x n ) + d ( x n - 1 , x n + 1 ) ] φ 1 2 [ d ( x n - 1 , x n ) + d ( x n , x n + 1 ) ] ,
and by the conditions (φ1) and (φ2), we can deduce that for each n ,
d ( x n + 1 , x n ) φ ( d ( x n , x n - 1 ) ) < d ( x n , x n - 1 )
and
d ( x n + 1 , x n ) φ ( d ( x n , x n - 1 ) ) φ n ( d ( x 1 , x 0 ) ) .

By the condition (φ3), {φ n (d(x0, x1))}nis decreasing, it must converges to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then, by the definition of the weaker Meir-Keeler type function, there exists δ > 0 such that for x0, x1 X with ηd(x0, x1) < δ + η, there exists n0 such that φ n 0 ( d ( x 0 , x 1 ) ) < η . Since limn→∞φ n (d(x0, x1)) = η, there exists m0 such that ηφ m (d(x0, x1)) < δ + η, for all mm0. Thus, we conclude that φ m 0 + n 0 ( d ( x 0 , x 1 ) ) < η . Hence, we get a contradiction. Hence, limn→∞φ n (d(x0, x1)) = 0, and hence, limn→∞d(x n , xn+1) = 0.

Next, we let c m = d(x m , xm+1), and we claim that the following result holds:

for each ε > 0, there is n0(ε) such that for all m , nn0(ε),
d ( x m , x m + 1 ) < ε . ( * * * * )
We shall prove (****) by contradiction. Suppose that (****) is false. Then, there exists some ε > 0 such that for all p N, there are m p , n p with m p > n p p satisfying:
  1. (i)

    m p is even and n p is odd,

     
  2. (ii)

    d ( x m p , x n p ) ε , and

     
  3. (iii)

    m p is the smallest even number such that the conditions (i), (ii) hold.

     
Since c m 0, by (ii), we have lim p d ( x m p , x n p ) = ε , and
ε d ( x m p , x n p ) H ( T x m p - 1 , T x n p - 1 ) φ 1 2 [ D ( x m p - 1 , T x n p - 1 ) + D ( x n p - 1 , T x m p - 1 ) ] φ 1 2 [ d ( x m p - 1 , x n p ) + d ( x n p - 1 , x m p ) ] φ 1 2 [ d ( x m p - 1 , x m p ) + 2 d ( x n p , x m p ) + d ( x n p - 1 , x n p ) ] .
Letting p → ∞. By the condition (φ4), we have
ε lim p φ 1 2 [ d ( x m p - 1 , x m p ) + 2 d ( x n p , x m p ) + d ( x n p - 1 , x n p ) ] < ε ,
a contradiction. Hence, {x n } is a Cauchy sequence. Since (X, d) is a complete metric space, there exists μ X such that limn→∞xn+1= μ. Therefore,
D ( μ , T μ ) = lim n D ( x n + 1 , T μ ) lim n H ( T x n , T μ ) lim n φ 1 2 [ D ( x n , T μ ) ) + D ( μ , T x n ) lim n φ 1 2 [ D ( x n , T μ ) ) + d ( μ , x n + 1 ) 1 2 D ( μ , T μ ) ,

and hence, D(μ, ) = 0, that is, μ , since is closed.

Declarations

Acknowledgements

This research was supported by the National Science Council of the Republic of China.

Authors’ Affiliations

(1)
Department of Applied Mathematics, National Hsinchu University of Education, Taiwan

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Copyright

© Chen; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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