Multivalued F-Contractions and Some Fixed Point...

Multivalued F-Contractions and Some Fixed PointResults

_ISHAK ALTUNInternational Summer Workshop in Applied Topology

ISWAT 2014Valencia, Spain

1-2 September 2014

_Ishak ALTUN ()Multivalued F-Contractions 1-2 September 2014 1 / 53

Introduction and preliminaries

Fixed point theory has various applications in many di�erent �elds of mathematics suchas nonlinear functional analysis, mathematical analysis, operator theory and generaltopology. The �xed point theory is divided into three major areas:

First is the topological �xed point theory, which attributed to the work of Brouwerin 1910, who proved that any continuous self-map of the closed unit ball of Rn hasa �xed point. The results of Schauder (1930), Darbo (1955), Krasnoselskii (1955)and M�onch (1980) are related to these directions.

Second is the discrete �xed point theory, which begins to the work of Kneser in1950, who proved that: Let (X ,�) be a partially ordered set and T be a selfmapping of X such that x � Tx for all x 2 X . If every chain in X has asupremum, then T has a �xed point. The results of Tarski (1955) and Aman(1977) are related to these directions.

Third is the metrical �xed point theory on contraction or contraction typemappings on complete metric spaces. The metrical �xed point theory based on theBanach Contraction Principle, published in 1922.


Let (X , d) be a metric space and T : X ! X be a mapping. Then T is said to be acontraction (ordinary contraction) mapping if there exists a constant L 2 [0, 1), called acontraction factor, such that

d(Tx ,Ty) � Ld(x , y) for all x , y 2 X . (1)

Banach Contraction Principle says that any contraction self-mappings on a completemetric space has a unique �xed point. This principle is one of a very power test forexistence and uniqueness of the solution of considerable problems arising inmathematics. Because of its importance for mathematical theory, Banach ContractionPrinciple has been extended and generalized in many directions.

One of the most interesting generalization of it was given by Wardowski [1]. First werecall the concept of F -contraction, which was introduced by Wardowski [1], later wewill men�tion his result.

1D. Wardowski, Fixed points of a new type of contractive mappings in completemetric spaces, Fixed Point Theory Appl. 2012, 2012:94, 6 pp.


Let F be the set of all functions F : (0,∞)! R satisfying the followingconditions:

(F1) F is strictly increasing, i.e., for all α, β 2 (0,∞) such that α < β,F (α) < F (β),(F2) For each sequence fαng of positive numbers limn!∞ αn = 0 if and only iflimn!∞ F (αn) = �∞,(F3) There exists k 2 (0, 1) such that limα!0+ αkF (α) = 0.

Some examples of the functions belonging F are F1(α) = ln α, F2(α) = α+ ln α,F3(α) = � 1p

α, F4(α) = ln

�α2 + α

�and F5(α) = � 1

arctan αt for t 2 (0, 1).

De�nition (Wardowski [1])

Let (X , d) be a metric space and T : X ! X be a mapping. Then T is said to be anF -contraction if there exist F 2 F and τ > 0 such that

τ + F (d(Tx ,Ty)) � F (d(x , y)) (2)

for all x , y 2 X with d(Tx ,Ty) > 0.



From (F1), we say that every F -contraction T is a contractive mapping, i.e.,

d(Tx ,Ty) < d(x , y), for all x , y 2 X ,Tx 6= Ty .

Thus, every F -contraction is a continuous mapping.

Also, it is easy to see that every ordinary contraction mapping is an F -contractionwith F1(α) = ln α.

If we consider F2(α) = α+ ln α. Then each self mappings T on a metric space(X , d) satisfying (2) is an F2-contraction such that

d(Tx ,Ty)

d(x , y)ed(Tx ,Ty)�d(x ,y) � e�τ, for all x , y 2 X ,Tx 6= Ty . (3)

Also, Wardowski concluded that if F ,G 2 F with F (α) � G (α) for all α > 0 andH = G � F is nondecreasing, then every F -contraction T is an G -contraction.


He noted that for the mappings F1(α) = ln α and F2(α) = α+ ln α, F1 < F2 and amapping F2 � F1 is strictly increasing. Hence, it obtained that every ordinarycontraction satis�es the contractive condition (3). On the other side, the followingexample, which is Example 2.5 in [1], shows that the mapping T is not F1-contraction(ordinary contraction), but still is an F2-contraction.

Example

Let X = fxn = n(n+1)2 : n 2 Ng and d(x , y) = jx � y j. De�ne the mapping T : X ! X

by T (x1) = x1 and T (xn) = xn�1 for n > 1. Since limn!∞d(Txn,Tx1)d(xn,x1)

= 1, the mapping

T is not ordinary contraction. But after the some calculation we can see that T is anF2-contraction with F2(α) = α+ ln α and τ = 1.



Thus, the following theorem, which was given by Wardowski, is a proper generalizationof Banach Contraction Principle.

Theorem (Wardowski [1])

Let (X , d) be a complete metric space and let T : X ! X be an F-contraction. Then Thas a unique �xed point in X .



By combining the ideas of Wardowski, �Ciri�c and Berinde, the following results for singlevalued mappings are obtained.

