Open Access

On generalized weakly directional contractions and approximate fixed point property with applications

Fixed Point Theory and Applications20122012:6

https://doi.org/10.1186/1687-1812-2012-6

Received: 6 August 2011

Accepted: 17 January 2012

Published: 17 January 2012

Abstract

In this article, we first introduce the concept of directional hidden contractions in metric spaces. The existences of generalized approximate fixed point property for various types of nonlinear contractive maps are also given. From these results, we present some new fixed point theorems for directional hidden contractions which generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem and some well-known results in the literature.

MSC: 47H10; 54H25.

Keywords

τ-functionτ0-metricReich's condition -function( T -function)directional hidden contractionapproximate fixed point propertygeneralized Mizoguchi-Takahashi's fixed point theoremgeneralized Berinde-Berinde's fixed point theorem

1 Introduction and preliminaries

Let (X, d) be a metric space. The open ball centered in x X with radius r > 0 is denoted by B(x, r). For each x X and A X, let d(x, A) = infyAd(x, y). Denote by N ( X ) the class of all nonempty subsets of X, C ( X ) the family of all nonempty closed subsets of X and C ( X ) the family of all nonempty closed and bounded subsets of X. A function : C ( X ) × C ( X ) [ 0 , ) defined by
( A , B ) = max sup x B d ( x , A ) , sup x A d ( x , B )

is said to be the Hausdorff metric on C ( X ) induced by the metric d on X. A point v in X is a fixed point of a map T if v = Tv (when T : XX is a single-valued map) or v Tv (when T : X N ( X ) is a multivalued map). The set of fixed points of T is denoted by ( T ) . Throughout this article, we denote by and , the sets of positive integers and real numbers, respectively.

The celebrated Banach contraction principle (see, e.g., [1]) plays an important role in various fields of applied mathematical analysis. It is known that Banach contraction principle has been used to solve the existence of solutions for nonlinear integral equations and nonlinear differential equations in Banach spaces and been applied to study the convergence of algorithms in computational mathematics. Since then a number of generalizations in various different directions of the Banach contraction principle have been investigated by several authors; see [136] and references therein. A interesting direction of research is the extension of the Banach contraction principle to multivalued maps, known as Nadler's fixed point theorem [2], Mizoguchi-Takahashi's fixed point theorem [3], Berinde-Berinde's fixed point theorem [5] and references therein. Another interesting direction of research led to extend to the multivalued maps setting previous fixed point results valid for single-valued maps with so-called directional contraction properties (see [2024]). In 1995, Song [22] established the following fixed point theorem for directional contractions which generalizes a fixed point result due to Clarke [20].

Theorem S[22]. Let L be a closed nonempty subset of X and T : L C ( X ) be a multivalued map. Suppose that
  1. (i)

    T is H-upper semicontinuous, that is, for every ε > 0 and every x L there exists r > 0 such that supyTx' d(y, Tx) <ε for every x' B(x, r);

     
  2. (ii)
    there exist α (0, 1] and γ [0, α) such that for every x L with x T x , there exists y L \ {x} satisfying
    α d ( x , y ) + d ( y , T x ) d ( x , T x )
     
and
sup z T x d ( z , T y ) γ d ( x , y ) .

Then ( T ) L .

Definition 1.1[23]. Let L be a nonempty subset of a metric space (X, d). A multivalued map T : L C ( X ) is called a directional multivalued k(·)-contraction if there exist λ (0, 1], a : (0, ∞) → [λ, 1] and k : (0, ∞) → [0, 1) such that for every x L with x T x , there is y L \ {x} satisfying the inequalities
a ( d ( x , y ) ) d ( x , y ) + d ( y , T x ) d ( x , T x )
and
sup z T x d ( z , T y ) k ( d ( x , y ) ) d ( x , y ) .

Subsequently Uderzo [23] generalized Song's result and some main results in [21] for directional multivalued k(·)-contractions.

Theorem U[23]. Let L be a closed nonempty subset of a metric space (X, d) and T : L C ( X ) be an u.s.c. directional multivalued k(·)-contraction. Assume that there exist x0 L and δ > 0 such that d(x0, Tx0) ≤ αδ and
sup t ( 0 , δ ] k ( t ) < inf t ( 0 , δ ] a ( t ) ,

where λ (0, 1], a and k are the constant and the functions occuring in the definition of directional multivalued k(·)-contraction. Then ( T ) L .

Recall that a function p : X × X → [0, ∞) is called a w-distance[1, 2530], if the following are satisfied:

(w 1) p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z X;

(w 2) for any x X, p(x, ·): X → [0, ∞) is l.s.c;

(w 3) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε.

A function p : X × X → [0, ∞) is said to be a τ-function[14, 26, 2830], first introduced and studied by Lin and Du, if the following conditions hold:

(τ 1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z X;

(τ 2) if x X and {y n } in X with limn→∞y n = y such that p(x, y n ) ≤ M for some M = M(x) > 0, then p(x, y) ≤ M;

(τ 3) for any sequence {x n } in X with limn→∞sup{p(x n , x m ): m >n} = 0, if there exists a sequence {y n } in X such that limn→∞p(x n , y n ) = 0, then limn→∞d(x n , y n ) = 0;

(τ 4) for x, y, z X, p(x, y) = 0 and p(x, z) = 0 imply y = z.

Note that not either of the implications p(x, y) = 0 x = y necessarily holds and p is nonsymmetric in general. It is well known that the metric d is a w-distance and any w-distance is a τ-function, but the converse is not true; see [26] for more detail.

The following result is simple, but it is very useful in this article.

Lemma 1.1. Let A be a nonempty subset of a metric space (X, d) and p : X × X → [0, ∞) be a function satisfying (τ 1). Then for any x X, p(x, A) ≤ p(x, z) + p(z, A) for all z X.

The following results are crucial in this article.

Lemma 1.2[14]. Let A be a closed subset of a metric space (X, d) and p : X × X → [0, ∞) be any function. Suppose that p satisfies (τ 3) and there exists u X such that p(u, u) = 0. Then p(u, A) = 0 if and only if u A, where p(u, A) = infaAp(u, a).

Lemma 1.3 [29, Lemma 2.1]. Let (X, d) be a metric space and p : X × X → [0, ∞) be a function. Assume that p satisfies the condition (τ 3). If a sequence {x n } in X with limn→∞sup{p(x n , x m ): m >n} = 0, then {x n } is a Cauchy sequence in X.

Recently, Du first introduced the concepts of τ0-functions and τ0-metrics as follows.

Definition 1.2[14]. Let (X, d) be a metric space. A function p : X × X → [0, ∞) is called a τ0-function if it is a τ-function on X with p(x, x) = 0 for all x X.

Remark 1.1. If p is a τ0-function, then, from (τ 4), p(x, y) = 0 if and only if x = y.

Example 1.1[14]. Let X = with the metric d(x, y) = |x - y| and 0 <a <b. Define the function p : X × X → [0, ∞) by
p ( x , y ) = max { a ( y - x ) , b ( x - y ) } .

