A note on fixed point theory for cyclic ϕ−contractions
Stojan Radenović
Faculty of Mathematics and Information Technology, Teacher Education, Dong Thap
University, Cao Lanch City, Dong Thap Province, Viet Nam
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Abstract: In this paper, we consider, discuss, improve and complement recent fixed points
results for mappings satisfying cyclical contractive conditions established by M. Pacurar, I. A. Rus
[M. Pacurar, I. A. Rus, Fixed point theory for cyclic ϕ−contractions, Nonlinear Anal., 72 (2010)
1181-1187] and S. Candok, M. Postolache [S. Candok, M. Postolache, Fixed point theorem for
weakly Chatterjea-type cyclic contractions, Fixed Point Theory Appl., 2013, 2013:28]. By using
our new Lemma we get much shorter and nicer proofs of some results with the new concept of
mappings.
Keywords: Cyclic ϕ−contraction, Picard operator, comparison function, (c)-comparison
function, weakly Chatterjea-type contraction.
MSC: 47H10, 54H25.
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1. INTRODUCTION AND PRELIMINARIES
It is well known that the Banach contraction principle is one of the fundamental result
in nonlinear analysis and fixed point theory, in general. It has various applications in many
branches of applied sciences. Also, there are several extensions and generalizations of this
principle. For example, W. A. Kirk et al. [4] obtained an extension of Banach principle
for mappings satisfying cyclical contractive conditions. They also proved in [4] Edelstein’s,
Boyd-Wong, Geraghty and Caristy type theorem for new concept of mappings. For some
other generalizations also see ([7], [11] and [12]).
Following [4], in 2010 [9], M. Pacurar and I. A. Rus proved a fixed point theorem for
cyclic ϕ−contractions. Also, very recent, following ideas from [4], S. Chandok and M.
Postolache [5] proved fixed point theorem for weakly Chatterjea-type cyclic contractions.
However, in the present paper, we show that all these results established in [9] and [5]
are in the fact equivalent with well known ordinary fixed point results in literature.
Let us start by recalling a definition from [4], [5] and [9].
Definition 1.1. [4] Let X be a non-empty set, p ∈ N, and f : X → X a map. Then
we say that ∪pi=1 Ai (where ∅ =
Ai ⊆ X for all i ∈ {1, 2, ..., p}) is a cyclic representation of
X with respect to f if and only if the following two conditions hold:
(I) X = ∪pi=1 Ai ;
(II) f (Ai ) ⊆ Ai+1 for 1 ≤ i ≤ p − 1, and f (Ap ) ⊆ A1 .
In 2003 [4] W. A. Kirk et al. proved the following result:
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E-mail address:
[email protected]; [email protected]; (S. Radenović).
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Theorem 1.2. [4] Let {Ai }pi=1 be non-empty closed subsets of a complete metric space,
and suppose F : ∪pi=1 Ai → ∪pi=1 Ai satisfies the following conditions (where Ap+1 = A1 ):
(1) F (Ai ) ⊂ Ai+1 for 1 ≤ i ≤ p;
(2) ∃k ∈ (0, 1) such that d (F (x) , F (y)) ≤ kd (x, y) ∀x ∈ Ai , y ∈ Ai+1 for 1 ≤ i ≤ p.
Then F has a unique fixed point.
Remark 1.3. It is not hard to see that Theorem 2.3. holds if k = 0. Also, easily follows
that Picard’s sequence {F n x} converges to unique fixed point of F for all x ∈ X.
Definition 1.4. [9] A function ϕ : [0, ∞) → [0, ∞) is called a comparison function if
it satisfies:
(i) ϕ ϕ is increasing, i.e., t1 ≤ t2 implies ϕ (t1 ) ≤ ϕ (t2 ) , for t1 , t2 ∈ [0, ∞);
(ii) ϕ (ϕn (t))n∈N converges to 0 as n → ∞, for all t ∈ (0, ∞) .
