New problems concerning with F-contractions and [620042]

New problems concerning with F-contractions and
(;p )-contractions on dierent spaces
Arslan Hojat Ansari1, Naeem Saleem2;3, Liliana Guran4
Abstract. The aim of this paper is to prove that for the cases of F-contractions and ( ;p)-
contractions, some results from the related literature are not new. This new type of contractions
can be reduced, in some cases, to the usual contraction dened by Banach in [ 6].
1. Introduction and preliminaries
In 1922, in [ 6] S. Banach dened contraction mapping usual metric space. Let us recall this
notion from literature.
Definition 1.1.Let(X;d)be a metric space. Then a mapping f:X!Xis called a
contraction mapping on Xif there exists k2[0;1)such that:
(1) d(f(x);f(y))kd(x;y);for allx;yinX:
Recently, in [ 32] D. Wardowski introduced a new class of contraction mappings called F-
contractions. The class Fconsists upon the functions F:R+!R, satisfying the following
conditions:
(F1)Fis strictly increasing;
(F2) for each sequence fxngof positive numbers, lim
n!1xn= 0 if and only if lim
n!1F(xn) =1;
(F3) there exists k2(0;1) such that lim
n!1xk
nF(xn) = 0.
Definition 1.2.[32]Let(X;d)be a metric space. A mapping f:X!Xis called an
F-contraction, if there exist F2Fand2R+such that:
(2) +F(d(f(x);f(y)))F(d(x;y));
for allx;y2Xwithd(f(x);f(y))>0.
In [18] we nd the denition for altering distance function as follows.
Definition 1.3.[18]A function : [0;+1)![0;+1)is called a altering distance function,
if the following properties are satised:
(i) (0) = 0 ;
(ii) is continuous and monotonically non-decreasing.
We denote with the set of altering distance functions above dened.
In [8] we have the following xed point result.
2010 Mathematics Subject Classication. 46T99; 47H10; 54H25.
Key words and phrases. xed point, metric space, F-contraction.
1

2 A. H. ANSARI, N. SALEEM, L. GURAN
Theorem 1.1.[8]Let(X;D )be a complete b-metric-like space with the constant s1and
letf:X!Xbe a mapping such that
D(f(x);f(y))D(x;y)
s (D(x;y));for allx;y2Xand 2 :
Thenfhas a unique xed point.
In 1982, Istr at escu [ 14] introduced the class of convex contraction mappings in the setting
of metric space and generalized the well known Banach's contraction principle [ 6]. Recently,
some work have appeared on generalization of such classes of mappings, in the settings of metric,
ordered metric, and cone metric spaces (for example, Alghamdi et al. [ 2], Ghorbanian et al. [ 9],
Miandaragh et al. [ 21], Miculescu [ 22] etc.).
Then, let us recall the denition of ( ;p)-contractions.
Definition 1.4.[18]Let(X;d)be a metric space. A self mapping fonXis said to be
(;p)-contraction, if for some 2(0;1)andp1, there exists 0k<1satisfying the following
inequality:
(3) dp(f(x);f(y)) + (1)dp(f2x;f2y)kdp(x;y);for allx;y2X:
The purpose of this paper is to prove that, if we use simple mathematical operations, then
F-contraction, which is used in some papers from related literature is an usual contraction. On
the same lines, this problem holds true for the cases of ( ;p)-contractions aswell.
2. Remarks concerning F-contractions
In literature, many papers concerning xed point results for F-contractions, used the mapping
F:R+!Rsatisfying the conditions that Fis strictly increasing and continuous. Concerning
this, we have the following lemma.
Lemma 2.1.Let(X;d)be a metric space and F:R+!Rbe a mapping strictly increasing
and continuous. If the following inequality
(4) +F(d(f(x);f(y)))F(d(x;y));
holds true, then for every sequence fxng2Xdened byxn=f(xn1), for alln2N, then we have
d(xn;xn1)!0andxnis a Cauchy sequence.
Proof. We dened a sequence fxng2Xbyxn=f(xn1), for alln2N. If there exists some
n02Nsuch thatxn0=xn01thenxn0is a xed point of f.
Therefore, assume that xn6=xn1;for alln2N. From (4), we get
d(xn+1;xn)<d(xn;xn1)!r0:
Then
r= 0 orr>0:
Ifr>0, forn!1 , we have
F(r)<+F(r)F(r);
which generates a contradiction. Therefore r= 0. 
Hussain et al. [ 12] generalized the results of Wordowski [ 32] by introducing  Gset of functions
G:R+4!R+which satisfy the following condition:
(G) for allt1;t2;t3;t42R+witht1t2t3t4= 0 there exists  >0 such that G(t1;t2;t3;t4) =.
Starting, from the inequality (2), we have the following remark.

