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

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R E S E A R C H

Open Access

On generalized weakly directional contractions

and approximate fixed point property with

applications

Wei-Shih Du

Correspondence: wsdu@nknucc. nknu.edu.tw

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

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-function,τ0-metric, Reich’s condition, ℛ-function(ℳT-function), direc-tional hidden contraction, approximate fixed point property, generalized Mizoguchi-Takahashi’s fixed point theorem, generalized Berinde-Berinde’s fixed point theorem

1 Introduction and preliminaries

Let (X, d) be a metric space. The open ball centered in x Î X with radius r > 0 is denoted byB(x,r). For eachxÎ XandA⊆ X, letd(x,A) = infyÎA d(x,y). Denote by

N(X) the class of all nonempty subsets ofX, C(X) the family of all nonempty closed subsets ofXand CB(X) the family of all nonempty closed and bounded subsets ofX. A function H:CB(X)×CB(X)→[0,∞) defined by

H(A,B) = max

sup x∈B

d(x,A), sup x∈A

d(x,B)

is said to be the Hausdorff metric on CB(X) induced by the metricdonX. A point

vinX is a fixed point of a mapTif v=Tv (whenT: X ®X is a single-valued map) orvÎTv(when T:X→N(X) is a multivalued map). The set of fixed points of Tis denoted by F(T). Throughout this article, we denote by Nandℝ, 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 equa-tions and nonlinear differential equaequa-tions 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

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been investigated by several authors; see [1-36] and references therein. A interesting direction of research is the extension of the Banach contraction principle to multiva-lued 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 con-traction properties (see [20-24]). 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]. LetLbe a closed nonempty subset of X and T:L→CB(X) be a

multivalued map. Suppose that

(i) T is H-upper semicontinuous, that is, for everyε > 0 and every x Î Lthere existsr > 0 such that supyÎTx’d(y,Tx) <εfor everyx’ÎB(x,r);

(ii) there exist a Î (0, 1] and gÎ [0, a) such that for everyx Î L with x∈Tx, there existsy ÎL\ {x} satisfying

αd(x,y) +d(y,Tx)≤d(x,Tx)

and

sup z∈Tx

d(z,Ty)≤γd(x,y).

Then F(T)∩L=0.

Definition 1.1[23]. LetLbe a nonempty subset of a metric space (X,d). A

multiva-lued map T:L→CB(X) is called adirectional multivalued k(·)-contraction if there exist lÎ (0, 1],a: (0, ∞)® [l, 1] andk: (0, ∞)® [0, 1) such that for everyx ÎL

with x∈Tx, there isyÎL\ {x} satisfying the inequalities

a(d(x,y))d(x,y) +d(y,Tx)≤d(x,Tx)

and

sup z∈Tx

d(z,Ty)≤k(d(x,y))d(x,y).

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

Theorem U [23]. Let Lbe a closed nonempty subset of a metric space (X, d) and

T:L→CB(X) be an u.s.c. directional multivaluedk(·)-contraction. Assume that there existx0 ÎLandδ> 0 such thatd(x0,Tx0)≤aδand

sup t∈(0,δ]

k(t)< inf t∈(0,δ]a(t),

where lÎ(0, 1], aandkare the constant and the functions occuring in the defini-tion of direcdefini-tional multivalued k(·)-contraction. Then F(T)∩L=0.

Recall that a function p: X × X ® [0, ∞) is called a w-distance [1,25-30], if the following are satisfied:

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(w3) for anyε> 0, there existsδ> 0 such thatp(z,x)≤δandp(z,y)≤δimplyd(x,y)≤ε. A functionp:X×X®[0,∞) is said to be aτ-function[14,26,28-30], first introduced and studied by Lin and Du, if the following conditions hold:

(τ1)p(x,z)≤p(x,y) +p(y,z) for allx,y,zÎX;

(τ2) ifxÎ X and {yn} inXwith limn_®_∞yn =y such thatp(x, yn)≤M for someM =

M(x) > 0, thenp(x,y)≤M;

(τ3) for any sequence {xn} inXwith limn_®_∞sup{p(xn,xm):m>n} = 0, if there exists a sequence {yn} inX such that limn_®_∞p(xn,yn) = 0, then limn_®_∞d(xn,yn) = 0;

(τ4) for x,y,zÎX,p(x,y) = 0 andp(x,z) = 0 implyy=z.

Note that not either of the implicationsp(x,y) = 0⇔x=y necessarily holds andpis nonsymmetric in general. It is well known that the metric dis aw-distance and any

w-distance is aτ-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. LetAbe a nonempty subset of a metric space (X,d) andp: X ×X ®

[0,∞) be a function satisfying (τ1). Then for anyxÎ X, p(x, A)≤p(x,z) + p(z,A) for allzÎX.

The following results are crucial in this article.

Lemma 1.2[14]. LetAbe a closed subset of a metric space (X, d) andp:X ×X®

[0,∞) be any function. Suppose thatpsatisfies (τ3) and there existsuÎ X such that

p(u,u) = 0. Thenp(u,A) = 0if and only if u ÎA, wherep(u,A) = infa_ÎA p(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 psatisfies the condition (τ3). If a sequence {xn} inX with limn_®_∞sup{p(xn, xm):m>n} = 0, then {xn} is a Cauchy sequence inX.

Recently, Du first introduced the concepts ofτ0-functions andτ0-metrics as follows.

Definition 1.2 [14]. Let (X, d) be a metric space. A functionp: X×X ® [0,∞) is

called aτ0-function if it is aτ-function onXwithp(x, x) = 0 for allxÎX.

Remark 1.1. Ifpis aτ0-function, then, from (τ4),p(x,y) = 0 if and only ifx=y.

Example 1.1 [14]. LetX =ℝwith the metricd(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)}.

Thenpis nonsymmetric and hencepis not a metric. It is easy to see thatpis aτ0 -function.