Theorem (M�nak et al. [2], Wardowski-Van Dung [3])

Let (X , d) be a complete metric space and T : X ! X be a �Ciri�c type generalizedF -contraction, that is, there exist F 2 F and τ > 0 such that

τ + F (d(Tx ,Ty)) � F (M(x , y))

for all x , y 2 X with d(Tx ,Ty) > 0, where

M(x , y) = maxfd(x , y), d(x ,Tx), d(y ,Ty), 12[d(x ,Ty) + d(y ,Tx)]g.

If T or F is continuous, then T has a unique �xed point in X .

2G. M�nak, A. Helvac� and I. Altun, �Ciri�c type generalized F -contractions oncomplete metric space and �xed point results, Filomat, Accepted.

3D. Wardowski and N. Van Dung, Fixed points of F -weak contractions on completemetric spaces, Demonstratio Mathematica, 47 (1) (2014), 146-155.


Theorem (M�nak et al. [2])

Let (X , d) be a complete metric space and T : X ! X be an almost F -contraction,that is, there exist F 2 F , τ > 0 and λ � 0 such that

τ + F (d(Tx ,Ty)) � F (d(x , y) + λd(y ,Tx))

for all x , y 2 X with d(Tx ,Ty) > 0. Then T has a �xed point in X .

We can �nd some detailed information about �Ciri�c type generalized F -contractions,almost F -contractions and some counter examples in [2,3].

2G. M�nak, A. Helvac� and I. Altun, �Ciri�c type generalized F -contractions oncomplete metric space and �xed point results, Filomat, Accepted.

3D. Wardowski and N. Van Dung, Fixed points of F -weak contractions on completemetric spaces, Demonstratio Mathematica, 47 (1) (2014), 146-155.


Fixed point theory for multivalued maps

In this section, we recall some fundamental �xed point theorems for multivaluedmappins on a complete metric space. Let (X , d) be a metric space.

P(X ) denotes the family of all nonempty subsets of X ,

C (X ) denotes the family of all nonempty, closed subsets of X ,

CB(X ) denotes the family of all nonempty, closed and bounded subsets of X ,

K (X ) denotes the family of all nonempty compact subsets of X .

It is clear thatK (X ) � CB(X ) � C (X ) � P(X ).

For A,B 2 C (X ), let

H(A,B) = max

(supx2A

d(x ,B), supy2B

d(y ,A)

),

where d(x ,B) = inf fd(x , y) : y 2 Bg. Then H is called generalizedPompeiu-Hausdor� distance on C (X ).

It is well known that H is a metric on CB(X ), which is called Pompeiu-Hausdor�metric induced by d .


In 1969, Nadler �rst extended the Banach contraction principle to multivaluedmappings, Nadler [4] �rst initiated the study of �xed point theorems for multivaluedcontraction mappings.

Theorem (Nadler [4])

Let (X , d) be a complete metric space and T : X ! CB(X ) be a multivaluedcontraction, that is, there there exists L 2 [0, 1) such that

H(Tx ,Ty) � Ld(x , y)

for all x , y 2 X . Then T has a �xed point in X .

Then many researchers studied on �xed points of multivalued contractive mappings,which some important of them as follows:

4S.B. Nadler, Multi-valued contraction mappings, Paci�c J. Math., 30 (1969),475-488.


Theorem (Reich [5])

Let (X , d) be a complete metric space and T : X ! K (X ). Assume that there exists amap ϕ : (0,∞)! (0, 1) such that

lim supt!s+

ϕ(t) < 1, 8s > 0;

andH(Tx ,Ty) � ϕ(d(x , y))d(x , y).

for all x , y 2 X with x 6= y. Then T has a �xed point in X .

In [6], Reich asked the question as if the above theorem is also true for the mapT : X ! CB(X ). The partial a�rmative answer was given by Mizoguchi and Takahashi[7]. They proved the following theorem.

5S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972),26-42.

6S. Reich, Some �xed point problems, Atti Acad. Naz. Lincei 57 (1974), 194-198.7N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on

complete metric spaces, J. Math. Anal. Appl.,141 (1989), 177-188._Ishak ALTUN ()Multivalued F-Contractions 1-2 September 2014 12 / 53

Theorem (Mizoguchi-Takahashi [7])

Let (X , d) be a complete metric space and T : X ! CB(X ). Assume that there exists amap ϕ : (0,∞)! (0, 1) such that

lim supt!s+

ϕ(t) < 1, 8s � 0;

andH(Tx ,Ty) � ϕ(d(x , y))d(x , y).

for all x , y 2 X with x 6= y. Then T has a �xed point in X .

In [8] Suzuki gave a simple proof of Mizoguchi Takahashi �xed point theorem and alsoan example to show that it is a real generalization of Nadler's.

7N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings oncomplete metric spaces, J. Math. Anal. Appl.,141 (1989), 177-188.

8T. Suzuki, Mizoguchi Takahashi's �xed point theorem is a real generalization ofNadler's, J. Math. Anal. Appl. 340 (2008), 752 755.


Following the above results, Berinde and Berinde [9] introduced a general class ofmultivalued contractions and proved the following �xed point theorems:

Theorem (Berinde-Berinde [9])

Let (X , d) be a complete metric space and T : X ! CB(X ) be a multivalued almostcontraction, that is, there exist two constants δ 2 (0, 1) and L � 0 such that

H(Tx ,Ty) � δd(x , y) + Ld(y ,Tx) (4)

for all x , y 2 X. Then T has a �xed point in X . .

