1. No category

Ansari et al. Journal of Inequalities and Applications 2014, 2014:429
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RESEARCH
Open Access
Fixed point theorems on b-metric spaces for
weak contractions with auxiliary functions
Arslan Hojat Ansari1 , Sumit Chandok2 and Cristiana Ionescu3*
*
Correspondence:
cristianaionescu58@yahoo.com
3
Department of Mathematics and
Informatics, University Politehnica of
Bucharest, Bucharest, 060042,
Romania
Full list of author information is
available at the end of the article
Abstract
In this paper, we obtain some fixed point results for generalized weakly contractive
mappings with some auxiliary functions in the framework of b-metric spaces. The
proved results generalize and extend the corresponding well-known results of the
literature. Some examples are also provided in order to show that these results are
more general than the well-known results existing in literature.
MSC: Primary 47H10; secondary 54H25
Keywords: fixed point; b-metric space; sequentially convergent mappings
1 Introduction
The Banach contraction principle [] is a basic result on fixed points for contractive-type
mappings. So far, there have been a lot of fixed point results dealing with mappings satisfying diverse types of contractive inequalities. Various researchers have worked on different types of inequalities having continuity on mapping or not on different abstract spaces
viz. metric spaces [–], convex metric spaces [], ordered metric spaces [], cone metric
spaces [, ], generalized metric spaces [, ], b-metric spaces [–] and many more
(see [–] and references cited therein).
In , Czerwik [] introduced the b-metric spaces. These form a nontrivial generalization of metric spaces and several fixed point results for single and multivalued mappings in such spaces have been obtained since then (see [, , , ] and references cited
therein).
Let (X, d) be a metric space and T : X → X. A mapping T is said to be a K -contraction
[] if there exists α ∈ (,  ) such that for all x, y ∈ X the following inequality holds:
d(Tx, Ty) ≤ α d(x, Tx) + d(y, Ty) .
In , Kannan [] proved that if (X, d) is a complete metric space, then every K contraction on X has a unique fixed point.
In , Chatterjea [] established a fixed point theorem for C-contractions mappings,
that is, a mapping T is said to be a C-contraction if there exists α ∈ (,  ) such that for all
x, y ∈ X the following inequality holds:
d(Tx, Ty) ≤ α d(x, Ty) + d(y, Tx) .
©2014 Ansari et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.
Ansari et al. Journal of Inequalities and Applications 2014, 2014:429
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Various researchers generalize and/or extend Kannan and Chatterjea type contraction
mappings to obtain fixed point results in abstract spaces (see [, , , , , , –]
and references cited therein). In this paper, we generalize and extend the Kannan and
Chatterjea type contractions with some auxiliary functions to obtain some new fixed point
results in the framework of b-metric spaces. The proved results generalize and extend the
corresponding well-known results of Chandok [–], Choudhury [], Filipović et al.
[], Harjani et al. [], Moradi [], Morales and Rojas [], Razani and Parvaneh [] and
of Shatanawi [].
2 Preliminaries
To begin with, we give some basic definitions and notations which will be used in the
sequel.
Definition . ([]) Let X be a (nonempty) set and s ≥  be a given real number. A function d : X ×X → R+ is a b-metric if, for all x, y, z ∈ X, the following conditions are satisfied:
(b ) d(x, y) =  iff x = y,
(b ) d(x, y) = d(y, x),
(b ) d(x, z) ≤ s[d(x, y) + d(y, z)].
The pair (X, d) is called a b-metric space.
It should be noted that the class of b-metric spaces is effectively larger than that of metric
spaces, since a b-metric is a metric if (and only if ) s = . We present an easy example to
show that in general a b-metric need not be a metric.
Example . Let (X, ρ) be a metric space, and d(x, y) = (ρ(x, y))p , where p ≥  is a real
number. Then d is a b-metric with s = p– . However, (X, d) is not necessarily a metric
space. For example, if X = R is the set of real numbers and ρ(x, y) = |x – y| is the usual
Euclidean metric, then d(x, y) = (x – y) is a b-metric on R with s = , but it is not a metric
on R.
It should also be noted that a b-metric might not be a continuous function (see Example  of []). Thus, while working in b-metric spaces, the following lemma is useful.
Lemma . ([]) Let (X, d) be a b-metric space with s ≥ , and suppose that {xn } and {yn }
are b-convergent to x, y, respectively. Then we have

