We first introduce the concept of manageable functions and then prove some new existence theorems related to approximate fixed point property for manageable functions and α-admissible multivalued maps. As applications of our results, some new fixed point theorems which
generalize and improve Du's fixed point theorem, Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and Nadler's fixed point theorem and some well-known results in the literature are given.

1. Introduction and Preliminaries

In 1922, Banach established the most famous fundamental fixed point theorem (so-called the Banach contraction principle [1]) which has played an important role in various fields of applied mathematical analysis. It is known that the Banach contraction principle has been extended and generalized in many various different directions by several authors; see [2–40] and references therein. An 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], and Berinde-Berinde’s fixed point theorem [5] and references therein.

Let us recall some basic notations, definitions, and well-known results needed in this paper. Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. Let (X,d) be a metric space. For each x∈X and A⊆X, let d(x,A)=infy∈Ad(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 𝒞ℬ(X) the family of all nonempty closed and bounded subsets of X. A function ℋ:𝒞ℬ(X)×𝒞ℬ(X)→[0,∞) defined by
(1)ℋ(A,B)=max{supx∈Bd(x,A),supx∈Ad(x,B)}
is said to be the Hausdorff metric on 𝒞ℬ(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:X→X 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 ℱ(T). The map T is said to have the approximate fixed point property [29–34] on X provided infx∈Xd(x,Tx)=0. It is obvious that ℱ(T)≠∅ implies that T has the approximate fixed point property, but the converse is not always true.

Definition 1 (see [<xref ref-type="bibr" rid="B6">6</xref>, <xref ref-type="bibr" rid="B13">13</xref>]).

A function φ:[0,∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if limsups→t+φ(s)<1 for all t∈[0,∞).

It is evident that if φ:[0,∞)→[0,1) is a nondecreasing function or a nonincreasing function, then φ is a ℳ𝒯-function. So the set of ℳ𝒯-functions is a rich class.

Recently, Du [6] first proved the following characterizations of ℳ𝒯-functions which are quite useful for proving our main results.

Theorem 2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let φ:[0,∞)→[0,1) be a function. Then the following statements are equivalent.

φis an ℳ𝒯-function.

For each t∈[0,∞), there exist rt(1)∈[0,1) and εt(1)>0 such that φ(s)≤rt(1) for all s∈(t,t+εt(1)).

For each t∈[0,∞), there exist rt(2)∈[0,1) and εt(2)>0 such that φ(s)≤rt(2) for all s∈[t,t+εt(2)].

For each t∈[0,∞), there exist rt(3)∈[0,1) and εt(3)>0 such that φ(s)≤rt(3) for all s∈(t,t+εt(3)].

For each t∈[0,∞), there exist rt(4)∈[0,1) and εt(4)>0 such that φ(s)≤rt(4) for all s∈[t,t+εt(4)).

For any nonincreasing sequence {xn}n∈ℕ in [0,∞), one has 0≤supn∈ℕφ(xn)<1.

φ is a function of contractive factor [15]; that is, for any strictly decreasing sequence {xn}n∈ℕ in [0,∞), one has 0≤supn∈ℕφ(xn)<1.

In 1989, Mizoguchi and Takahashi [3] proved a famous generalization of Nadler’s fixed point theorem which gives a partial answer of Problem 9 in Reich [4].

Theorem 3 (Mizoguchi and Takahashi [<xref ref-type="bibr" rid="B3">3</xref>]).

Let (X,d) be a complete metric space, let φ:[0,∞)→[0,1) be an ℳ𝒯-function, and let T:X→𝒞ℬ(X) be a multivalued map. Assume that
(2)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y),
for all x,y∈X. Then ℱ(T)≠∅.

In 2007, M. Berinde and V. Berinde [5] proved the following interesting fixed point theorem which generalized and extended Mizoguchi-Takahashi’s fixed point theorem.

Theorem 4 (M. Berinde and V. Berinde [<xref ref-type="bibr" rid="B5">5</xref>]).

