A NOTE ON RECENT CYCLIC FIXED POINT RESULTS IN DISLOCATED
QUASI-b-METRIC SPACES
DIANA DOLIĆANIN-D̄EKIĆ1 , TATJANA DOŠENOVIĆ2 , HUAPING HUANG3 , STOJAN RADENOVIĆ4,5
Abstract. The purpose of this paper is to establish some fixed point results for cyclic contractions in the setting of dislocated quasi-b-metric spaces. We verify that some previous cyclic
contraction results in dislocated quasi-b-metric spaces are just equivalent to the non-cyclic ones in
the same spaces. Moreover, by using two examples, we highlight the superiority of the obtained
results.
1. INTRODUCTION AND PRELIMINARIES
French mathematician H. Poinćare firstly used the concept of fixed point in “Poinćare’s final
theorem” during the period of 1895 to 1900, from restricting the existence of periodic solution for
three body problem to the existence of fixed point under some conditions of planar continuous
transformations. In 1910, L. E. J. Brouwer proved that there exists at least one fixed point for the
polyhedron continuous map in finite dimensional space, and then it opened the situation of fixed
point theory research. Particularly in 1922, Polish mathematician S. Banach innovated Banach
contraction mapping principle by using Picard iteration method. Due to its beautiful assertion
and successful way of solving the implicit function existence theorem, the existence of solution for
differential equation with initial value condition, fixed point theory catches the eyes of scholars
and it sparkles people’s inspirations towards in-depth and extensive research. Especially in recent
decades, with the development of computer, many people cope with numerous applications by
utilizing a variety of iteration methods to approach the fixed point and hence they make this
subject into breakthrough and gradual perfection. Nowadays fixed point theory plays a crucial
role in nonlinear functional analysis. Just because of this, in this paper, we start fixed point
investigation based on some previous work.
To start this article, we firstly recall some basic knowledge.
In 1922, S. Banach [3] introduced Banach contraction mapping principle as follows:
A self-map T on a metric space (X, d) has a unique fixed point if there exists k ∈ [0, 1) such
that
d (T x, T y) ≤ kd (x, y) , ∀x, y ∈ X.
After that, based on this finding, a large number of fixed point results have appeared in recent
years. Generally speaking, there usually have two generalizations on them. One is from mappings.
The other is from spaces.
Concretely, for one thing, from mappings, for example, the concept of Kannan contraction
mapping was introduced in 1969 by Kannan [7] as follows:
A self-map T on a metric space (X, d) is called a Kannan contraction mapping if there exists
k ∈ [0, 21 ) such that
d (T x, T y) ≤ k [d (x, T x) + d (y, T y)] , ∀x, y ∈ X.
In this case, T has a unique fixed point.
2000 Mathematics Subject Classification. 54H25, 47H10.
Key words and phrases. Fixed point; dislocated quasi-b-metric space; Banach contraction; Kannan contraction.
The corresponding author:[email protected] (Stojan Radenović).
1
2
D. DOLIĆANIN-D̄EKIĆ, T. DOŠENOVIĆ, H. HUANG, S. RADENOVIĆ
Recently, cyclic contraction mapping is popular for research activities (see [5], [9-13], [15-17]).
Let A and B be nonempty subsets of a metric space (X, d) and let T : A ∪ B → A ∪ B be a
mapping. T is called a cyclic map if and only if T (A) ⊆ B and T (B) ⊆ A. In 2003, Kirk et al.
[9] introduced cyclic contraction mapping as follows:
A cyclic map T : A ∪ B → A ∪ B is said to be a cyclic contraction mapping if there exists
k ∈ [0, 1) such that
d (T x, T y) ≤ kd (x, y) , ∀x ∈ A, ∀y ∈ B.
Then T has a unique fixed point in A ∩ B.
In 2010, Karapınar and Erhan [8] introduced Kannan type cyclic contraction mapping as follows:
Let A and B be nonempty subsets of a metric space (X, d). A mapping T : A ∪ B → A ∪ B is
called a Kannan type cyclic contraction mapping if there exists k ∈ [0, 21 ) such that
d (T x, T y) ≤ k [d (x, f x) + d (y, f y)] , ∀x ∈ A, ∀y ∈ B.
If (X, d) is still complete, then T has a unique fixed point in A ∩ B ([8], [9]).