Then p is nonsymmetric and hence p is not a metric. It is easy to see that p is a τ0-function.

Definition 1.3[14]. Let (X, d) be a metric space and p be a τ0-function. For any A , B C ( X ) , define a function D p : C ( X ) × C ( X ) [ 0 , ) by
D p ( A , B ) = max { δ p ( A , B ) , δ p ( B , A ) } ,

where δ p (A, B) = supxAp(x, B), then D p is said to be the τ0-metric on C ( X ) induced by p.

Clearly, any Hausdorff metric is a τ0-metric, but the reverse is not true. It is known that every τ0-metric D p is a metric on C ( X ) ; see [14] for more detail.

Let f be a real-valued function defined on . For c , we recall that
lim sup x c f ( x ) = inf ε > 0 sup 0 < | x - c | < ε f ( x )
and
lim sup x c + f ( x ) = inf ε > 0 sup 0 < x - c < ε f ( x ) .
Definition 1.4. A function α : [0, ∞) → [0, 1) is said to be a Reich's function ( -function, for short) if
lim sup s t + α ( s ) < 1 for all t [ 0 , ) .
(1.1)

Remark 1.2. In [1419, 30], a function α : [0, ∞) → [0, 1) satisfying the property (1.1) was called to be an T -function. But it is more appropriate to use the terminology -function instead of T -function since Professor S. Reich was the first to use the property (1.1).

It is obvious that if α : [0, ∞) → [0, 1) is a nondecreasing function or a nonincreasing function, then α is a -function. So the set of -functions is a rich class. It is easy to see that α : [0, ∞) → [0, 1) is a -function if and only if for each t [0, ∞), there exist r t [0, 1) and ε t > 0 such that α(s) ≤ r t for all s [t, t + ε t ); for more details of characterizations of -functions, one can see [19, Theorem 2.1].

In [14], the author established some new fixed point theorems for nonlinear multivalued contractive maps by using τ0-function, τ0-metrics and -functions. Applying those results, the author gave the generalizations of Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Nadler's fixed point theorem, Banach contraction principle, Kannan's fixed point theorems and Chatterjea's fixed point theorems for nonlinear multivalued contractive maps in complete metric spaces; for more details, we refer the reader to [14].

This study is around the following Reich's open question in [35] (see also [36]): Let (X, d) be a complete metric space and T : X C ( X ) be a multivalued map. Suppose that
( T x , T y ) φ ( d ( x , y ) ) d ( x , y ) for all x , y X ,

where ϕ : [0, ∞) → [0, 1) satisfies the property (*) except for t = 0. Does T have a fixed point? In this article, our some new results give partial answers of Reich's open question and generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem and some well-known results in the literature.

The article is divided into four sections. In Section 2, in order to carry on the development of metric fixed point theory, we first introduce the concept of directional hidden contractions in metric spaces. In Section 3, we present some new existence results concerning p-approximate fixed point property for various types of nonlinear contractive maps. Finally, in Section 4, we establish several new fixed point theorems for directional hidden contractions. From these results, new generalizations of Berinde-Berinde's fixed point theorem and Mizoguchi-Takahashi's fixed point theorem are also given.

2 Directional hidden contractions

Let (X, d) be a metric space and p : X × X → [0, ∞) be any function. For each x X and A X, let
p ( x , A ) = inf y A p ( x , y ) .
Recall that a multivalued map T : X N ( X ) is called
  1. (1)
    a Nadler's type contraction (or a multivalued k-contraction [3]), if there exists a number 0 <k < 1 such that
    ( T x , T y ) k d ( x , y ) for all x , y X .
     
  2. (2)
    a Mizoguchi-Takahashi's type contraction, if there exists a -function α : [0, ∞) → [0, 1) such that
    ( T x , T y ) α ( d ( x , y ) ) d ( x , y ) for all x , y X ;
     
  3. (3)
    a multivalued (θ, L)-almost contraction [57], if there exist two constants θ (0, 1) and L ≥ 0 such that
    ( T x , T y ) θ d ( x , y ) + L d ( y , T x ) for all x , y X .
     
  4. (4)
    a Berinde-Berinde's type contraction (or a generalized multivalued almost contraction [57]), if there exist a -function α : [0, ∞) → [0, 1) and L ≥ 0 such that
    ( T x , T y ) α ( d ( x , y ) ) d ( x , y ) + L d ( y , T x ) for all x , y X .
     

Mizoguchi-Takahashi's type contractions and Berinde-Berinde's type contractions are relevant topics in the recent investigations on metric fixed point theory for contractive maps. It is quite clear that any Mizoguchi-Takahashi's type contraction is a Berinde-Berinde's type contraction. The following example tell us that a Berinde-Berinde's type contraction may be not a Mizoguchi-Takahashi's type contraction in general.

Example 2.1. Let be the Banach space consisting of all bounded real sequences with supremum norm d and let {e n } be the canonical basis of . Let {τ n } be a sequence of positive real numbers satisfying τ1 = τ2 and τn+1<τ n for n ≥ 2 (for example, let τ 1 = 1 2 and τ n = 1 n for n with n ≥ 2). Thus {τ n } is convergent. Put v n = τ n e n for n and let X = {v n }nbe a bounded and complete subset of . Then (X, d) be a complete metric space and d(v n , v m ) = τ n if m >n.

Let T : L C ( X ) be defined by
T v n : = { v 1 , v 2 } , if n { 1 , 2 } , X \ { v 1 , v 2 , , v n , v n + 1 } , if n 3 .
and define φ : [0, ∞) → [0, 1) by
φ ( t ) : = τ n + 2 τ n , if t = τ n for some n , 0 , otherwise .
Then the following statements hold.
  1. (a)

    T is a Berinde-Berinde's type contraction;

     
  2. (b)

    T is not a Mizoguchi-Takahashi's type contraction.

     
Proof. Observe that lim sup s t + φ ( s ) = 0 < 1 for all t [0, ∞), so φ is a -function. It is not hard to verify that
( T v 1 , T v m ) = τ 1 > τ 3 = φ ( d ( v 1 , v m ) ) d ( v 1 , v m ) for all m 3 .
Hence T is not a Mizoguchi-Takahashi's type contraction. We claim that T is a Berinde-Berinde's type contraction with L ≥ 1; that is,
( T x , T y ) φ ( d ( x , y ) ) d ( x , y ) + L d ( y , T x ) for all x , y X ,
where is the Hausdorff metric induced by d. Indeed, we consider the following four possible cases:
  1. (i)

    φ ( d ( v 1 , v 2 ) ) d ( v 1 , v 2 ) + L d ( v 2 , T v 1 ) = τ 3 > 0 = ( T v 1 , T v 2 ) .

     
  2. (ii)
    For any m ≥ 3, we have
    φ ( d ( v 1 , v m ) ) d ( v 1 , v m ) + L d ( v m , T v 1 ) = τ 3 + L τ 2 > τ 1 = ( T v 1 , T v m ) .
     
  3. (iii)
    For any m ≥ 3, we obtain
    φ ( d ( v 2 , v m ) ) d ( v 2 , v m ) + L d ( v m , T v 2 ) = τ 4 + L τ 2 > τ 1 = ( T v 2 , T v m ) .
     