There is a rich literature referring to ϕ−contractions, i.e., self-operator defined on a
metric space, which satisfy
d (f (x) , f (y)) ≤ ϕ (d (x, y))
(1.1)
for any x, y ∈ X, where ϕ is a comparison function.
Definition 1.5. [9] A function ϕ : [0, ∞) → [0, ∞) is called a (c)-comparison function
if it satisfies:
(i) ϕ ϕ is increasing, i.e., t1 ≤ t2 implies ϕ (t1 ) ≤ ϕ (t2 ) , for t1 , t2 ∈ [0, ∞);
∞
vk
(iii) ϕ there exist k0 ∈ N, a ∈ (0, 1) and a convergent series of nonnegative terms
k=1
such that
ϕk+1 (t) ≤ aϕk (t) + vk ,
(1.2)
for k ≥ k0 and any t ∈ (0, ∞) .
The following result is enough known:
Lemma 1.6. [9] If ϕ : [0, ∞) → [0, ∞) is a (c)-comparison function, then the following
hold:
(i) ϕ is a comparison function;
(ii) ϕ (t) < t, for any t ∈ (0, ∞) ;
(iii) ϕ is continuous at 0;
∞
(iv) the series
ϕk (t) converges for any t ∈ (0, ∞) .
k=0
Let Pcl (X) denotes the collection of non-empty closed subsets of X. Then we have:
Definition 1.7. [9] Let (X, d) be a metric space, m a positive integer, A1 , ..., Am ∈
Pcl (X) , Y = ∪m
i=1 Ai and f : Y → Y an operator. If
(i) ∪m
A
is
a cyclic representation of Y w.r.t f ;
i=1 i
(ii) there exists a comparison function ϕ : [0, ∞) → [0, ∞) such that
d (f (x) , f (y)) ≤ ϕ (d (x, y))
for any x ∈ Ai , y ∈ Ai+1 , where Am+1 = A1 , then f is a cyclic ϕ−contraction.
2
(1.3)
The main result of [9] is the following:
Theorem 1.8. [9] Let (X, d) be a complete metric space, m a positive integer, A1 , ..., Am ∈
Pcl (X) , Y = ∪m
i=1 Ai , ϕ : [0, ∞) → [0, ∞) a (c)-comparison function and f : Y → Y an
operator. Assume that:
(i) ∪m
i=1 Ai is a cyclic representation of Y with respect to f ;
(ii) f is a cyclic ϕ−contraction.
Then:
(1) f has a unique fixed point x∗ ∈ ∩m
i=1 Ai and the Picard iteration {xn }n≥0 given by
∗
xn = f (xn−1 ) , n ≥ 1 converges to x for any starting point x0 ∈ Y.
(2) the following estimates hold:
∗
d (xn , x ) ≤
∞
ϕn (d (x0 , x1 )) ;
(1.4)
ϕ (d (xn , xn+1 )) ;
(1.5)
ϕk (d (x, f (x))) .
(1.6)
n=1
d (xn , x∗ ) ≤
∞
n=1
(3) for any x ∈ Y :
∗
d (x, x ) ≤
∞
n=1
It is worth to notice that Theorem 1.8. is a cyclical-type extension of the following
ordinary fixed point theorem:
Theorem 1.9. Let (X, d) be a complete metric space, and f : X → X an operator.
Assume that there exists a (c)-comparison function ϕ : [0, ∞) → [0, ∞) such that
d (f (x) , f (y)) ≤ ϕ (d (x, y))
foa all x, y ∈ X.
Then:
(1) f has a unique fixed point x∗ ∈ X and the Picard iteration {xn }n≥0 given by
xn = f (xn−1 ) , n ≥ 1 converges to x∗ for any starting point x0 ∈ X.