For another thing, from spaces, there are too many generalizations of metric spaces. For instance,
b-metric space, quasi-metric space, quasi-b-metric space, dislocated metric space (or metric-like
space), dislocated b-metric space (or b-metric-like space), dislocated quasi-metric space (or quasimetric-like space), dislocated quasi-b-metric space (or quasi-b−metric-like space) (see [1], [2], [4-6],
[10], [15-20]). Their definitions are as follows:
Let X be a nonempty set and s ≥ 1 a real number. Let d : X × X → [0, ∞) be a mapping and
x, y, z ∈ X:
(d1) d(x, y) = 0 ⇔ x = y;
(d2) d(x, y) = 0 ⇒ x = y;
(d3) d(x, y) = 0 = d(y, x) ⇒ x = y;
(d4) d(x, y) = d(y, x);
(d5) d(x, z) ≤ d(x, y) + d(y, z);
(d6) d(x, z) ≤ s[d(x, y) + d(y, z)], s ≥ 1.
Then
(1) (X, d) is called a metric space if (d1), (d4) and (d5) hold;
(2) (X, d) is called a b-metric space if (d1), (d4) and (d6) hold;
(3) (X, d) is called a quasi-metric space if (d1) and (d5) hold;
(4) (X, d) is called a quasi-b-metric space if (d1) and (d6) hold;
(5) (X, d) is called a dislocated metric space if (d2), (d4) and (d5) hold;
(6) (X, d) is called a dislocated b-metric space if (d2), (d4) and (d6) hold;
(7) (X, d) is called a dislocated quasi-metric space if (d3) and (d5) hold;
(8) (X, d) is called a dislocated quasi-b-metric space if (d3) and (d6) hold.
Despite the fact that the given examples previously known, we thought it is useful for easy
reference to give a full review.
Example 1.1. a) Let X = R and d : X × X → [0, ∞) be defined as
x − y, x ≥ y
d(x, y) =
1,
otherwise.
Then (X, d) is a quasi metric space, but it is not a metric space.
b) Let X = [0, ∞) and d : X × X → [0, ∞) be defined as
0,
x=y
d(x, y) =
(x + y)2 , otherwise.
FIXED POINT RESULTS IN DISLOCATED QUASI-b-METRIC SPACES
3
Then (X, d) is a b-metric space, but it is not a metric space.
c) Let X = C([0, 1], R) with the usual partial ordering, and let d : X × X → R+ be defined as
( R
1
(g(t) − f (t))3 dt, f ≤ g
R01
d(f, g) =
(f (t) − g(t))3 dt, f ≥ g.
0
Then (X, d) is a quasi b-metric space, but it is not a quasi-metric space and b-metric space.
d) Let X = R+ and d : X × X → R+ be defined as d(x, y) = max{x, y}. Then (X, d) is a
dislocated metric space, but it is not a metric space.
e) Let X = [0, 1] and d : X × X → R+ be defined as d(x, y) = |x − y| + x. Then (X, d) is a
dislocated quasi-metric space, but it is not a dislocated metric space, and it is not a quasi-metric
space.
f) Let X = [0, ∞) and d : X × X → [0, ∞) be defined as d(x, y) = (x + y)2 . Then (X, d) is a
dislocated b-metric space, but it is not a b-metric space.
|y|
g) Let X = R and d : X × X → [0, ∞) be defined as d(x, y) = |x − y|2 + |x|
n + m , where
n, m ∈ N \ {1}, n 6= m. Then (X, d) is a dislocated quasi b-metric space, but it is not a quasi
b-metric space, dislocated b-metric space and dislocated quasi-metric space.
So, we have the following process diagram, where arrows stand for generalization relationships.
Also, scholars are interested in dislocated quasi-b-metric spaces since they are more general
spaces. Based on this fact, we consider fixed point results in such spaces.
For the sake of reader, we recall the following concepts and results.
Definition 1.2. ([10]) Let (X, d) be a dislocated quasi-b-metric space and {xn }n∈N be a sequence
in X. Then we call
(i) {xn }n∈N converges to x ∈ X if
lim d (xn , x) = 0 = lim d (x, xn ) .
n→∞
n→∞
In this case x is called a dqb-limit of {xn }, and we write it as xn → x (n → ∞).