Definition 1.3 [14]. Let (X, d) be a metric space and p be aτ0-function. For any

A,B∈CB(X), define a function Dp:CB(X)×CB(X)→[0,∞) by

Dp(A,B) = max{δp(A,B),δp(B,A)},

where δp(A, B) = supxÎA p(x, B), then Dp is said to be the τ0-metric on CB(X) induced byp.

Clearly, any Hausdorff metric is a τ0-metric, but the reverse is not true. It is known that everyτ0-metric Dp is a metric onCB(X); see [14] for more detail.

Let fbe a real-valued function defined onℝ. ForcÎℝ, we recall that

lim sup x→c

f(x) = inf

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and

lim sup x→c+

f(x) = inf

ε>0₀_<sup_x₋_c_<εf(x).

Definition 1.4. A function a : [0, ∞) ® [0, 1) is said to be a Reich’s function

(_R-function, for short) if

lim sup

s→t+ α(s)<1 for allt∈[0,∞). (1:1)

Remark 1.2. In [14-19,30], a functiona: [0, ∞)® [0, 1) satisfying the property (1.1)

was called to be an _MT-function. But it is more appropriate to use the terminology

R-function instead of _MT-function since Professor S. Reich was the first to use the property (1.1).

It is obvious that if a: [0,∞)®[0, 1) is a nondecreasing function or a nonincreas-ing function, then a is a R-function. So the set of R-functions is a rich class. It is easy to see that a: [0,∞)® [0, 1) is a R-functionif and only if for eachtÎ [0,∞), there exist rt Î [0, 1) and εt> 0 such that a(s)≤ rtfor all s Î [t, t +εt); for more details of characterizations of R-functions, one can see [19, Theorem 2.1].

In [14], the author established some new fixed point theorems for nonlinear multiva-lued contractive maps by using τ0-function, τ0-metrics and _R-functions. Applying those results, the author gave the generalizations of Berinde-Berinde’s fixed point theo-rem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, Banach contraction principle, Kannan’s fixed point theorems and Chatterjea’s fixed point theo-rems 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:L→CB(X) be a multivalued map. Suppose that

H(Tx,Ty)≤ϕ(d(x,y))d(x,y) for allx,y∈X,

where j: [0, ∞)® [0, 1) satisfies the property (*) except fort= 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 devel-opment 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 andp: X×X ®[0,∞) be any function. For eachxÎ X

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p(x,A) = inf y∈Ap(x,y).

Recall that a multivalued map T:X→N(X) is called

(1) aNadler’s type contraction(or amultivalued k-contraction[3]), if there exists a number 0 <k< 1 such that

H(Tx,Ty)≤kd(x,y) for allx,y∈X.

(2) aMizoguchi-Takahashi’s type contraction, if there exists aR-functiona: [0,∞)

®[0, 1) such that

H(Tx,Ty)≤α(d(x,y))d(x,y) for allx,y∈X;

(3) amultivalued(θ,L)-almost contraction[5-7], if there exist two constants θÎ (0, 1) andL≥0 such that

H(Tx,Ty)≤θd(x,y) +Ld(y,Tx) for allx,y∈X.

(4) aBerinde-Berinde’s type contraction(or ageneralized multivalued almost con-traction[5-7]), if there exist a R-functiona: [0, ∞)®[0, 1) and L≥0 such that

H(Tx,Ty)≤α(d(x,y))d(x,y) +Ld(y,Tx) for allx,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-Berinde’s type contraction may be not a Mizoguchi-Takahashi’s type contraction in general.

Example 2.1.Let ℓ∞ be the Banach space consisting of all bounded real sequences

with supremum norm d_∞ and let {en} be the canonical basis of ℓ∞. Let {τn} be a sequence of positive real numbers satisfyingτ1=τ2and τn+1<τnforn≥2 (for example,

let _τ1=

1

2 and τn= 1

n fornÎNwithn≥2). Thus {τn} is convergent. Putvn=τnenfor nÎ Nand letX = {vn}nÎNbe a bounded and complete subset of ℓ∞. Then (X,d∞) be a complete metric space and d∞(vn,vm) =τnifm>n.

Let T:L→CB(X) be defined by

Tvn:=

{v1,v2}, ifn∈ {1, 2},

X\{v1,v2,. . .,vn,vn+1}, ifn≥3.

and define : [0,∞)®[0, 1) by

ϕ(t) :=

_τ_n₊₂ τn

, ift=τnfor somen∈N,

0, otherwise.

Then the following statements hold.

(a)Tis a Berinde-Berinde’s type contraction;

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Proof. Observe that lim sup

s→t+ ϕ

(s) = 0<1 _{for all} _t_Î_[0, _∞_{), so} _{is a} _R_{-function. It}

is not hard to verify that

H∞(Tv1,Tvm) =τ1> τ3=ϕ(d∞(v1,vm))d∞(v1,vm) for allm≥3.

Hence Tis not a Mizoguchi-Takahashi’s type contraction. We claim that Tis a Ber-inde-Berinde’s type contraction withL≥1; that is,

H∞(Tx,Ty)≤ϕ(d∞(x,y))d∞(x,y) +Ld∞(y,Tx) for allx,y∈X,

where H_∞ is the Hausdorff metric induced byd∞. Indeed, we consider the following four possible cases:

(i) ϕ(d(v1,v2))d∞(v1,v2) +Ld∞(v2,Tv1) =τ3>0 =H∞(Tv1,Tv2). (ii) For anym≥3, we have

ϕ(d∞(v1,vm))d∞(v1,vm) +Ld∞(vm,Tv1) =τ3+Lτ2> τ1=H∞(Tv1,Tvm).

(iii) For anym≥3, we obtain

ϕ(d_∞(v2,vm))d∞(v2,vm) +Ld∞(vm,Tv2) =τ4+Lτ2> τ1=H∞(Tv2,Tvm).