  4. (iv)
    For any n ≥ 3 and m >n, we get
    φ ( d ( v n , v m ) ) d ( v n , v m ) + L d ( v m , T v n ) = τ n + 2 = ( T v n , T v m ) .
     

Hence, by (i)-(iv), we prove that T is a Berinde-Berinde's type contraction with L ≥ 1.

In order to carry on such development of classic metric fixed point theory, we first introduce the concept of directional hidden contractions as follows. Using directional hidden contractions, we will present some new fixed point results and show that several already existent results could be improved.

Definition 2.1. Let L be a nonempty subset of a metric space (X, d), p : X × X → [0, ∞) be any function, c (0, 1), η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) be functions. A multivalued map T : L N ( X ) is called a directional hidden contraction with respect to p, c, η and ϕ ((p, c, η, ϕ)-DHC, for short) if for any x L with x T x , there exist y L \ {x} and z Tx such that
p ( z , T y ) ϕ ( p ( x , y ) ) p ( x , y )
and
η ( p ( x , y ) ) p ( x , y ) + p ( y , z ) p ( x , T x ) .

In particular, if pd, then we use the notation (c, η, ϕ)-DHC instead of (d, c, η, ϕ)-DHC.

Remark 2.1. We point out the fact that the concept of directional hidden contractions really generalizes the concept of directional multivalued k(·)-contractions. Indeed, let T be a directional multivalued k(·)-contraction. Then there exist λ (0, 1], a : (0, ∞) → [λ, 1] and k : (0, ∞) → [0, 1) such that for every x L with x T x , there is y L \ {x} satisfying the inequalities
a ( d ( x , y ) ) d ( x , y ) + d ( y , T x ) d ( x , T x )
(2.1)
and
sup z T x d ( z , T y ) k ( d ( x , y ) ) d ( x , y ) .
(2.2)
Note that xy and hence d(x, y) > 0. We consider the following two possible cases:
  1. (i)
    If λ = 1, then a(t) = 1 for all t (0, ∞). Choose c 1, r (0, 1) with c 1 <r. By (2.1), we have
    r d ( x , y ) + d ( y , T x ) < d ( x , T x ) ,
     
which it is thereby possible to find z r Tx such that
r d ( x , y ) + d ( y , z r ) < d ( x , T x ) .
Define η1 : [0, ∞) → (c1, 1] by
η 1 ( t ) = r
and let ϕ1 : [0, ∞) → [0, 1) be defined by
ϕ 1 ( t ) = 0 , if t = 0 , k ( t ) , if t ( 0 , ) .
Hence T is a (c1, η1, ϕ1)-DHC.
  1. (ii)
    If λ (0, 1), we choose c 2 satisfying 0 <c 2 <λ. Then
    c 2 < λ + c 2 2 a ( t ) + c 2 2 < a ( t ) 1 for all t ( 0 , ) .
     
So we can define η2 : [0, ∞) → (c2, 1] by
η 2 ( t ) = 0 , if t = 0 , a ( t ) + c 2 2 , if t ( 0 , ) .
Since η2(t) <a(t) for all t (0, ∞), the inequality (2.1) admits that there exists z Tx such that
η ( d ( x , y ) ) d ( x , y ) + d ( y , z ) < d ( x , T x ) .

Let ϕ2 = ϕ1. Therefore T is a (c2, η2, ϕ2)-DHC.

The following example show that the concept of directional hidden contractions is indeed a proper extension of classic contractive maps.

Example 2.2. Let X = [0, 1] with the metric d(x, y) = |x - y| for x, y X. Let T : X C ( X ) be defined by
T x = { 0 , 1 } , if x = 0 , { 1 2 x 4 , 1 } , if x ( 0 , 1 4 ] , { 0 , 1 2 x 4 } , if x ( 1 4 , 1 ) , { 1 } , if x = 1 .
Define η : [ 0 , ) ( 1 2 , 1 ] and ϕ : [0, ∞) → [0, 1) by
η ( s ) = 3 4 for all s [ 0 , )
and
ϕ ( t ) = 2 t , if t [ 0 , 1 2 ) , 0 , if t [ 1 2 , ) ,
respectively. It is not hard to verify that T is a ( 1 2 , η , ϕ )-DHC. Notice that
( T ( 0 ) , T ( 1 ) ) = 1 = d ( 0 , 1 ) ,

so T is not a Mizoguchi-Takahashi's type contraction (hence it is also not a Nadler's type contraction).

We now present some existence theorems for directional hidden contractions.

Theorem 2.1. Let (X, d) be a metric space, p be a τ0-function, T : X C ( X ) be a multivalued map and γ [0, ∞). Suppose that

( P ) there exists a function φ : (0, ∞) → [0, 1) such that
lim sup s γ + φ ( s ) < 1
and for each x X with x T x , it holds
p ( y , T y ) φ ( p ( x , y ) ) p ( x , y ) for all y T x .
(2.3)
Then there exist c (0, 1) and functions η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) such that
  1. (a)

    lim sup s γ + ϕ ( s ) < lim inf s γ + η ( s ) ;

     
  2. (b)

    T is a (p, c, η, ϕ)-DHC.

     
Proof. Set LX. Let ϕ : [0, ∞) → [0, 1) be defined by
ϕ ( s ) : = 0 , if s = 0 , φ ( s ) , if s ( 0 , ) .
By ( P ), there exists c (0, 1) such that
lim sup s γ + φ ( s ) < c < 1 .
Put α = c + 1 2 . Then 0 <c <α < 1. Define η : [0, ∞) → (c, 1] by η(s) = α for all s [0, ∞). So we obtain
lim sup s γ + ϕ ( s ) < α = lim inf s γ + η ( s ) .
Given x X with x T x . Since p is a τ0-function and Tx is a closed set in X, by Lemma 1.2, p(x, Tx) > 0. Since p ( x , T x ) < p ( x , T x ) α , there exists y Tx, such that
p ( x , y ) < p ( x , T x ) α .
(2.4)
Clearly, yx. Let z = y Tx. Since p is a τ0-function, we have p(y, z) = 0. From (2.3) and (2.4), we obtain
p ( z , T y ) ϕ ( p ( x , y ) ) p ( x , y )
and
η ( p ( x , y ) ) p ( x , y ) + p ( y , z ) p ( x , T x ) ,

which show that T is a (p, c, η, ϕ)-DHC.   □

If we put pd in Theorem 2.1, then we have the following result.

Theorem 2.2. Let (X, d) be a metric space, T : X C ( X ) be a multivalued map and γ [0, ∞). Suppose that

( P d ) there exists a function φ : (0, ∞) → [0, 1) such that
lim sup s γ + φ ( s ) < 1
and for each x X with x T x , it holds
d ( y , T y ) φ ( d ( x , y ) ) d ( x , y ) for all y T x .
Then there exist c (0, 1) and functions η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) such that
  1. (a)

    lim sup s γ + ϕ ( s ) < lim inf s γ + η ( s ) ;

     
  2. (b)

    T is a (c, η, ϕ)-DHC.