(2) the following estimates hold:
d (xn , x∗ ) ≤
∞
ϕn (d (x0 , x1 )) ;
(1.7)
ϕ (d (xn , xn+1 )) ;
(1.8)
n=1
∗
d (xn , x ) ≤
∞
n=1
3
(3) for any x ∈ X :
∗
d (x, x ) ≤
∞
ϕk (d (x, f (x))) .
(1.9)
n=1
Definition 1.10. [5] Let Φ denote the set of all monotone increasing continuous
functions µ : [0, ∞) → [0, ∞), with µ(t) = 0, if and only if t = 0, and let Ψ denote
the set of all lower semi-continuous functions ψ : [0, ∞)2 → [0, ∞),with ψ(t1 , t2 ) > 0, for
t1 , t2 ∈ (0, ∞) and ψ(0, 0) = 0.
Definition 1.11. [5] Let (X, d) be a metric space, m be a natural number, A1 , A2 , ..., Am ,be
non-empty subsets of X and Y = ∪m
i=1 Ai . An operator T : Y → Y is called a Chatterjeatype cyclic weakly contraction if:
(1) ∪m
of Y with respect
to T ;
i=1 Ai is a cyclic representation
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(2) µ (d (T x, T y)) ≤ µ 2 (d (x, T y) + d (y, T x)) − ψ (d (x, T y) , d (y, T x)) for all x ∈
Ai , y ∈ Ai+1 , i = 1, 2, ..., m, where Am+1 = A1 , µ ∈ Φ and ψ ∈ Ψ.
In [5] authors proved the following result:
Theorem 1.12.[5] Let (X, d) be a complete metric space, m ∈ N, A1 , A2 , ..., Am be nonempty closed subsets of X and Y = ∪m
i=1 Ai . Suppose that T is a Chatterjea-type cyclic
weakly contraction. Then T has a fixed point z ∈ ∩m
i=1 Ai .
It is obvious that Theorem 1.12. is a cyclical-type extension of the following ordinary
fixed point theorem:
Theorem 1.13. Let (X, d) be a complete metric space, T : X → X be (so-called
Chatterjea-type contraction) such that
1
µ (d (T x, T y)) ≤ µ
(d (x, T y) + d (y, T x)) − ψ (d (x, T y) , d (y, T x)) ,
(1.10)
2
for all x, y ∈ X, µ ∈ Φ and ψ ∈ Ψ. Then T has a unique fixed point.
2. MAIN RESULTS
Here we will prove and use the following (new, useful and very significant) result in
proofs of cyclic-type results (see also the proof of [7], Theorem 2.1).
Lemma 2.1. Let (X, d) be a metric space, f : X → X be a mapping and let X =
p
∪i=1 Ai be a cyclic representation of X w.r.t. f. Assume that
lim d (xn , xn+1 ) = 0,
n→∞
(2.1)
where xn+1 = f xn , x1 ∈ A1 . If {xn } is not a Cauchy sequence then there exist a ε > 0 and
two sequences {n (k)} and {m (k)} of positive integers such that the following sequences
tend to ε+ when k → ∞ :
d xm(k)−j(k) , xn(k) , d xm(k)−j(k)+1 , xn(k) , d xm(k)−j(k) , xn(k)+1 , d xm(k)−j(k)+1 , xn(k)+1 ,
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where j (k) ∈ {1, 2, ..., p} is chosen so that n (k) − (m (k) − j (k)) ≡ 1 (mod p) , for each
k ∈ N.
Proof. If {xn } is not a Cauchy sequence, then there exist ε > 0 and sequences {n (k)}
and {m (k)} of positive integers such that
n (k) > m (k) > k, d xm(k) , xn(k)−1 < ε, d xm(k) , xn(k) ≥ ε
(2.2)
for all positive integers k. Then using (2.2) and the triangle inequality, we get
ε ≤ d xm(k) , xn(k) ≤ d xm(k) , xn(k)−1 + d xn(k)−1 , xn(k) < ε + d xn(k)−1 , xn(k) . (2.3)
Passing to the limit as k → ∞ in the above inequality and using (2.1), we obtain
lim d xm(k) , xn(k) = ε+ .
k→∞
(2.4)
Note that, by the way j (k) were chosen, xm(k)−j(k) and xn(k) (for k large enough, m (k) >
j (k)) lie in different adjacently labelled sets Ai and Ai+1 for certain i ∈ {1, 2, ..., p} . This
will be used further in the proof of Theorems 2.3. and 2.7.