(iv) For anyn≥3 andm>n, we get

ϕ(d_∞(vn,vm))d_∞(vn,vm) +Ld_∞(vm,Tvn) =τn+2=H∞(Tvn,Tvm).

Hence, by (i)-(iv), we prove that Tis a Berinde-Berinde’s type contraction withL≥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 sev-eral already existent results could be improved.

Definition 2.1.LetLbe a nonempty subset of a metric space (X,d),p:X×X®[0,∞)

be any function,cÎ(0, 1),h: [0,∞)®(c, 1] andj: [0,∞)®[0, 1) be functions. A multi-valued map T:X→N(X) is called adirectional hidden contractionwith respect top,c, handj((p,c,h,j)-DHC, for short) if for anyxÎLwith x∈Tx, there existyÎL\ {x} andzÎTxsuch that

p(z,Ty)≤φ(p(x,y))p(x,y)

and

η(p(x,y))p(x,y) +p(y,z)≤p(x,Tx).

In particular, ifp≡d, then we use the notation (c,h,j)-DHCinstead of (d, c,h,

j)-DHC.

Remark 2.1.We point out the fact that the concept of directional hidden contractions

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a(d(x,y))d(x,y) +d(y,Tx)≤d(x,Tx) (2:1)

and

sup z∈Tx

d(z,Ty)≤k(d(x,y))d(x,y). ₍₂_:₂₎

Note thatx≠y and henced(x,y) > 0. We consider the following two possible cases:

(i) Ifl= 1, thena(t) = 1 for alltÎ(0,∞). Choosec1,rÎ(0, 1) withc1 <r. By (2.1),

we have

rd(x,y) +d(y,Tx)<d(x,Tx),

which it is thereby possible to findzrÎ Txsuch that

rd(x,y) +d(y,zr)<d(x,Tx).

Defineh1: [0, ∞)®(c1, 1] by

η1(t) =r

and let j1 : [0,∞)® [0, 1) be defined by

φ1(t) =

0, ift= 0, k(t), ift∈(0,∞).

HenceT is a (c1, h1,j1)-DHC.

(ii) IflÎ(0, 1), we choosec2satisfying 0 <c2<l. Then

c2< λ

+c2

2 ≤

a(t) +c2

2 <a(t)≤1 for allt∈(0,∞).

So we can defineh2: [0,∞)®(c2, 1] by

η2(t) = ⎧ ⎨ ⎩

0, ift= 0, a(t) +c2

2 , ift∈(0,∞).

Since h2(t) <a(t) for alltÎ(0,∞), the inequality (2.1) admits that there existszÎTx

such that

η(d(x,y))d(x,y) +d(y,z)<d(x,Tx).

Let j2=j1. ThereforeTis a (c2,h2,j2)-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 Î X. Let

T:X→C(X) be defined by

Tx=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

{0, 1}, ifx= 0,

{1 2x

4_{, 1}_}_{, if}_x_∈_(0,1 4], {0,1

2x

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Define η: [0,∞)→(1

2, 1] andj: [0,∞)®[0, 1) by

η(s) = 3

4 for alls∈[0,∞)

and

φ(t) =

2t, ift∈[0,1 2),

0, ift∈[1 2,∞),

respectively. It is not hard to verify thatT is a (1

2,η,φ)-DHC. Notice that

H(T(0),T(1)) = 1 =d(0, 1),

so Tis 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,pbe a τ0-function, T:X→C(X) be a

multivalued map andgÎ[0,∞). Suppose that

(P) there exists a function: (0, ∞)®[0, 1) such that

lim sup

s→γ+ ϕ(s)<1

and for each xÎ Xwith x∈Tx, it holds

p(y,Ty)≤ϕ(p(x,y))p(x,y) for ally∈Tx. (2:3)

Then there exist cÎ (0, 1) and functionsh: [0,∞)® (c, 1] andj: [0,∞)® [0, 1) such that

(a) lim sup

s→γ+ φ(s)<lim infs→γ+ η(s);

(b)Tis a (p,c,h,j)-DHC.

Proof. SetL≡X. Letj: [0,∞)®[0, 1) be defined by

φ(s) :=

0, ifs= 0,

ϕ(s), ifs∈(0,∞).

By (P), there exists cÎ (0, 1) such that

lim sup

s→γ+ ϕ(s)<c<1.

Put _α= c+ 1

2 . Then 0 <c<a< 1. Defineh: [0,∞)® (c, 1] byh(s) =afor allsÎ[0, ∞). So we obtain

lim sup

s→γ+ φ(s)< α= lim infs→γ+ η(s).

Given xÎ X with x∈Tx. Since pis a τ0-function and Txis a closed set in X, by

Lemma 1.2,p(x,Tx) > 0. Since p(x,Tx)< p(x,Tx)

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p(x,y)< p(x,Tx)

α . (2:4)

Clearly, y ≠x. Letz =y Î Tx. Sincepis a τ0-function, we have p(y, z) = 0. From (2.3) and (2.4), we obtain

p(z,Ty)≤φ(p(x,y))p(x,y)

and

η(p(x,y))p(x,y) +p(y,z)≤p(x,Tx),

which show thatTis a (p,c,h,j)-DHC. □

If we putp≡din 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

gÎ [0,∞). Suppose that

(Pd) there exists a function: (0,∞)® [0, 1) such that

lim sup

s→γ+ ϕ(s)<1

d(y,Ty)≤ϕ(d(x,y))d(x,y) for ally∈Tx.

(a) lim sup

s→γ+ φ

(s)<lim inf s→γ+ η(s);

(b)Tis a (c, h,j)-DHC.

