Fixed-point theorems for integral-type contractions on partial metric spaces
We present some fixed-point results for single-valued mappings on partial metric spaces satisfying a contractive condition of integral type.
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| author | Altun, I. Masiha, H. P. Sabetghadam, F. Алтун, І. Масиха, Х. П. Сабетхадам, Ф. |
| author_facet | Altun, I. Masiha, H. P. Sabetghadam, F. Алтун, І. Масиха, Х. П. Сабетхадам, Ф. |
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| description | We present some fixed-point results for single-valued mappings on partial metric spaces satisfying a contractive condition
of integral type. |
| first_indexed | 2026-03-24T02:14:30Z |
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UDC 517.9
F. Sabetghadam, H. P. Masiha (K. N. Toosi Univ. Technology, Tehran, Iran),
I. Altun (King Saud Univ., Saudi Arabia and Kirikkale Univ., Turkey)
FIXED-POINT THEOREMS FOR INTEGRAL-TYPE CONTRACTIONS
ON PARTIAL METRIC SPACES
TЕОРЕМИ ПРО НЕРУХОМУ ТОЧКУ ДЛЯ СТИСКIВ
IНТЕГРАЛЬНОГО ТИПУ НА ЧАСТИННИХ МЕТРИЧНИХ ПРОСТОРАХ
We present some fixed-point results for single-valued mappings on partial metric spaces satisfying a contractive condition
of integral type.
Наведено деякi теореми про нерухому точку для однозначних вiдображень на частинних метричних просторах, що
задовольняють умову стиску iнтегрального вигляду.
1. Introduction and preliminaries. The Banach contraction mapping principle is one of the
pivotal results of analysis. It is widely considered as the source of metric fixed-point theory and its
significance lies in its vast applicability in a number of branches of mathematics. There are a lot of
generalization of Banach contraction mapping principle in the literature. One of the most important
and interesting generalization of it was given by Branciari [1]. He obtained the existence of fixed
points for mappings f : X \rightarrow X defined on a complete metric space (X, d) satisfying contractive
condition of integral type as follows:
d(fx,fy)\int
0
\varphi (t)dt \leq c
d(x,y)\int
0
\varphi (t)dt,
where c \in [0, 1) and \varphi : [0,+\infty ) \rightarrow [0,+\infty ) is a Lebesgue-integrable mapping which is summable
on each compact subset of [0,+\infty ), nonnegative and such that for each \epsilon > 0,
\int \epsilon
0
\varphi (t)dt > 0.
After that in [2], the author proved two fixed-point theorems involving more general contractive
condition of integral type.
Recently, there is a trend to weaken the requirement on the contraction by considering metric
spaces endowed with partial order. In [3, 4] the Banach contraction principle was discussed in a
metric space endowed with partial order. Also, existence of fixed point in partially ordered sets has
been considered recently in [5 – 15]. The study on the existence of fixed points for single-valued
increasing operators is successful, the results obtained are widely used to investigate the existence
of solutions to the ordinary and partial differential equations (see [9, 12]). Recently Bhaskar and
Lakshmikantham [6], Nieto and Lopez [3, 14], Lakshmikantham and Ciric [13], Ran and Reurings
[4] and Agarwal, El-Gebeily and O’Regan [5] presented some new results for contraction in partially
ordered metric spaces. Bhaskar and Lakshmikantham [6] noted that their theorem can be used to
investigate a large class of problems and have discussed the existence and uniqueness of solution for
a periodic boundary-value problem.
On the other hand, after the definitions of partial metric space by Matthews [16], fixed-point
theory on this interesting space is rapidly developed. For example in [16 – 24], the authors presented
c\bigcirc F. SABETGHADAM, H. P. MASIHA, I. ALTUN, 2016
826 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
FIXED-POINT THEOREMS FOR INTEGRAL-TYPE CONTRACTIONS ON PARTIAL METRIC SPACES 827
some fixed-point theorems for generalized contractive type mappings on partial metric spaces. Then,
Altun and Erduran [25] gave some fixed-point theorems on ordered partial metric spaces.
In this paper, we give fixed-point theorems for mappings satisfying a general contractive condition
of integral type on ordered partial metric spaces.