Let (X,d) be a complete metric space, let φ:[0,∞)→[0,1) be an ℳ𝒯-function, let T:X→𝒞ℬ(X) be a multivalued map, and L≥0. Assume that
(3)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+Ld(y,Tx),
for all x,y∈X. Then ℱ(T)≠∅.

In 2012, Du [6] established the following fixed point theorem which is an extension of Berinde-Berinde’s fixed point theorem and hence Mizoguchi-Takahashi’s fixed point theorem.

Let (X,d) be a complete metric space, let T:X→𝒞ℬ(X) be a multivalued map, let φ:[0,∞)→[0,1) be a ℳ𝒯-function, and let h:X→[0,∞) be a function. Assume that
(4)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X.
Then T has a fixed point in X.

The paper is organized as follows. In Section 2, we first introduce the concept of manageable function and give some examples of it. Section 3 is dedicated to the study of some new existence theorems related to approximate fixed point property for manageable functions and α-admissible multivalued maps. As applications of our results, some new fixed point theorems which generalize and improve Du’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, and Nadler’s fixed point theorem and some well-known results in the literature are given in Section 4. Consequently, some of our results in this paper are original in the literature, and we obtain many results in the literature as special cases.

2. Manageable Functions

In this paper, we first introduce the concept of manageable functions.

Definition 6.

A function η:ℝ×ℝ→ℝ is called manageable if the following conditions hold:

η(t,s)<s-t for all s,t>0;

for any bounded sequence {tn}⊂(0,+∞) and any nonincreasing sequence {sn}⊂(0,+∞), it holds that
(5)limsupn→∞tn+η(tn,sn)sn<1.

We denote the set of all manageable functions by Man(ℝ)^.

Here, we give simple examples of manageable function.

Example A.

Let γ∈[0,1). Then ηγ:ℝ×ℝ→ℝ defined by
(6)ηγ(t,s)=γs-t
is a manageable function.

Example B.

Let f:ℝ×ℝ→ℝ be any function. Then the function η:ℝ×ℝ→ℝ defined by
(7)η(t,s)={ss+9ln(s+10)-t,if(t,s)∈[0,+∞)×[0,+∞),f(t,s),otherwise,
is a manageable function. Indeed, let
(8)g(x)=ln(x+10)x+9∀x>-9.
Then g(s)<1 for all s>0, and
(9)η(t,s)={sg(s)-t,if(t,s)∈[0,+∞)×[0,+∞),f(t,s),otherwise.
For any s,t>0, we have
(10)η(t,s)=sg(s)-t<s-t,
so (η1) holds. Let {tn}⊂(0,+∞) be a bounded sequence and let {sn}⊂(0,+∞) be a nonincreasing sequence. Then limn→∞sn=infn∈ℕsn=a for some a∈[0,+∞). Since g is continuous, we get
(11)limsupn→∞tn+η(tn,sn)sn=limn→∞g(sn)=g(a)<1,
which means that (η2) holds. Hence, η∈Man(ℝ)^.

Example C.

Let f:ℝ×ℝ→ℝ be any function and let φ:[0,∞)→[0,1) be an ℳ𝒯-function. Define η:ℝ×ℝ→ℝ by
(12)η(t,s)={sφ(s)-t,if(t,s)∈[0,+∞)×[0,+∞),f(t,s),otherwise.
Then η is a manageable function. Indeed, one can verify easily that (η1) holds. Next, we verify that η satisfies (η2). Let {tn}⊂(0,+∞) be a bounded sequence and let {sn}⊂(0,+∞) be a nonincreasing sequence. Then limn→∞sn=infn∈ℕsn=a for some a∈[0,+∞). Since φ is an ℳ𝒯-function, by Theorem 2, there exist ra∈[0,1) and εa>0 such that φ(s)≤ra for all s∈[a,a+εa). Since limn→∞sn=infn∈ℕsn=a, there exists na∈ℕ, such that
(13)a≤sn<a+εa∀n∈ℕwithn≥na.
Hence, we have
(14)limsupn→∞tn+η(tn,sn)sn=limsupn→∞φ(sn)≤ra<1,
which means that (η2) holds. So we prove η∈Man(ℝ)^.