d(x, y) ≤ lim inf d(xn , yn ) ≤ lim sup d(xn , yn ) ≤ s d(x, y).
n→∞
s
n→∞
In particular, if x = y, then we have limn→∞ d(xn , yn ) = . Moreover, for each z ∈ X, we have

d(x, z) ≤ lim inf d(xn , z) ≤ lim sup d(xn , z) ≤ sd(x, z).
n→∞
s
n→∞
Definition . Let (X, d) be a metric space. A mapping T : X → X is said to be sequentially convergent [] (respectively, subsequentially convergent) if, for every sequence {xn }
in X for which {Txn } is convergent, {xn } is also convergent (respectively, {xn } has a convergent subsequence).
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3 Main results
We denote by the family of functions ψ : [, ∞) → [, ∞) such that ψ is continuous,
strictly increasing and ψ – ({}) = .
Also we denote by the family of functions ϕ : [, ∞)  → [, ∞) such that ϕ(, ) ≥ ,
ϕ(x, y) >  if (x, y) = (, ), and ϕ(lim infn→∞ an , lim infn→∞ bn ) ≤ lim infn→∞ ϕ(an , bn ).
Theorem . Let (X, d) be a complete b-metric space with parameter s ≥ , T, f : X → X
be such that, for some ψ ∈ , ϕ ∈ , and all x, y ∈ X,
ψ sd(Tfx, Tfy) ≤
)
ψ( d(Tx,Tfy)+d(Ty,Tfx)
s+
,
 + ϕ(d(Tx, Tfy), d(Ty, Tfx))
(.)
and let T be one-to-one and continuous. Then:
() For every x ∈ X the sequence {Tf n x } is convergent.
() If T is subsequentially convergent then f has a unique fixed point.
() If T is sequentially convergent then, for each x ∈ X the sequence {f n x } converges to
the fixed point of f .
n+
x ,
Proof Let x ∈ X be arbitrary. Consider the sequence {xn }∞
n= given by xn+ = fxn = f
for n ≥ .
Step I. First, we will prove that limn→∞ d(Txn , Txn+ ) = .
Using (.), we obtain
ψ sd(Txn+ , Txn ) = ψ sd(Tfxn , Tfxn– )
≤
)+d(Txn– ,Tfxn )
ψ( d(Txn ,Tfxn–s+
)
 + ϕ(d(Txn , Tfxn– ), d(Txn– , Tfxn ))
=
n– ,Txn+ )
)
ψ( d(Txn ,Txn )+d(Tx
s+
.
 + ϕ(d(Txn , Txn ), d(Txn– , Txn+ ))
(.)
Since ϕ is nonnegative and ψ is an increasing function and using the triangular inequality
we have
d(Txn– , Txn+ )
ψ sd(Txn+ , Txn ) ≤ ψ
s+
s d(Txn– , Txn ) + d(Txn , Txn+ ) .
≤ψ
s+
Again, since ψ is increasing, we have
d(Txn+ , Txn ) ≤
 d(Txn– , Txn ) + d(Txn , Txn+ ) ,
s+
wherefrom

d(Txn+ , Txn ) ≤ d(Txn , Txn– ) ≤ d(Txn , Txn– ).
s
Thus, {d(Txn+ , Txn )} is a decreasing sequence of nonnegative real numbers and hence it
is convergent.
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Assume that limn→∞ d(Txn+ , Txn ) = r ≥ . From the above argument we have

d(Txn– , Txn+ )
s+
s d(Txn– , Txn ) + d(Txn , Txn+ )
≤
s+
s
≤ d(Txn– , Txn ) + d(Txn , Txn+ ) .