First, we recall some definitions and properties of partial metric space. Further detailed informa-
tion about partial metric spaces can be found in [16 – 19].
A partial metric on a nonempty set X is a function \scrP : X\times X \rightarrow \BbbR + such that for all x, y, z \in X :
(p1) x = y \leftrightarrow \scrP (x, x) = \scrP (x, y) = \scrP (y, y),
(p2) \scrP (x, x) \leq \scrP (x, y),
(p3) \scrP (x, y) = \scrP (y, x),
(p4) \scrP (x, y) \leq \scrP (x, z) + \scrP (z, y) - \scrP (z, z).
A partial metric space is a pair (X,\scrP ) such that X is a nonempty set and \scrP is a partial metric
on X. It is clear that, if \scrP (x, y) = 0, then from (p1) and (p2) x = y. But if x = y, \scrP (x, y) may not
be 0. A basic example of a partial metric space is the pair (\BbbR +,\scrP ), where \scrP (x, y) = \mathrm{m}\mathrm{a}\mathrm{x}\{ x, y\} for
all x, y \in \BbbR +. Other examples of partial metric spaces which are interesting from a computational
point of view may be found in [16].
Each partial metric \scrP on X generates a T0 topology \tau P on X which has as a base the family of
open \scrP -balls \{ Bp(x, \epsilon ) : x \in X, \epsilon > 0\} , where Bp(x, \epsilon ) = \{ y \in X : \scrP (x, y) < \scrP (x, x) + \epsilon \} for all
x \in X and \epsilon > 0.
If \scrP is a partial metric on X, then the function \scrP s : X \times X \rightarrow \BbbR + given by
\scrP s(x, y) = 2\scrP (x, y) - \scrP (x, x) - \scrP (y, y)
is a metric on X.
Let (X,\scrP ) be a partial metric space. Then:
a sequence \{ xn\} in a partial metric space (X,\scrP ) converges to a point x \in X if and only if
\scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (x, xn);
a sequence \{ xn\} in a partial metric space (X,\scrP ) is called a Cauchy sequence if there exists (and
is finite) \mathrm{l}\mathrm{i}\mathrm{m}n,m\rightarrow \infty \scrP (xn, xm);
a partial metric space (X,\scrP ) is said to be complete if every Cauchy sequence \{ xn\} in X
converges, with respect to \tau p, to a point x \in X such that \scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}n,m\rightarrow \infty \scrP (xn, xm);
a mapping f : X \rightarrow X is said to be continuous at x0 \in X, if for every \epsilon > 0, there exists \delta > 0
such that F (Bp(x0, \delta )) \subseteq Bp(Fx0, \epsilon ).
Lemma 1. Let (X,\scrP ) be a partial metric space.
(1) \{ xn\} is a Cauchy sequence in (X,\scrP ) if and only if it is a Cauchy sequence in the metric
space (X,\scrP s).
(2) A partial metric space (X,\scrP ) is complete if and only if the metric space (X,\scrP s) is complete.
Furthermore, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP s(xn, x) = 0 if and only if
\scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrP (xn, x) = \mathrm{l}\mathrm{i}\mathrm{m}
n,m\rightarrow \infty
\scrP (xn, xm).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
828 F. SABETGHADAM, H. P. MASIHA, I. ALTUN
Now, we give some notations, which will be used in this paper. Let \varphi : [0,+\infty ) \rightarrow [0,+\infty ) and
\psi : [0,+\infty ) \rightarrow [0,+\infty ) be two functions. For convenience, we consider the following properties of
these functions:
(\varphi 1) \varphi is nonincreasing on [0,\infty );
(\varphi 2) \varphi is Lebesgue integrable;
(\varphi 3) for each \epsilon > 0,
\int \epsilon
0
\varphi (t)dt > 0.
and
(\psi 1) \psi is nondecreasing on [0,+\infty );
(\psi 2)
\sum \infty
n=1
\psi n(t) <\infty for each t > 0.
Denote by \Phi the family of all functions \varphi : [0,+\infty ) \rightarrow [0,+\infty ) satisfying (\varphi 1), (\varphi 2) and denote
by \Psi the family of all functions \psi : [0,+\infty ) \rightarrow [0,+\infty ) satisfying (\psi 1), (\psi 2). Note that \Psi is called
the family of (c)-comparison functions in the literature and if \psi \in \Psi , then we have \psi (t) < t for
each t > 0.