The following result is quite obvious.

Proposition 7.

Let ζ:ℝ×ℝ→ℝ be a function. If there exists η∈
Man
(ℝ)^ such that ζ(t,s)≤η(t,s) for all s,t>0, then ζ∈
Man
(ℝ)^.

Proposition 8.

Let {ηi}i∈ℕ⊂
Man
(ℝ)^. Then the following statements hold.

For eachk∈ℕ, the functionη(k)min:ℝ×ℝ→ℝ, defined by(15)η(k)min(t,s)=min{η1(t,s),η2(t,s),…,ηk(t,s)},is a manageable function(i.e.,η(k)min∈
Man
(ℝ)^foranyk∈ℕ).

For eachk∈ℕ, the functionη(k)¯:ℝ×ℝ→ℝ, defined by(16)η(k)¯(t,s)=1k∑i=1kηi(t,s),is a manageable function(i.e.,η(k)¯∈
Man
(ℝ)^foranyk∈ℕ).

Proof.

Since η(k)min(t,s)≤η1(t,s) for all t,s>0, the conclusion (a) is a direct consequence of Proposition 7. Next, we prove the conclusion (b). Let k∈ℕ be given. It is obvious that η(k)¯(t,s)<s-t for all s,t>0. Let {tn}⊂(0,+∞) be a bounded sequence and let {sn}⊂(0,+∞) be a nonincreasing sequence. For any n∈ℕ, we have
(17)tn+η(k)¯(tn,sn)sn=1sn(tn+1k∑i=1kηi(t,s))=1k∑i=1ktn+ηi(tn,sn)sn.
Because each ηi satisfies (η2), we get
(18)limsupn→∞tn+η(k)¯(tn,sn)sn=1klimsupn→∞(∑i=1ktn+ηi(tn,sn)sn)≤1k∑i=1k(limsupn→∞tn+ηi(tn,sn)sn)<1.
Hence, for each k∈ℕ, the function η(k)¯ is a manageable function.

3. New Existence Results for Manageable Functions and Approximate Fixed Point Property

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

a Nadler’s type contraction (or a multivalued k-contraction [3, 33]), if there exists a number 0<k<1 such that
(19)ℋ(Tx,Ty)≤kd(x,y)∀x,y∈X;

a Mizoguchi-Takahashi’s type contraction [33], if there exists an ℳ𝒯-function α:[0,∞)→[0,1) such that
(20)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)∀x,y∈X;

a multivalued (θ,L)-almost contraction [28, 29, 33], if there exist two constants θ∈(0,1) and L≥0 such that
(21)ℋ(Tx,Ty)≤θd(x,y)+Ld(y,Tx)∀x,y∈X;

a Berinde-Berinde’s type contraction [33] (or a generalized multivalued almost contraction [28, 29, 33]), if there exists an ℳ𝒯-function α:[0,∞)→[0,1) and L≥0 such that
(22)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)+Ld(y,Tx)∀x,y∈X;

a Du’s strong type contraction, if there exist an ℳ𝒯-function α:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(23)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X;

a Du’s weak type contraction, if there exist an ℳ𝒯-function α:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(24)d(y,Ty)≤α(d(x,y))d(x,y)∀y∈Tx.

Definition 9 (see [<xref ref-type="bibr" rid="B36">36</xref>–<xref ref-type="bibr" rid="B39">39</xref>]).

Let (X,d) be a metric space and let T:X→𝒩(X) be a multivalued map. One says that T is α-admissible, if there exists a function α:X×X→[0,+∞) such that for each x∈X and y∈Tx with α(x,y)≥1, one has α(y,z)≥1 for all z∈Ty.

The following existence theorem is one of the main results of this paper.

Theorem 10.