sd(Txn+ , Txn ) ≤
On taking the limit n → ∞, we obtain
lim d(Txn– , Txn+ ) = s(s + )r.
n→∞
From (.), we have
ψ sd(Txn+ , Txn ) ≤
n– ,Txn+ )
)
ψ( +d(Txs+
.
 + ϕ(, d(Txn– , Txn+ ))
On letting n → ∞ and using the continuity of ψ and the properties of ϕ we get
ψ(sr) ≤
ψ(sr)
,
 + ϕ(, s(s + )r)
and consequently, ψ(sr) = . Hence using the properties of ψ , we have
r = lim d(Txn , Txn+ ) = .
n→∞
(.)
Step II. Now in next step we will show that {Txn } is a b-Cauchy sequence.
Suppose that {Txn } is not a b-Cauchy sequence. Then there exists ε >  for which we
can find subsequences {Txmk } and {Txnk } of {Txn } with nk is the smallest index for which
nk > mk > k such that
d(Txmk , Txnk ) ≥ ε
(.)
d(Txmk , Txnk – ) < ε.
(.)
and
From (.), (.), and using the triangular inequality, we have
ε ≤ d(Txmk , Txnk ) ≤ s d(Txmk , Txnk – ) + d(Txnk – , Txnk )
< sε + sd(Txnk – , Txnk ).
On letting k → ∞, and using (.), we obtain
ε ≤ lim sup d(Txmk , Txnk ) ≤ sε.
k→∞
(.)
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Further, we have
d(Txmk , Txnk ) ≤ s d(Txmk , Txnk – ) + d(Txnk – , Txnk ) .
Now using (.) and (.), we get
ε
≤ lim sup d(Txnk – , Txmk ) ≤ ε.
s
k→∞
(.)
Consider
d(Txmk , Txnk ) ≤ s d(Txmk , Txmk – ) + d(Txmk – , Txnk )
and
d(Txmk – , Txnk ) ≤ s d(Txmk – , Txmk ) + d(Txmk , Txnk ) .
Using (.) and (.), we get
ε
≤ lim sup d(Txmk – , Txnk ) ≤ s ε.
s
k→∞
(.)
Similarly, we can show that
ε
≤ lim inf d(Txnk – , Txmk ) ≤ ε
k→∞
s
(.)
and
ε
≤ lim inf d(Txmk – , Txnk ) ≤ s ε.
k→∞
s
Since
s +
s+
(.)
≤ s and using (.) and (.)-(.), we have
ψ(sε) ≤ ψ s lim sup d(Txmk , Txnk )
k→∞
= ψ s lim sup d(Tfxmk – , Tfxnk – )
k→∞
d(Txm – ,Tfxn – )+d(Txn – ,Tfxm – )
k
k
k
k
)
lim supk→∞ ψ(
s+
≤
 + lim infk→∞ ϕ(d(Txmk – , Tfxnk – ), d(Txnk – , Tfxmk – ))
d(Txm – ,Txn )+d(Txn – ,Txm )
k
k
k
k
)
ψ(lim supk→∞
s+
≤
 + ϕ(lim infk→∞ d(Txmk – , Txnk ), lim infk→∞ d(Txnk – , Txmk ))

ε+ε
)
ψ( s s+
≤
 + ϕ(lim infk→∞ d(Txmk – , Txnk ), lim infk→∞ d(Txnk – , Txmk ))
≤
ψ(sε)
.
 + ϕ(lim infk→∞ d(Txmk – , Txnk ), lim infk→∞ d(Txnk – , Txmk ))
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Hence, we obtain
ϕ lim inf d(Txmk – , Txnk ), lim inf d(Txnk – , Txmk ) ≤ .
k→∞
k→∞
By our assumption about ϕ, we have
lim inf d(Txmk – , Txnk ) = lim inf d(Txnk – , Txmk ) = ,
k→∞
k→∞
which contradicts (.) and (.).
Since (X, d) is b-complete and {Txn } = {Tf n x } is a b-Cauchy sequence, there exists v ∈ X
such that
lim Tf n x = v.
(.)
n→∞
Step III. Now in the last step, first we will prove that f has a unique fixed point by assuming that T is subsequentially convergent.
As T is subsequentially convergent, {f n x } has a b-convergent subsequence. Hence,
there exist u ∈ X and a subsequence {ni } such that
lim f ni x = u;
(.)
i→∞
using (.) and continuity of T, we obtain
lim Tf ni x = Tu.
(.)
i→∞
From (.) and (.) we have Tu = v.
From Lemma . and using (.), we have