2. Main results. Let (X,\leq ) be a partially ordered set and T : X \rightarrow X be a mapping, then T is
said to be nondecreasing if x, y \in X with x \leq y implies Tx \leq Ty and T is said to be nondecreasing
if x, y \in X with x \leq y implies Ty \leq Tx.
Theorem 1. Let (X,\leq ) be a partially ordered set and suppose that there is a partial metric \scrP
on X such that (X,\scrP ) is a complete partial metric space. Suppose T : X \rightarrow X is a continuous and
nondecreasing mapping such that
\scrP (Tx,Ty)\int
0
\varphi (t)dt \leq \psi
\left( M(x,y)\int
0
\varphi (t)dt
\right) (1)
for all x, y \in X with x \leq y, where \varphi \in \Phi , \psi \in \Psi , and
M(x, y) := \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\scrP (x, y),\scrP (x, Tx),\scrP (y, Ty),
[\scrP (x, Ty) + \scrP (y, Tx)]
2
\biggr\}
.
If there exists an x0 \in X with x0 \leq Tx0, then there exists x \in X such that \scrP (x, Tx) = 0, in
particular, x = Tx and \scrP (x, x) = 0.
Proof. Let xn = Txn - 1 for n = 1, 2, . . . . Suppose that xn+1 = xn for some n. Then xn = Txn
and xn is a fixed point, therefore the proof is finished. Now, assume that xn \not = xn+1 for all n. Since
x0 < Tx0 and T is a nondecreasing mapping, we obtain by induction that
x0 < x1 \leq x2 \leq . . . \leq xn \leq xn+1 \leq . . . .
From (1) and, as the elements xn and xn+1 are comparable, we get
\scrP (xn,xn+1)\int
0
\varphi (t)dt =
\scrP (Txn - 1,Txn)\int
0
\varphi (t)dt \leq \psi
\left( M(xn - 1,xn)\int
0
\varphi (t)dt
\right) , (2)
where
M(xn - 1, xn) = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\scrP (xn - 1, xn),\scrP (xn - 1, xn),\scrP (xn, xn+1),
[\scrP (xn - 1, xn+1) + \scrP (xn, xn)]
2
\biggr\}
,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
FIXED-POINT THEOREMS FOR INTEGRAL-TYPE CONTRACTIONS ON PARTIAL METRIC SPACES 829
and, as
\scrP (xn - 1, xn+1) + \scrP (xn, xn)
2
\leq \scrP (xn - 1, xn) + \scrP (xn, xn+1)
2
\leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ \scrP (xn - 1, xn),\scrP (xn, xn+1)\} ,
we obtain
M(xn - 1, xn) = \mathrm{m}\mathrm{a}\mathrm{x}\{ \scrP (xn, xn - 1),\scrP (xn, xn+1)\} .
Now, if
\mathrm{m}\mathrm{a}\mathrm{x}\{ \scrP (xn, xn - 1),\scrP (xn, xn+1)\} = \scrP (xn, xn+1),
for some n, then using (2), (\varphi 3) and the condition \scrP (xn, xn+1) > 0, we have
\scrP (xn,xn+1)\int
0
\varphi (t)dt \leq \psi
\left( \scrP (xn,xn+1)\int
0
\varphi (t)dt
\right) <
\scrP (xn,xn+1)\int
0
\varphi (t)dt,
that is a contradiction. Thus
M(xn - 1, xn) = \mathrm{m}\mathrm{a}\mathrm{x}\{ \scrP (xn, xn - 1),\scrP (xn, xn+1)\} = \scrP (xn, xn - 1) for all n.
Then taking into account (2) and (\psi 1), we get
\scrP (xn,xn+1)\int
0
\varphi (t)dt \leq \psi
\left( \scrP (xn - 1,xn)\int
0
\varphi (t)dt
\right) \leq . . . \leq \psi n(d), (3)
where
d :=
\scrP (x0,x1)\int
0
\varphi (t)dt.