Let (X,d) be a metric space, let T:X→𝒩(X) be an α-admissible multivalued map, and η∈
Man
(ℝ)^. Let
(25)Ω={(α(x,y)d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.
If η(t,s)≥0 for all (t,s)∈Ω and there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, then the following statements hold.

There exists a Cauchy sequence {wn}n∈ℕ in X such that

wn+1∈Twn for all n∈ℕ,

α(wn,wn+1)≥1 for all n∈ℕ,

limn→∞d(wn,wn+1)=infn∈ℕd(wn,wn+1)=0.

infx∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.

Proof.

By our assumption, there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1. If x1=x0, then x0∈Tx0 and
(26)infx∈Xd(x,Tx)≤d(x0,Tx0)≤d(x0,x0)=0,
which implies infx∈Xd(x,Tx)=0. Let wn=x0 for all n∈ℕ. Then {wn}n∈ℕ is a Cauchy sequence in X and
(27)limn→∞d(wn,wn+1)=infn∈ℕd(wn,wn+1)=d(x0,x0)=0.
Clearly, α(wn,wn+1)=α(x0,x1)≥1 for all n∈ℕ. Hence, the conclusions (a) and (b) hold in this case. Assume x1∉x0 or d(x0,x1)>0. If x1∈Tx1, then, following a similar argument as above, we can prove the conclusions (a) and (b) by taking a Cauchy sequence {wn}n∈ℕ with w1=x0 and wn=x1 for all n≥2. Suppose x1∉Tx1. Thus d(x1,Tx1)>0. Define λ:ℝ×ℝ→ℝ by
(28)λ(t,s)={t+η(t,s)s,if(t,s)∈Ω∖{(0,0)},0,otherwise.
By (η1), we know that
(29)0<λ(t,s)<1∀(t,s)∈Ω∖{(0,0)}.
Since η∈Man(ℝ)^ and η(t,s)≥0 for all (t,s)∈Ω, we have
(30)0<t≤sλ(t,s)∀(t,s)∈Ω∖{(0,0)}.
Clearly, (α(x0,x1)d(x1,Tx1),d(x0,x1))∈Ω∖{(0,0)}. So, by (29), we obtain
(31)0<λ(α(x0,x1)d(x1,Tx1),d(x0,x1))<1.
Let
(32)ϵ1=(α(x0,x1)λ(α(x0,x1)d(x1,Tx1),d(x0,x1))-1)×d(x1,Tx1).
Taking into account α(x0,x1)≥1, d(x1,Tx1)>0, and the last inequality, we get ϵ1>0. Since
(33)d(x1,Tx1)<d(x1,Tx1)+ϵ1=α(x0,x1)λ(α(x0,x1)d(x1,Tx1),d(x0,x1))×d(x1,Tx1),
there exists x2∈Tx1 such that x2≠x1 and
(34)d(x1,x2)<α(x0,x1)λ(α(x0,x1)d(x1,Tx1),d(x0,x1))×d(x1,Tx1).
If x2∈Tx2, then the proof can be finished by a similar argument as above. Otherwise, we have d(x2,Tx2)>0. Since T is α-admissible, we obtain α(x1,x2)≥1. By taking
(35)ϵ2=(α(x1,x2)λ(α(x1,x2)d(x2,Tx2),d(x1,x2))-1)×d(x2,Tx2),
then there exists x3∈Tx2 with x3≠x2 such that
(36)d(x2,x3)<α(x1,x2)λ(α(x1,x2)d(x2,Tx2),d(x1,x2))×d(x2,Tx2).
By induction, if xk-1,xk,xk+1∈X is known satisfying xk-1∈Txk, xk+1∈Txk+2, d(xk,Txk)>0, α(xk-1,xk)≥1, and
