Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces
We prove some fixed-point results for multivalued generalized nonlinear contractive mappings in partial metric spaces, which generalize and improve the corresponding recent fixed-point results due to Ćirić [L. B. Ćirić, “Multivalued nonlinear contraction mappings,” Nonlin. Anal., 71, 2716–2723 (2009...
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| author | Aghajani, A. Allahyari, R. Агажані, А. Аллахярі, Р. |
| author_facet | Aghajani, A. Allahyari, R. Агажані, А. Аллахярі, Р. |
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| description | We prove some fixed-point results for multivalued generalized nonlinear contractive mappings in partial metric spaces, which generalize and improve the corresponding recent fixed-point results due to Ćirić [L. B. Ćirić, “Multivalued nonlinear contraction mappings,” Nonlin. Anal., 71, 2716–2723 (2009)] and Klim and Wardowski [D. Klim and D. Wardowski, “Fixed-point theorems for set-valued contractions in complete metric spaces,” J. Math. Anal. Appl., 334, 132–139 (2007)]. |
| first_indexed | 2026-03-24T02:18:49Z |
| format | Article |
| fulltext |
UDC 517.9
A. Aghajani, R. Allahyari (Karaj Branch, Islamic Azad Univ., Karaj, Iran)
FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED
NONLINEAR CONTRACTIVE MAPS IN PARTIAL METRIC SPACES
ТЕОРЕМИ ПРО НЕРУХОМУ ТОЧКУ ДЛЯ БАГАТОЗНАЧНИХ
УЗАГАЛЬНЕНИХ НЕЛIНIЙНИХ СТИСКАЮЧИХ ВIДОБРАЖЕНЬ
В ЧАСТКОВО МЕТРИЧНИХ ПРОСТОРАХ
We prove some fixed point results for multivalued generalized nonlinear contractive mappings in partial metric spaces
which generalize and improve the corresponding recent fixed point results due to Ćirić [Ćirić L. B. Multivalued nonlinear
contraction mappings // Nonlinear Anal. – 2009. – 71. – P. 2716 – 2723], and Klim and Wardowski [Klim D., Wardowski
D. Fixed point theorems for set-valued contractions in complete metric spaces // J. Math. Anal. and Appl. – 2007. – 334. –
P. 132 – 139].
Доведено деякi теореми про нерухому точку в частково метричних просторах, що узагальнюють та покращують
вiдповiднi новi результати про нерухому точку, отриманi Чiрiчем (Ćirić L. B. Multivalued nonlinear contraction
mappings // Nonlinear Anal. – 2009. – 71. – P. 2716 – 2723) та Клiмом i Вардовським (Klim D., Wardowski D. Fixed point
theorems for set-valued contractions in complete metric spaces // J. Math. Anal. and Appl. – 2007. – 334. – P. 132 – 139).
1. Introduction and preliminaries. Let (X, d) be a metric space, 2X the collection of nonempty
subsets of X, Cl(X) be the subcollection of nonempty closed subsets of X. Investigations on the
existence of fixed points of multivalued contractions in metric spaces were initiated by Nadler [15]
and has been extended in many different directions by many authors, in particular, by Reich [19], Mi-
zoguchi, Takahashi [14] and Feng, Liu [8]. In this context Klim and Wardowski proved the following
nice result:
Theorem 1.1 [12]. Let (X, d) be a complete metric space and let T : X −→ Cl(X). Assume
that the following conditions hold:
(i) there exist a number b ∈ (0, 1) and a function k : [0,∞) −→ [0, b) such that for each
t ∈ [0,∞),
lim sup
r−→t+
k(r) < b,
(ii) for any x ∈ X there is y ∈ T (x) satisfying
bd(x, y) ≤ d(x, T (x))
and
d(y, T (y)) ≤ k(d(x, y))d(x, y).