Using the triangle inequality, we get
d xm(k)−j(k) , xn(k) − d xm(k) , xn(k) ≤ d xm(k)−j(k) , xm(k)
j(k)−1
≤
l=0
d xm(k)−j(k)+l , xm(k)−j(k)+l+1 → 0 as k → ∞
(from (2.4)), which, by (2.5) implies that
lim d xm(k)−j(k) , xn(k) = ε.
k→∞
(2.5)
(2.6)
Using (2.1), we have that
lim d xm(k)−j(k)+1 , xm(k)−j(k) = 0 and lim d xn(k)+1 , xn(k) = 0.
k→∞
k→∞
Again using the triangle inequality, we get
d xm(k)−j(k) , xn(k)+1 − d xm(k)−j(k) , xn(k) ≤ d xn(k)+1 , xn(k) .
Passing to the limit as k → ∞ in the above inequality, and using (2.1) and (2.6), we get
lim d xm(k)−j(k) , xn(k)+1 = ε.
k→∞
Similarly, we have
lim d xm(k)−j(k)+1 , xn(k)+1 = ε. k→∞
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If p = 1 we obtain the following known result:
Corollary 2.2. [10] Let (X, d) be a metric space and let {xn } be a sequence in X such
that
lim d (xn+1 , xn ) = 0.
n→∞
If {xn } is not a Cauchy sequence, then there exist an ε > 0 and two sequences {n (k)} and
{m (k)} of positive integers such that the following sequences tend to ε+ when k → ∞ :
d xm(k) , xn(k) , d xm(k) , xn(k)+1 , d xm(k)−1 , xn(k) ,
d xm(k)−1 , xn(k)+1 , d xm(k)+1 , xn(k)+1 , d xm(k)+1 , xn(k) , ...
Now, we announce our first result (probably new because using the new Lemma 2.1)
with comparison function. It generalize several known results in existing literature ([2],
[3], [6], [8]).
Theorem 2.3. Let (X, d) be a complete metric space, ϕ : [0, ∞) → [0, ∞) a comparison
function and f : X → X. If
d (f (x) , f (y)) ≤ ϕ (d (x, y))
(2.5)
for all x, y ∈ X, then f has a unique fixed point z ∈ X and all sequences of Picard iterates
defined by f converges to z.
Proof. If x0 is an arbitrary point from X we can consider Picard sequence as follows:
xn+1 = f (xn ) , n = 0, 1, 2, ...
If xn+1 = xn for some n, then xn is a fixed point of f. So, we will suppose that xn+1 = xn
for all n.
Now, applying (2.5) putting x = xn and y = xn−1 , we obtain
d (xn+1 , xn ) = d (f (xn ) , f (xn−1 )) ≤ ϕ (d (xn , xn−1 ))
≤ ϕ2 (d (xn−1 , xn−2 )) ≤ ... ≤ ϕn (d (x1 , x0 )) → 0, as n → ∞.
Since d (xn+1 , xn ) → 0 we can prove that {xn } is a Cauchy sequence in the space (X, d) by
using Corollary 2.2. Indeed, suppose that {xn } is not a Cauchy sequence. Then, Corollary
2.2. implies that there exist ε > 0 and two sequences n (k) and m (k) of positive integer
such that
n (k) > m (k) > k, d xm(k) , xn(k)−1 < ε, d xm(k) , xn(k) ≥ ε.