9M. Berinde and V. Berinde, On a general class of multi-valued weakly Picardmappings, J. Math. Anal. Appl., 326 (2007), 772-782.


Theorem (Berinde-Berinde [9])

Let (X , d) be a complete metric space and T : X ! CB(X ) be a multivalued nonlinearalmost contraction, that is, there exist a constant L � 0 and a functionϕ : [0,∞)! [0, 1) satisfying

lim supt!s+

ϕ(t) < 1, 8s � 0, (5)

such thatH(Tx ,Ty) � ϕ(d(x , y))d(x , y) + Ld(y ,Tx) (6)

for all x , y 2 X. Then T has a �xed point in X .

TheoremA function ϕ : [0,∞)! [0, 1) satisfying (5) is called Mizoguchi-Takahashi function(MT-function) in the literature.

9M. Berinde and V. Berinde, On a general class of multi-valued weakly Picardmappings, J. Math. Anal. Appl., 326 (2007), 772-782.


On the other hand, without using the Pompeiu-Hausdor� metric H, many �xed pointresults for multivalued mappings were obtained. Here we will mention some importantof them. For the sake of conformity we denote a set

I xb = fy 2 Tx : bd(x , y) � d(x ,Tx)g,

where b is a real constant and T is a multivalued mapping on a metric space X . Notethat the mapping T is de�ned from X to C (X ) in the following three theorems.

Theorem (Feng-Liu [10])

Let (X , d) be a complete metric space and T : X ! C (X ). Assume that the followingconditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exist b, c 2 (0, 1) with c < b such that for any x 2 X there is y 2 I xb satisfying

d(y ,Ty) � cd(x , y).

Then T has a �xed point in X .

10Y. Feng and S. Liu, Fixed point theorems for multi-valued contractive mappings andmulti-valued Caristi type mappings, J. Math. Anal. Appl., 317 (2006), 103-112.


Then Klim and Wardowski [11] generalized the Feng-Liu's result as follows:

Theorem (Klim-Wardowski [11])

Let (X , d) be a complete metric space and T : X ! C (X ). Assume that the followingconditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exists b 2 (0, 1) and a function ϕ : [0,∞)! [0, b) satisfying

lim supt!s+

ϕ(t) < b, 8s � 0

and for any x 2 X , there is y 2 I xb satisfying

d(y ,Ty) � ϕ(d(x , y))d(x , y).

Then T has a �xed point in X .

11D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions incomplete metric spaces, J. Math. Anal. Appl., 334 (2007), 132-139.


Considering the same direction, in 2009, �Ciri�c [12] introduced new multivalued nonlinearcontractions and established a few nice �xed point theorems for such mappings, one ofthem is as follows:

Theorem (�Ciri�c [12])

Let (X , d) be a complete metric space and T : X ! C (X ). Assume that the followingconditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exists a function ϕ : [0,∞)! [a, 1) , 0 < a < 1, satisfying

lim supt!s+

ϕ(t) < 1, 8s � 0;

(iii) for any x 2 X , there is y 2 Tx satisfyingqϕ(d(x ,Tx))d(x , y) � d(x ,Tx)

andd(y ,Ty) � ϕ(d(x ,Tx))d(x , y).

Then T has a �xed pointin X .

12Lj. B. �Ciri�c, Multi-valued nonlinear contraction mappings, Nonlinear Anal., 71(2009), 2716-2723.


Analyzing the proofs of above all theorems, we can observe that the mentionedmappings on complete metric spaces are multivalued weakly Picard (MWP)operators. We know that, a multivalued map T on a metric space is MWP operator ifthere exists a sequence fxng in X such that xn+1 2 Txn for any initial point x0,converges to a �xed point of T .


Multivalued F-contractions

In this section we consider the Wardowski's technique for multivalued mappings.

De�nition

Let (X , d) be a metric space and T : X ! CB(X ). Then T is said to be a multivaluedF -contraction if there exist F 2 F and τ > 0 such that

τ + F (H(Tx ,Ty)) � F (d(x , y)) (7)

for all x , y 2 X with H(Tx ,Ty) > 0.

When we consider F (α) = ln α, we can say that every multivalued contraction is alsomultivalued F -contraction.

Theorem (Altun et al. [13])

(Theorem MF1)Let (X , d) be a complete metric space and T : X ! K (X ) be amultivalued F -contraction, then T has a �xed point in X .

13I. Altun, G. M�nak and H. Da�g, Multivalued F -contractions on complete metricspace, Journal of Nonlinear and Convex Analysis, Accepted.


In the proof of this theorem we use the following important property: Let A be acompact subset of a metric space (X , d) and x 2 X , then there exists a 2 A such thatd(x , a) = d(x ,A).