ψ s · d(Tfu, Tu) ≤ ψ lim sup sd Tfu, Tf n+ x
s
n→∞
= ψ lim sup sd(Tfu, Tfxn )
n→∞
≤
)+d(Txn ,Tfu)
)
ψ(lim supn→∞ d(Tu,Tfxns+
 + lim infn→∞ ϕ(d(Tu, Tfxn ), d(Txn , Tfu))
≤
ψ( sd(Tu,Tu)+sd(Tu,Tfu)
)
s+
 + ϕ(lim infn→∞ d(Tu, Tfxn ), lim infn→∞ d(Txn , Tfu))
≤
ψ(d(Tu, Tfu))
.
 + ϕ(, lim infn→∞ d(Txn , Tfu))
Using the properties of ϕ ∈ , we have lim infn→∞ d(Txn , Tfu) = . By the triangular inequality we get
d(Tfu, Tu) ≤ s d(Tfu, Txn ) + d(Txn , Tu) .
On letting n → ∞ in above inequality, we have d(Tfu, Tu) = . Hence, Tfu = Tu. As T is
one-to-one, fu = u. Therefore, f has a fixed point.
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Now assume that w is another fixed point of f . From inequality (.), we have
ψ sd(Tu, Tw) = ψ sd(Tfu, Tfw)
≤
ψ( d(Tu,Tfw)+d(Tw,Tfu)
)
s+
 + ϕ(d(Tu, Tfw), d(Tw, Tfu))
=
)
ψ( d(Tu,Tw)+d(Tw,Tu)
s+
 + ϕ(d(Tu, Tw), d(Tw, Tu))
≤
ψ(sd(Tu, Tw))
,
 + ϕ(d(Tu, Tw), d(Tw, Tu))

since s+
≤ s and ψ is increasing. Hence, d(Tu, Tw) = . As T is one-to-one, u = w. Therefore, f has a unique fixed point.
Finally, if mapping T is sequentially convergent, replacing {n} with {ni } we conclude that
limn→∞ f n x = u.
Theorem . Let (X, d) be a complete b-metric space with parameter s ≥ , T, f : X → X
be such that, for some ψ, ϕ ∈ , l >  and all x, y ∈ X,

+ϕ(d(Tx,Tfy),d(Ty,Tfx))
d(Tx, Tfy) + d(Ty, Tfx)
ψ sd(Tfx, Tfy) ≤ ψ
+l
– l,
s+
(.)
and let T be one-to-one and continuous. Then:
() For every x ∈ X the sequence {Tf n x } is convergent.
() If T is subsequentially convergent then f has a unique fixed point.
() If T is sequentially convergent then, for each x ∈ X the sequence {f n x } converges to
the fixed point of f .
n+
Proof Let x ∈ X be arbitrary. Consider the sequence {xn }∞
x ,
n= given by xn+ = fxn = f
for n ≥ .
Step I. First, we will prove that limn→∞ d(Txn , Txn+ ) = .
Using (.), we obtain
ψ sd(Txn+ , Txn )
= ψ sd(Tfxn , Tfxn– )

+ϕ(d(Txn ,Tfxn– ),d(Txn– ,Tfxn ))
d(Txn , Tfxn– ) + d(Txn– , Tfxn )
+l
–l
≤ ψ
s+

+ϕ(d(Txn ,Txn ),d(Txn– ,Txn+ ))
d(Txn , Txn ) + d(Txn– , Txn+ )
= ψ
+l
– l.
s+
(.)
Since ϕ is nonnegative and ψ is an increasing function and using the triangular inequality
we have
d(Txn– , Txn+ )
ψ sd(Txn+ , Txn ) ≤ ψ
s+
s d(Txn– , Txn ) + d(Txn , Txn+ ) .
≤ψ
s+
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Again, since ψ is increasing, we have
d(Txn+ , Txn ) ≤
 d(Txn– , Txn ) + d(Txn , Txn+ ) ,
s+
wherefrom

d(Txn+ , Txn ) ≤ d(Txn , Txn– ) ≤ d(Txn , Txn– ).
s
Thus, {d(Txn+ , Txn )} is a decreasing sequence of nonnegative real numbers and hence it
is convergent.
Assume that limn→∞ d(Txn+ , Txn ) = r ≥ . Using similar steps to Theorem ., we obtain
lim d(Txn– , Txn+ ) = s(s + )r.
n→∞
From (.), we have

+ϕ(,d(Txn– ,Txn+ ))
 + d(Txn– , Txn+ )
ψ sd(Txn+ , Txn ) ≤ ψ
+l
– l.
s+
On letting n → ∞ and using the continuity of ψ and the properties of ϕ we have