Then Fix(T ) 6= ∅ provided the real-valued function g on X, g(x) = d(x, T (x)) is lower semicon-
tinuous.
c© A. AGHAJANI, R. ALLAHYARI, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1 3
4 A. AGHAJANI, R. ALLAHYARI
Recently, Ćirić [6] extended the above result and proved the following two interesting theorems:
Theorem 1.2. Let (X, d) be a complete metric space and let T : X −→ Cl(X). Assume that the
following conditions hold:
(i) the map f : X −→ R, defined by f(x) = d(x, T (x)) is lower semicontinuous,
(ii) there exists a function ϕ : [0,∞) −→ [b, 1), 0 < b < 1, satisfying
lim sup
r−→t+
ϕ(r) < 1 for each t ∈ [0,∞),
(iii) for any x ∈ X, there is y ∈ T (x) satisfying√
ϕ(f(x))d(x, y) ≤ f(x)
such that
f(y) ≤ ϕ(f(x))d(x, y).
Then, there exists z ∈ X such that z ∈ T (z).
Theorem 1.3. Let (X, d) be a complete metric space and let T : X −→ Cl(X). Assume that the
following conditions hold:
(i) the map f : X −→ R, defined by f(x) = d(x, T (x)) is lower semicontinuous,
(ii) there exists a function ϕ : [0,∞) −→ [b, 1), 0 < b < 1, satisfying
lim sup
r−→t+
ϕ(r) < 1 for each t ∈ [0,∞),
(iii) for any x ∈ X, there is y ∈ T (x) satisfying√
ϕ(d(x, y)d(x, y) ≤ f(x)
such that
f(y) ≤ ϕ(d(x, y))d(x, y).
Then, there exists z ∈ X such that z ∈ T (z).
Note that Theorem 1.3 is a generalization of Theorem 1.1.
The aim of this paper is to prove generalized corresponding theorems in the setting of partial
metric space which includes as a special case the standard metric space. Also we present an example
to show that our results improve on the above results.
Here we present some definitions and properties of partial metric spaces, for more detail see [4,
7, 9, 13, 16, 17].
A partial metric space is a pair (X, p), where X is a nonempty set and p is a partial metric on
X. A partial metric is a function p : X ×X −→ R+ such that for all x, y, z ∈ X we have:
(p1) x = y ⇔ p(x, x) = p(x, y) = p(y, y),
(p2) p(x, x) ≤ p(x, y),
(p3) p(x, y) = p(y, x),
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FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED NONLINEAR CONTRACTIVE MAPS . . . 5
(p4) p(x, y) ≤ p(x, z) + p(z, y)− p(z, z).
It is clear that, if p(x, y) = 0, then from (p1) and (p2), x = y. But if x = y, p(x, y) may not
be 0. A basic example of a partial metric space is the pair (R+, p), where p(x, y) = max{x, y} for
all x, y ∈ R+. Additional references, mainly from the computational point of view, may be found in
[1 – 3, 5, 7, 9 – 11, 17, 18, 20].
Each partial metric p on X induces a T0 topology τp on X which has as a base the family open
p-balls {Bp(x, ε) : x ∈ X, ε > 0}, where Bp(x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X
and ε > 0 (see [13]).
Let (X, p) be a partial metric space, then we have the following:
(i) A sequence {xn} in a partial metric space (X, p) converges to a point x ∈ X if and only if
p(x, x) = limn−→∞ p(x, xn).
(ii) A sequence {xn} in a partial metric space (X, p) is called a Cauchy sequence if there exists
(and is finite) limn,m−→∞ p(xm, xn) ([13], Definition 5.2).
(iii) A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn} in X
converges, with respect to τp, to a point x ∈ X such that p(x, x) = limn,m−→∞ p(xm, xn) ([13],
Definition 5.3).
(iv) A map f : X −→ X is τp-continuous if lim n−→∞ xn = x with respect to τp implies
lim n−→∞ fxn = fx with respect to τp.
It is easy to see that, every closed subset of a complete partial metric space is complete. The
following lemma is crucial for the proofs of our results.
Lemma 1.1 [4]. Let (X, p) be a partial metric space, A ⊂ X and x0 ∈ X. Define p(x0, A) =
= inf{p(x0, x) : x ∈ A}. Then a ∈ A if and only if p(a,A) = p(a, a).
