some generalizations of mizoguchi-takahashi’s fixed point theorem with new local constraints

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Du et al. Fixed Point Theory and Applications 2014, 2014:31 http://www.fixedpointtheoryandapplications.com/content/2014/1/31 RESEARCH Open Access Some generalizations of Mizoguchi-Takahashi’s fixed point theorem with new local constraints Wei-Shih Du 1* , Farshid Khojasteh 2,3 and Yung-Nan Chiu 1 * Correspondence: [email protected] 1 Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan Full list of author information is available at the end of the article Abstract In this paper, motivated by Kikkawa-Suzuki’s fixed point theorem, we establish some new generalizations of Mizoguchi-Takahashi’s fixed point theorem with new local constraints on discussion maps. MSC: 47H10; 54C60; 54H25; 55M20 Keywords: MT -function (or R-function); fixed point; approximate fixed point property; manageable function; transmitted function; Kikkawa-Suzuki’s fixed point theorem; Mizoguchi-Takahashi’s fixed point theorem; Nadler’s fixed point theorem; Banach contraction principle 1 Introduction and preliminaries Let (X, d) be a metric space. Denote by N (X) the family of all nonempty subsets of X, C (X) the class of all nonempty closed subsets of X and CB(X) the family of all nonempty closed and bounded subsets of X. For each x X and A X, let d(x, A)= inf yA d(x, y). A function H : CB(X) × CB(X) [, ) defined by H(A, B)= max sup xB d(x, A), sup xA d(x, B) is said to be the Hausdorff metric on CB(X) induced by the metric d on X. Let T : X N (X) be a multivalued map. A point v in X is said to be a fixed point of T if v Tv. The set of fixed points of T is denoted by F (T ). The map T is said to have the approximate fixed point property [–] on X provided inf xX d(x, Tx) = . It is obvious that F (T ) = implies that T has the approximate fixed point property. The symbols N and R are used to denote the sets of positive integers and real numbers, respectively. A function ϕ : [, ) [,) is said to be an MT -function (or R-function) [–] if lim sup st + ϕ(s) < for all t [, ). It is evident that if ϕ : [, ) [, ) is a nonde- creasing function or a nonincreasing function, then ϕ is a MT -function. So the set of MT -functions is a rich class. Recently, Du [] first proved the following characterizations of MT -functions. Theorem . ([]) Let ϕ : [, ) [, ) be a function. Then the following statements are equivalent. (a) ϕ is an MT -function. ©2014 Du et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Du et al. Fixed Point Theory and Applications 2014, 2014:31http://www.fixedpointtheoryandapplications.com/content/2014/1/31

RESEARCH Open Access

Some generalizations ofMizoguchi-Takahashi’s fixed point theoremwith new local constraintsWei-Shih Du1*, Farshid Khojasteh2,3 and Yung-Nan Chiu1

*Correspondence:[email protected] of Mathematics,National Kaohsiung NormalUniversity, Kaohsiung 824, TaiwanFull list of author information isavailable at the end of the article

AbstractIn this paper, motivated by Kikkawa-Suzuki’s fixed point theorem, we establish somenew generalizations of Mizoguchi-Takahashi’s fixed point theorem with new localconstraints on discussion maps.MSC: 47H10; 54C60; 54H25; 55M20

Keywords: MT -function (orR-function); fixed point; approximate fixed pointproperty; manageable function; transmitted function; Kikkawa-Suzuki’s fixed pointtheorem; Mizoguchi-Takahashi’s fixed point theorem; Nadler’s fixed point theorem;Banach contraction principle

1 Introduction and preliminariesLet (X,d) be ametric space. Denote byN (X) the family of all nonempty subsets of X, C(X)the class of all nonempty closed subsets of X and CB(X) the family of all nonempty closedand bounded subsets ofX. For each x ∈ X andA ⊆ X, let d(x,A) = infy∈A d(x, y). A functionH : CB(X)× CB(X)→ [,∞) defined by