ψ(sr) ≤ ψ(sr) + l +ϕ(,s(s+)r) – l,
and consequently, ψ(sr) = . Hence using the properties of ψ , we have
r = lim d(Txn , Txn+ ) = .
n→∞
(.)
Step II. Now in next step we will show that {Txn } is a b-Cauchy sequence.
Suppose that {Txn } is not a b-Cauchy sequence. Then there exists ε >  for which we can
find subsequences {Txmk } and {Txnk } of {Txn } with nk being the smallest index for which
nk > mk > k such that
d(Txmk , Txnk ) ≥ ε
(.)
d(Txmk , Txnk – ) < ε.
(.)
and
From (.), (.), and using the triangular inequality, we have
ε ≤ d(Txmk , Txnk ) ≤ s d(Txmk , Txnk – ) + d(Txnk – , Txnk )
< sε + sd(Txnk – , Txnk ).
On letting k → ∞, and using (.), we obtain
ε ≤ lim sup d(Txmk , Txnk ) ≤ sε.
k→∞
(.)
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Further, we have
d(Txmk , Txnk ) ≤ s d(Txmk , Txnk – ) + d(Txnk – , Txnk ) .
Now using (.) and (.), we get
ε
≤ lim sup d(Txnk – , Txmk ) ≤ ε.
s
k→∞
(.)
Consider
d(Txmk , Txnk ) ≤ s d(Txmk , Txmk – ) + d(Txmk – , Txnk )
and
d(Txmk – , Txnk ) ≤ s d(Txmk – , Txmk ) + d(Txmk , Txnk ) .
Using (.) and (.), we get
ε
≤ lim sup d(Txmk – , Txnk ) ≤ s ε.
s
k→∞
(.)
Similarly, we can show that
ε
≤ lim inf d(Txnk – , Txmk ) ≤ ε
k→∞
s
(.)
ε
≤ lim inf d(Txmk – , Txnk ) ≤ s ε.
k→∞
s
(.)
and
Since
s +
s+
≤ s and using (.) and (.)-(.), we have
ψ(sε)
≤ ψ s lim sup d(Txmk , Txnk )
k→∞
= ψ s lim sup d(Tfxmk – , Tfxnk – )
k→∞
≤ lim sup ψ
k→∞
d(Txmk – , Tfxnk – ) + d(Txnk – , Tfxmk – )
s+
+l

+lim infk→∞ ϕ(d(Txm – ,Tfxn – ),d(Txn – ,Tfxm – ))
k
k
k
k
–l

+ϕ(lim infk→∞ d(Txm – ,Txnk ),lim infk→∞ d(Txn – ,Txmk ))
d(Txmk – , Txnk ) + d(Txnk – , Txmk )
k
k
≤ ψ lim sup
+l
–l
s+
k→∞


+ϕ(lim infk→∞ d(Txm – ,Txnk ),lim infk→∞ d(Txn – ,Txmk ))
s ε+ε
k
k
≤ ψ
+l
–l
s+

≤ ψ(sε) + l +ϕ(lim infk→∞ d(Txmk – ,Txnk ),lim infk→∞ d(Txnk – ,Txmk )) – l.
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Hence, we have
ϕ lim inf d(Txmk – , Txnk ), lim inf d(Txnk – , Txmk ) ≤ .
k→∞
k→∞
By our assumption about ϕ, we have
lim inf d(Txmk – , Txnk ) = lim inf d(Txnk – , Txmk ) = ,
k→∞
k→∞
which contradicts (.) and (.).
Since (X, d) is b-complete and {Txn } = {Tf n x } is a b-Cauchy sequence, there exists v ∈ X
such that
lim Tf n x = v.
(.)
n→∞
Step III. Now, in the last step, first we will prove that f has a unique fixed point by assuming that T is subsequentially convergent.
As T is subsequentially convergent, {f n x } has a b-convergent subsequence. Hence,
there exist u ∈ X and a subsequence {ni } such that
lim f ni x = u;
(.)
i→∞
using (.) and continuity of T, we obtain
lim Tf ni x = Tu.
(.)
i→∞
From (.) and (.) we have Tu = v.
From Lemma . and using (.), we have

ψ s · d(Tfu, Tu)
s
≤ ψ lim sup sd Tfu, Tf n+ x
n→∞
= ψ lim sup sd(Tfu, Tfxn )
n→∞