Proof.Let x0 2 X . As Tx is nonempty for all x 2 X , we can choose x1 2 Tx0. If x1 2 Tx1,then x1 is a �xed point of T and so the proof is complete. Let x1 /2 Tx1. Then, sinceTx1 is closed, d(x1,Tx1) > 0. On the other hand, from d(x1,Tx1) � H(Tx0,Tx1) and(F1)

F (d(x1,Tx1)) � F (H(Tx0,Tx1)).From (7), we can write that

F (d(x1,Tx1)) � F (H(Tx0,Tx1)) � F (d(x1, x0))� τ (8)

Since Tx1 is compact, we obtain that there exists x2 2 Tx1 such thatd(x1, x2) = d(x1,Tx1). Then, from (8)

F (d(x1, x2)) � F (H(Tx0,Tx1)) � F (d(x1, x0))� τ

If we continue recursively, then we obtain a sequence fxng in X such that xn+1 2 Txnand

F (d(xn, xn+1)) � F (d(xn, xn�1))� τ (9)

for all n = 0, 1, 2, � � � .


Proof.If there exists n0 2 N for which xn0 2 Txn0 , then xn0 is a �xed point of T and so theproof is complete. Thus, suppose that for every n 2 N, xn /2 Txn. Denotean = d(xn, xn+1), for n = 0, 1, 2, � � � . Then an > 0 for all n 2 N and, using (9), thefollowing holds:

F (an) � F (an�1)� τ � F (an�2)� 2τ � � � � � F (a0)� nτ. (10)

From (10), we get limn!∞ F (an) = �∞. Thus, from (F2), we have

limn!∞

an = 0.

From (F3) there exists k 2 (0, 1) such that

limn!∞

aknF (an) = 0.

By (10), the following holds for all n 2 N

aknF (an)� aknF (a0) � �aknnτ � 0. (11)

Letting n! ∞ in (11), we obtain that

limn!∞

nakn = 0. (12)


Proof.

From (12), there exits n1 2 N such that nakn � 1 for all n � n1. So, we have, for alln � n1

an �1

n1/k . (13)

In order to show that fxng is a Cauchy sequence consider m, n 2 N such thatm > n � n1. Using the triangular inequality for the metric and from (13), we have

d(xn, xm) � d(xn, xn+1) + d(xn+1, xn+2) + � � �+ d(xm�1, xm)= an + an+1 + � � �+ am�1

=m�1∑i=n

ai �∞

∑i=n

ai �∞

∑i=n

1

i1/k

By the convergence of the series∞∑i=1

1i1/k , passing to limit n! ∞, we get

d(xn, xm) ! 0. This yields that fxng is a Cauchy sequence in (X , d). Since (X , d) is acomplete metric space, the sequence fxng converges to some point z 2 X , thatis,limn!∞ xn = z . Taking into accout (7), we have

d(xn+1,Tz) � H(Txn,Tz) � d(xn, z).

Passing to limit n! ∞, we obtain d(z ,Tz) = 0.Thus, we get z 2 Tz = Tz .


Note that in this theorem, Tx is compact for all x 2 X . Thus, we can present thefollowing problem: Can we replace CB(X ) instead of K (X ) in this theorem. In thefollowing example shows that this is not possible with the same conditions.

Example (Atun et al. [14])

Let X = [0, 1] and

d(x , y) =

8<:0 , x = y

1+ jx � y j , x 6= y,

then it is clear that (X , d) is complete metric space, which is also bounded. Since τd isdiscrete topology, all subsets of X are closed. Therefore all subsets of X are closed andbounded. De�ne T : X ! CB(X ) as:

Tx =

8<:A , x 2 B

B , x 2 A,

where A is the set of all rational numbers in X and B is the set of all irrational numbersin X . Therefore T has no �xed point.

14I. Altun, G. Durmaz, G. M�nak and S. Romaguera, Multivalued almostF -contractions on complete metric spaces, Filomat, Accepted.


Example

Now, de�ne F : (0,∞)! R by

F (α) =

8<:ln α , α � 1

α , α > 1,

then we can see that F 2 F . After the some calculation, we can see that T ismultivalued F -contraction. Consequently all conditions of the above theorem except forTx is compact are satis�ed, but T has no �xed point.

Here, if we consider the following condition on F , we can take CB(X ) instead ofK (X ) in this theorem.

(F4) F (inf A) = inf F (A) for all A � (0,∞) with inf A > 0.

Note that if F satis�es (F1), then it satis�es (F4) if and only if it is rightcontinuous.

We denote by F� be the set of all functions F satisfying (F1)-(F4). For example,let F (α) = ln α for α � 1 and F (α) = 2α for α > 1, then it is clear that F 2 FnF�.



(Theorem MF2)Let (X , d) be a complete metric space and T : X ! CB(X ) be amultivalued F -contraction with F 2 F�, then T has a �xed point in X .

Proof.As in the proof of the above theorem we can write that

F (d(x1,Tx1)) � F (H(Tx0,Tx1)) � F (d(x1, x0))� τ. (14)

From (F4) we obtain (note that d(x1,Tx1) > 0 )

F (d(x1,Tx1)) = infy2Tx1

F (d(x1, y)),

and so from (14) we have

infy2Tx1

F (d(x1, y)) � F (d(x1, x0))� τ < F (d(x1, x0))�τ

2. (15)

Then, from (15) there exists x2 2 Tx1 such that

F (d(x1, x2)) � F (d(x1, x0))�τ

2.

The rest of the proof can be completed as in the proof of previous theorem.


In the light of the Wardowski's example, we can give the following. This example showsthat T is a multivalued F -contraction but it is not multivalued contraction.