(37)0<d(xk,xk+1)<α(xk-1,xk)λ(α(xk-1,xk)d(xk,Txk),d(xk-1,xk))×d(xk,Txk),k∈ℕ,
then, by taking
(38)ϵk=(α(xk-1,xk)λ(α(xk-1,xk)d(xk,Txk),d(xk-1,xk))-1)×d(xk,Txk),
one can obtain xk+2∈Txk+1 with xk+2≠xk+1 such that
(39)d(xk+1,xk+2)<α(xk,xk+1)λ(α(xk,xk+1)d(xk+1,Txk+1),d(xk,xk+1))×d(xk+1,Txk+1).
Hence, by induction, we can establish sequences {xn} in X satisfying, for each n∈ℕ,
(40)xn∈Txn-1,d(xn-1,xn)>0,d(xn,Txn)>0,α(xn-1,xn)≥1,d(xn,xn+1)<α(xn-1,xn)λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))×d(xn,Txn).
By (30), we have
(41)α(xn-1,xn)d(xn,Txn)≤d(xn-1,xn)λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))nnnnnnnnnnnnnnnnnnnnnnnnnnnnforeachn∈ℕ.
Hence, for each n∈ℕ, by combining (40) and (41), we get
(42)d(xn,xn+1)<λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))×d(xn-1,xn),
which means that the sequence {d(xn-1,xn)}n∈ℕ is strictly decreasing in (0,+∞). So
(43)γ:=limn→∞d(xn,xn+1)=infn∈ℕd(xn,xn+1)≥0exists.
By (41), we have
(44)α(xn-1,xn)d(xn,Txn)≤d(xn-1,xn)∀n∈ℕ,
which means that {α(xn-1,xn)d(xn,Txn)}n∈ℕ is a bounded sequence. By (η2), we have
(45)limsupn→∞λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))<1.
Now, we claim γ=0. Suppose γ>0. Then, by (45) and taking lim sup in (42), we get
(46)γ≤limsupn→∞λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))γ<γ,
a contradiction. Hence we prove
(47)limn→∞d(xn,xn+1)=infn∈ℕd(xn,xn+1)=0.
To complete the proof of (a), it suffices to show that {xn}n∈ℕ is a Cauchy sequence in X. For each n∈ℕ, let
(48)ρn:=λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn)).
Then ρn∈(0,1) for all n∈ℕ. By (42), we obtain
(49)d(xn,xn+1)<ρnd(xn-1,xn)∀n∈ℕ.
From (45), we have limsupn→∞ρn<1, so there exist c∈[0,1) and n0∈ℕ, such that
(50)ρn≤c∀n∈ℕwithn≥n0.
For any n≥n0, since ρn∈(0,1) for all n∈ℕ and c∈[0,1), taking into account (49) and (50) concludes that
(51)d(xn,xn+1)<ρnd(xn-1,xn)<⋯<ρnρn-1ρn-2⋯ρn0d(x0,x1)≤cn-n0+1d(x0,x1).
Put αn=(cn-n0+1/(1-c))d(x0,x1), n∈ℕ. For m,n∈ℕ with m>n≥n0, we have from the last inequality that
(52)d(xn,xm)≤∑j=nm-1d(xj,xj+1)<αn.
Since c∈[0,1), limn→∞αn=0. Hence
(53)limn→∞sup{d(xn,xm):m>n}=0.
So {xn} is a Cauchy sequence in X. Let wn=xn-1 for all n∈ℕ. Then {wn}n∈ℕ is the desired Cauchy sequence in (a).