NEW PROBLEMS CONCERNING WITH F-CONTRACTIONS AND 3
Remark 2.1.From (2), for d(x;y)>0we obtain that
F(d(f(x);f(y)))F(d(x;y))F(d(x;y))d(x;y)
1 +d(x;y):
Then, for (t) =F(t)and'(t) =t
1+t;where ;'2 are two alterig distance functions, we
have:
(d(f(x);f(y))) (d(x;y)) (d(x;y))'(d(x;y)):
Also,
(1)for (t) =F(t);we have'(t) =G(t1;t2;t3;t4)t
1+t, see ( [5]).
(2)for (t) =F(t);we have'(t) =tG(t). If0t <1, then'(t) =G(t). Ift >1, see
[8],


,[26].
Therefore, if Fis a continuous mapping then it is not a Wardowski type contractions. Then,
we conclude that the papers [1, 5, 7, 10, 11, 13, 15, 17, 19, 25, 26, 27, 28, 30, 33 ]are not
new.
In [3], we nd the denition of the multiplicative F-contractions in a multiplicative metric
space .
Definition 2.1.Let(X;d)be a multiplicative metric space and a self mapping fonX. Then
fis said to be multiplicative F-contraction, if there exists k2(0;1)such that for all x;y2X,
(5) d(f(x);f(y))>0)F(d(f(x);f(y)))[F(d(x;y))]k
whereF: (1;+1)!(1;+1)is a mapping satisfying the following conditions:
(F1)Fis strictly increasing;
(F2)for each sequence fng1
n=1(1;1),lim
n!1n= 1+1if and only if lim
n!1F(n) = 1
(F3)F is continuous.
One of the main results of the paper [ 3] is Theorem 6, as follows.
Theorem 2.1.[3]Let(X;d)be a complete multiplicative metric space and f:X!Xbe
multiplicative F-contraction. Then fhas a unique xed point and for every x02X, the sequence
ff(x0)gn2Nconverges to z.
Remark 2.2.In the previous theorem by [3], we have
(6) F(d(f(x);f(y)))F(d(x;y))kwherek2[0;1);
whereFis continuous and monotonically non-decreasing.
The existence of the xed point can be obtained without condition (F2)from the Denition
(2.1). Then the xed point theorem for muliplicative F-contractions, Theorem (2.1), is not new.
Example 2.1.In[3]we found the Example 6. We will prove that under same conditions of
that example, the inequality (7) holds.
LetX=f0;1;3gandd:RR!R+be dened as: d=ejxyj. Letf:X!Xis dened as
follows
f(x) =
1;ifx2f0;1g;
0;ifx= 3:
andF(x) =px:In[3]Example 6, x= 3 is the unique xed point of mapping fandy2f0;1g.
Case 1 . Forx= 3 andy= 0, after simple calculations, we have
d(x;y) =ej03j=e3;d(f(x);f(y)) =eandF(d(f(x);f(y))) =pe:
From inequality (6),
F(d(f(x);f(y)))F(d(x;y))k;

4 A. H. ANSARI, N. SALEEM, L. GURAN
after simple calculation, we get e1
2e3
2k. Then inequality (6) holds if we take k2[1
3;1).
Next, fork=1
3, we prove that the inequality (7) is true. Then, replacing in (7), we get that
e1
3e3, which shows that inequality (7) holds.
Case 2 . Forx= 3 andy= 1 we have:
d(x;y) =ej13j=e2;d(f(x);f(y)) =eandF(d(f(x);f(y))) =pe:
Replacing in (6), we have
e1
2ek:
In this case, we have k2[1
2;1)
If we replace in (7), k=1
2then we get e1
2e2, which also holds true.
As a conclusion, under the same conditions of Example 6 from [ 3], the inequality (7) is true.
Then the Theorem (2.1) is not new.
In [23] we nd the denition of C ri c type generalized F-contraction, with respect to a w-
distance as follows.
Definition 2.2.Let(X;d)be a metric space equipped with a w-distance p. A mapping f:
X!Xis called the C ri c type generalized F-contraction (for short, the CF-contraction) if, for
allx;y2X, there exist F2
ForF2F and >0such that
p(fx;fy )>0implies+F(p(fx;fy ))F(Mp(x;y))
for allx;y2X, where 0<1, the set
Fdenote all the functions F:R+!Rincreasing and
continuous on (0;1)and
Mp(x;y) = maxfp(x;y);p(x;fx );p(y;fy );p(x;fy );p(y;fx )g:
If we generalize this inequality for the case of an usual metric we get the following result.
Remark 2.3.By the following inequality
+F(d(f(x);f(y)))F(kd(x;y))
wherek2(0;1);we get that:
F(d(f(x);f(y)))+F(d(f(x);f(y)))F(kd(x;y)):
Result that:
(7) d(f(x);f(y))kd(x;y)):
ThenFis not a Wardowski type contractions. Then, our conclusion is that the papers [4, 16,
20, 23, 24, 29, 31 ]are not new works.
Remark 2.4.If we take particular cases, as F(t) =1
t+t,F(t) = lntandF(t) =t+ lnt,
then aFcontraction become an ordinary contraction.
Proof. (1) ForF(t) =1
t+t, then we have
+ (1
d(f(x);f(y))) +d(f(x);f(y))1
d(x;y)+d(x;y):
After simplication, we have
d(f(x);f(y))d(x;y)d(f(x);f(y)) + 1
d(x;y) +d(x;y)d(f(x);f(y)) + 1d(x;y):