(2.6)
Now, using (2.5) with x = xm(k) and y = xn(k)−1 , we get
d xm(k)+1 , xn(k) ≤ ϕ d xm(k) , xn(k)−1 ≤ ϕ (ε) < ε
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(2.7)
(for each comparison function ϕ holds: ϕ (t) < t for all t > 0).
Taking limit in (2.7) as k → ∞ and using Corollary 2.2., we get
ε = lim d xm(k)+1 , xn(k) ≤ ϕ (ε) < ε,
k→∞
which is a contradiction. Hence, {xn } is a Cauchy sequence. Therefore, there exists z ∈ X
such that xn → z. Since, according (2.5) f is a continuous, we obtain that f (z) = z is a
fixed point of f. Uniqueness follows easy from (2.5). Theorem 2.3. is proved. Remark 2.4. It is clear that Theorem 2.3. holds if ϕ is also (c)-comparison function.
It this case, it follows that the main result from [9]-Theorem 1.8. above is cyclic type
extension of our Theorem 2.3. where ϕ is a (c)-comparison function. Also, if ϕ is a ccomparison function in our Theorem 2.3 we have the same error estimates (1.4)-(1.6) as in
[9]-Theorem 1.8.
Now, we shall prove that Theorem 1.8. and Theorem 2.3. with (c)-comparison function
ϕ are equivalent, that is, Theorem 1.8. holds if and only if Theorem 2.3. holds.
Theorem 2.5. Theorem 1.8. and Theorem 2.3. with (c)-comparison function are
equivalent.
Proof. First of all, Theorem 1.8. implies Theorem 2.3. Indeed, putting in Theorem 1.8.
Ai = X for all i = 1, 2, ..., m we obtain result.
Conversely, we shall show that Theorem 2.3. implies Theorem 1.8. If x0 ∈ ∪m
i=1 Ai
then Picard’s sequence {f n x0 } is obviously a Cauchy sequence (the proof as in [9]) and
n
converges to some z ∈ ∪m
i=1 Ai . Also, it is clear that infinitely many terms of {f x0 } lie
m
in each Ai . From this it follows that ∩i=1 Ai = ∅, because z lie in each Ai . Further, since
m
(according to (1)) f : ∩m
of f on ∩m
i=1 Ai → ∩i=1 Ai ,the restriction f |∩m
i=1 Ai satisfies (2.5)
i=1 Ai
m
and (∩i=1 Ai , d) is a complete metric space, then Theorem 2.3. where ϕ is a (c)-comparison
function, implies that the mapping f has a unique fixed point in ∩m
i=1 Ai , that is, Theorem
1.8. holds. Theorem 2.5. is proved. m
Remark 2.6. From (1I) of Definition 1.1. follows that f : ∩m
i=1 Ai → ∩i=1 Ai in the case
n
m
m
that ∩m
i=1 Ai = ∅. Further, if Picard’s sequence {f x} , where f : ∪i=1 Ai → ∪i=1 Ai , x ∈
m
∪m
i=1 Ai is a Cauchy, then ∩i=1 Ai = ∅. Also, if such f has a fixed point say z, then by (1I )
of Definition 1.1. follows that z ∈ ∩m
i=1 Ai , that is,
F ix (f ) = ∅ ⇒ ∩m
i=1 Ai = ∅.
(2.8)
This follows directly from the conditions f (Ai ) ⊆ Ai+1 , i = 1, 2, ..., m.
If a certain (non-cyclic) fixed point result is known, in order to obtain the respective
cyclic-type fixed point result, it is enough to prove that the respective cyclic contractive
m
condition implies that ∩m
i=1 Ai = ∅. Indeed, in this case (∩i=1 Ai , d) is a complete metric
m
space and the restriction of f to ∩i=1 Ai satisfies the given standard condition.
By using Lemma 2.7. we prove the following result:
Theorem 2.7. Theorem 1.12. and Theorem 1.13. are equivalent.