Theorem 2.3. Let (X,d) be a metric space, pbe aτ0-function, Dp be aτ0-metric on

CB(X) induced byp, T:X→CB(X) be a multivalued map, h: X×X ®[0,∞) be a function and gÎ[0,∞). Suppose that

(A) there exists a function: (0,∞)® [0, 1) such that

lim sup s→γ+ ϕ

(s)<1

and

Dp(Tx,Ty)≤ϕ(p(x,y))p(x,y) +h(x,y)p(y,Tx) for allx,y∈Xwithx=y. (2:5)

(a) lim sup

s→γ+ φ(s)<lim infs→γ+ η(s);

Proof.LetxÎ Xwith x∈Tx and lety ÎTxbe given. So x≠y. By Lemma 1.2,p(y,

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Theorem 2.4. Let (X,d) be a metric space, T:X→CB(X) be a multivalued map,

h:X ×X® [0,∞) be a function andgÎ[0,∞). Suppose that (Ad) there exists a function: (0, ∞)®[0, 1) such that

lim sup

s→γ+ ϕ(s)<1

and

H(Tx,Ty)≤ϕ(d(x,y))d(x,y) +h(x,y)d(y,Tx) for allx,y∈Xwithx=y.

(a) lim sup

s→γ+ φ

(s)<lim inf s→γ+ η(s);

(b)Tis a (c, h,j)-DHC.

The following result is immediate from Theorem 2.4.

Theorem 2.5. Let (X, d) be a metric space and T:X→CB(X) be a multivalued

map. Assume that one of the following conditions holds.

(1)Tis a Berinde-Berinde’s type contraction; (2)Tis a multivalued (θ,L)-almost contraction; (3)Tis a Mizoguchi-Takahashi’s type contraction;

(4)Tis a Nadler’s type contraction.

Then there exist cÎ (0, 1) and functionsh: [0,∞)® (c, 1] andj: [0,∞)® [0, 1) such thatTis a (c,h,j)-DHC.

3 Nonlinear conditions forp-approximate fixed point property

Let Kbe a nonempty subset of a metric space (X, d). Recall that a multivalued map

T:K →N(X) is said to have theapproximate fixed point property[7] in Kprovided

inf

x∈Kd(x,Tx) = 0. Clearly, F(T)=0 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 Kbe a nonempty subset of a metric space (X, d) and pbe a

τ-function. A multivalued map T:K→N(X) is said to have the p-approximate fixed point property inKprovided inf_x_∈_Kp(x,Tx) = 0.

Lemma 3.1.Let: (0,∞)®[0, 1) be a function andgÎ (0,∞). If lim sup

s→γ+ ϕ (s)<1_,

then for any strictly decreasing sequence {ξn}n_Î_N in (0,∞) with _nlim_→∞ξn=γ, we have

0≤sup n∈N

ϕ(ξn)<1_.

Proof. Since lim sup_s_→_γ+ ϕ(s)<1, there existsε> 0 such that

sup

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By the denseness of ℝ, there existsaÎ [0, 1) such that

sup

γ <s<γ+εϕ(s)≤α <1.

Hence (s)≤afor alls Î(g,g+ε). Let {ξn}n_Î_N be a strictly decreasing sequence in (0, ∞) with _nlim_→∞ξn=γ. Then

γ = lim

n→∞ξn= infn∈Nξn≥0. (3:1)

Since {ξn}n_Î_N is strictly decreasing, it is obvious that ξn>gfor all nÎ N. By (3.1), there existsℓÎ N, such that

γ < ξn< γ +ε for alln∈Nwithn≥.

Hencej(ξn)≤afor alln≥ℓ. Let

ς:= max{ϕ(ξ1),ϕ(ξ2),. . .,ϕ(ξ−1),α}<1.

Then j(ξn)≤ζfor allnÎNand hence 0≤sup

n∈N

ϕ(ξn)≤ς <1_. _□

Theorem 3.1.Let (X,d) be a metric space,pbe aτ0-function and T:X→N(X) be

a multivalued map. Suppose that

(R) there exists a function: (0, ∞)®[0, 1) satisfying Reich’s condition; that is

lim sup

s→t+ ϕ(s)<1 for allt∈(0,∞)

p(y,Ty)≤ϕ(p(x,y))p(x,y) for ally∈Tx. (3:2)

(a) There exists a Cauchy sequence {xn}n_Î_NinXsuch that (i)xn+1ÎTxnfor eachnÎ N;

(ii) _ninf_∈_Np(xn,xn+1) = lim

n→∞p(xn,xn+1) = limn→∞d(xn,xn+1) = infn∈Nd(xn,xn+1) = 0.

(b) inf

x∈Xp(x,Tx) = infx∈Xd(x,Tx) = 0; that is T have the p-approximate fixed point

property and approximate fixed point property inX.

Proof. Letx1ÎX withx1∈Tx1 andx2ÎTx1. Thenx1≠x2. Sincepis aτ0-function,

p(x1,x2) > 0. By (3.2), we have

p(x2,Tx2)≤ϕ(p(x1,x2))p(x1,x2). (3:3)

Ifx2 ÎTx2, then x2∈F(T). Since

inf

x∈Xp(x,Tx)≤p(x2,Tx2)≤p(x2,x2) = 0

and

inf

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we have inf_x_∈_Xp(x,Tx) = inf_x_∈_Xd(x,Tx) = 0. Let {zn} be a sequence defined by zn =x2 for

all nÎ N. Then {zn} is Cauchy and (a) holds. Hence the proof is finished in this case.

Suppose x2∈Tx2. Define : (0,∞)®[0, 1) by κ(t) = ϕ(t) + 1

2 . Then(t) <(t) and 0

<(t) < 1 for alltÎ(0,∞). By (3.3), there existsx3Î Tx2such that

p(x2,x3)< κ(p(x1,x2))p(x1,x2).

Since x2≠x3,p(x2,x3) > 0. By (3.2) again, we obtain

p(x3,Tx3)≤ϕ(p(x2,x3))p(x2,x3).

If x3Î Tx3, then, following a similar argument as above, we finish the proof.

Other-wise, there existsx4 ÎTx3 such that

p(x3,x4)< κ(p(x2,x3))p(x2,x3).