NEW PROBLEMS CONCERNING WITH F-CONTRACTIONS AND 5
(2) ForF(t) = lnt, then
+ lnd(f(x);f(y))lnd(x;y)
we get:
d(f(x);f(y))1
ed(x;y):
(3) ForF(t) =t+ lnt, then
+d(f(x);f(y)) + lnd(f(x);f(y))d(x;y) + lnd(x;y)
we get:
d(f(x);f(y))ed(f(x);f(y))1
ed(x;y)ed(x;y);
If we choose (t) =tet;then
(d(f(x);f(y)))1
e (d(x;y)):

Open problem 1: Does there exists some functions F:R+!Rfor which a F-contraction
cannot be transformed in a ordinary contraction? ( Fbe continuous).
Open problem 2: Does there exists some functions F:R+!Rfor which a F-contraction
cannot be transformed in a ordinary contraction? ( Fbe discontinuous).
3. Remarks concerning (;p)-contractions
In this section, we will show that ( ;p)-contraction (resp. ( ;p)-convex contraction) mappings
and some approximate xed point and xed point theorems in the settings of metric spaces are
not new.
Theorem 3.1.[18]Let(X;d)be a metric space and T:X!Xbe a(;p)- contraction such
thatk+<1. Then,Thas the approximate xed point property. Further, if (X;d)is a complete
metric space, then Thas a unique xed point.
Remark 3.1.Recall the inequality (3)
dp(Tx;Ty ) + (1)dp(T2x;T2y)kdp(x;y):
We observe that
(8) dp(Tx;Ty )k
dp(x;y);
or
(9) dp(T2x;T2y)k
(1)dp(x;y):
Result that both are Banach contractions, then Theorem (3.1) is not new.
Definition 3.1.[18]Let(X;d)be a metric space. A self mapping TonXis said to be
(;p)-convex contraction, if for some 2(0;1)andp1, and for all j2f1;2;:::;5g, such thatP5
j=1kj<1satisfying the following inequality
dp(Tx;Ty ) + (1)dp(T2x;T2y)k1dp(x;y) +k2dp(x;Tx ) +k3dp(Tx;T2x) (10)
+k4dp(y;Ty ) +k5dp(Ty;T2y);
for allx;y2X.

6 A. H. ANSARI, N. SALEEM, L. GURAN
Theorem 3.2.[18]Let(X;d)be a metric space and T:X!Xbe a(;p)- convex contraction
such that (P5
j=1kj) +<1. Then,Thas the approximate xed point property. Further, if (X;d)
is a complete metric space, then Thas a unique xed point.
Remark 3.2.Recall the inequality (10)
dp(Tx;Ty ) + (1)dp(T2x;T2y)k1dp(x;y) +k2dp(x;Tx ) +k3dp(Tx;T2x)
+k4dp(y;Ty ) +k5dp(Ty;T2y):
We observe that
dp(Tx;Ty )1
[k1dp(x;y) +k2dp(x;Tx ) +k3dp(Tx;T2x)
+k4dp(y;Ty ) +k5dp(Ty;T2y)];
or
dp(T2x;T2y)1
(1)[k1dp(x;y) +k2dp(x;Tx ) +k3dp(Tx;T2x)
+k4dp(y;Ty ) +k5dp(Ty;T2y)]:
Then, Theorem (3.2) is not new.
4. Conclusions
In this paper we make some remarks concerning some papers from the related literature. We
prove, for particular cases of contractions, that in dierent conditions they are only Banach usual
contractions. Then, we discuss the case of Wardowski type contractions and we prove that if Fis a
continuous mapping then it is not a Wardowski type contractions, is only a Banach contraction. In
the same frame we discuss the case of multipicative F-contractions giving an illustrative example
and the case of Ciri c type generalized F-contraction. In both cases we prove that the new type of
given contractions can be reduced to a Banach contraction.
Particular cases are the cases of ( ;p)-contractions and ( ;p)-convex contractions. Then, using
sample mathematical calculation, we get by the particular contraction conditions the usual Banach
contraction condition.
The nal conclusion of our paper is that, in the view of our results, many papers given in the
literature of xed points are not new.
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8 A. H. ANSARI, N. SALEEM, L. GURAN
A. H. Ansari,1Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj,
Iran.
Email address :analsisamirmath2@gmail.com, mathanalsisamir4@gmail.com.
Naeem Saleem,2Nonlinear Analysis Research Group,Ton Duc Thang University, Ho Chi Minh
City, Vietnam.,3Department of Mathematics and Statistics, Ton Duc Thang University, Ho Chi
Minh City, Vietnam.
Email address :naeem.saleem@tdtu.edu.vn.
Liliana Guran,4Department of Pharmaceutical Sciences, "Vasile Goldis " Western University
of Arad, L. Rebreanu Street, no. 86, 310414, Arad, Romania.
Email address :lguran@uvvg.ro, gliliana.math@gmail.com.