Proof. It is clear that Theorem 1.12. is true, if Theorem 1.11. is such. Conversely, we
will prove that Theorem 1.22. implies Theorem 1.11. Indeed, if x0 ∈ ∪m
i=1 Ai then for Picard
sequence xn = f (xn−1 ) , xn = xn−1 , n = 1, 2, ... we have as in [5] that d (xn+1 , xn ) ↓ 0.
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Now, putting in (2) of Definition 1.11. x = xm(k)−j(k) , y = xn(k) we obtain that
1 µ d xm(k)−j(k)+1 , xn(k)+1 ≤ µ
d xm(k)−j(k) , xn(k)+1 + d xm(k)−j(k)+1 , xn(k)
2
−ψ d xm(k)−j(k) , xn(k)+1 , d xm(k)−j(k)+1 , xn(k) .
(2.9)
Taking limit in (2.9) as k → ∞, we get by Lemma 2.1. that
1
(ε + ε) − ψ (ε, ε) = µ (ε) − ψ (ε, ε) ,
µ (ε) ≤ µ
2
that is ψ (ε, ε) = 0. According to the property of the function ψ, follows that, ε = 0. A
contradiction. Hence, Picard sequence xn = f (xn−1 ) is a Cauchy. Further, according to
Remark 2.6. we obtain that Theorem 1.13. implies Theorem 1.12. The proof of Theorem
2.7. is completed. An open question:
Prove or disprove the following:
• Let {Ai }m
i=1 be non-empty closed subsets of a complete metric space, and suppose
m
f : ∪m
A
→
∪
i=1 i
i=1 Ai satisfies the following conditions (where Am+1 = A1 ):
(1) f (Ai ) ⊆ Ai+1 for 1 ≤ i ≤ m and f (Am ) ⊆ A1 ;
(2) there exists comparison function ϕ : [0, ∞) → [0, ∞) such that
d (f (x) , f (y)) ≤ ϕ (d (x, y)) ,
for all x ∈ Ai , y ∈ Ai+1 , 1 ≤ i ≤ m.
Then f has a unique fixed point x∗ ∈ ∩m
i=1 Ai and Picard iteration {xn }n≥1 given by
∗
xn = f (xn−1 ) converge to x for any starting point x0 ∈ ∪m
i=1 Ai .
References
[1] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux
equations integrales, Fund. Math. 3 (1922) 133-181.
[2] F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Indag. Math. 30 (1968), 27-35
[3] D. W. Boyd and J. S. Wong, On nonlinear contractions, Pros. Amer. Math. Soc. 20
(1969), 458-464
[4] W. A. Kirk, P. S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying
cyclical contractive conditions, Fixed Point Theory, Volume 4, No. 1, 2003, 79-89.
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[5] S. Candok, M. Postolache, Fixed point theorem for weakly Chatterjea-type cyclic contractions, Fixed Point Theory Appl. 2013, 2013:28.
[6] Janusz Matkowski, Integrable solutions of functional equations, Dissertationes Math.
(Rozprawa Mat.) 127 (1975), 68 pp.
[7] H. K. Nashine, Cyclic generalized ψ−weakly contractive mappings and fixed point
results with applications to integral equations, Nonlinear Anal., 75 (2012) 6160-6169.
[8] I. A. Rus, Generalized ϕ−contractions, Mathematica 24 (1982) 175-178
[9] M. Pacurar, I. A. Rus, Fixed point theory for cyclic ϕ−contractions, Nonlinear Anal.,
72 (2010)1181-1187.
[10] S. Radenović, Z. Kadelburg, D. Jandrlić and A. Jandrlić, Some results on weak contraction maps, Bull. Iranian Math. Soc.38, 3 (2012), 625-645.
[11] S. Radenović, Some remarks on mappings satisfying cyclical contractive conditions,
Afr.Mat DOI 10. 1007/s13370-015-0327-6
[12] S. Radenović, A note on fixed point theory for cyclic weaker Meir-Keeler function in
complete metric spaces, In. J. Anal. Appl. 7 (1) (2015), 16-21
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