To see (b), since xn∈Txn-1 for each n∈ℕ, we have
(54)infx∈Xd(x,Tx)≤d(xn,Txn)≤d(xn,xn+1)∀n∈ℕ.
Combining (47) and (54) yields
(55)infx∈Xd(x,Tx)=0.
The proof is completed.

Applying Theorem 10, we can establish the following new existence theorem related to approximate fixed point property for α-admissible multivalued maps.

Theorem 11.

Let (X,d) be a metric space and let T:X→𝒩(X) be an α-admissible multivalued map. Suppose that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that
(56)α(x,y)d(y,Ty)≤φ(d(x,y))d(x,y)∀y∈Tx.
If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, then the following statements hold.

There exists η∈
Man
(ℝ)^ such that η(t,s)≥0 for all (t,s)∈Ω, where
(57)Ω={(α(x,y)d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.

There exists a Cauchy sequence {wn}n∈ℕ in X such that

wn+1∈Twn for all n∈ℕ,

α(wn,wn+1)≥1 for all n∈ℕ,

limn→∞d(wn,wn+1)=infn∈ℕd(wn,wn+1)=0.

infx∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.

Proof.

Define η:ℝ×ℝ→ℝ by
(58)η(t,s)={sφ(s)-t,if(t,s)∈[0,+∞)×[0,+∞),0,otherwise.
By Example C, we know η∈Man(ℝ)^. By (56), we obtain η(t,s)≥0 for all (t,s)∈Ω. Therefore (a) is proved. It is obvious that the desired conclusions (b) and (c) follow from Theorem 10 immediately.

The following interesting results are immediate from Theorem 11.

Corollary 12.

Let (X,d) be a metric space and let T:X→𝒞ℬ(X) be an α-admissible multivalued map. Assume that one of the following conditions holds.

There exist an ℳ𝒯-function φ:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(59)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X;

there exist an ℳ𝒯-function φ:[0,∞)→[0,1) and L≥0 such that
(60)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+Ld(y,Tx)∀x,y∈X;

there exist two constants θ∈(0,1) and L≥0 such that
(61)α(x,y)ℋ(Tx,Ty)≤θd(x,y)+Ld(y,Tx)∀x,y∈X;

there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that
(62)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)∀x,y∈X;

there exists a number 0<k<1 such that
(63)α(x,y)ℋ(Tx,Ty)≤kd(x,y)∀x,y∈X.

If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, then the following statements hold.

There exists η∈
Man
(ℝ)^ such that η(t,s)≥0 for all (t,s)∈Ω, where
(64)Ω={(α(x,y)d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.

There exists a Cauchy sequence {wn}n∈ℕ in X such that

wn+1∈Twn for all n∈ℕ,

α(wn,wn+1)≥1 for all n∈ℕ,

limn→∞d(wn,wn+1)=infn∈ℕd(wn,wn+1)=0.

infx∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.

Proof.

It suffices to verify the conclusion under (L1). Note first that, for each x∈X, d(y,Tx)=0 for all y∈Tx. So, for each x∈X, by (L1), we obtain
(65)α(x,y)d(y,Ty)≤φ(d(x,y))d(x,y)∀y∈Tx,
which means (56) holds. Therefore, the conclusion follows from Theorem 11.

In Corollary 12, if we take α:X×X→[0,+∞) by α(x,y)=1 for all x,y∈X, then we obtain the following existence theorem.

Corollary 13.

Let (X,d) be a metric space and let T:X→𝒞ℬ(X) be a multivalued map. Assume that one of the following conditions holds.

T is a Du’s weak type contraction;

T is a Du’s strong type contraction;

T is a Berinde-Berinde’s type contraction;

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

T is a Mizoguchi-Takahashi’s type contraction;

T is a Nadler’s type contraction.

Then the following statements hold.

There exists η∈
Man
(ℝ)^ such that η(t,s)≥0 for all (t,s)∈𝒟, where
(66)𝒟={(d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.

There exists a Cauchy sequence {wn}n∈ℕ in X such that

wn+1∈Twn for all n∈ℕ,

limn→∞d(wn,wn+1)=infn∈ℕd(wn,wn+1)=0.

infx∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.

4. Some Applications to Fixed Point TheoryDefinition 14 (see [<xref ref-type="bibr" rid="B36">36</xref>–<xref ref-type="bibr" rid="B39">39</xref>]).

Let (X,d) be a metric space and let α:X×X→[0,+∞) be a function. α is said to have the property (B) if any sequence {xn} in X with α(xn,xn+1)≥1 for all n∈ℕ and limn→∞xn=v, we have α(xn,v)≥1 for all n∈ℕ.

Theorem 15.