4
D. DOLIĆANIN-D̄EKIĆ, T. DOŠENOVIĆ, H. HUANG, S. RADENOVIĆ
(ii) {xn }n∈N is a dqb-Cauchy sequence if
lim d (xn , xm ) = 0 =
n,m→∞
lim d (xm , xn ) .
n,m→∞
(iii) (X, d) is dqb-complete if every dqb-Cauchy sequence is convergent in X.
Definition 1.3. ([10, Definition 2.8]) Let A and B be nonempty subsets of a dislocated quasi-bmetric space (X, d). A cyclic map T : A ∪ B → A ∪ B is said to be a dislocated quasi-b-metriccyclic-Banach contraction if there exists k ∈ [0, 1s ), s ≥ 1, such that
d (T x, T y) ≤ kd (x, y)
(1.1)
for all x ∈ A, y ∈ B.
Definition 1.4. Definition 1.3.([10, Definition 2.11]) Let A and B be nonempty subsets of a
dislocated quasi-b-metric space (X, d). A cyclic map T : A ∪ B → A ∪ B is said to be a dislocated
1
), s ≥ 1, such that
quasi-b-metric-cyclic-Kannan contraction if there exists k ∈ [0, 2s
(1.2)
d (T x, T y) ≤ k [d (x, f x) + d (y, f y)]
for all x ∈ A, y ∈ B.
In [10], authors proved the following main results:
Theorem 1.5. ([10, Theorem 2.9]) Let A and B be nonempty closed subsets of a dqb-complete
dislocated quasi-b-metric space (X, d). Let T be a cyclic mapping that satisfies the condition of a
dislocated quasi-b-metric-cyclic-Banach contraction. Then T has a unique fixed point in A ∩ B.
Theorem 1.6. ([10, Theorem 2.12]) Let A and B be nonempty closed subsets of a dqb-complete
dislocated quasi-b-metric space (X, d). Let T be a cyclic mapping that satisfies the condition of a
dislocated quasi-b-metric-cyclic-Kannan contraction. Then T has a unique fixed point in A ∩ B.
2. MAIN RESULTS
In this section, we consider and generalize some previous results. We also prove that the results
from Theorems 1.5 and Theorem 1.6 are just equivalent to the respective ordinary fixed point
results in the same framework.
Firstly, we recall the following lemma [13, Remark 2.13].
Lemma 2.1. If some ordinary fixed point theorem in the framework of metric (resp. b-metric)
spaces has a true cyclic-type extension, then these both theorems are equivalent.
Now we announce the following:
Theorem 2.2. Theorem 1.5 (that is, [10, Theorem 2.9]) is equivalent with the following claim:
(Claim 1) Let (X, d) be a dqb-complete dislocated quasi-b-metric space with s ≥ 1, and let
T : X → X be a mapping. Assume that there exists k ∈ [0, 1s ) such that for all x, y ∈ X,
(2.1)
d (T x, T y) ≤ kd (x, y) .
Then T has a unique fixed point in X.
Theorem 2.3. Theorem 1.6 (that is, [10, Theorem 2.12]) is equivalent with the following claim:
(Claim 2) Let (X, d) be a dqb-complete dislocated quasi-b-metric space with s ≥ 1, and let
1
T : X → X be a mapping. Assume that there exists k ∈ [0, 2s
) such that for all x, y ∈ X,
(2.2)
d (T x, T y) ≤ k [d (x, f x) + d (y, f y)] .
Then T has a unique fixed point in X.
In order to prove the above two theorems, we use the following crucial lemma:
FIXED POINT RESULTS IN DISLOCATED QUASI-b-METRIC SPACES
5
p
Lemma 2.4. Let {Ai }i=1 be nonempty closed subsets of a dqb-complete dislocated quasi-b-metric
space (X, d), and suppose that T : ∪pi=1 Ai → ∪pi=1 Ai satisfies the following conditions:
(1) T (Ai ) ⊆ Ai+1 for 1 ≤ i ≤ p where Ap+1 = A1 ;
(2) there exists k ∈ [0, 1s ) such that for all x ∈ ∪pi=1 Ai
(2.3)
d T 2 x, T x ≤ kd (T x, x) , d T x, T 2 x ≤ kd (x, T x) .
Then ∩pi=1 Ai 6= ∅.