2. Main results. In this section, we prove some fixed point results for multivalued generalized
nonlinear contractive maps in partial metric space which generalize and improve the mentioned
corresponding results of Ćirić [6], Klim and Wardowski [12] and several authors.
Theorem 2.1. Let (X, p) be a complete partial metric space. Let T : X −→ Cl(X). Assume
that the following conditions hold:
(i) there exist a constant b ∈ (0, 1) and two functions α : [0,∞) −→ [b,∞) and β : [0,∞) −→
−→ (0,∞) such that for each t ∈ [0,∞), β(t) ≤ α(t) and
lim sup
r−→t+
β(r)
α(r)
< 1,
(ii) the map f : X −→ R, defined by f(x) = p(x, T (x)) is lower semicontinuous,
(iii) for any x ∈ X, there exists y ∈ T (x) satisfying
α(f(x))p(x, y) ≤ f(x)
and
f(y) ≤ β(f(x))p(x, y).
Then there exists v0 ∈ X such that f(v0) = 0. Further, if p(v0, v0) = 0, then v0 ∈ T (v0).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
6 A. AGHAJANI, R. ALLAHYARI
Remark 2.1. Notice that a complete metric space (X, d) is a complete partial metric space
(since every metric d is a partial metric).
Therefore, taking β(t) = ϕ(t), α(t) =
√
ϕ(t) and p(x, y) = d(x, y) in Theorem 2.1, we get
Theorem 1.2.
Proof of Theorem 2.1. Let x0 ∈ X. From condition (iii) there exists x1 ∈ T (x0) such that
α(f(x0))p(x0, x1) ≤ f(x0) (1)
and
f(x1) ≤ β(f(x0))p(x0, x1). (2)
From (1) and (2), we get
f(x1) ≤
β(f(x0))
α(f(x0))
f(x0). (3)
Similarly, there exists x2 ∈ T (x1) such that
α(f(x1))p(x1, x2) ≤ f(x1)
and
f(x2) ≤ β(f(x1))p(x1, x2),
which imply f(x2) ≤
β(f(x1))
α(f(x1))
f(x1).
By induction we get an orbit {xn} of T in X such that for all integers n ≥ 0
α(f(xn))p(xn, xn+1) ≤ f(xn) (4)
and
f(xn+1) ≤
β(f(xn))
α(f(xn))
f(xn). (5)
Since by condition (i) we have
β(f(x))
α(f(x))
≤ 1 for all x ∈ X.
From (5), we get f(xn+1) ≤ f(xn).
Thus {f(xn)} is a nonincreasing sequence of real numbers which is bounded from below by 0,
so is convergent to a L ≥ 0. We show that L = 0. Suppose to the contrary that L > 0. From (5) and
taking the limit when n −→∞ we obtain
L ≤ lim sup
f(xn)−→L+
β(f(xn))
α(f(xn))
L < L,
which is a contradiction, so L = 0.
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FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED NONLINEAR CONTRACTIVE MAPS . . . 7
Now, from condition (i), we can choose q such that
lim sup
f(xn)−→0+
β(f(xn))
α(f(xn))
< q < 1.
So
β(f(xn))
α(f(xn))
< q for all n ≥ n0, for some n0 ∈ N.
Again using (5), we get f(xn+1) ≤ qf(xn) for all n ≥ n0. Hence, by induction, we have
f(xn+1) ≤ qn+1−n0f(xn0) for all n ≥ n0. (6)
From (4), (6) and the fact that α(t) ≥ b > 0 for all t ≥ 0, we obtain
p(xn, xn+1) ≤
1
b
qn−n0f(xn0) for all n ≥ n0. (7)
In this step of the proof, we show that {xn} is a Cauchy sequence. For any m,n ∈ N,m > n ≥ n0,
and using property (p4) of the partial metric p we get
p(xn, xm) ≤
m−1∑
k=n
p(xk, xk+1) ≤
1
b
m−1∑
k=n
qk−n0f(xn0) ≤
1
b
(
qn−n0
1− q
)
f(xn0).