H(A,B) = max{

supx∈B

d(x,A), supx∈A

d(x,B)}

is said to be the Hausdorff metric on CB(X) induced by the metric d on X. Let T : X →N (X) be a multivalued map. A point v in X is said to be a fixed point of T if v ∈ Tv. The setof fixed points of T is denoted by F (T). The map T is said to have the approximate fixedpoint property [–] on X provided infx∈X d(x,Tx) = . It is obvious that F (T) �= ∅ impliesthat T has the approximate fixed point property. The symbolsN and R are used to denotethe sets of positive integers and real numbers, respectively.A function ϕ : [,∞) → [, ) is said to be an MT -function (or R-function) [–] if

lim sups→t+ ϕ(s) < for all t ∈ [,∞). It is evident that if ϕ : [,∞) → [, ) is a nonde-creasing function or a nonincreasing function, then ϕ is a MT -function. So the set ofMT -functions is a rich class.Recently, Du [] first proved the following characterizations ofMT -functions.

Theorem . ([]) Let ϕ : [,∞)→ [, ) be a function. Then the following statements areequivalent.(a) ϕ is anMT -function.

©2014 Du et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 2 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

(b) For each t ∈ [,∞), there exist r()t ∈ [, ) and ε()t > such that ϕ(s)≤ r()t for all

s ∈ (t, t + ε()t ).

(c) For each t ∈ [,∞), there exist r()t ∈ [, ) and ε()t > such that ϕ(s)≤ r()t for all

s ∈ [t, t + ε()t ].

(d) For each t ∈ [,∞), there exist r()t ∈ [, ) and ε()t > such that ϕ(s)≤ r()t for all

s ∈ (t, t + ε()t ].

(e) For each t ∈ [,∞), there exist r()t ∈ [, ) and ε()t > such that ϕ(s)≤ r()t for all

s ∈ [t, t + ε()t ).

(f ) For any nonincreasing sequence {xn}n∈N in [,∞), we have ≤ supn∈N ϕ(xn) < .(g) ϕ is a function of contractive factor; that is, for any strictly decreasing sequence

{xn}n∈N in [,∞), we have ≤ supn∈N ϕ(xn) < .

In , Mizoguchi and Takahashi [] proved a famous generalization of Nadler’s fixedpoint theorem, which gives a partial answer of Problem in Reich [].

Theorem . (Mizoguchi and Takahashi []) Let (X,d) be a complete metric space, ϕ :[,∞) → [, ) be a MT -function and T : X → CB(X) be a multivalued map. Assumethat

H(Tx,Ty)≤ ϕ(d(x, y)

)d(x, y),

for all x, y ∈ X. Then F (T) �= ∅.

A number of generalizations in various different directions of research of Mizoguchi-Takahashi’s fixed point theoremwere investigated by several authors; see, e.g., [–, –]and references therein.In , Suzuki [] presented a new type of generalization of the celebrated Banach

contraction principle [] which characterized the metric completeness.

Theorem . (Suzuki []) Define a nonincreasing function θ from [, ) onto ( , ] by

θ (r) =

⎧⎪⎪⎨⎪⎪⎩, if ≤ r ≤

(√ – ),

–rr , if

(√ – )≤ r ≤ √

,+r , if √

≤ r < .

Then for a metric space (X,d), the following are equivalent:() X is complete.() Every mapping T on X satisfying the following has a fixed point:

• There exists r ∈ [, ) such that θ (r)d(x,Tx)≤ d(x, y) implies d(Tx,Ty)≤ rd(x, y)for all x, y ∈ X .

() There exists r ∈ [, ) such that every mapping T on X satisfying the following has afixed point:•

,d(x,Tx)≤ d(x, y) implies d(Tx,Ty)≤ rd(x, y) for all x, y ∈ X .

Remark . ([]) For every r ∈ [, ), θ (r) is the best constant.

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 3 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

Later, Kikkawa and Suzuki [] proved an interesting generalization of both Theorem .and Nadler’s fixed point theorem. In fact, Kikkawa-Suzuki’s fixed point theorem can beregarded as a generalization of Nadler fixed point theorem with a local constraint on thediscussion map.

Theorem . (Kikkawa and Suzuki []) Define a strictly decreasing function η from [, )onto ( , ] by

η(r) = + r

.

Let (X,d) be a complete metric space and let T be a map from X into CB(X). Assume thatthere exists r ∈ [, ) such that

η(r)d(x,Tx)≤ d(x, y) implies H(Tx,Ty)≤ rd(x, y)

for all x, y ∈ X. Then F (T) �= ∅.