+lim infn→∞ ϕ(d(Tu,Tfxn ),d(Txn ,Tfu))
d(Tu, Tfxn ) + d(Txn , Tfu)
+l
≤ ψ lim sup
–l
s+
n→∞

+ϕ(lim infn→∞ d(Tu,Tfxn ),lim infn→∞ d(Txn ,Tfu))
sd(Tu, Tu) + sd(Tu, Tfu)
≤ ψ
+l
–l
s+

≤ ψ d(Tu, Tfu) + l +ϕ(,lim infn→∞ d(Txn ,Tfu)) – l.
Using the properties of ϕ ∈ , we have lim infn→∞ d(Txn , Tfu) = . By the triangular inequality we have
d(Tfu, Tu) ≤ s d(Tfu, Txn ) + d(Txn , Tu) .
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On letting n → ∞ in above inequality, we have d(Tfu, Tu) = . Hence, Tfu = Tu. As T is
one-to-one, fu = u. Therefore, f has a fixed point.
Now assume that w is another fixed point of f . From inequality (.), we have
ψ sd(Tu, Tw) = ψ sd(Tfu, Tfw)

+ϕ(d(Tu,Tfw),d(Tw,Tfu))
d(Tu, Tfw) + d(Tw, Tfu)
+l
–l
≤ ψ
s+

+ϕ(d(Tu,Tw),d(Tw,Tu))
d(Tu, Tw) + d(Tw, Tu)
+l
= ψ
–l
s+

≤ ψ sd(Tu, Tw) + l +ϕ(d(Tu,Tw),d(Tw,Tu)) – l,
since

s+
≤ s and ψ is increasing. Hence, d(Tu, Tw) = . As T is one-to-one, u = w. There-
fore, f has a unique fixed point.
Finally, if T is sequentially convergent, replacing {n} with {ni } we conclude that
limn→∞ f n x = u.
If we take ψ(t) = t and ϕ(t, u) =
s
(s+)a
– , a ∈ (, ), in Theorem ., we obtain the follow-
ing result which is an extended Chatterjea fixed point theorem in the setting of b-metric
spaces.
Corollary . Let (X, d) be a complete b-metric space and T, f : X → X be mappings such
that T is continuous, one-to-one and subsequentially convergent. If a ∈ (, ) and
d(Tfx, Tfy) ≤
a d(Tx, Tfy) + d(Ty, Tfx) ,
s(s + )
for all x, y ∈ X, then f has a unique fixed point. Moreover, if T is sequentially convergent,
then for every x ∈ X the sequence of iterates f n x converges to this fixed point.
Remark .
() If we take Tx = x, in Corollary ., then we obtain the result of Jovanovic [,
Corollary ..◦ ] (the case g = f ), which is Chatterjea’s Theorem [] in the
framework of b-metric spaces.
() By taking Tx = x and ψ(t) = t in Theorem ., we obtain an extension of
Choudhury’s [] main result to the setup of b-metric spaces.
() If s = , in Theorem ., we obtain the corresponding result of [].
Example . Let X = {} ∪ {/n | n ∈ N}, and let d(x, y) = (x – y) for x, y ∈ X. Then d is a
b-metric with the parameter s =  and (X, d) is a complete b-metric space. Consider the
mappings f , T : X → X given by
f () = ,
f


,
=
n
n+
T() = ,
T


= n,
n
n
n ∈ N.
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We will show that the mappings f , T satisfy the conditions of Corollary . with α =


= s+
. Indeed, for m, n ∈ N, m > n, we have



d Tf , Tf
n
m


=
–
(n + )n+ (m + )m+


<
(n + )n+


<

.
It is easy to prove that, for n ∈ N,
 


<
–
.
(n + )n+  nn (n + )n+
It follows that


d Tf , Tf
n
m

 

<
–
.
 nn (n + )n+
Now, m > n implies that m ≥ n +  and n +  ≤ m + . It follows that /(n + )n+ ≥ /(m +
)m+ , and hence


 