Proof. If k = 0, then for all x ∈ ∪pi=1 Ai , by (2.3) and (d3), we have T (T x) = T x, i.e., T x is a
p
p
fixed point of T . Thus
by (1), it pis not hard to verify that T x ∈ ∩i=1 Ai . That is, ∩i=1 Ai 6= ∅.
1
Now, let k ∈ 0, s and x ∈ ∪i=1 Ai . Then by (2.3), on the one hand, we arrive at
d T n+1 x, T n x = d T 2 T n−1 x, T T n−1 x
≤ kd T n x, T n−1 x = kd T 2 T n−2 x, T T n−2 x
(2.4)
≤ k 2 d T n−1 x, T n−2 x ≤ · · · ≤ k n d (T x, x) .
On the other hand, we have
(2.5)
d T n x, T n+1 x = d T T n−1 x, T 2 T n−1 x
≤ kd T n−1 x, T n x = kd T T n−2 x, T 2 T n−2 x
≤ k 2 d T n−2 x, T n−1 x ≤ · · · ≤ k n d (x, T x) .
Hence by (2.4) and (2.5), we easily speculate that
lim d (T n x, T m x) = 0,
n,m→∞
lim d (T m x, T n x) = 0.
n,m→∞
In other words, we demonstrate that {T n x} is a dqb-Cauchy sequence in dislocated quasi-b-metric
space (∪pi=1 Ai , d). Note that (∪pi=1 Ai , d) is dqb-complete, it establishes that {T n x} converges to
some z ∈ ∪pi=1 Ai . Whereas, in view of (1), T n x has infinite terms lying in each Ai , i ∈ {1, 2, ..., p}.
As Ai is closed for all i ∈ {1, 2, ..., p} , we claim that z ∈ ∩pi=1 Ai . Therefore, ∩pi=1 Ai 6= ∅.
The proof of Theorem 2.2 Putting Ai = X for i ∈ {1, 2, ..., p} in Theorem 1.5, we obtain
Claim 1. Conversely, let Claim 1 holds. We shall prove that Theorem 1.5 also holds. Indeed, if
n
x ∈ ∪m
i=1 Ai , then by virtue of Lemma 2.4, it establishes that {T x} is a dqb-Cauchy sequence,
p
p
further, ∩i=1 Ai 6= ∅. Now that (∩i=1 Ai , d) is a dqb-complete dislocated quasi-b-metric space, we
restrict T to (∩pi=1 Ai , d) and hence the condition (2.1) holds for all x, y ∈ ∩pi=1 Ai , then Claim 1
implies that T has a unique fixed point in ∩pi=1 Ai . Accordingly, Theorem 1.5 is satisfied.
The proof of Theorem 2.3 We use the same method as in the proof of Theorem 2.2 and
hence the proof is omitted.
Further, we announce the result for the existence of fixed point under cyclical consideration in
the framework following dislocated quasi-b-metric spaces:
Lemma 2.5. Let (X = ∪pi=1 Ai , d) be a dqb-complete dislocated quasi-b-metric space. If T : X → X
satisfies (1) of Lemma 2.4, and for all x ∈ X = ∪pi=1 Ai , the corresponding Picard sequence {T n x}
is a dqb-Cauchy sequence, then ∩pi=1 Ai 6= ∅.
Proof. Since (X, d) is dqb-complete, then dqb-Cauchy sequence {T n x} converges to some z ∈ X.
We shall prove that z ∈ ∩pi=1 Ai . Actually, observing that T (Ai ) ⊆ Ai+1 for all i ∈ {1, 2, ..., p} and
T (Ap+1 ) ⊆ A1 , we conclude that {T n x} has infinite terms in Ai for all i ∈ {1, 2, ..., p}. As Ai is
closed for all i ∈ {1, 2, ..., p}, we claim that z ∈ ∩pi=1 Ai . Consequently, ∩pi=1 Ai 6= ∅.
The following two examples support our Theorems 2.2 and Theorem 2.3.
6
D. DOLIĆANIN-D̄EKIĆ, T. DOŠENOVIĆ, H. HUANG, S. RADENOVIĆ
Example 2.6. ([10, Example 2.10]) Let X = [−1, 1] and T x = − x5 . Suppose that A = [−1, 0] and
B = [0, 1]. Define the function d : X × X → [0, ∞) by
|x| |y|
+
.