In this paper, motivated by Kikkawa-Suzuki’s fixed point theorem, we establish somenew generalizations of Mizoguchi-Takahashi’s fixed point theorem with new local con-straints on discussion maps. Our new results generalize and improve Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and Banach contractionprinciple.

2 Main resultsVery recently, Du andKhojasteh [] first introduced the concept ofmanageable functions.

Definition . ([]) A function η :R×R →R is calledmanageable if the following con-ditions hold:

(η) η(t, s) < s – t for all s, t > .(η) For any bounded sequence {tn} ⊂ (, +∞) and any nonincreasing sequence {sn} ⊂

(, +∞), we have

lim supn→∞

tn + η(tn, sn)sn

< .

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

Remark . If η ∈ Man(R), then η(t, t) < for all t > .

Example . Let γ ∈ [, ) and a ≥ . Then the function ηγ : R × R → R defined byηγ (t, s) = γ s – t – a is manageable.

Example . ([]) Let f :R×R→R be any function and ϕ : [,∞)→ [, ) be anMT -function. Define η :R×R →R by

η(t, s) =

⎧⎨⎩sϕ(s) – t, if (t, s) ∈ [, +∞)× [, +∞),f (t, s), otherwise.

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 4 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

Then η is a manageable function. Indeed, one can verify easily that (η) holds. Next, weverify that η satisfies (η). Let {tn} ⊂ (, +∞) be a bounded sequence and {sn} ⊂ (, +∞)be a nonincreasing sequence. Then limn→∞ sn = infn∈N sn = a for some a ∈ [, +∞). Sinceϕ is anMT -function, by Theorem ., there exist ra ∈ [, ) and εa > such that ϕ(s)≤ rafor all s ∈ [a,a + εa). Since limn→∞ sn = infn∈N sn = a, there exists na ∈N, such that

a ≤ sn < a + εa for all n ∈N with n≥ na.

Hence we have

lim supn→∞

tn + η(tn, sn)sn

= lim supn→∞

ϕ(sn)≤ ra < ,

which means that (η) holds. Thus we prove η ∈ Man(R).

In this paper, we first introduce the concepts of weakly transmitted functions and (λ)-strongly transmitted functions.

Definition . A function ξ :R×R →R is called(i) weakly transmitted if ξ (t, s) > s – t for all s, t > ;(ii) (λ)-strongly transmitted if there exists λ > , such that ξ (t, s)≥ s –

λt for all s, t ≥ .

We denote by TRA(w) and TRA(λ), the sets of all weakly transmitted functions and (λ)-strongly transmitted functions, respectively. It is quite obvious that TRA(λ)⊆ TRA(w) forall λ > .

Example . Let f : R × R → R, g : [, +∞) → [, +∞) be functions and λ > . Defineξ :R×R →R by

ξ (t, s) =

⎧⎨⎩sg(s) –

λt, if (t, s) ∈ [, +∞)× [, +∞),

f (t, s), otherwise.

Then ξ ∈ TRA(λ).

The following simple example shows that there exists a weakly transmitted functionwhich is not (λ)-strongly transmitted for all λ > . In other words, TRA(λ) � TRA(w) forall λ > .

Example . Let ξ :R×R →R be defined by

ξ (t, s) =

⎧⎨⎩s – t, if (t, s) ∈ [, +∞)× [, +∞),, otherwise.

Then ξ is a weakly transmitted function which is not (λ)-strongly transmitted for all λ > .

The following result is simple, but it is very crucial in our proofs.

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 5 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

Lemma . Let (X,d) be a metric space, W be a nonempty subset of X × X and T : X →CB(X) be a multivalued map. Suppose that there exists η ∈ Man(R) such that

η(H(Tx,Ty),d(x, y)

) ≥ for all (x, y) ∈W .

If (p,q) ∈W with p �= q, then H(Tp,Tq) < d(p,q).