In what follows we will show that \{ xn\} is a Cauchy sequence in the metric space (X,\scrP s). By
definition of \scrP s we have
\scrP s(xn, xn+1) = 2\scrP (xn, xn+1) - \scrP (xn, xn) - \scrP (xn+1, xn+1) \leq 2\scrP (xn, xn+1)
or, equivalently,
1
2
\scrP s(xn, xn+1) \leq \scrP (xn, xn+1). (4)
Let m,n \in \BbbN , m > n. Then using the triangle inequality and (4)
1
2
\scrP s(xn, xm) \leq 1
2
m - 1\sum
i=n
\scrP s(xi, xi+1) \leq
m - 1\sum
i=n
\scrP (xi, xi+1).
Since \varphi is nonincreasing we obtain
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
830 F. SABETGHADAM, H. P. MASIHA, I. ALTUN
a+b\int
0
\varphi (t)dt \leq
a\int
0
\varphi (t)dt+
b\int
0
\varphi (t)dt for all a, b \geq 0.
Now, we can get
1
2
\scrP s(xn,xm)\int
0
\varphi (t)dt \leq
\sum m - 1
i=n \scrP (xi,xi+1)\int
0
\varphi (t)dt \leq
m - 1\sum
i=n
\scrP (xi,xi+1)\int
0
\varphi (t)dt. (5)
Using (3) and (5),
1
2
\scrP s(xn,xm)\int
0
\varphi (t)dt \leq
m - 1\sum
i=n
\psi i(d) \leq
\infty \sum
i=n
\psi i(d).
By the convergence of the series
\sum \infty
i=1
\psi i(d), passing to the limit as n,m \rightarrow \infty , we have\int 1/2\scrP s(xn,xm)
0
\varphi (t)dt \rightarrow 0. Using condition (\varphi 3), it follows that \scrP s(xn, xm) \rightarrow 0, that is, \{ xn\} is
a Cauchy sequence in the metric space (X,\scrP s). Since (X,\scrP ) is complete, by Lemma 1, (X,\scrP s) is
complete and the sequence \{ xn\} is convergent in X, say \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP s(xn, x) = 0. From Lemma 1,
we get
\scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrP (xn, x) = \mathrm{l}\mathrm{i}\mathrm{m}
n,m\rightarrow \infty
\scrP (xn, xm).
Now we show that \scrP (x, Tx) = 0. Suppose \scrP (x, Tx) > 0. Since T is continuous, then given \epsilon > 0,
there exists \delta > 0 such that T (Bp(x, \delta )) \subseteq Bp(Tx, \epsilon ). Since \scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (xn, x), then
there exists k \in \BbbN such that \scrP (xn, x) < \scrP (x, x)+\delta for all n \geq k. Therefore, we have xn \in Bp(x, \delta )
for all n \geq k. Thus Txn \in T (Bp(x, \delta )) \subseteq Bp(Tx, \epsilon ) and so \scrP (Txn, Tx) < \scrP (Tx, Tx) + \epsilon for all
n \geq k. This shows that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (xn+1, Tx) = \scrP (Tx, Tx). Then we have
\scrP (x, Tx) \leq \scrP (x, xn+1) + \scrP (xn+1, Tx) - \scrP (xn+1, xn+1),
and letting n\rightarrow \infty , we get
\scrP (x, Tx) \leq \scrP (Tx, Tx).
Now, by using inequality (1) and (\varphi 3) and (\psi 2) we obtain
\scrP (x,Tx)\int
0
\varphi (t)dt \leq
\scrP (Tx,Tx)\int
0
\varphi (t)dt \leq
\leq \psi
\left( M(x,x)\int
0
\varphi (t)dt
\right) =
= \psi
\left( max\{ \scrP (x,x),\scrP (x,Tx)\} \int
0
\varphi (t)dt
\right) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
FIXED-POINT THEOREMS FOR INTEGRAL-TYPE CONTRACTIONS ON PARTIAL METRIC SPACES 831
= \psi
\left( \scrP (x,Tx)\int
0
\varphi (t)dt
\right) <
\scrP (x,Tx)\int
0
\varphi (t)dt,
that is a contradiction. Thus \scrP (x, Tx) = 0.
Theorem 1 is proved.