By induction, we can obtain a sequence {xn} inX satisfyingxn+1Î Txn,p(xn,xn+1) >

0 and

p(xn+1,xn+2)< κ(p(xn,xn+1))p(xn,xn+1) for eachn∈N. (3:4)

Since(t) < 1 for alltÎ(0,∞), the sequence {p(xn,xn+1)} is strictly decreasing in (0,∞).

Then

γ := lim

n→∞p(xn,xn+1) = infn∈Np(xn,xn+1)≥0 exists.

We claim that g = 0. Assume to the contrary that g > 0. By (R), we have

lim sup s→γ+ ϕ

(s)<1_{. Applying Lemma 3.1,}

0≤sup n∈N

ϕ(p(xn,xn+1))<1.

By exploiting the last inequality we obtain

0<sup n∈Nκ

(p(xn,xn+1)) =

1

2 1 + supn∈Nϕ

(p(xn,xn+1))

<1.

Let λ:= sup

n∈Nκ(p(xn,xn+1)). SolÎ(0, 1). It follows from (3.4) that

p(xn+1,xn+2)< κ(p(xn,xn+1))p(xn,xn+1) ≤λp(xn,xn+1)

≤ · · · ≤λn_p(x

1,x2) for eachn∈N.

Taking the limit in the last inequality as n ® ∞ yields _nlim_→∞p(xn,xn+1) = 0 which

leads to a contradiction. Thus it must be

γ = lim

n→∞p(xn,xn+1) = infn∈Np(xn,xn+1) = 0.

Now, we show that {xn} is indeed a Cauchy sequence in X. Let

αn= λ n−1

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p(xn,xm)≤ m−1

j=n

p(xj,xj+1)< αn.

Since lÎ(0, 1), limn_®_∞an= 0 and hence

lim

n→∞sup{p(xn,xm) :m>n}= 0.

Applying Lemma 1.3, we show that {xn} is a Cauchy sequence in X. Hence

lim

n→∞d(xn,xn+1) = 0. Since ninf∈Nd(xn,xn+1)≤d(xm,xm+1) for all m Î N and lim

m→∞d(xm,xm+1) = 0, one also obtain

lim

n→∞d(xn,xn+1) = infn∈Nd(xn,xn+1) = 0.

Since xn+1ÎTxnfor eachnÎN,

inf

x∈Xp(x,Tx)≤p(xn,Txn)≤p(xn,xn+1) (3:5)

and

inf

x∈Xd(x,Tx)≤d(xn,Txn)≤d(xn,xn+1) (3:6)

for all nÎN. Since _nlim_→∞p(xn,xn+1) = lim

n→∞d(xn,xn+1) = 0, by (3.5) and (3.6), we get

inf

x∈Xp(x,Tx) = infx∈Xd(x,Tx) = 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

(Rd) there exists a function: (0,∞)® [0, 1) satisfying Reich’s condition and for eachxÎXwith x∈Tx, it holds

d(y,Ty)≤ϕ(d(x,y))d(x,y) for ally∈Tx.

(ii) inf

n∈Nd(xn,xn+1) = limn→∞d(xn,xn+1) = 0.

(b)Thave the approximate fixed point property inX.

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, pbe aτ0-function, Dp be aτ0-metric on

CB(X) induced by pT:X→CB(X) be a multivalued map andh: X×X® [0,∞) be a function. Suppose that

(L) there exists a function: (0, ∞)®[0, 1) satisfying Reich’s condition and

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(a) There exists a Cauchy sequence {xn}nÎNinXsuch that (i)xn+1ÎTxnfor eachnÎ N;

(b)T have the p-approximate fixed point property and approximate fixed point property inX.

Proof. Letx ÎX with x∈Tx and letyÎ 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→CB(X) be a multivalued map

and h: X×X ®[0,∞) be a function. Suppose that

(Ld) there exists a function: (0,∞)® [0, 1) satisfying Reich’s condition and

H(Tx,Ty)≤ϕ(d(x,y))d(x,y) +h(x,y)d(y,Tx) for allx,y∈Xwithx=y.

(a) There exists a Cauchy sequence {xn}nÎNinXsuch that (i)xn+1ÎTxnfor eachnÎ N;

(ii) _nlim_→∞d(xn,xn+1) = inf

n∈Nd(xn,xn+1) = 0.

Theorem 3.5. Let (X, d) be a metric space and T:X→CB(X) be a multivalued

map. Assume that one of the following conditions holds.

(1)Tis a Berinde-Berinde’s type contraction;

(2)Tis a multivalued (θ,L)-almost contraction; (3)Tis a Mizoguchi-Takahashi’s type contraction;

(4)Tis a Nadler’s type contraction.

(ii) _nlim_→∞d(xn,xn+1) = inf

n∈Nd(xn,xn+1) = 0.

Let Ωdenote the class of functionsμ: [0, ∞)®[0,∞) satisfying

•μ(0) = 0;

•0 <μ(t)≤tfor all t> 0; •μis l.s.c. from the right; • lim sup

s→0+ s

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Examples of such functions are μ(t) = t

t+ 1,μ(t) = ln(1 +t) andμ(t) =ct, where c Î (0, 1), for allt≥0.

Theorem 3.6.Let (X,d) be a metric space,pbe aτ0-function and T:X→C(X) be

a multivalued map. Suppose that

(Δ) there existsμÎΩsuch that for eachxÎ Xwith x∈Tx, it holds

p(y,Ty)≤p(x,y)−μ(p(x,y)) for ally∈Tx. (3:8)

(a) There exists a functiona from [0, ∞) into [0, 1) such thatais a R-function andp(y,Ty)≤a(p(x,y))p(x,y) for ally ÎTx.

(b) There exists a Cauchy sequence {xn}nÎNinX such that (i)xn+1ÎTxnfor eachnÎ N;

(c)T have the p-approximate fixed point property and approximate fixed point property inX.