Example (Altun et al. [13])

Let X = fxn = n(n+1)2 : n 2 Ng and d(x , y) = jx � y j . Then (X , d) is a complete

metric space. De�ne the mapping T : X ! CB(X ) by the formulae:

Tx =

8<: fx1g , x = x1

fx1, x2, � � � , xn�1g , x = xn

.

Then T is a multivalued F -contraction with respect to F (α) = α+ ln α and τ = 1, butit is not multivalued contraction.

13I. Altun, G. M�nak and H. Da�g, Multivalued F -contractions on complete metricspace, Journal of Nonlinear and Convex Analysis, Accepted.


Now we consider τ as a function of d(x , y) in the de�nition of multivaluedF -contraction and de�ne a new concept of multivalued nonlinear F -contraction. Thenwe give some �xed point results for mappings of this type on complete metric spaces. Ina special case, we obtain the Mizoguchi-Takahashi �xed point theorem.

De�nition

Let (X , d) be a metric space, T : X ! CB(X ) and τ : (0,∞)! (0,∞) be twomappings. Given F 2 F , we say that T is a multivalued nonlinear F -contraction suchthat

τ(d(x , y)) + F (H(Tx ,Ty)) � F (d(x , y)) (16)



Theorem (Olgun et al. [15])

(Theorem MF3)Let (X , d) be a complete metric space and T : X ! K (X ) be amultivalued nonlinear F -contraction. If τ satis�es

lim inft!s+

τ(t) > 0, for all s � 0,

then T has a �xed pointin X .

By considering the condition (F4) we can prove the following:

Theorem (Olgun et al. [15])

(Theorem MF4)Let (X , d) be a complete metric space and T : X ! CB(X ) be amultivalued nonlinear F -contraction with F 2 F�. If τ satis�es

lim inft!s+

τ(t) > 0, for all s � 0,

then T has a �xed point in X .

15M. Olgun, G. M�nak and I. Altun, A new approach to Mizoguchi-Takahashi type�xed point theorem, Journal of Nonlinear and Convex Analysis, Accepted.


If we take F (α) = ln α in the last theorem, we have the following corollaries.

Corollary

Let (X , d) be a complete metric space. Suppose that T : X ! CB(X ) satis�es

H(Tx ,Ty) � e�τ(d(x ,y))d(x , y),

for all x , y 2 X, x 6= y, where τ : (0,∞)! (0,∞) satisfying lim inft!s+

τ(t) > 0 for all

s � 0. Then T has a �xed point in X .

Corollary (Mizoguchi-Takahashi)

Let (X , d) be a complete metric space. Suppose that T : X ! CB(X ) satis�es

H(Tx ,Ty) � ϕ(d(x , y))d(x , y),

for all x , y 2 X, x 6= y, where ϕ : (0,∞)! (0, 1) satisfying lim supt!s+

ϕ(t) < 1 for all

s � 0. Then T has a �xed point in X .

Proof.De�ne τ(t) = � ln ϕ(t). Since lim sup

t!s+ϕ(t) < 1 for all s � 0, then lim inf

t!s+τ(t) > 0 for all

s � 0. Therefore, by the previous corollary, the proof is complete._Ishak ALTUN ()Multivalued F-Contractions 1-2 September 2014 30 / 53

By considering the almost contraction method, we introduce some new concept ofmultivalued almost F -contraction and multivalued nonlinear almost F -contraction. Thenwe give some �xed point results for mappings of these type on complete metric spaces.In a special case, we obtain the Berinde-Berinde �xed point theorem.

De�nition

Let (X , d) be a metric space and T : X ! CB(X ). We say that T is a multivaluedalmost F -contraction if there exist F 2 F , τ > 0 and λ � 0 such that

τ + F (H(Tx ,Ty)) � F ((d(x , y) + λd(y ,Tx)) (17)



(Theorem MF5)Let (X , d) be a complete metric space and T : X ! CB(X ) be amultivalued almost F -contraction with F 2 F�, then T is an MWP operator.



RemarkTaking into account the example in [14], we can say that the condition (F4) on F cannot be removed in this theorem. But, if we take T : X ! K (X ), we can remove thecondition (F4) on F .

RemarkIf there exist δ 2 (0, 1) and L � 0 satisfying (4) (multivalued almost contractioncondition), then (17) is satis�ed with F (α) = ln α, τ = � ln δ and λ = L

δ . Therefore,the result of Berinde-Berinde is a special case of this theorem.

RemarkIf there exist τ > 0 and F 2 F� satisfying (7) (multivalued F -contraction condition),then (17) is satis�ed with λ = 0. Therefore, the result of Altun et al is a special case ofthis theorem.



Now we consider the following two examples.



metric space. De�ne a mapping T : X ! CB(X ) by:

Tx =

8<: fx1g , x = x1

fx1, x2, � � � , xn�1g , x = xn

.

Then, as shown in previous example, T is multivalued almost F -contraction with respectto F (α) = α+ ln α, τ = 1 and λ � 0. Thus T is an MWP operator. But it can beshown that T is not multivalued almost contraction.