In what follows we will prove a variant of Theorem 1, where we will use the following properties
of the space X instead of the continuity of T :
(H1) if \{ xn\} is a nondecreasing sequence in X such that \scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (x, xn), then
xn \leq x for all n \in \BbbN .
Theorem 2. Let (X,\leq ) be a partially ordered set and suppose \scrP is a partial metric on X such
that (X,\scrP ) is a complete partial metric space. Assume that X satisfies (H1). Let T : X \rightarrow X be a
nondecreasing mapping such that
\scrP (Tx,Ty)\int
0
\varphi (t)dt \leq \psi
\left( M(x,y)\int
0
\varphi (t)dt
\right) (6)
for all x, y \in X with x \leq y, where \varphi \in \Phi , \psi \in \Psi , and
M(x, y) := \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\scrP (x, y),\scrP (x, Tx),\scrP (y, Ty),
[\scrP (x, Ty) + \scrP (y, Tx)]
2
\biggr\}
.
If there exists an x0 \in X with x0 \leq Tx0, then there exists x \in X such that \scrP (x, Tx) = \scrP (x, x).
Remark 1. For the point x, which is the limit of iterative sequence of \{ xn\} , in Theorem 2, if
\scrP (x, x) = 0, then x is a fixed point of T. We know that the condition \scrP (x, x) = 0 is not strong. For
example, if (X,\scrP ) is a 0-complete partial metric space, then \scrP (x, x) = 0, because x is the limit of
iterative sequence of \{ xn\} . We can find some detailed information about 0-complete partial metric
space in [23]. If (X,\scrP ) is an ordinary metric, then \scrP (x, x) = 0 is also satisfied.
Proof. Following the proof of Theorem 1 we only have to check that \scrP (x, Tx) = \scrP (x, x).
Suppose that \scrP (x, Tx) > \scrP (x, x), as \scrP (x, x) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (xn, x) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (xn, xn+1), there
exists k \in \BbbN such that
\scrP (xn, x) < \scrP (x, x) +
\scrP (x, Tx) - \scrP (x, x)
2
=
\scrP (x, Tx) + \scrP (x, x)
2
and
\scrP (xn, xn+1) <
\scrP (x, Tx) + \scrP (x, x)
2
for all n \geq k. Then for n \geq k we have
1
2
[\scrP (x, Txn) + \scrP (xn, Tx)] \leq
\leq 1
2
[\scrP (x, xn+1) + \scrP (xn, x) + \scrP (x, Tx) - \scrP (x, x)] < \scrP (x, Tx).
Now from (H1), we get xn \leq x for all n \in \BbbN and so we can use the inequality (6). Therefore, for
n \geq k we obtain
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
832 F. SABETGHADAM, H. P. MASIHA, I. ALTUN
\scrP (Tx,xn+1)\int
0
\varphi (t)dt =
\scrP (Tx,Txn)\int
0
\varphi (t)dt \leq \psi
\left( M(x,xn)\int
0
\varphi (t)dt
\right) , (7)
where
M(x, xn) = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\scrP (x, xn),\scrP (x, Tx),\scrP (xn, Txn),
1
2
[\scrP (x, Txn) + \scrP (xn, Tx)]
\biggr\}
= \scrP (x, Tx).
Therefore from (7) we get
\scrP (Tx,xn+1)\int
0
\varphi (t)dt \leq \psi
\left( \scrP (x,Tx)\int
0
\varphi (t)dt
\right) . (8)
On the other hand
\scrP (x, Tx) \leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrP (x, xn+1) + \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrP (xn+1, Tx) - \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrP (xn+1, xn+1) =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrP (xn+1, Tx).
Therefore by (8) and \psi (t) < t, we have
\scrP (x,Tx)\int
0
\varphi (t)dt \leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\left( \scrP (Tx,xn+1)\int
0
\varphi (t)dt
\right) \leq
\leq \psi
\left( \scrP (x,Tx)\int
0
\varphi (t)dt
\right) <
\scrP (x,Tx)\int
0
\varphi (t)dt,
that is a contradiction. Thus \scrP (x, Tx) = \scrP (x, x).
Theorem 2 is proved.
Remark 2. If X is a totally ordered set in Theorem 1, then the fixed point of T is unique.