Proof Since p �= q, d(p,q) > . IfH(Tp,Tq) = d(p,q) > , then, by Remark ., we have ≤η(H(Tp,Tq),d(p,q)) < , a contradiction. IfH(Tp,Tq) > d(p,q) > , then, by (η), we have

≤ η(H(Tp,Tq),d(p,q)

)< d(p,q) –H(Tp,Tq) < ,

which also leads a contradiction. ThereforeH(Tp,Tq) < d(p,q). �

Now, we establish an existence theorem for approximate fixed point property and fixedpoints by using manageable functions and transmitted functions which is one of the mainresults of this paper.

Theorem . Let (X,d) be a metric space and T : X → CB(X) be a multivalued map.Assume that there exist ξ ∈ TRA(w) and η ∈ Man(R) such that

η(H(Tx,Ty),d(x, y)

) ≥ for all (x, y) ∈W , (.)

where

W ={(x, y) ∈ X ×X : ξ

(d(x,Tx),d(x, y)

) ≥ }.

Then T has the approximate fixed property on X .Moreover, if (X,d) is complete and ξ ∈ TRA(λ), then F (T) �= ∅.

Proof Let x ∈ X. If x ∈ Tx, then x is a fixed point of T and we are done. Suppose thatx /∈ Tx. Then d(x,Tx) > . Since Tx �= ∅, we can find x ∈ Tx with x �= x. Thus

d(x,x) > .

Since ξ ∈ TRA(w), we get

ξ(d(x,Tx),d(x,x)

)> d(x,x) – d(x,Tx)≥ ,

which means that (x,x) ∈W . Therefore, by (.), we obtain

η(H(Tx,Tx),d(x,x)

) ≥ .

By Lemma ., we have

H(Tx,Tx) < d(x,x).

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If x ∈ Tx, then we have nothing to prove. So we assume that x /∈ Tx. Hence we have

< d(x,Tx)≤H(Tx,Tx). (.)

Define h :R×R →R by

h(t, s) =

⎧⎨⎩

t+η(t,s)s , if t, s > ,

, otherwise.

By (η), we know that

< h(t, s) < for all t, s > . (.)

SinceH(Tx,Tx) > and d(x,x) > , by the definition of h and (.), we have

d(x,Tx)≤H(Tx,Tx)≤ d(x,x)h(H(Tx,Tx),d(x,x)

). (.)

Take

ε =(

√h(H(Tx,Tx),d(x,x))

– )d(x,Tx).

Then ε > . Since

d(x,Tx) < d(x,Tx) + ε

=√

h(H(Tx,Tx),d(x,x))d(x,Tx),

there exists x ∈ Tx such that x �= x and

d(x,x) <√

h(H(Tx,Tx),d(x,x))d(x,Tx)

≤ d(x,x)√h(H(Tx,Tx),d(x,x)

).

If x ∈ Tx, then the proof is finished. Otherwise, we have

< d(x,Tx)≤H(Tx,Tx). (.)

By (.) and x �= x, we get

ξ(d(x,Tx),d(x,x)

) ≥ d(x,x) – d(x,Tx)≥ ,

which implies (x,x) ∈W . By (.), we obtain

η(H(Tx,Tx),d(x,x)

) ≥ .

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 7 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

By Lemma . and (.), we have

<H(Tx,Tx) < d(x,x). (.)

Taking into account (.), (.) and the definition of h conclude that

d(x,Tx)≤ d(x,x)h(H(Tx,Tx),d(x,x)

).

By taking

ε =(

√h(H(Tx,Tx),d(x,x))

– )d(x,Tx),

there exists x ∈ Tx with x �= x such that

d(x,x) < d(x,x)√h(H(Tx,Tx),d(x,x)

).

Hence, by induction, we can establish a sequences {xn} in X satisfying for each n ∈ N,

xn ∈ Txn–,

d(xn–,xn) > ,

< d(xn,Txn)≤H(Txn–,Txn) < d(xn–,xn),

η(H(Txn–,Txn),d(xn–,xn)

) ≥ ,

(.)

and

d(xn,xn+) < d(xn–,xn)√h(H(Txn–,Txn),d(xn–,xn)

). (.)

We claim that {xn}n∈N is a Cauchy sequence in X. For each n ∈N, let

ρn :=√h(H(Txn–,Txn),d(xn–,xn)

).

By (.), we know that

< h(H(Txn–,Txn),d(xn–,xn)

)< for all n ∈N, (.)

so, from (.) and (.), we obtain ρn ∈ (, ) and

d(xn,xn+) < ρnd(xn–,xn) for all n ∈N. (.)