     

Theorem 2.3. Let (X, d) be a metric space, p be a τ0-function, D p be a τ0-metric on C ( X ) induced by p, T : X C ( X ) be a multivalued map, h : X × X → [0, ∞) be a function and γ [0, ∞). Suppose that

( A ) there exists a function φ : (0, ∞) → [0, 1) such that
lim sup s γ + φ ( s ) < 1
and
D p ( T x , T y ) φ ( p ( x , y ) ) p ( x , y ) + h ( x , y ) p ( y , T x ) for all x , y X with x y .
(2.5)
Then there exist c (0, 1) and functions η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) such that
  1. (a)

    lim sup s γ + ϕ ( s ) < lim inf s γ + η ( s ) ;

     
  2. (b)

    T is a (p, c, η, ϕ)-DHC.

     

Proof. Let x X with x T x and let y Tx be given. So xy. By Lemma 1.2, p(y, Tx) = 0. It is easy to see that (2.5) implies (2.3). Therefore the conclusion follows from Theorem 2.1.   □

Theorem 2.4. Let (X, d) be a metric space, T : X C ( X ) be a multivalued map, h : X × X → [0, ∞) be a function and γ [0, ∞). Suppose that

( A d ) there exists a function φ : (0, ∞) → [0, 1) such that
lim sup s γ + φ ( s ) < 1
and
( T x , T y ) φ ( d ( x , y ) ) d ( x , y ) + h ( x , y ) d ( y , T x ) for all x , y X with x y .
Then there exist c (0, 1) and functions η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) such that
  1. (a)

    lim sup s γ + ϕ ( s ) < lim inf s γ + η ( s ) ;

     
  2. (b)

    T is a (c, η, ϕ)-DHC.

     

The following result is immediate from Theorem 2.4.

Theorem 2.5. Let (X, d) be a metric space and T : X C ( X ) be a multivalued map. Assume that one of the following conditions holds.
  1. (1)

    T is a Berinde-Berinde's type contraction;

     
  2. (2)

    T is a multivalued (θ, L)-almost contraction;

     
  3. (3)

    T is a Mizoguchi-Takahashi's type contraction;

     
  4. (4)

    T is a Nadler's type contraction.

     

Then there exist c (0, 1) and functions η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) such that T is a (c, η, ϕ)-DHC.

3 Nonlinear conditions for p-approximate fixed point property

Let K be a nonempty subset of a metric space (X, d). Recall that a multivalued map T : K N ( X ) is said to have the approximate fixed point property[7] in K provided inf x K d ( x , T x ) = 0 . Clearly, ( T ) implies that T has the approximate fixed point property. A natural generalization of the approximate fixed point property is defined as follows.

Definition 3.1. Let K be a nonempty subset of a metric space (X, d) and p be a τ-function. A multivalued map T : K N ( X ) is said to have the p-approximate fixed point property in K provided inf x K p ( x , T x ) = 0 .

Lemma 3.1. Let φ : (0, ∞) → [0, 1) be a function and γ (0, ∞). If lim sup s γ + φ ( s ) < 1 , then for any strictly decreasing sequence {ξ n }nin (0, ∞) with lim n ξ n = γ , we have 0 sup n φ ( ξ n ) < 1 .

Proof. Since lim sup s γ + φ ( s ) < 1 , there exists ε > 0 such that
sup γ < s < γ + ε φ ( s ) < 1 .
By the denseness of , there exists α [0, 1) such that
sup γ < s < γ + ε φ ( s ) α < 1 .
Hence φ(s) ≤ α for all s (γ, γ + ε). Let {ξ n }nbe a strictly decreasing sequence in (0, ∞) with lim n ξ n = γ . Then
γ = lim n ξ n = inf n ξ n 0 .
(3.1)
Since {ξ n }nis strictly decreasing, it is obvious that ξ n >γ for all n . By (3.1), there exists , such that
γ < ξ n < γ + ε for all n with n .
Hence ϕ(ξ n ) ≤ α for all n. Let
ς : = max { φ ( ξ 1 ) , φ ( ξ 2 ) , , φ ( ξ - 1 ) , α } < 1 .

Then ϕ(ξ n ) ≤ ζ for all n and hence 0 sup n φ ( ξ n ) ς < 1 .   □

Theorem 3.1. Let (X, d) be a metric space, p be a τ0-function and T : X N ( X ) be a multivalued map. Suppose that

( ) there exists a function φ : (0, ∞) → [0, 1) satisfying Reich's condition; that is
lim sup s t + φ ( s ) < 1 for all t ( 0 , )
and for each x X with x T x , it holds
p ( y , T y ) φ ( p ( x , y ) ) p ( x , y ) for all y T x .
(3.2)
Then the following statements hold.
  1. (a)

    There exists a Cauchy sequence {x n }nin X such that

     
  2. (i)

    x n+1 Tx n for each n ;

     
  3. (ii)

    inf n p ( x n , x n + 1 ) = lim n p ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

     
  4. (b)

    inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 ; that is T have the p-approximate fixed point property and approximate fixed point property in X.