Suppose that there exist z, y \in X which are two fixed points of T. Since X is totally ordered
set, then z is comparable to y, and so Tnz = z is comparable to Tny = y for all n \in \BbbN . Therefore
from (1) we get \scrP (z, y) = 0 since M(y, z) = \scrP (z, y).
Remark 3. In the following theorem we present parallel results to Theorems 1 and 2 for nonin-
creasing functions.
Theorem 3. Let (X,\leq ) be a partially ordered set and suppose that there is a partial metric \scrP
on X such that (X,\scrP ) is a complete partial metric space. Suppose T : X \rightarrow X is a nonincreasing
mapping such that
\scrP (Tx,Ty)\int
0
\varphi (t)dt \leq \psi
\left( M(x,y)\int
0
\varphi (t)dt
\right)
for all x, y \in X with x \leq y, where \varphi \in \Phi , \psi \in \Psi , and
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
FIXED-POINT THEOREMS FOR INTEGRAL-TYPE CONTRACTIONS ON PARTIAL METRIC SPACES 833
M(x, y) := \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\scrP (x, y),\scrP (x, Tx),\scrP (y, Ty),
[\scrP (x, Ty) + \scrP (y, Tx)]
2
\biggr\}
.
If T is continuous and there exists x0 \in X with x0 \leq Tx0, then T has a fixed point.
Proof. If Tx0 = x0 then the existence of a fixed point is proved. Suppose that x0 \not = Tx0, by our
assumption Tn+1x0 and Tnx0 are comparable, for every n = 0, 1, 2, . . . . Using the same argument
that in Theorem 1 we prove that \{ Tnx0\} is convergent to certain x \in X and by the continuity of T,
we see that x is fixed point of T.
Theorem 3 is proved.
We can consider the following condition instead of the continuity of T in Theorem 3:
(H2) if \{ xn\} is a sequence in X whose consecutive terms are comparable and such that \scrP (x, x) =
= \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrP (x, xn), then there exists a subsequences \{ xnk
\} of \{ xn\} such that every term is
comparable to x for all n \in \BbbN .
Remark 4. Suppose that all conditions of Theorem 3 are satisfied except for the continuity of T.
If the condition (H2) hold, then there exists x \in X such that \scrP (x, Tx) = \scrP (x, x).
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Received 11.06.12,
after revision — 03.01.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
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| id | umjimathkievua-article-1881 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:30Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/b5/904cb82cbd6abfbdc48253178d331db5.pdf |
| spelling | umjimathkievua-article-18812019-12-05T09:30:37Z Fixed-point theorems for integral-type contractions on partial metric spaces Теореми про нерухому точку для стискiв iнтегрального типу на частинних метричних просторах Altun, I. Masiha, H. P. Sabetghadam, F. Алтун, І. Масиха, Х. П. Сабетхадам, Ф. We present some fixed-point results for single-valued mappings on partial metric spaces satisfying a contractive condition of integral type. Наведено деякi теореми про нерухому точку для однозначних вiдображень на частинних метричних просторах, що задовольняють умову стиску iнтегрального вигляду. Institute of Mathematics, NAS of Ukraine 2016-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1881 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 6 (2016); 826-834 Український математичний журнал; Том 68 № 6 (2016); 826-834 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1881/863 Copyright (c) 2016 Altun I.; Masiha H. P.; Sabetghadam F. |
| spellingShingle | Altun, I. Masiha, H. P. Sabetghadam, F. Алтун, І. Масиха, Х. П. Сабетхадам, Ф. Fixed-point theorems for integral-type contractions on partial metric spaces |
| title | Fixed-point theorems for integral-type contractions
on partial metric spaces |
| title_alt | Теореми про нерухому точку для стискiв iнтегрального типу на частинних метричних просторах |
| title_full | Fixed-point theorems for integral-type contractions
on partial metric spaces |
| title_fullStr | Fixed-point theorems for integral-type contractions
on partial metric spaces |
| title_full_unstemmed | Fixed-point theorems for integral-type contractions
on partial metric spaces |
| title_short | Fixed-point theorems for integral-type contractions
on partial metric spaces |
| title_sort | fixed-point theorems for integral-type contractions
on partial metric spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1881 |
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