Hence the sequence {d(xn–,xn)}n∈N is strictly decreasing in (,+∞). Thus

limn→∞d(xn,xn+) = inf

n∈Nd(xn,xn+)≥ exists. (.)

By (.), we get

H(Txn–,Txn)≤ d(xn–,xn) for all n ∈N,

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 8 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

which means that {H(Txn–,Txn)}n∈N is a bounded sequence. By (η) and the definition ofh, we have

lim supn→∞

h(H(Txn–,Txn),d(xn–,xn)

)< ,

which implies lim supn→∞ ρn < . So, there exists c ∈ [, ) and n ∈N, such that

ρn ≤ c for all n ∈N with n≥ n. (.)

For any n ≥ n, since ρn ∈ (, ) for all n ∈ N and c ∈ [, ), taking into account (.) and(.), we conclude

d(xn,xn+) < ρnd(xn–,xn)

< · · ·< ρnρn–ρn– · · ·ρnd(x,x)

≤ cn–n+d(x,x).

Put αn = cn–n+–c d(x,x), n ∈ N. For m,n ∈ N with m > n ≥ n, from the last inequality, we

have

d(xn,xm)≤m–∑j=n

d(xj,xj+) < αn.

Since c ∈ [, ), limn→∞ αn = and hence

limn→∞ sup

{d(xn,xm) :m > n

}= . (.)

So {xn} is a Cauchy sequence in X. Combining (.) and (.), we get

infn∈N

d(xn,xn+) = limn→∞d(xn,xn+) = . (.)

Since xn ∈ Txn– for each n ∈ N, we have

infx∈X d(x,Tx)≤ d(xn,Txn)≤ d(xn,xn+) for all n ∈N. (.)

Combining (.) and (.) yields

infx∈X d(x,Tx) = ,

which means that T has the approximate fixed property on X.Now, we assume that (X,d) is complete and ξ ∈ TRA(λ). Since {xn} is a Cauchy sequence

in X, by the completeness of X, there exists v ∈ X such that xn → v as n → ∞. We willproceed with the following claims to prove v ∈ F (T).Claim . d(v,Tx)≤ d(v,x) for all x ∈ X�{v}.

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Given x ∈ X with x �= v. Let

S = {n ∈N : xn = x}.

Suppose that �(S) = ∞, where �(S) is the cardinal number of S. Then there exists {xnj} ⊂{xn} such that xnj = x for all j ∈N. So xnj → x as j → ∞. By the uniqueness of the limit, weget x = v, a contradiction. Hence �(S) <∞ which deduces that there exists ∈ N such thatxn �= x for all n ∈N with n≥ . For any n ∈N, put

wn = xn+ –.

Thus we have• wn �= x for all n ∈N;• wn+ ∈ Twn for all n ∈N;• wn → v as n→ ∞.Since d(x, v) > , there exists n > such that

d(v,wn)≤ d(x, v) for all n ∈N with n≥ n. (.)

For n ∈N with n≥ n, from (.), we have

ξ(d(wn,Twn),d(wn,x)

)> d(wn,x) – d(wn,Txn)

≥ d(x, v) – d(wn, v) – d(wn,Twn)

≥ d(x, v) – d(wn, v) – d(wn,wn+)

≥ d(x, v) – d(wn, v) – d(wn, v) – d(wn+, v)

= d(x, v) – d(wn, v) – d(wn+, v)

> d(x, v) – d(x, v) –

d(x, v)

= ,

which implies that (wn,x) ∈W . Applying Lemma .,

H(Twn,Tx) < d(wn,x) for all n ∈N with n ≥ n.

Since wn → v as n→ ∞ and

d(wn+,Tx)≤H(Twn,Tx) < d(wn,x) for all n ∈N with n≥ n, (.)

by taking the limit from both sides of (.), we get

d(v,Tx)≤ d(v,x).

Claim . v ∈F (T).

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We first prove (x, v) ∈ W for all x ∈ X�{v}. Suppose that there exists u ∈ X with u �= vsuch that (u, v) /∈W . So

ξ(d(u,Tu),d(u, v)

)< .