10
11
Then (X, d) is a dislocated quasi-b-metric space with the coefficient s = 2. As a consequence of
d (1, 1) 6= 0, so (X, d) is not a quasi-b-metric space. Also, it is not a dislocated b-metric space based
on the fact that d (1, 0) 6= d (0, 1). Clearly, X = A ∪ B and T (A) ⊆ B, T (B) ⊆ A. We shall prove
(2.1), where k ∈ [ 51 , 12 ) ⊆ [0, 1s ).
Indeed, for all x, y ∈ X, we have
x y
d (T x, T y) = d − , −
5
5
x y 2 |x| |y|
= − + +
+
5
50
55 5
1 1
|x|
|y|
2
=
|x − y| +
+
5 5
10
11
1
|x|
|y|
2
≤
|x − y| +
+
5
10
11
≤ kd (x, y) ,
2
d (x, y) = |x − y| +
where k ∈ [ 15 , 21 ) ⊆ [0, 1s ). Hence, all the conditions of Claim 1 are satisfied, that is, T has a
unique fixed point in X. Now, using Theorem 2.2, we obtain that T has a unique fixed point in
A ∩ B = {0}, i.e., 0 is the unique fixed point of T .
Example 2.7. ([10, Example 2.13]) Let X = [−1, 1] and T x = − x7 . Suppose that A = [−1, 0] and
B = [0, 1]. Define the function d : X × X → [0, ∞) by
2
d (x, y) = |x − y| + 3 |x| + 2 |y| .
Then (X, d) is a dislocated quasi-b-metric space with the coefficient s = 2. Owing to d (1, 1) 6= 0,
hence (X, d) is not a quasi-b-metric space. Also, it is not a dislocated b-metric space because
of d (1, 0) 6= d (0, 1). Further, similar as in the preceding example, we can show (2.2), where
2 1
k ∈ [ 23
, 2 ) ⊆ [0, 1s ). Thereupon all the conditions of Claim 2 are satisfied, it ensures us that T
has a unique fixed point in X. Now, by Theorem 1.6, T has a unique fixed point in A ∩ B = {0}.
Finally by Theorem 2.3, 0 is a unique fixed point of T .
The following lemmas are useful in proving of all main results in the framework of dislocated
quasi-b-metric spaces. The proofs are almost same as in [14] for the case of b-metric spaces and
hence we omit them.
Lemma 2.8. Let (X, d) be a dislocated quasi-b-metric space with s ≥ 1, and suppose that {xn }
and {yn } are dqb-convergent to x, y, respectively. Then we have
1
d (x, y) ≤ lim inf d (xn , yn ) ≤ lim sup d (xn , yn ) ≤ s2 d (x, y)
n→∞
s2
n→∞
and
1
d (y, x) ≤ lim inf d (yn , xn ) ≤ lim sup d (yn , xn ) ≤ s2 d (y, x) .
n→∞
s2
n→∞
In particular, if x = y, then limn→∞ d (xn , yn ) = 0 and limn→∞ d (yn , xn ) = 0. Moreover, for each
z ∈ X, it follows that
1
d (x, z) ≤ lim inf d (xn , z) ≤ lim sup d (xn , z) ≤ sd (x, z)
n→∞
s
n→∞
FIXED POINT RESULTS IN DISLOCATED QUASI-b-METRIC SPACES
and
7
1
d (z, x) ≤ lim inf d (z, xn ) ≤ lim sup d (z, xn ) ≤ sd (z, x) .
n→∞
s
n→∞
Lemma 2.9. Let (X, d) be a dislocated quasi-b-metric space with s ≥ 1 and let {xn } be a sequence
in X such that
lim d (xn , xn+1 ) = 0.