Let X = [0, 1] [ f2, 3g and d(x , y) = jx � y j, then (X , d) is complete metric space.De�ne a map T : X ! CB(X ),

Tx =

8<:�1�x3 , 1�x2

�, x 2 [0, 1]

fxg , x 2 f2, 3g.

Since H(T2,T3) = 1 = d(2, 3), then for all F 2 F and τ > 0 we have

τ + F (H(T2,T3)) > F (d(2, 3)).

Therefore, T is not a multivalued F -contraction. But after the some calculation, we cansee that T is multivalued almost F -contraction with F (α) = ln α, τ = ln 2 and λ = 10.



By considering τ as a function of d(x , y) in the de�nition of multivalued almostF -contraction, we give a new concept of multivalued nonlinear almost F -contraction.

De�nition

Let (X , d) be a metric space and T : X ! CB(X ). We say that T is a multivaluednonlinear almost F -contraction with F 2 F if there exist a constant λ � 0 and afunction τ : (0,∞)! (0,∞) such that

lim inft!s+

τ(t) > 0, for all s � 0 (18)

satisfyingτ(d(x , y)) + F (H(Tx ,Ty)) � F ((d(x , y) + λd(y ,Tx)) (19)


RemarkIt is clear that every multivalued almost F -contraction is also multivalued nonlinearalmost F -contraction.


RemarkEvery multivalued nonlinear almost contraction is also multivalued nonlinear almostF -contraction with a special F . Indeed, let (X , d) be metric space and T be amultivalued nonlinear almost contraction. Then, there exist a constant L � 0 and anMT-function ϕ satisfying

H(Tx ,Ty) � ϕ(d(x , y))d(x , y) + Ld(y ,Tx) (20)

for all x , y 2 X. De�ne β(t) =1+ϕ(t)2 , then β is also an MT-function. Therefore from

(20),H(Tx ,Ty) � β(d(x , y))[d(x , y) + 2Ld(y ,Tx)]

for all x , y 2 X with H(Tx ,Ty) > 0. Thus, we get

� ln(β(d(x , y))) + ln(H(Tx ,Ty)) � ln(d(x , y) + 2Ld(y ,Tx)) (21)

for all x , y 2 X with H(Tx ,Ty) > 0. Now, de�ne τ(t) = � ln β(t). Since β is anMT-function, then

lim inft!s+

τ(t) > 0 for all s � 0.

Therefore, from (21), T is a multivalued nonlinear almost F -contraction with

F (α) = ln α, λ = 2L and τ(t) = � ln�1+ϕ(t)2

�.



(Theorem MF6)Let (X , d) be a complete metric space and T : X ! CB(X ) be amultivalued nonlinear almost F -contraction with F 2 F�, then T is an MWP operator.

Example (M�nak et al. [16])

Consider the complete metric space (X , d), where X = f 1n2

: n 2 N, n � 2g [ f0g andd : X � X ! [0,∞) is given by d(x , y) = jx � y j . De�ne T : X ! CB(X ) by

Tx =

8><>:n0, 1(n+1)2

o, x = 1

n2, n > 2

fxg , x =�0, 14

.

Since H(T0,T 14 ) =

14 = d(0,

14 ), then for all F 2 F� and τ : (0,∞)! (0,∞) satisfying

inequality (18), we have

τ(d(0,1

4)) + F (H(T0,T

1

4)) > F (d(0,

1

4)).

16G. M�nak, I. Altun and S. Romaguera, Recent developments about multivaluedweakly Picard operators, Submitted.


ExampleTherefore T is not multivalued nonlinear F -contraction. We can also see that T is notmultivalued nonlinear almost contraction. But T is multivalued nonlinear almostF -contraction with λ = 1, τ = ln 10081 and

F (α) =

8><>:ln αp

α, 0 < α < e2

2αe3

, α � e2.


Fixed point results without using Pompeiu-Hausdor�metric

Let T : X ! P(X ) be a multivalued map, F 2 F and σ � 0. For x 2 X withd(x ,Tx) > 0, de�ne the set F xσ � X as

F xσ = fy 2 Tx : F (d(x , y)) � F (d(x ,Tx)) + σg.We need to consider the following cases:

If T : X ! K (X ), then for all σ � 0 and x 2 X with d(x ,Tx) > 0, we haveF xσ 6= ∅. Indeed, since Tx is compact, for every x 2 X we have y 2 Tx such thatd(x , y) = d(x ,Tx). Therefore, for every x 2 X with d(x ,Tx) > 0, we haveF (d(x , y)) = F (d(x ,Tx)). Thus y 2 F xσ for all σ � 0.If T : X ! C (X ), then F xσ may be empty for some x 2 X and σ > 0. Forexample, let F (α) = ln α for α � 1 and F (α) = 2α for α > 1 and letX = f0g [ (1, 2) with the usual metric. De�ne T : X ! C (X ) by T0 = (1, 2) andTx = f0g for x 2 (1, 2). Then, for x = 0 we have (note that d(0,T0) = 1 > 0)

F 01 = fy 2 T0 : F (d(0, y)) � F (d(0,T0)) + 1g= fy 2 (1, 2) : F (y) � F (1) + 1g= fy 2 (1, 2) : 2y � 1g= ∅.