<
–
d Tf , Tf
n
m
 nn (m + )m+




α
d T , Tf
+ d T , TF
.
≤
s
n
m
m
n
If one of the points is equal to , the proof is even simpler.
By Corollary ., it follows that f has a unique fixed point (which is u = ).
Theorem . Let (X, d) be a complete b-metric space with the parameter s ≥ , T, f : X →
X be such that for some ψ ∈ , ϕ ∈ , and all x, y ∈ X,
ψ d(Tfx, Tfy) ≤
ψ( d(Tx,Tfx)+d(Ty,Tfy)
)
s+
,
 + ϕ(d(Tx, Tfx), d(Ty, Tfy))
(.)
and let T be one-to-one and continuous. Then:
() For every x ∈ X the sequence {Tf n x } is convergent.
() If T is subsequentially convergent then f has a unique fixed point.
() If T is sequentially convergent then, for each x ∈ X the sequence {f n x } converges to
the fixed point of f .
n+
x ,
Proof Let x ∈ X be arbitrary. Consider the sequence {xn }∞
n= given by xn+ = fxn = f
n ≥ .
In the first step, we will prove that limn→∞ d(Txn , Txn+ ) = .
Using (.), we obtain
ψ d(Txn+ , Txn ) = ψ d(Tfxn , Tfxn– ) ≤
=
n– ,Tfxn– )
)
ψ( d(Txn ,Tfxn )+d(Tx
s+
 + ϕ(d(Txn , Tfxn ), d(Txn– , Tfxn– ))
)+d(Txn– ,Txn )
)
ψ( d(Txn ,Txn+s+
.
 + ϕ(d(Txn , Tfxn ), d(Txn– , Tfxn– ))
(.)
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Since ϕ is nonnegative and ψ is increasing, we have
d(Txn+ , Txn ) ≤
d(Txn , Txn+ ) + d(Txn– , Txn )
,
s+
that is,

d(Txn+ , Txn ) ≤ d(Txn , Txn– ) ≤ d(Txn , Txn– ).
s
Thus, {d(Txn+ , Txn )} is a decreasing sequence of nonnegative real numbers and hence it
is convergent.
Assume that limn→∞ d(Txn+ , Txn ) = r. On letting n → ∞ in (.) and using the properties of ψ and ϕ we obtain
ψ(r) ≤
r
ψ( s+
)
ψ(r)
≤
,
 + ϕ(r, r)  + ϕ(r, r)
which is possible only if
r = lim d(Txn , Txn+ ) = .
n→∞
Now, we will show that {Txn } is a b-Cauchy sequence.
Suppose that this is not true. Then there exists ε >  for which we can find subsequences
{Txmk } and {Txnk } of {Txn } with nk is the smallest index for which nk > mk > k such that
d(Txmk , Txnk ) ≥ ε. This means that
d(Txmk , Txnk – ) < ε.
Again, as in Step II of Theorem . one can prove that
ε ≤ lim sup d(Txmk , Txnk ) ≤ sε.
(.)
k→∞
Using (.) we have
ψ d(Txmk , Txnk ) = ψ d(Tfxmk – , Tfxnk – )
d(Txm – ,Tfxm – )+d(Txn – ,Tfxn – )
≤
k
k
k
k
)
ψ(
s+
 + ϕ(d(Txmk – , Tfxmk – ), d(Txnk – , Tfxnk – ))
d(Txm – ,Txm )+d(Txn – ,Txn )
k
k
k
k
)
ψ(
s+
.
=
 + ϕ(d(Txmk – , Txmk ), d(Txnk – , Txnk ))
Passing to the upper limit as k → ∞ in the above inequality and using (.), we have
ψ(ε) ≤
ψ()
= ,
 + ϕ(, )
and so ψ(ε) = . By our assumptions about ψ , we have ε = , which is a contradiction.
Therefore as in Step II of Theorem ., we obtain {Txn } is a b-Cauchy sequence.
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Since (X, d) is b-complete and {Txn } = {Tf n x } is a b-Cauchy sequence, there exists v ∈ X
such that
lim Tf n x = v.
(.)
n→∞
Now, if T is subsequentially convergent, then {f n x } has a convergent subsequence. Hence,
there exist a point u ∈ X and a sequence {ni } such that
lim f ni x = u.
(.)
i→∞
Using (.) and continuity of T, we obtain
lim Tf ni x = Tu.
(.)
i→∞
By using (.) and (.), we obtain Tu = v.
Using Lemma . and inequality (.), we have