Let (X,d) be a complete metric space and let T:X→𝒞ℬ(X) be an α-admissible multivalued map. Suppose that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that
(67)α(x,y)d(y,Ty)≤φ(d(x,y))d(x,y)∀y∈Tx.
If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, and one of the following conditions is satisfied:

T is ℋ-continuous (i.e., xn→v implies ℋ(Txn,Tv)→0 as n→∞);

T is closed (i.e., GrT:={(x,y)∈X×X:y∈Tx}; the graph of T is a closed subset of X×X);

the map g:X→[0,∞) defined by g(x)=d(x,Tx) is l.s.c.;

for any sequence {zn} in X with α(zn,zn+1)≥1, zn+1∈Tzn, n∈ℕ, and limn→∞zn=c, one has limn→∞d(zn,Tc)=0,

then T admits a fixed point in X.
Proof.

Applying Theorem 11, there exists a Cauchy sequence {wn}n∈ℕ in X such that
(68)wn+1∈Twn,α(wn,wn+1)≥1∀n∈ℕ.
By the completeness of X, there exists v∈X such that wn→v as n→∞.

Now, we verify v∈ℱ(T). If (H1) holds, since T is ℋ-continuous on X, wn+1∈Twn for each n∈ℕ, and wn→v as n→∞, we get
(69)d(v,Tv)=limn→∞d(wn+1,Tv)≤limn→∞ℋ(Twn,Tv)=0,
which implies d(v,Tv)=0. By the closeness of Tv, we have v∈Tv. If (H2) holds, since T is closed, wn+1∈Twn for each n∈ℕ, and wn→v as n→∞, we have v∈ℱ(T). Suppose that (H3) holds. Since {wn}n∈ℕ is convergent in X, we have
(70)limn→∞d(wn,wn+1)=0.
Since
(71)d(v,Tv)=g(v)≤liminfn→∞g(wn)≤limn→∞d(wn,wn+1)=0,
we obtain d(v,Tv)=0, and hence v∈ℱ(T). Finally, assume (H4) holds. Then we obtain
(72)d(v,Tv)=limn→∞d(wn,Tv)=0.
Hence v∈Tv. Therefore, in any case, we prove v∈ℱ(T). This completes the proof.

Theorem 16.

Let (X,d) be a complete metric space and let T:X→𝒞ℬ(X) be an α-admissible multivalued map. Suppose that there exist an ℳ𝒯-function φ:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(73)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X.
If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, and one of the following conditions is satisfied:

T is ℋ-continuous;

T is closed;

the map g:X→[0,∞) defined by g(x)=d(x,Tx) is l.s.c.;

the function α has the property (B),

then T admits a fixed point in X.
Proof.

It is obvious that (73) implies (67). If one of the conditions (S1), (S2), and (S3) is satisfied, then the desired conclusion follows from Theorem 15 immediately. Suppose that (S4) holds. We claim that (H4) as in Theorem 15 is satisfied. Let {zn} be in X with α(zn,zn+1)≥1, zn+1∈Tzn, n∈ℕ, and limn→∞zn=c. Since α has the property (B), α(zn,c)≥1 for all n∈ℕ. So, it follows from (73) that
(74)limn→∞d(zn+1,Tc)≤limn→∞ℋ(Tzn,Tc)≤limn→∞α(zn,c)ℋ(Tzn,Tc)≤limn→∞{φ(d(zn,c))d(zn,c)+h(c)d(c,zn+1)}=0,
which implies limn→∞d(zn,Tc)=0. Hence (H4) holds. By Theorem 15, we also prove ℱ(T)≠∅. The proof is completed.

Applying Theorem 16, we can give a short proof of Du’s fixed point theorem.

Let (X,d) be a complete metric space, let T:X→𝒞ℬ(X) be a multivalued map, let φ:[0,∞)→[0,1) be a ℳ𝒯-function, let and h:X→[0,∞) be a function. Assume that
(75)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X.
Then ℱ(T)≠∅.

Proof.