     
Proof. Let x1 X with x 1 T x 1 and x2 Tx1. Then x1x2. Since p is a τ0-function, p(x1, x2) > 0. By (3.2), we have
p ( x 2 , T x 2 ) φ ( p ( x 1 , x 2 ) ) p ( x 1 , x 2 ) .
(3.3)
If x2 Tx2, then x 2 ( T ) . Since
inf x X p ( x , T x ) p ( x 2 , T x 2 ) p ( x 2 , x 2 ) = 0
and
inf x X d ( x , T x ) d ( x 2 , x 2 ) = 0 ,
we have inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 . Let {z n } be a sequence defined by z n = x2 for all n . Then {z n } is Cauchy and (a) holds. Hence the proof is finished in this case. Suppose x 2 T x 2 . Define κ : (0, ∞) → [0, 1) by κ ( t ) = φ ( t ) + 1 2 . Then φ(t) <κ(t) and 0 <κ(t) < 1 for all t (0, ∞). By (3.3), there exists x3 Tx2 such that
p ( x 2 , x 3 ) < κ ( p ( x 1 , x 2 ) ) p ( x 1 , x 2 ) .
Since x2x3, p(x2, x3) > 0. By (3.2) again, we obtain
p ( x 3 , T x 3 ) φ ( p ( x 2 , x 3 ) ) p ( x 2 , x 3 ) .
If x3 Tx3, then, following a similar argument as above, we finish the proof. Otherwise, there exists x4 Tx3 such that
p ( x 3 , x 4 ) < κ ( p ( x 2 , x 3 ) ) p ( x 2 , x 3 ) .
By induction, we can obtain a sequence {x n } in X satisfying xn+1 Tx n , p(x n , xn+1) > 0 and
p ( x n + 1 , x n + 2 ) < κ ( p ( x n , x n + 1 ) ) p ( x n , x n + 1 ) for each n .
(3.4)
Since κ(t) < 1 for all t (0, ∞), the sequence {p(x n , xn+1)} is strictly decreasing in (0, ∞). Then
γ : = lim n p ( x n , x n + 1 ) = inf n p ( x n , x n + 1 ) 0 exists .
We claim that γ = 0. Assume to the contrary that γ > 0. By ( ), we have lim sup s γ + φ ( s ) < 1 . Applying Lemma 3.1,
0 sup n φ ( p ( x n , x n + 1 ) ) < 1 .
By exploiting the last inequality we obtain
0 < sup n κ ( p ( x n , x n + 1 ) ) = 1 2 1 + sup n φ ( p ( x n , x n + 1 ) ) < 1 .
Let λ : = sup n κ ( p ( x n , x n + 1 ) ) . So λ (0, 1). It follows from (3.4) that
p ( x n + 1 , x n + 2 ) < κ ( p ( x n , x n + 1 ) ) p ( x n , x n + 1 ) λ p ( x n , x n + 1 ) λ n p ( x 1 , x 2 ) for each n .
Taking the limit in the last inequality as n → ∞ yields lim n p ( x n , x n + 1 ) = 0 which leads to a contradiction. Thus it must be
γ = lim n p ( x n , x n + 1 ) = inf n p ( x n , x n + 1 ) = 0 .
Now, we show that {x n } is indeed a Cauchy sequence in X. Let α n = λ n - 1 1 - λ p ( x 1 , x 2 ) , n . For m, n with m >n, we obtain
p ( x n , x m ) j = n m - 1 p ( x j , x j + 1 ) < α n .
Since λ (0, 1), limn→∞α n = 0 and hence
lim n sup { p ( x n , x m ) : m > n } = 0 .
Applying Lemma 1.3, we show that {x n } is a Cauchy sequence in X. Hence lim n d ( x n , x n + 1 ) = 0 . Since inf n d ( x n , x n + 1 ) d ( x m , x m + 1 ) for all m and lim m d ( x m , x m + 1 ) = 0 , one also obtain
lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .
Since xn+1 Tx n for each n ,
inf x X p ( x , T x ) p ( x n , T x n ) p ( x n , x n + 1 )
(3.5)
and
inf x X d ( x , T x ) d ( x n , T x n ) d ( x n , x n + 1 )
(3.6)
for all n . Since lim n p ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = 0 , by (3.5) and (3.6), we get
inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 .

The proof is completed.   □

Theorem 3.2. Let (X, d) be a metric space and T : X N ( X ) be a multivalued map. Suppose that

( d ) there exists a function φ : (0, ∞) → [0, 1) satisfying Reich's condition and for each x X with x T x , it holds
d ( y , T y ) φ ( d ( x , y ) ) d ( x , y ) for all y T x .
Then the following statements hold.
  1. (a)

    There exists a Cauchy sequence {x n }nin X such that

     
  2. (i)

    x n+1 Tx n for each n ;

     
  3. (ii)

    inf n d ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = 0 .

     
  4. (b)

    T have the approximate fixed point property in X.

     

Remark 3.1. [23, Proposition 3.1] is a special case of Theorems 3.1 and 3.2.

Theorem 3.3. Let (X, d) be a metric space, p be a τ0-function, D p be a τ0-metric on C ( X ) induced by p ,T : X C ( X ) be a multivalued map and h : X × X → [0, ∞) be a function. Suppose that

( ) there exists a function φ : (0, ∞) → [0, 1) satisfying Reich's condition and
D p ( T x , T y ) φ ( p ( x , y ) ) p ( x , y ) + h ( x , y ) p ( y , T x ) for all x , y X with x y .
(3.7)
Then the following statements hold.
  1. (a)

    There exists a Cauchy sequence {x n }nin X such that

     
  2. (i)

    x n+1 Tx n for each n ;

     
  3. (ii)

    inf n p ( x n , x n + 1 ) = lim n p ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

     
  4. (b)

    T have the p-approximate fixed point property and approximate fixed point property in X.

     

Proof. Let x X with x T x and let y Tx be given. By Lemma 1.2, p(y, Tx) = 0 and hence (3.7) implies (3.2). Therefore the conclusion follows from Theorem 3.1.   □

Theorem 3.4. Let (X, d) be a metric space, T : X C ( X ) be a multivalued map and h : X × X → [0, ∞) be a function. Suppose that

( d ) there exists a function φ : (0, ∞) → [0, 1) satisfying Reich's condition and
( T x , T y ) φ ( d ( x , y ) ) d ( x , y ) + h ( x , y ) d ( y , T x ) for all x , y X with x y .
Then the following statements hold.
  1. (a)

    There exists a Cauchy sequence {x n }nin X such that

     
  2. (i)

    x n+1 Tx n for each n ;

     
  3. (ii)

    lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

     
  4. (b)

    T have the approximate fixed point property in X.

     
Theorem 3.5. Let (X, d) be a metric space and T : X C ( X ) be a multivalued map. Assume that one of the following conditions holds.
  1. (1)

    T is a Berinde-Berinde's type contraction;

     
  2. (2)

    T is a multivalued (θ, L)-almost contraction;

     
  3. (3)

    T is a Mizoguchi-Takahashi's type contraction;

     
  4. (4)

    T is a Nadler's type contraction.

     
Then the following statements hold.
  1. (a)

    There exists a Cauchy sequence {x n }nin X such that

     
  2. (i)

    x n+1 Tx n for each n ;

     
  3. (ii)

    lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

     
  4. (b)

    T have the approximate fixed point property in X.

     

Let Ω denote the class of functions μ : [0, ∞) → [0, ∞) satisfying

  • μ(0) = 0;

  • 0 <μ(t) ≤ t for all t > 0;

  • μ is l.s.c. from the right;

  • lim sup s 0 + s μ ( s ) < .

Examples of such functions are μ ( t ) = t t + 1 , μ ( t ) = ln ( 1 + t ) and μ(t) = ct, where c (0, 1), for all t ≥ 0.

Theorem 3.6. Let (X, d) be a metric space, p be a τ0-function and T : X N ( X ) be a multivalued map. Suppose that

(Δ) there exists μ Ω such that for each x X with x T x , it holds
p ( y , T y ) p ( x , y ) - μ ( p ( x , y ) ) for all y T x .
(3.8)
Then the following statements hold.
  1. (a)

    There exists a function α from [0, ∞) into [0, 1) such that α is a -function and p(y, Ty) ≤ α(p(x, y))p(x, y) for all y Tx.

     
  2. (b)

    There exists a Cauchy sequence {x n }nin X such that

     
  3. (i)

    x n+1 Tx n for each n ;

     
  4. (ii)

    inf n p ( x n , x n + 1 ) = lim n p ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

     
  5. (c)

    T have the p-approximate fixed point property and approximate fixed point property in X.