ψ d(Tfu, Tu) ≤ ψ lim sup d Tfu, Tf n+ x
s
n→∞
= ψ lim sup d(Tfu, Tfxn )
n→∞
≤
n ,Tfxn )
)
ψ(lim supn→∞ d(Tu,Tfu)+d(Tx
s+
 + lim infn→∞ ϕ(d(Tu, Tfu), d(Txn , Tfxn ))
=
)
ψ( d(Tu,Tfu)+
s+
 + ϕ(d(Tu, Tfu), )
≤
ψ( d(Tu,Tfu)
)
s
.
 + ϕ(d(Tu, Tfu), )
Using the properties of ϕ ∈ , d(Tu, Tfu) = . As T is one-to-one, fu = u. Therefore, f has
a fixed point.
Uniqueness of the fixed point can be proved similarly to Theorem ..
Finally, if T is sequentially convergent, replacing {n} with {ni } we conclude that
limn→∞ f n x = u.

– , a ∈ (, ) in Theorem ., an extended Kannan
Taking ψ(t) = t and ϕ(t, u) = (s+)a
fixed point theorem in the setting of b-metric spaces has been obtained.
Corollary . Let (X, d) be a complete b-metric space with the parameter s ≥ , T, f : X →

) and all x, y ∈ X,
X be such that for some a ∈ (, s+
d(Tfx, Tfy) ≤ a d(Tx, Tfx) + d(Ty, Tfy)
and let T be one-to-one and continuous. Then:
() For every x ∈ X the sequence {Tf n x } is convergent.
() If T is subsequentially convergent then f has a unique fixed point.
(.)
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() If T is sequentially convergent then, for each x ∈ X the sequence {f n x } converges to
the fixed point of f .
Remark .
() If we take Tx = x, in Corollary ., then we obtain the result of Jovanović et al. [,
Corollary ..◦ ] (the case g = f ).
() If s = , in Corollary ., then we obtain the main result of Moradi (i.e., [,
Theorem .]).
() If both of these conditions are fulfilled, we get just the classical result of Kannan [].
Example . ([]) Let X = {a, b, c} and d : X × X → R be defined by d(x, x) =  for x ∈ X,
d(a, b) = d(b, c) = , d(a, c) =  , d(x, y) = d(y, x) for x, y ∈ X. It is easy to check that (X, d)
is a b-metric space (with s =  > ) which is not a metric space. Consider the mapping
f : X → X given by
a
f=
a
b
a
c
.
b
We first note that the b-metric version of the classical weak Kannan theorem is not
satisfied in this example. Indeed, for x = b, y = c, we have d(fx, fy) = d(a, b) =  and
d(x, fx) + d(y, fy) = d(b, a) + d(c, b) = , hence the inequality
d(x, fx) + d(y, fy)
– ϕ d(x, fx), d(y, fy)
ψ d(fx, fy) ≤ ψ
s+
cannot hold, whatever ψ ∈ and ϕ ∈ are chosen.
Take now T : X → X defined by
a
T=
b
b
c
c
.
a
Obviously, all the properties of T given in Corollary . are fulfilled. We will check that
the contractive condition (.) holds true if α is chosen such that



<α<
=
.

 s + 
Only the following cases are nontrivial:
◦ x = a, y = c. Then (.) reduces to
d(Tfa, Tfc) = d(b, c) =  =
 
· < α d(b, b) + d(a, c) = α d(Ta, Tfa) + d(Tc, Tfc) .
 
◦ x = b, y = c. Then (.) reduces to
d(Tfb, Tfc) = d(b, c) =  <
 
·
< α d(c, b) + d(a, c) = α d(Tb, Tfb) + d(Tc, Tfc) .
 
All the conditions of Corollary . are satisfied and f has a unique fixed point (u = a).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. 2 Department of Mathematics, Khalsa
College of Engineering & Technology, Punjab Technical University, Amritsar, Punjab 143001, India. 3 Department of
Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, 060042, Romania.
Acknowledgements
The work has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the
Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/132395.
Received: 29 July 2014 Accepted: 16 October 2014 Published: 31 Oct 2014
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arXiv:0903.1504v1 [math.FA]
10.1186/1029-242X-2014-429
Cite this article as: Ansari et al.: Fixed point theorems on b-metric spaces for weak contractions with auxiliary
functions. Journal of Inequalities and Applications 2014, 2014:429
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