Note that for any n ∈ N, there exists zn ∈ Tu such that

d(v, zn) < d(v,Tu) +nd(v,u).

Thus, for any n ∈N, by Claim , we have

d(u,Tu) ≤ d(u, zn)

≤ d(u, v) + d(v, zn)

≤ d(u, v) + d(v,Tu) +nd(v,u)

≤( +

n

)d(v,u).

Hence we get d(u,Tu) ≤ d(v,u). Since ξ ∈ TRA(λ), there exists λ > , such that ξ (t, s) ≥s –

λt for all s, t ≥ . So, we obtain

> ξ(d(u,Tu),d(u, v)

)

≥ d(u, v) –λd(u,Tu)

> d(u, v) – d(u,Tu)

≥ ,

a contradiction. Therefore (x, v) ∈W for all x ∈ X�{v}. By Lemma ., we have

H(Tx,Tv) < d(x, v) for all x ∈ X�{v}.

Therefore,

H(Tx,Tv)≤ d(x, v) for all x ∈ X. (.)

From (.), we obtain

d(xn+,Tv)≤H(Txn,Tv)≤ d(xn, v) for all n ∈ N. (.)

By taking limit from both side of (.), we get d(v,Tv) = . By the closedness of Tv, wehave v ∈F (T). The proof is completed. �

Theorem. Let (X,d) be a complete metric space, T : X → CB(X) be amultivaluedmapand λ > . Assume that there exist an MT -function α : [,∞) → [, ) and a function

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β : [, +∞)→ [, +∞) such that, for x, y ∈ X,

λd(x,Tx)≤ β

(d(x, y)

)d(x, y) implies H(Tx,Ty)≤ α

(d(x, y)

)d(x, y). (.)

Then F (T) �= ∅.

Proof Define ξ :R×R →R and η :R×R →R, respectively, by

ξ (t, s) =

⎧⎨⎩sβ(s) –

λt, if (t, s) ∈ [, +∞)× [, +∞),

, otherwise,

and

η(t, s) =

⎧⎨⎩sα(s) – t, if (t, s) ∈ [, +∞)× [, +∞),, otherwise.

Thus ξ ∈ TRA(λ). By Example ., we know η ∈ Man(R). By (.), we obtain

η(H(Tx,Ty),d(x, y)

) ≥ for all (x, y) ∈W ,

where

W ={(x, y) ∈ X ×X : ξ

(d(x,Tx),d(x, y)

) ≥ }.

Therefore the desired conclusion follows from Theorem . immediately. �

In Theorem ., if we take β(t) = for all t ≥ , then we obtain the following new gen-eralization of Mizoguchi-Takahashi’s fixed point theorem.

Theorem . Let (X,d) be a complete metric space, T : X → CB(X) be a multivaluedmap and λ > . Assume that there exists anMT -function α : [,∞)→ [, ) such that forx, y ∈ X,

d(x,Tx)≤ λd(x, y) implies H(Tx,Ty)≤ α(d(x, y)

)d(x, y).

Then F (T) �= ∅.

Remark . Theorems ., . and . generalize and improve Mizoguchi-Takahashi’sfixed point theorem, Nadler’s fixed point theorem, and Banach contraction principle.

Finally, a question arises naturally.

Question Can we give new generalizations of Mizoguchi-Takahashi’s fixed point theo-rem with other new local constraints which also extend Kikkawa-Suzuki’s fixed point the-orem?

Du et al. Fixed Point Theory and Applications 2014, 2014:31 Page 12 of 12http://www.fixedpointtheoryandapplications.com/content/2014/1/31

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details1Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan. 2Department ofMathematics, Arak Branch, Islamic Azad University, Arak, Iran. 3School of Mathematics, Institute for Studies in TheoreticalPhysics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran.

AcknowledgementsThe first author was supported by grant no. NSC 102-2115-M-017-001 of the National Science Council of the Republic ofChina.

Received: 2 November 2013 Accepted: 22 January 2014 Published: 07 Feb 2014

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10.1186/1687-1812-2014-31Cite this article as: Du et al.: Some generalizations of Mizoguchi-Takahashi’s fixed point theorem with new localconstraints. Fixed Point Theory and Applications 2014, 2014:31