     
Proof. Set
α ( t ) = 1 - μ ( t ) t , t > 0 , 0 , t = 0 .
Since 0 <μ(t) ≤ t for all t > 0, we have α(t) [0, 1) for all t [0, ∞). Hence α is a function from [0, ∞) into [0, 1). Let x X with x T x be given. Since p is a τ0-function, p(x, y) > 0 for all y Tx. Hence (3.8) implies
p ( y , T y ) α ( p ( x , y ) ) p ( x , y ) for all y T x .
We claim that α is a -function. Indeed, by (Δ), the function t μ ( t ) t is l.s.c. from the right and hence
lim sup s t + α ( s ) = 1 - lim inf s t + μ ( s ) s < 1 - μ ( t ) t < 1 for all t > 0 .
On the other hand, since t μ ( t ) 1 for all t > 0 and lim sup s 0 + s μ ( s ) < , it follows that
lim sup s 0 + α ( s ) = 1 - lim inf s 0 + μ ( s ) s = 1 - 1 lim sup s 0 + s μ ( s ) < 1 .

So we prove lim sup s t + α ( s ) < 1 for all t [0, ∞) which say that α : [0, ∞) → [0, 1) is a -function and (a) is true. The conclusions (b) and (c) follows from Theorem 3.1.   □

Theorem 3.7. Let (X, d) be a metric space and T : X N ( X ) be a multivalued map. Suppose that

d ) there exists μ Ω such that for each x X with x T x , it holds
d ( y , T y ) d ( x , y ) - μ ( d ( x , y ) ) for all y T x .
Then the following statements hold.
  1. (a)

    There exists a function α from [0, ∞) into [0, 1) such that α is a -function and d(y, Ty) ≤ α(d(x, y))d(x, y) for all y Tx.

     
  2. (b)

    There exists a Cauchy sequence {x n }nin X such that

     
  3. (i)

    x n+1 Tx n for each n ;

     
  4. (ii)

    lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

     
  5. (c)

    T have the approximate fixed point property in X.

     

4 Some applications in fixed point theory

The following existence theorem is a τ-function variant of generalized Ekeland's variational principle.

Lemma 4.1. Let (X, d) be a complete metric space, f : X → (-∞, ∞] be a proper l.s.c. and bounded below function. Let p be a τ-function and ε > 0. Suppose that there exists u X such that p(u, ·) is l.s.c, f(u) < ∞ and p(u, u) = 0. Then there exists v X such that
  1. (a)

    εp(u, v) ≤ f(u) - f(v);

     
  2. (b)

    εp(v, x) >f(v) - f(x) for all x X with xv.

     
Proof. Let
Y = { x X : ε p ( u , x ) f ( u ) - f ( x ) } .
Clearly, u Y. By the completeness of X and the lower semicontinuity of f and p(u, ·), we know that (Y, d) is a nonempty complete metric space. Applying a generalization version of Ekeland's variational principle due to Lin and Du (see, for instance, [26, 28]), there exists v Y such that εp(v, x) >f(v) - f(x) for all x Y with xv. Hence (a) holds from v Y. For any x X \ Y, since
ε [ p ( u , v ) + p ( v , x ) ] ε p ( u , x ) > f ( u ) - f ( x ) ε p ( u , v ) + f ( v ) - f ( x ) ,

it follows that εp(v, x) >f(v) - f(x) for all x X \ Y. Therefore εp(v, x) >φ(f(v))(f(v) - f(x)) for all x X with xv. The proof is completed.   □

Theorem 4.1. Let L be a nonempty closed subset of a complete metric space (X, d), p be a τ0-function and T : L C ( X ) be a (p, c, η, ϕ)-DHC. Suppose that
  1. (i)
    there exist u L and δ > 0 such that p(u, ·) is l.s.c,
    p ( u , T u ) c δ ,
    (4.1)
     
and
sup t ( 0 , δ ) ( ϕ ( t ) - η ( t ) ) < 0 ,
(4.2)
  1. (ii)

    the function f : L → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c.

     

Then ( T ) L .

Proof. Since L is a nonempty closed subset in X, (L, d) is also a complete metric space. By (4.1), f(u) ≤ < ∞. From (4.2), there exists γ > 0 such that
sup t ( 0 , δ ) ( ϕ ( t ) - η ( t ) ) - γ .
(4.3)
Applying Lemma 4.1 for u and γ 2 , there exists v L, such that
γ 2 p ( u , v ) f ( u ) - f ( v ) ;
(4.4)
γ 2 p ( v , x ) > f ( v ) - f ( x ) for all x L with x v .
(4.5)
So f(v) ≤ f(u) from (4.4). We claim that v Tv, or equivalent, p(v, Tv) = 0. On the contrary, suppose that f(v) = p(v, Tv) > 0. Since T is a (p, c, η, ϕ)-DHC, there exists y v L \ {v} and z v Tv such that
p ( z v , T y v ) ϕ ( p ( v , y v ) ) p ( v , y v )
(4.6)
and
η ( p ( v , y v ) ) p ( v , y v ) + p ( y v , z v ) f ( v ) .
(4.7)
Since y v v, c <η(p(v, y v )) and f(v) ≤ f(u), by (4.1) and (4.7), we have
0 < p ( v , y v ) < c - 1 f ( v ) c - 1 f ( u ) δ .
(4.8)
Combining (4.3) and (4.8), we get
ϕ ( p ( v , y v ) ) - η ( p ( v , y v ) ) - γ .
(4.9)
By Lemma 1.1, (4.6), (4.7), and (4.9), one obtains
f ( y v ) = p ( y v , T y v ) p ( y v , z v ) + p ( z v , T y v ) f ( v ) + [ ϕ ( p ( v , y v ) ) - η ( p ( v , y v ) ) ] p ( v , y v ) f ( v ) - γ p ( v , y v ) .
On the other hand, since y v L \ {v}, it follows from (4.5) and the last inequality that
f ( v ) < f ( y v ) + γ 2 p ( v , y v ) f ( v ) + γ 2 - γ p ( v , y v ) = f ( v ) - γ 2 p ( v , y v ) < f ( v ) ,

which yields a contradiction. Hence it must be f(v) = p(v, Tv) = 0. Since Tv is closed, by Lemma 1.2, we get v Tv which means that v ( T ) L . The proof is completed.   □

Remark 4.1.
  1. (a)

    Let K be a nonempty subset of a metric space (X, d) and T : X C ( X ) be u.s.c. Then the function f : K → [0, ∞) defined by f(x) = d(x, Tx) is l.s.c. For more detail, one can see, e.g., [31, Lemma 3.1] and [32, Lemma 2].

     
  2. (b)

    [23, Theorem 2.1] is a special case of Theorem 4.1.

     
Theorem 4.2. Let L be a nonempty closed subset of a complete metric space (X, d) and T : L C ( X ) be a (c, η, ϕ)-DHC. Suppose that
  1. (i)
    there exist u L and δ > 0 such that
    d ( u , T u ) c δ ,
     
and
sup t ( 0 , δ ) ( ϕ ( t ) - η ( t ) ) < 0 ,
  1. (ii)

    the function f : L → [0, ∞) defined by f(x) = d(x, Tx) is l.s.c.

     

Then ( T ) L .