Since q < 1, we conclude that the sequence {xn} is a Cauchy sequence. Due to the completeness
of X, there exists v0 ∈ X such that limn−→∞ xn = v0, with respect to τp. Since f is lower
semicontinuous and have in mind that {f(xn)} is convergent to 0, we have
0 ≤ f(v0) ≤ lim inf
n−→∞
f(xn) = 0,
and thus, f(v0) = p(v0, T (v0)) = 0. Now if p(v0, v0) = 0, since T (v0) is closed, it follows from
Lemma 1.4 that v0 ∈ T (v0).
Theorem 2.1 is proved.
Theorem 2.2. Let (X, p) be a complete partial metric space. Let T : X −→ Cl(X). Assume
that the following conditions hold:
(i) there exist a constant b ∈ (0, 1) and two functions α : [0,∞) −→ [b,∞) and β : [0,∞) −→
−→ (0,∞) such that for each t ∈ [0,∞), β(t) ≤ α(t) and
lim sup
r−→t+
β(r)
α(r)
< 1,
(ii) assume that inf{p(x, v) + p(x, T (x)) : x ∈ X} > 0, for every v ∈ X with v 6∈ T (v),
(iii) for any x ∈ X, there exists y ∈ T (x) satisfying
α(f(x))p(x, y) ≤ f(x)
and
f(y) ≤ β(f(x))p(x, y).
Then Fix(T ) 6= ∅.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
8 A. AGHAJANI, R. ALLAHYARI
Proof. As in the proof of Theorem 2.1, we can obtain a Cauchy sequence {xn} with xn ∈
∈ T (xn−1). Since (X, p) is a complete partial metric space, there exists some v0 ∈ X such that
limn−→∞ xn = v0, with respect to τp. Also by property (p4) of partial metric p on X
p(xn, v0) ≤ p(xn, xm) + p(xm, v0)− p(xm, xm).
By taking the limit when m −→∞ we get
p(xn, v0) ≤ lim
m−→∞
p(xn, xm) + lim
m−→∞
p(xm, v0)− lim
m−→∞
p(xm, xm).
Now, since xm −→ v0 ∈ X, with respect to τp, we have
lim
m−→∞
p(xm, v0) = lim
m−→∞
p(xm, xm) = p(v0, v0).
Therefore, it follows that for all n ≥ n0
p(xn, v0) ≤ lim
m−→∞
p(xn, xm) ≤
1
b
(
qn−n0
1− q
)
f(xn0)
and
p(xn, T (xn)) ≤ p(xn, xn+1) ≤
qn−n0
b
f(xn0).
Assume that v0 6∈ T (v0). Then, we obtain
0 < inf{p(x, v0) + p(x, T (x)) : x ∈ X} ≤
≤ inf{p(xn, v0) + p(xn, T (xn)) : n ≥ n0} ≤
≤ inf
{
1
b
(
qn−n0
1− q
)
f(xn0) +
1
b
qn−n0f(xn0) : n ≥ n0
}
=
=
{
1
b
(
2− q
1− q
)
f(xn0)
}
inf{qn−n0 : n ≥ n0} = 0,
which is impossible and hence v0 ∈ Fix(T ).
Theorem 2.2 is proved.
Theorem 2.3. Let (X, p) be a complete partial metric space. Let T : X −→ Cl(X). Assume
that the following conditions hold:
(i) there exist a constant b ∈ (0, 1) and two functions α : [0,∞) −→ [b,∞) and β : [0,∞) −→
−→ (0,∞) such that for each t ∈ [0,∞), β(t) ≤ min{α(t), α2(t)} and
lim sup
r−→t+
β(r)
α(r)
< 1,
(ii) the map f : X −→ R, defined by f(x) = p(x, T (x)) is lower semicontinuous,
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FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED NONLINEAR CONTRACTIVE MAPS . . . 9
(iii) for any x ∈ X, there exists y ∈ T (x) satisfying
α(P (x, y))p(x, y) ≤ p(x, T (x))
and
p(y, T (y)) ≤ β(p(x, y))p(x, y).
Then there exists v0 ∈ X such that f(v0) = 0. Further, if p(v0, v0) = 0, then v0 ∈ T (v0).