If T : X ! C (X ) (even if T : X ! P(X )) and F 2 F�, then for all σ > 0 andx 2 X with d(x ,Tx) > 0, we have F xσ 6= ∅. Indeed, by (F4), we have

F xσ = fy 2 Tx : F (d(x , y)) � F (d(x ,Tx)) + σg= fy 2 Tx : F (d(x , y)) � F (inffd(x , y) : y 2 Txg) + σg= fy 2 Tx : F (d(x , y)) � inffF (d(x , y)) : y 2 Txg+ σg6= ∅.


By considering the above facts we give the following theorems:


(Theorem MF7)Let (X , d) be a complete metric space, T : X ! K (X ) be amultivalued map and F 2 F . If there exists τ > 0 such that for any x 2 X withd(x ,Tx) > 0, there exists y 2 F xσ satisfying

τ + F (d(y ,Ty)) � F (d(x , y)),

then T has a �xed point in X provided σ < τ and x ! d(x ,Tx) is lowersemi-continuous.


(Theorem MF8)Let (X , d) be a complete metric space, T : X ! C (X ) and F 2 F�. Ifthere exists τ > 0 such that for any x 2 X with d(x ,Tx) > 0, there exists y 2 F xσsatisfying

τ + F (d(y ,Ty)) � F (d(x , y))then T has a �xed point in X provided 0 < σ < τ and x ! d(x ,Tx) is lowersemi-continuous.

17G. M�nak, M. Olgun and I. Altun, A new approach to �xed point theorems formultivalued contractive maps Carpathian Journal of Mathematics, Accepted.


Corollary (Feng-Liu)

Let (X , d) be a complete metric space and T : X ! C (X ). If there exists c 2 (0, 1)such that for any x 2 X, there exists y 2 I xb (b 2 (0, 1)) satisfying

d(y ,Ty) � cd(x , y),

then T has a �xed point in X provided c < b and x ! d(x ,Tx) is lowersemi-continuous.

RemarkTheorem MF7 is a generalization of Theorem MF1. In fact, let T satis�es theconditions of Theorem MF1. Since every multivalued F -contractions are multivaluednonexpansive and every multivalued nonexpansive maps are upper semi-continuous, thenT is upper semi-continuous. Therefore, the function x ! d(x ,Tx) is lowersemi-continuous (see the Proposition 4.2.6 of [18]). On the other hand, for any x 2 Xwith d(x ,Tx) > 0 and y 2 F xσ , we have

τ + F (d(y ,Ty)) � τ + F (H(Tx ,Ty)) � F (d(x , y)).

Hence T satis�es conditions of Theorem MF7, the existence of a �xed point has beenproved. There is the similar relation between Theorem MF2 and Theorem MF8.

18R. P. Agarwal, D. O'Regan and D. R. Sahu, Fixed Point Theory forLipschitzian-type Mappings with Applications, Springer, New York, 2009.



Let X = f 12n�1 : n 2 Ng [ f0g with the usual metric d , then (X , d) is a complete

metric space. De�ne a mapping T : X ! C (X ) as

Tx =

8<:f 12n , 1g , x = 1

2n�1

f0, 12g , x = 0.

Since H(T 12 ,T0) =

12 = d(

12 , 0), then for all F 2 F and τ > 0 we have

τ + F (H(T1

2,T0)) > F (d(

1

2, 0)).

Thus T is not multivalued F -contraction. Therefore Theorem MF1 and Theorem MF2can not be applied to this example.On the other hand, it is easy to compute that all conditions of Theorem MF7 andTheorem MF8 are satis�ed and so T has a �xed point.



In the following theorem we replace P(X ) by C (X ), but we need to add an extracondition.


(Theorem MF9)Let (X , d) be a complete metric space, T : X ! P(X ) and F 2 F�.Suppose there exists τ > 0 such that for any x 2 X with d(x ,Tx) > 0, there existsy 2 F xσ satisfying d(y ,Ty) > 0 and

τ + F (d(y ,Ty)) � F (d(x , y)).

If there exists x0 2 X with d(x0,Tx0) > 0 such that for all convergent sequence fxngwith xn+1 2 Txn, we have T (lim xn) is closed, then T has a �xed point in X providedσ < τ and x ! d(x ,Tx) is lower semi-continuous.

Corollary

Let (X , d) be a complete metric space and T : X ! P(X ). Suppose there existsc 2 (0, 1) such that for any x 2 X with d(x ,Tx) > 0 there exists y 2 I xb (b 2 (0, 1))satisfying

0 < d(y ,Ty) � cd(x , y). (22)

If there exists x0 2 X with d(x0,Tx0) > 0 such that for all convergent sequence fxngwith xn+1 2 Txn, we have T (lim xn) is closed, then T has a �xed point in X providedc < b and x ! d(x ,Tx) is lower semi-continuous.




Let X = [0, 2] with the usual metric. De�ne T : X ! P(X ) as

Tx =

8<:( x4 ,

x2 ] , x 2 (0, 1]

f x2 g , x 2 f0g [ (1, 2].

Since Tx is not closed for some x 2 X , both Nadler and Feng-Liu's results can not beapplied to this example. On the other hand if we take 1

2 � c < b and x0 2 (0, 2], thenall conditions of the above Corollary are satis�ed. Therefore T has a �xed point.