n→∞
If {xn } is not a dqb-Cauchy sequence, then there exist ε > 0 and two sequences {m (k)} and {n (k)}
of positive integers such that for the following sequences
d xm(k) , xn(k) , d xm(k) , xn(k)+1 , d xm(k)+1 , xn(k) , d xm(k)+1 , xn(k)+1 ,
and
d xn(k) , xm(k) , d xn(k)+1 , xm(k) , d xn(k) , xm(k)+1 , d xn(k)+1 , xm(k)+1 ,
it holds:
ε ≤ lim inf d xm(k) , xn(k) ≤ lim sup d xm(k) , xn(k) ≤ sε,
n→∞
n→∞
ε
≤ lim inf d xm(k) , xn(k)+1 ≤ lim sup d xm(k) , xn(k)+1 ≤ s2 ε,
n→∞
s
n→∞
ε
≤ lim inf d xm(k)+1 , xn(k) ≤ lim sup d xm(k)+1 , xn(k) ≤ s2 ε,
n→∞
s
n→∞
ε
≤ lim inf d xm(k)+1 , xn(k)+1 ≤ lim sup d xm(k)+1 , xn(k)+1 ≤ s3 ε,
2
n→∞
s
n→∞
and
ε ≤ lim inf d xn(k) , xm(k) ≤ lim sup d xn(k) , xm(k) ≤ sε,
n→∞
n→∞
ε
≤ lim inf d xn(k)+1 , xm(k) ≤ lim sup d xn(k)+1 , xm(k) ≤ s2 ε,
n→∞
s
n→∞
ε
≤ lim inf d xn(k) , xm(k)+1 ≤ lim sup d xn(k) , xm(k)+1 ≤ s2 ε,
n→∞
s
n→∞
ε
≤ lim inf d xn(k)+1 , xm(k)+1 ≤ lim sup d xn(k)+1 , xm(k)+1 ≤ s3 ε.
n→∞
s2
n→∞
Finally, we shall announce a generalization of the recent Geraghty type result from [4, Theorem
2.1] used in the setting of b-metric spaces. In order to use our ideas in the framework of dislocated
quasi-b-metric spaces, we shall consider the class of functions Bs , where β ∈ Bs satisfying β :
[0, +∞) → [0, 1s ) and has the property
1
implies tn → 0.
s
An example such as β is given by β (t) = 1s e−t for t > 0 and β (0) ∈ [0, 1s ).
The following result is a generalization of [4, Theorem 2.1] since we do not assume that the
dislocated quasi-b-metric d is continuous. With regard to its proof, we omit it because it is almost
same as the counterpart in [4, Theorem 2.1] for b-metric spaces without using Lemma 2.8 and
Lemma 2.9., however, many other papers have to use them.
β (tn ) →
Theorem 2.10. Let (X, d) be a dislocated quasi-b-metric space with s > 1 and let f, g : X → X
be two maps such that f (X) ⊆ g (X) and one of these two subsets of X is dqb-complete. If for
some function β ∈ Bs and for all x, y ∈ X, it holds
d (f x, f y) ≤ β (d (gx, gy)) d (gx, gy) ,
then f and g have a unique point of coincidence ω. Moreover, for each x0 ∈ X, a corresponding
Jungck sequence {yn } can be chosen such that limn→∞ yn = ω. In addition, if f and g are weakly
compatible, then they have a unique common fixed point.
8
D. DOLIĆANIN-D̄EKIĆ, T. DOŠENOVIĆ, H. HUANG, S. RADENOVIĆ
Taking g = IX (identity mapping of X) in Theorem 2.10, we obtain that the following Geraghty
type theorem in dislocated quasi-b-metric spaces:
Corollary 2.11. Let (X, d) be a dqb-complete dislocated quasi-b-metric space with s > 1. Suppose
that a mapping f : X → X satisfies
d (f x, f y) ≤ β (d (x, y)) d (x, y) ,
for all x, y ∈ X and some β ∈ Bs . Then f has a unique fixed point z ∈ X, and for each x ∈ X,
the Picard sequence {f n x} converges to z in X.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Acknowledgment
The first and second authors are thankful to the Ministry of Education, Science and Technological Development of Serbia. The third author is thankful to the science and technology research
project of education department in Hubei province of China (B2015137).
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FIXED POINT RESULTS IN DISLOCATED QUASI-b-METRIC SPACES
9
Diana Dolićanin-D̄ekić1 , Faculty of Technical Sciences, University of Priština-Kosovska Mitrovica,
Kneza Miloša 7, Serbia
E-mail address:
[email protected]
Tatjana Došenović2 , Faculty of Technology, University of Novi Sad, Bulevar cara Lazara 1, Serbia
E-mail address: [email protected]
Huaping Huang3 , School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002,
China
E-mail address: [email protected]
Stojan Radenović, 4 Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh
City, Vietnam, 5 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City,
Vietnam
E-mail address: [email protected]