Proof. Set

α(t) =

⎧ ⎨ ⎩1−

μ(t) t ,t>0, 0, t= 0.

Since 0 <μ(t)≤tfor allt> 0, we have a(t)Î [0, 1) for alltÎ [0,∞). Hencea is a function from [0, ∞) into [0, 1). LetxÎ Xwith x∈Tx be given. Sincepis aτ0 -func-tion, p(x,y) > 0 for allyÎTx. Hence (3.8) implies

p(y,Ty)≤ϕ(p(x,y))p(x,y) for ally∈Tx.

We claim that a is a R-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 infs→t+

μ(s) s <1−

μ(t)

t <1 for allt>0.

On the other hand, since t

μ(t) ≥1 for allt > 0 and lim sups→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 alltÎ[0,∞) which say thata: [0,∞)®[0, 1) is a R-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→C(X) be a multivalued map.

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(Δd) there existsμÎΩsuch that for eachxÎXwith x∈Tx, it holds

d(y,Ty)≤d(x,y)−μ(d(x,y)) for ally∈Tx.

(a) There exists a functiona from [0, ∞) into [0, 1) such thatais a R-function andd(y,Ty)≤a(d(x,y))d(x,y) for ally ÎTx.

(b) There exists a Cauchy sequence {xn}nÎNinX such that (i)xn+1ÎTxnfor eachnÎ N;

(ii) _nlim_→∞d(xn,xn+1) = inf

n∈Nd(xn,xn+1) = 0.

(c)Thave the approximate fixed point property inX.

4 Some applications in fixed point theory

The following existence theorem is aτ-function variant of generalized Ekeland’s varia-tional principle.

Lemma 4.1.Let (X,d) be a complete metric space,f:X®(-∞,∞] be a proper l.s.c. and

bounded below function. Letpbe aτ-function andε> 0. Suppose that there existsuÎX

such thatp(u, ·) is l.s.c,f(u) <∞andp(u,u) = 0. Then there existsvÎXsuch that

(a)εp(u,v)≤f(u) -f(v);

(b)εp(v, x) >f(v) -f(x) for allxÎ Xwithx≠v.

Proof. Let

Y ={x∈X:εp(u,x)≤f(u)−f(x)}.

Clearly,uÎY. By the completeness ofXand the lower semicontinuity offandp(u, ·), we know that (Y,d) is a nonempty complete metric space. Applying a generalization ver-sion of Ekeland’s variational principle due to Lin and Du (see, for instance, [26,28]), there existsvÎYsuch thatεp(v,x) >f(v) -f(x) for allxÎYwithx≠v. Hence (a) holds fromv ÎY. For anyxÎX\Y, since

ε[p(u,v) +p(v,x)]≥εp(u,x)

>f(u)−f(x)

≥εp(u,v) +f(v)−f(x),

it follows that εp(v,x) >f(v) -f(x) for allxÎX \Y. Thereforeεp(v,x) >(f(v))(f(v) - f

(x)) for allxÎXwithx≠v. The proof is completed. □

Theorem 4.1.LetLbe a nonempty closed subset of a complete metric space (X,d),

pbe aτ0-function and T:L→C(X) be a (p,c,h,j)-DHC. Suppose that

(i) there existuÎLandδ> 0 such thatp(u, ·) is l.s.c,

p(u,Tu)≤cδ, (4:1)

and

sup t∈(0,δ)

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(ii) the functionf:L®[0,∞) defined byf(x) =p(x,Tx) is l.s.c.

Then F(T)∩L=0.

Proof. Since Lis a nonempty closed subset in X, (L, d) is also a complete metric

space. By (4.1),f(u)≤cδ<∞. From (4.2), there existsg> 0 such that

sup t∈(0,δ)

(φ(t)−η(t))≤ −γ. ₍₄_:₃₎

Applying Lemma 4.1 foruand γ

2, there existsvÎL, such that

γ

2p(u,v)≤f(u)−f(v); (4:4)

γ

2p(v,x)>f(v)−f(x) for allx∈Lwithx=v. (4:5)

So f(v)≤f(u) from (4.4). We claim thatvÎ Tv, or equivalent,p(v, Tv) = 0. On the contrary, suppose that f(v) = p(v, Tv) > 0. Since T is a (p, c, h,j)-DHC, there exists

yvÎ L\ {v} andzvÎ Tvsuch that

p(zv,Tyv)≤φ(p(v,yv))p(v,yv) (4:6)

and

η(p(v,yv))p(v,yv) +p(yv,zv)≤f(v). (4:7)

Since yv≠v,c<h(p(v,yv)) andf(v)≤f(u), by (4.1) and (4.7), we have

0<p(v,yv)<c−1f(v)≤c−1f(u)≤δ. (4:8)

Combining (4.3) and (4.8), we get

φ(p(v,yv))−η(p(v,yv))≤ −γ. (4:9)

By Lemma 1.1, (4.6), (4.7), and (4.9), one obtains

f(yv) =p(yv,Tyv)≤p(yv,zv) +p(zv,Tyv)

≤f(v) + [φ(p(v,yv))−η(p(v,yv))]p(v,yv)

≤f(v)−γp(v,yv).

On the other hand, since yv Î L\ {v}, it follows from (4.5) and the last inequality that

f(v)<f(yv) +γ 2p(v,yv)

≤f(v) +

_γ

2 −γ

p(v,yv)

=f(v)− γ 2p(v,yv)

<f(v),

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Remark 4.1.

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

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

Theorem 4.2. LetLbe a nonempty closed subset of a complete metric space (X, d)

and T:L→C(X) be a (c,h,j)-DHC. Suppose that

(i) there existuÎLandδ> 0 such that

d(u,Tu)≥cδ,

and

sup t∈(0,δ)

(φ(t)−η(t))<0,

(ii) the functionf:L®[0,∞) defined byf(x) =d(x,Tx) is l.s.c.

Then F(T)∩L=0.