(Theorem MF10)Let (X , d) be a complete metric space, T : X ! C (X ) and F 2 F�.Assume that the following conditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exist σ > 0 and a function τ : (0,∞)! (σ,∞) such that

lim inft!s+

τ(t) > σ for all s � 0

and for any x 2 X with d(x ,Tx) > 0, there exists y 2 F xσ satisfying

τ(d(x , y)) + F (d(y ,Ty)) � F (d(x , y)).

Then T has a �xed point.

19I. Altun, G. M�nak and M. Olgun, Fixed points of multivalued nonlinearF -contractions on complete metric spaces, Submitted.


In the following example, we show that there are some multivalued mappings such thatTheorem MF10 can be applied but Klim-Wardowski's result can not.

Example


metric space. De�ne a mapping T : X ! C (X ) by the formulae:

Tx =

8<: fx1g , x = x1

fx1, xn�1g , x = xn

.

Then, since τd is discrete topology, the map x ! d(x ,Tx) is continuous. Now we claimthat the condition (ii) of Klim-Wardowski's is not satis�ed. Indeed, let x = xn for n > 1,then Tx = fx1, xn�1g. In this case, for all b 2 (0, 1), there exists n0(b) 2 N such thatfor all n � n0(b), I xnb = fxn�1g. Thus, for n � n0(b) we have

d(y ,Ty) = n� 1, d(x , y) = n.

Therefore sinced(y ,Ty)d(x ,y)

= n�1n , we can not �nd a function ϕ : [0,∞)! [0, b) satisfying

d(y ,Ty) � ϕ(d(x , y))d(x , y).

On the other hand the condition (ii) of Theorem MF10 is satis�ed with F (α) = α+ ln α,σ = 1

2and τ(t) = 1t +

12 .


RemarkIf we take K (X ) instead of C (X ) in Theorem MF10, we can remove the condition (F4)on F . Further, by taking into account F xσ , we can take σ � 0. Therefore, the proof ofthe following theorem is obvious.


(Theorem MF11)Let (X , d) be a complete metric space and T : X ! K (X ). Assumethat the following conditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exist σ � 0, F 2 F and a function τ : (0,∞)! (σ,∞) such that

lim inft!s+

τ(t) > σ for all s � 0

and for any x 2 X with d(x ,Tx) > 0, there exists y 2 F xσ satisfying

τ(d(x , y)) + F (d(y ,Ty)) � F (d(x , y)).

Then T has a �xed point.

19I. Altun, G. M�nak and M. Olgun, Fixed points of multivalued nonlinearF -contractions on complete metric spaces, Submitted.



(Theorem MF12)Let (X , d) be a complete metric space, T : X ! C (X ) and F 2 F�.Assume that the following conditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exists a function τ : (0,∞)! (0, σ], σ > 0 such that

lim inft!s+

τ(t) > 0, 8s � 0; (23)

(iii) for any x 2 X with d(x ,Tx) > 0, there is y 2 Tx satisfying

F (d(x , y)) � F (d(x ,Tx)) + τ(d(x ,Tx))

2(24)

andτ(d(x ,Tx)) + F (d(y ,Ty)) � F (d(x , y)). (25)

Then T is a MWP operator.

20I. Altun, M. Olgun and G. M�nak, On a new class of multivalued weakly Picardoperators on complete metric spaces, Taiwanese Journal of Mathematics, Accepted.



(Theorem MF13)Let (X , d) be a complete metric space, T : X ! K (X ) and F 2 F .Assume that the following conditions hold:(i) the map x ! d(x ,Tx) is lower semi-continuous;(ii) there exists a function τ : (0,∞)! (0, σ], σ > 0 such that

lim inft!s+

τ(t) > 0, 8s � 0;

(iii) for any x 2 X with d(x ,Tx) > 0, there is y 2 Tx satisfying

F (d(x , y)) � F (d(x ,Tx)) + τ(d(x ,Tx))

2

andτ(d(x ,Tx)) + F (d(y ,Ty)) � F (d(x , y)).

Then T is a MWP operator.

20I. Altun, M. Olgun and G. M�nak, On a new class of multivalued weakly Picardoperators on complete metric spaces, Taiwanese Journal of Mathematics, Accepted.


Taking into account our results, T is a MWP operator in the following nontrivialexample. We also show that all mentioned theorems except for Theorems MF12 andMF13 can not be applied to this example.

Example

Let X = f 1n2

: n 2 Ng [ f0g and d(x , y) = jx � y j, then (X , d) is complete metricspace. Let T : X ! CB(X ) be de�ned by

Tx =

8><>:n0, 1(n+1)2

o, x = 1

n2

fxg , x 2 f0, 1g.

It is easy to see that

d(x ,Tx) =

8><>:0 , x 2 f0, 1g

2n+1n2(n+1)2

, x = 1n2, n � 2

and it is lower semi-continuous.


Example

Let τ(t) = ln 2 and σ = 4, then the condition (ii) of Theorem MF12 is satis�ed.We can see that the condition (iii) of Theorem MF12 is satis�ed with

F (α) =

8><>:ln αp

α, 0 < α < e2

α� e2 + 2e , α � e2

.

Thus all conditions of Theorem MF12 are satis�ed and so T has a �xed point in X .On the other hand, after some calculation we can see that the other all mentionedtheorems can not be applied to this example.


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