Theorem 4.3. Let L be a nonempty closed subset of a complete metric space (X, d) and p be a τ0-function. Let T : L C ( X ) be a (p, c, η, ϕ)-DHC satisfying
lim sup s 0 + ϕ ( s ) < lim inf s 0 + η ( s ) ,
(4.10)

and it has the p-approximate fixed point property in L. Suppose that there exists u X such that p(u, ·) is l.s.c. and the function f : L → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c, then ( T ) L .

Proof. First, we note that (4.10) implies that the existences of δ1 > 0 and δ2 > 0 such that
sup t ( 0 , δ 1 ) ϕ ( t ) < inf t ( 0 , δ 2 ) η ( t ) .
(4.11)
Let δ = min{δ1, δ2} > 0. Thus (4.11) implies
sup t ( 0 , δ ) ( ϕ ( t ) - η ( t ) ) sup t ( 0 , δ ) ϕ ( t ) - inf t ( 0 , δ ) η ( t ) sup t ( 0 , δ 1 ) ϕ ( t ) - inf t ( 0 , δ 2 ) η ( t ) < 0 .

Since T has the p-approximate fixed point property in L, we have inf x L p ( x , T x ) = 0 < c δ and hence there exists u L such that p(u, Tu) <. So all the hypotheses of Theorem 4.1 are fulfilled. It is therefore possible to apply Theorem 4.1 to get the thesis.   □

Theorem 4.4. Let L be a nonempty closed subset of a complete metric space (X, d). Let T : L C ( X ) be a (c, η, ϕ)-DHC satisfying
lim sup s 0 + ϕ ( s ) < lim inf s 0 + η ( s ) ,

and it has the approximate fixed point property in L. Suppose that the function f : L → [0, ∞) defined by f(x) = d(x, Tx) is l.s.c, then ( T ) L .

Theorem 4.5. Let (X, d) be a complete metric space and p be a τ0-function and T : X C ( X ) be a multivalued map. Suppose that

( V ) there exists a -function α : [0, ∞) → [0, 1) such that for each x X with x T x , it holds
p ( y , T y ) α ( p ( x , y ) ) p ( x , y ) for all y T x .

If there exists u X such that p(u, ·) is l.s.c. and the function f : X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c, then ( T ) .

Proof. First, we observe that the condition ( V ) implies the condition ( P ) as in Theorem 2.1. So we can apply Theorem 2.1 to know that there exist c (0, 1) and functions η : [0, ∞) → (c, 1] and ϕ : [0, ∞) → [0, 1) such that
  1. (a)

    lim sup s 0 + ϕ ( s ) < lim inf s 0 + η ( s ) ;

     
  2. (b)

    T is a (p, c, η, ϕ)-DHC.

     

On the other hand, the condition ( V ) also implies the condition ( ) as in Theorem 3.1. Hence T have the p-approximate fixed point property by using Theorem 3.1. Therefore the thesis follows from Theorem 4.3.   □

Theorem 4.6. Let (X, d) be a complete metric space and T : X C ( X ) be a multivalued map. Suppose that

( V d ) there exists a -function α : [0, ∞) → [0, 1) such that for each x X with x T x , it holds
d ( y , T y ) α ( d ( x , y ) ) d ( x , y ) for all y T x .

If the function f : X → [0, ∞) defined by f(x) = d(x, Tx) is l.s.c, then ( T ) .

Theorem 4.7. Let (X, d) be a complete metric space, p be a τ0-function, D p be a τ0-metric on C ( X ) induced by p , T : X C ( X ) be a multivalued map and h : X × X → [0, ∞) be a function. Suppose that

( W ) there exists a -function α : [0, ∞) → [0, 1) such that
D p ( T x , T y ) α ( p ( x , y ) ) p ( x , y ) + h ( x , y ) p ( y , T x ) for all x , y X .

If there exists u X such that p(u, ·) is l.s.c. and the function f : X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c, then ( T ) .

The following result is a generalization of Berinde-Berinde's fixed point theorem. It is worth observing that the following generalized Berinde-Berinde's fixed point theorem does not require the lower semicontinuity assumption on the function f(x) = d(x, Tx).

Theorem 4.8. Let (X, d) be a complete metric space, T : X C ( X ) be a multivalued map and g : X → [0, ∞) be a function. Suppose that there exists a -function α : [0, ∞) → [0, 1) such that
( T x , T y ) α ( d ( x , y ) ) d ( x , y ) + g ( y ) d ( y , T x ) for all x , y X .
(4.12)

Then ( T ) .

Proof. Observe that the condition (4.12) implies that for each x X with x T x , it holds
d ( y , T y ) φ ( d ( x , y ) ) d ( x , y ) for all y T x .

It is therefore possible to apply Theorem 3.2 to obtain a Cauchy sequence {x n }nin X satisfying

  • xn+1 Tx n , n ,

  • lim n d ( x n , x n + 1 ) = inf n d ( x n , x n + 1 ) = 0 .

By the completeness of X, there exists v X such that x n v as n → ∞. It follows from (4.12) again that
lim n d ( x n + 1 , T v ) lim n ( T x n , T v ) lim n { α ( d ( x n , v ) ) d ( x n , v ) + g ( v ) d ( v , x n + 1 ) } = 0 ,

which implies limn→∞d(x n , Tv) = 0. By the continuity of d(·, Tv) and x n v as n → ∞, d(v, Tv) = 0. By the closedness of Tv, we get v Tv or v ( T ) .   □

Remark 4.2.
  1. (a)

    Theorem 4.8 generalizes [7, Theorem 2.6], Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem and references therein.

     
  2. (b)

    In [7, Theorem 2.6], the authors shown that a generalized multivalued almost contraction T in a metric space (X, d) have ( T ) provided either (X, d) is compact and the function f(x) = d(x, Tx) is l.s.c. or T is closed and compact. But reviewing Theorem 4.8, we know that the conditions in [7, Theorem 2.6] are redundant.

     
Corollary 4.1. (M. Berinde and V. Berinde[5]). Let (X, d) be a complete metric space, T : X C ( X ) be a multivalued map and L ≥ 0. Suppose that there exists a -function α : [0, ∞) → [0, 1) such that
( T x , T y ) α ( d ( x , y ) ) d ( x , y ) + L d ( y , T x ) for all x , y X .

Then ( T )

Corollary 4.2[3]. Let (X, d) be a complete metric space and T : X N ( X ) be a multivalued map. Suppose that there exists a -function α : [0, ∞) → [0, 1) such that
( T x , T y ) α ( d ( x , y ) ) d ( x , y ) for all x , y X .

Then ( T ) .

Theorem 4.9. Let (X, d) be a complete metric space, p be a τ0-function and T : X C ( X ) be a multivalued map. Suppose that

(Δ) there exists μ Ω such that for each x X with x T x , it holds
p ( y , T y ) p ( x , y ) - μ ( p ( x , y ) ) for all y T x .

and further assume that there exists u X such that p(u, ·) is l.s.c. and the function f : X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c, then ( T ) .

Proof. The conclusion follows from Theorems 3.6 and 4.5.   □

Theorem 4.10. Let (X, d) be a complete metric space and T : X C ( X ) be a multivalued map. Suppose that

d ) there exists μ Ω such that for each x X with x T x , it holds
d ( y , T y ) d ( x , y ) - μ ( d ( x , y ) ) for all y T x .

and further assume that the function f : X → [0, ∞) defined by f(x) = d(x, Tx) is l.s.c, then ( T ) .