Theorem 4.3. LetLbe a nonempty closed subset of a complete metric space (X, d)

and pbe aτ0-function. Let T:L→C(X) be a (p,c,h,j)-DHCsatisfying

lim sup s→0+ φ

(s)<lim inf

s→0+ η(s), (4:10)

and it has thep-approximate fixed point property inL. Suppose that there existsuÎ X such thatp(u, ·) is l.s.c. and the functionf:L®[0,∞) defined byf(x) =p(x,Tx) is l. s.c, then F(T)∩L=0.

Proof. First, we note that (4.10) implies that the existences ofδ1> 0 andδ2> 0 such

that

sup t∈(0,δ1)

φ(t)< inf t∈(0,δ2)

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)≤_t_∈sup_(0,_δ₁₎φ(t)−t∈inf(0,δ2)

η(t)<0.

Since T has the p-approximate fixed point property in L, we have

inf

x∈Lp(x,Tx) = 0<cδ and hence there existsu Î Lsuch thatp(u,Tu) <cδ. 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.LetLbe a nonempty closed subset of a complete metric space (X,d).

Let T:L→C(X) be a (c,h,j)-DHCsatisfying

lim sup s→0+ φ

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and it has the approximate fixed point property in L. Suppose that the functionf:

L ®[0,∞) defined by f(x) = d(x,Tx) is l.s.c, then F(T)∩L=0.

Theorem 4.5. Let (X, d) be a complete metric space and pbe a τ0-function and

T:X→C(X) be a multivalued map. Suppose that

(V) there exists a R-function a : [0,∞)® [0, 1) such that for each xÎ X with

x∈Tx, it holds

p(y,Ty)≤α(p(x,y))p(x,y) for ally∈Tx.

If there existsuÎ Xsuch that p(u, ·) is l.s.c. and the functionf:X ®[0, ∞) defined by f(x) =p(x,Tx) is l.s.c, then F(T)=0.

Proof. First, we observe that the condition (V) implies the condition (P) as in

The-orem 2.1. So we can apply TheThe-orem 2.1 to know that there exist cÎ (0, 1) and func-tionsh: [0,∞)® (c, 1] and j: [0,∞)®[0, 1) such that

(a) lim sup

s→0+ φ(s)<lim infs→0+ η(s);

On the other hand, the condition (V) also implies the condition (R) 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

multiva-lued map. Suppose that

(Vd) there exists a _R-function a : [0,∞)® [0, 1) such that for each x Î X with

x∈Tx, it holds

d(y,Ty)≤α(d(x,y))d(x,y) for ally∈Tx.

If the functionf:X ®[0,∞) defined byf(x) =d(x, Tx) is l.s.c, then F(T)=0.

Theorem 4.7. Let (X, d) be a complete metric space, p be aτ0-function, Dp be a

τ0

-metric on CB(X) induced by p,T:X→CB(X) be a multivalued map andh: X ×

X ® [0,∞) be a function. Suppose that

(W) there exists a R-functiona: [0,∞)®[0, 1) such that

Dp(Tx,Ty)≤α(p(x,y))p(x,y) +h(x,y)p(y,Tx) for allx,y∈X.

If there existsuÎ Xsuch that p(u, ·) is l.s.c. and the functionf:X ®[0, ∞) defined by f(x) =p(x,Tx) is l.s.c, then F(T)=0.

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 functionf(x) =d(x,Tx).

Theorem 4.8.Let (X,d) be a complete metric space, T:X→CB(X) be a

multiva-lued map andg:X ®[0,∞) be a function. Suppose that there exists a _R-functiona: [0, ∞)®[0, 1) such that

H(Tx,Ty)≤α(d(x,y))d(x,y) +g(y)d(y,Tx) for allx,y∈X. (4:12)

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Proof. Observe that the condition (4.12) implies that for eachxÎ Xwith x∈Tx, it holds

d(y,Ty)≤ϕ(d(x,y))d(x,y) for ally∈Tx.

It is therefore possible to apply Theorem 3.2 to obtain a Cauchy sequence {xn}nÎNin

X satisfying

•xn+1Î Txn,nÎN,

• lim

n→∞d(xn,xn+1) = infn∈Nd(xn,xn+1) = 0.

By the completeness ofX, there existsvÎ Xsuch that xn®vas n®∞. It follows from (4.12) again that

lim

n→∞d(xn+1,Tv)≤nlim→∞H(Txn,Tv) ≤ lim

n→∞{ϕ(d(xn,v))d(xn,v) +h(v)d(v,xn+1)}= 0,

which implies limn_®_∞d(xn,Tv) = 0. By the continuity ofd(·,Tv) andxn®vasn®∞,

d(v,Tv) = 0. By the closedness ofTv, we getvÎTvor v∈F(T). □

Remark 4.2.

(a) Theorem 4.8 generalizes [7, Theorem 2.6], Berinde-Berinde’s fixed point theo-rem, Mizoguchi-Takahashi’s fixed point theorem and references therein.

(b) In [7, Theorem 2.6], the authors shown that a generalized multivalued almost contractionT in a metric space (X, d) have F(T)=0 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→CB(X) be a multivalued map and L≥ 0. Suppose that there exists a

R-function a: [0,∞)® [0, 1) such that

H(Tx,Ty)≥α(d(x,y))d(x,y) +Ld(y,Tx) for allx,y∈X.

Then F(T)=0

Corollary 4.2 [3]. Let (X, d) be a complete metric space and T:X→CB(X) be a

multivalued map. Suppose that there exists a R-function a: [0,∞)® [0, 1) such that

H(Tx,Ty)≤α(d(x,y))d(x,y) for allx,y∈X.

Then F(T)=0.

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 existsμÎΩsuch that for eachxÎ Xwith x∈Tx, it holds

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and further assume that there exists uÎ Xsuch thatp(u, ·) is l.s.c. and the functionf

: X®[0,∞) defined byf(x) =p(x,Tx) is l.s.c, then F(T)=0.