Take α:X×X→[0,+∞) by α(x,y)=1 for all x,y∈X. Then (75) implies (73). Moreover, T is an α-admissible multivalued map and the function α has the property (B). Therefore the conclusion follows from Theorem 16.

Remark 18.

Theorems 15 and 16 and Corollary 17 all generalize and improve Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and Banach contraction principle.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author was supported by grant no. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China. The second author would like to express his sincere thanks to the Arak branch of Islamic Azad University for supporting this work.

BanachS.Sur les opérations dans les ensembles abstraits et leurs applications aux équations integralesNadlerS. B.Jr.Multi-valued contraction mappingsMizoguchiN.TakahashiW.Fixed point theorems for multivalued mappings on complete metric spacesReichS.Some problems and results in fixed point theoryBerindeM.BerindeV.On a general class of multi-valued weakly Picard mappingsDuW.-S.On coincidence point and fixed point theorems for nonlinear multivalued mapsTakahashiW.UderzoA.Fixed points for directional multi-valued k(·)-contractionsPetruşelA.SîntămărianA.Single-valued and multi-valued Caristi type operatorsFengY.LiuS.Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappingsHuangL.-G.ZhangX.Cone metric spaces and fixed point theorems of contractive mappingsDuW.-S.A note on cone metric fixed point theory and its equivalenceDuW.-S.Some new results and generalizations in metric fixed point theoryDuW.-S.Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi's condition in quasiordered metric spacesDuW.-S.Nonlinear contractive conditions for coupled cone fixed point theoremsDuW.-S.New cone fixed point theorems for nonlinear multivalued maps with their applicationsDuW.-S.ZhengS.-X.Nonlinear conditions for coincidence point and fixed point theoremsDuW.-S.On Caristi type maps and generalized distances with applicationsDuW.-S.KarapınarE.ShahzadN.The study of fixed point theory for various multivalued non-self-mapsLinI. J.ChenT. H.New existence theorems of coincidence points approach to generalizations of Mizoguchi-Takahashi's fixed point theoremKhojastehF.ShuklaS.RadenovicS.A new approach to the study of fixed point theory for simulation functionFilomat. In pressEldredA. A.VeeramaniP.Existence and convergence of best proximity pointsKarapinarE.Generalizations of Caristi Kirk's theorem on partial metric spacesJleliM.KarapinarE.SametB.On best proximity points under the P-property on partially ordered metric spacesKarapınarE.On best proximity point of ψ-Geraghty contractionsJleliM.KarapınarE.SametB.Best proximity points for generalized α-ψ-proximal contractive type mappingsDuW.-S.LakzianH.Nonlinear conditions for the existence of best proximity pointsDuW. S.KarapinarE.A note on Caristi-type cyclic maps: related results and applicationsHussainN.Amini-HarandiA.ChoY. J.Approximate endpoints for set-valued contractions in metric spacesKhamsiM. A.On asymptotically nonexpansive mappings in hyperconvex metric spacesDuW.-S.On approximate coincidence point properties and their applications to fixed point theoryDuW.-S.HeZ.ChenY. L.New existence theorems for approximate coincidence point property and approximate fixed point property with applications to metric fixed point theoryDuW.-S.On generalized weakly directional contractions and approximate fixed point property with applicationsDuW.-S.New existence results and generalizations for coincidence points and fixed points without global completenessSametB.VetroC.VetroP.Fixed point theorems for α-ψ-contractive type mappingsAslJ. H.RezapourS.ShahzadN.On fixed points of α-ψ-contractive multifunctionsMohammadiB.RezapourS.ShahzadN.Some results on fixed points of α-ψ-Ciric generalized multifunctionsAlikhaniH.RezapourSh.ShahzadN.Fixed points of a new type of contractive mappings and multifunctionsAlikhaniH.RakocevicV.RezapourSh.ShahzadN.Fixed points of proximinal valued
β-ψ-contractive multifunctionsJournal of Nonlinear and Convex Analysis. In pressXuH. -K.Metric fixed point theory for multivalued mappings