Declarations

Acknowledgements

The author would like to express his sincere thanks to the anonymous referee for their valuable comments and useful suggestions in improving the article. This research was supported partially by grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.

Authors’ Affiliations

(1)
Department of Mathematics, National Kaohsiung Normal University

References

  1. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000.Google Scholar
  2. Nadler SB Jr: Multi-valued contraction mappings. Volume 30. Pacific J Math; 1969:475–488.Google Scholar
  3. Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188. 10.1016/0022-247X(89)90214-XMATHMathSciNetView ArticleGoogle Scholar
  4. Daffer PZ, Kaneko H: Fixed points of generalized contractive multi-valued mappings. J Math Anal Appl 1995, 192: 655–666. 10.1006/jmaa.1995.1194MATHMathSciNetView ArticleGoogle Scholar
  5. Berinde M, Berinde V: On a general class of multi-valued weakly Picard mappings. J Math Anal Appl 2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016MATHMathSciNetView ArticleGoogle Scholar
  6. Hussain N, Cho YJ: Weak contraction, common fixed points and invariant approximations. J Inequal Appl 2009. 2009, Article ID 390634, 10Google Scholar
  7. Hussain N, Amini-Harandi A, Cho YJ: Approximate endpoints for set-valued contractions in metric spaces. Fixed Point Theory Appl 2010., 13: 2010, Article ID 614867,Google Scholar
  8. Kamran T: Multivalued f -weakly Picard mappings. Nonlinear Anal 2007, 67: 2289–2296. 10.1016/j.na.2006.09.010MATHMathSciNetView ArticleGoogle Scholar
  9. Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087MATHMathSciNetView ArticleGoogle Scholar
  10. Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl 2008, 341: 416–420. 10.1016/j.jmaa.2007.09.070MATHMathSciNetView ArticleGoogle Scholar
  11. Rezapour Sh, Hamlbarani R: Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings". J Math Anal Appl 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049MATHMathSciNetView ArticleGoogle Scholar
  12. Berinde V, Păcurar M: Fixed points and continuity of almost contractions. Fixed Point Theory 2008, 9: 23–34.MATHMathSciNetGoogle Scholar
  13. Du W-S: Fixed point theorems for generalized Hausdorff metrics. Int Math Forum 2008, 3: 1011–1022.MATHMathSciNetGoogle Scholar
  14. Du W-S: Some new results and generalizations in metric fixed point theory. Nonlinear Anal 2010, 73: 1439–1446. 10.1016/j.na.2010.05.007MATHMathSciNetView ArticleGoogle Scholar
  15. Du W-S: Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi's condition in quasiordered metric spaces. Fixed Point Theory and Applications 2010., 9: 2010, Article ID 876372,Google Scholar
  16. Du W-S: Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems. Fixed Point Theory and Applications 2010., 16: 2010, Article ID 190606,Google Scholar
  17. Du W-S: New cone fixed point theorems for nonlinear multivalued maps with their applications. Appl Math Lett 2011, 24: 172–178. 10.1016/j.aml.2010.08.040MATHMathSciNetView ArticleGoogle Scholar
  18. Du W-S, Zheng S-X: Nonlinear conditions for coincidence point and fixed point theorems. Taiwanese J Math 2012, 16(3):857–868.MATHMathSciNetGoogle Scholar
  19. Du W-S: On coincidence point and fixed point theorems for nonlinear multivalued maps. Topol Appl 2012, 159: 49–56. 10.1016/j.topol.2011.07.021MATHView ArticleGoogle Scholar
  20. Clarke FH: Pointwise contraction criteria for the existence of fixed points. Can Math Bull 1978, 21(1):7–11. 10.4153/CMB-1978-002-8MATHView ArticleGoogle Scholar
  21. Sehgal VM, Smithson RE: A fixed point theorem for weak directional contraction multifunction. Math Japon 1980, 25(3):345–348.MATHMathSciNetGoogle Scholar
  22. Song W: A generalization of Clarke's fixed point theorem. Appl Math J Chin Univ Ser B 1995, 10(4):463–466. 10.1007/BF02662502MATHView ArticleGoogle Scholar
  23. Uderzo A: Fixed points for directional multi-valued k (·)-contractions. J Global Optim 2005, 31: 455–469. 10.1007/s10898-004-0571-zMATHMathSciNetView ArticleGoogle Scholar
  24. Frigon M: Fixed point results for multivalued maps in metric spaces with generalized inwardness conditions. Fixed Point Theory Appl 2010., 19: 2010, Article ID 183217,Google Scholar
  25. Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math Japon 1996, 44: 381–391.MATHMathSciNetGoogle Scholar
  26. Lin L-J, Du W-S: Ekeland's variational principle, minimax theorems and existence of noncon-vex equilibria in complete metric spaces. J Math Anal Appl 2006, 323: 360–370. 10.1016/j.jmaa.2005.10.005MATHMathSciNetView ArticleGoogle Scholar
  27. Lin L-J, Du W-S: Some equivalent formulations of generalized Ekeland's variational principle and their applications. Nonlinear Anal 2007, 67: 187–199. 10.1016/j.na.2006.05.006MATHMathSciNetView ArticleGoogle Scholar
  28. Lin L-J, Du W-S: On maximal element theorems, variants of Ekeland's variational principle and their applications. Nonlinear Anal 2008, 68: 1246–1262. 10.1016/j.na.2006.12.018MATHMathSciNetView ArticleGoogle Scholar
  29. Du W-S: Critical point theorems for nonlinear dynamical systems and their applications. Fixed Point Theory Appl 2010., 16: 2010, Article ID 246382,Google Scholar
  30. He Z, Du W-S, Lin I-J: The existence of fixed points for new nonlinear multivalued maps and their applications. Fixed Point Theory and Applications 2011 2011, 84.Google Scholar
  31. Ding XP, He YR: Fixed point theorems for metrically weakly inward set-valued mappings. J Appl Anal 1999, 5(2):283–293.MATHMathSciNetView ArticleGoogle Scholar
  32. Downing D, Kirk WA: Fixed point theorems for set-valued mappings in metric and Banach spaces. Math Japon 1977, 22: 99–112.MATHMathSciNetGoogle Scholar
  33. Reich S: A fixed point theorem for locally contractive multivalued functions. Rev Roumaine Math Pures Appl 1972, 17: 569–572.MATHMathSciNetGoogle Scholar
  34. Reich S: Fixed points of contractive functions. Boll Un Mat Ital 1972, 5(4):26–42.MATHMathSciNetGoogle Scholar
  35. Reich S: Some problems and results in fixed point theory. Contemp Math 1983, 21: 179–187.MATHView ArticleGoogle Scholar
  36. Xu H-K: Metric fixed point theory for multivalued mappings. Dissertationes Mathematicae 2000, 389: 39.View ArticleGoogle Scholar

Copyright

© Du; licensee Springer. 2012

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.