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

multi-valued map. Suppose that

(Δd) there existsμÎΩsuch that for eachxÎXwith x∈Tx, it holds

d(y,Ty)≤d(x,y)−μ(d(x,y)) for ally∈Tx.

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

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.

Competing interests

The author declares that they have no competing interests.

Received: 6 August 2011 Accepted: 17 January 2012 Published: 17 January 2012

References

1. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan (2000) 2. Nadler, SB Jr: Multi-valued contraction mappings. pp. 475–488. Pacific J Math30, (1969)

3. Mizoguchi, N, Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl.141, 177–188 (1989). doi:10.1016/0022-247X(89)90214-X

4. Daffer, PZ, Kaneko, H: Fixed points of generalized contractive multi-valued mappings. J Math Anal Appl.192, 655–666 (1995). doi:10.1006/jmaa.1995.1194

5. Berinde, M, Berinde, V: On a general class of multi-valued weakly Picard mappings. J Math Anal Appl.326, 772–782 (2007). doi:10.1016/j.jmaa.2006.03.016

6. Hussain, N, Cho, YJ: Weak contraction, common fixed points and invariant approximations. J Inequal Appl (2009).2009, Article ID 390634, 10

7. Hussain, N, Amini-Harandi, A, Cho, YJ: Approximate endpoints for set-valued contractions in metric spaces. Fixed Point Theory Appl13(2010).2010, Article ID 614867,

8. Kamran, T: Multivaluedf-weakly Picard mappings. Nonlinear Anal.67, 2289–2296 (2007). doi:10.1016/j.na.2006.09.010 9. Huang, L-G, Zhang, X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl.332,

1468–1476 (2007). doi:10.1016/j.jmaa.2005.03.087

10. Abbas, M, Jungck, G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl.341, 416–420 (2008). doi:10.1016/j.jmaa.2007.09.070

11. Rezapour, Sh, Hamlbarani, R: Some notes on the paper“Cone metric spaces and fixed point theorems of contractive mappings_”. J Math Anal Appl.345, 719_–724 (2008). doi:10.1016/j.jmaa.2008.04.049

12. Berinde, V, Păcurar, M: Fixed points and continuity of almost contractions. Fixed Point Theory.9, 23–34 (2008) 13. Du, W-S: Fixed point theorems for generalized Hausdorff metrics. Int Math Forum.3, 1011_–1022 (2008)

14. Du, W-S: Some new results and generalizations in metric fixed point theory. Nonlinear Anal.73, 1439–1446 (2010). doi:10.1016/j.na.2010.05.007

15. Du, W-S: Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi’s condition in quasiordered metric spaces. Fixed Point Theory and Applications9(2010).2010, Article ID 876372, 16. Du, W-S: Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems. Fixed Point Theory and

Applications16(2010).2010, Article ID 190606,

17. Du, W-S: New cone fixed point theorems for nonlinear multivalued maps with their applications. Appl Math Lett.24, 172–178 (2011). doi:10.1016/j.aml.2010.08.040

18. Du, W-S, Zheng, S-X: Nonlinear conditions for coincidence point and fixed point theorems. Taiwanese J Math.

16(3):857–868 (2012)

19. Du, W-S: On coincidence point and fixed point theorems for nonlinear multivalued maps. Topol Appl.159, 49_–56 (2012). doi:10.1016/j.topol.2011.07.021

20. Clarke, FH: Pointwise contraction criteria for the existence of fixed points. Can Math Bull.21(1):7–11 (1978). doi:10.4153/ CMB-1978-002-8

21. Sehgal, VM, Smithson, RE: A fixed point theorem for weak directional contraction multifunction. Math Japon.

25(3):345_–348 (1980)

22. Song, W: A generalization of Clarke’s fixed point theorem. Appl Math J Chin Univ Ser B.10(4):463–466 (1995). doi:10.1007/BF02662502

(22)

24. Frigon, M: Fixed point results for multivalued maps in metric spaces with generalized inwardness conditions. Fixed Point Theory Appl19(2010).2010, Article ID 183217,

25. Kada, O, Suzuki, T, Takahashi, W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math Japon.44, 381_–391 (1996)

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.323, 360_–370 (2006). doi:10.1016/j.jmaa.2005.10.005

27. Lin, L-J, Du, W-S: Some equivalent formulations of generalized Ekeland’s variational principle and their applications. Nonlinear Anal.67, 187_–199 (2007). doi:10.1016/j.na.2006.05.006

28. Lin, L-J, Du, W-S: On maximal element theorems, variants of Ekeland’s variational principle and their applications. Nonlinear Anal.68, 1246–1262 (2008). doi:10.1016/j.na.2006.12.018

29. Du, W-S: Critical point theorems for nonlinear dynamical systems and their applications. Fixed Point Theory Appl16

(2010).2010, Article ID 246382,

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. 84 (2011)

31. Ding, XP, He, YR: Fixed point theorems for metrically weakly inward set-valued mappings. J Appl Anal.5(2):283_–293 (1999)

32. Downing, D, Kirk, WA: Fixed point theorems for set-valued mappings in metric and Banach spaces. Math Japon.22, 99–112 (1977)

33. Reich, S: A fixed point theorem for locally contractive multivalued functions. Rev Roumaine Math Pures Appl.17, 569–572 (1972)

34. Reich, S: Fixed points of contractive functions. Boll Un Mat Ital.5(4):26–42 (1972)

35. Reich, S: Some problems and results in fixed point theory. Contemp Math.21, 179_–187 (1983) 36. Xu, H-K: Metric fixed point theory for multivalued mappings. Dissertationes Mathematicae.389, 39 (2000)

doi:10.1186/1687-1812-2012-6

Cite this article as:Du: On generalized weakly directional contractions and approximate fixed point property

with applications.Fixed Point Theory and Applications20122012:6.

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