Proof. Let x0 ∈ X. From condition (iii) there exists x1 ∈ T (x0) such that
α(p(x0, x1))p(x0, x1) ≤ p(x0, T (x0)) (8)
and
p(x1, T (x1)) ≤ β(p(x0, x1))p(x0, x1). (9)
From (8) and (9), we get
p(x1, T (x1)) ≤
β(p(x0, x1))
α(p(x0, x1))
p(x0, T (x0)).
Similarly, there exists x2 ∈ T (x1) such that
α(p(x1, x2))p(x1, x2) ≤ p(x1, T (x1))
and
p(x2, T (x2)) ≤ β(p(x1, x2))p(x1, x2),
which imply
p(x2, T (x2)) ≤
β(p(x1, x2))
α(p(x1, x2))
p(x1, T (x1)).
By induction we get an orbit {xn} of T in X such that for all integers n ≥ 0
α(p(xn, xn+1))p(xn, xn+1) ≤ p(xn, T (xn)) (10)
and
p(xn+1, T (xn+1)) ≤
β(p(xn, xn+1))
α(p(xn, xn+1))
p(xn, T (xn)). (11)
Since by condition (i) we have
β(t)
α(t)
≤ 1 for all t ∈ [0,∞),
from (11), we get
p(xn+1, T (xn+1)) ≤ p(xn, T (xn)). (12)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
10 A. AGHAJANI, R. ALLAHYARI
Thus {p(xn, T (xn))} is a nonincreasing sequence of real numbers which is bounded from below
by 0, so is convergent to an L ≥ 0. From (10) and the fact that α(t) ≥ b > 0 for all t ≥ 0, we
obtain
p(xn, xn+1) ≤
1
b
p(xn, T (xn)). (13)
By using (13) and the convergence of {p(xn, T (xn))}, we have that {p(xn, xn+1)} is bounded.
Therefore, there exists K ≥ 0 such that
lim inf
n−→∞
p(xn, xn+1) = K. (14)
Since xn+1 ∈ T (xn), we have p(xn, xn+1) ≥ p(xn, T (xn)) for each n ≥ 0, which implies that
K ≥ L. We claim that K = L. First suppose that L = 0. Again using (13) and the convergence of
{p(xn, T (xn))} we get limn−→∞ p(xn, xn+1) = 0.
Hence, if L = 0, then K = L. Now, suppose that L > 0 and assume to the contrary that K > L.
Then K − L > 0. So from (14) and the fact that {p(xn, T (xn))} is convergent to an L ≥ 0 there
exists an n0 ∈ N such that
L < p(xn, T (xn)) < L+
K − L
4
for all n ≥ n0 (15)
and
K − K − L
4
< p(xn, xn+1) for all n ≥ n0. (16)
From (10), (15) and (16) we have
α(p(xn, xn+1))
(
K − K − L
4
)
< α(p(xn, xn+1))p(xn, xn+1) ≤
≤ p(xn, T (xn)) < L+
K − L
4
.
This implies that
α(p(xn, xn+1)) ≤
K + 3L
3K + L
for all n ≥ n0. (17)
Take h =
K + 3L
3K + L
. Now, From (11), (17) and condition (i) we get
p(xn+1, T (xn+1)) ≤
β(p(xn, xn+1))
α2(p(xn, xn+1))
α(p(xn, xn+1))p(xn, T (xn)) ≤
≤ β(p(xn, xn+1))
min{α2(p(xn, xn+1)), α(p(xn, xn+1))}
α(p(xn, xn+1))p(xn, T (xn)) ≤
≤ α(p(xn, xn+1))p(xn, T (xn)) ≤ h(p(xn, T (xn))) for all n ≥ n0. (18)
By using (15) and (18), we have for any δ ≥ 1
L < p(xn0+δ, T (xn0+δ)) ≤ hδp(xn0 , T (xn0)). (19)
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FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED NONLINEAR CONTRACTIVE MAPS . . . 11
Since L > 0 and h =
K + 3L
3K + L
=
(
1− 2(K − L)
3K + L
)
< 1, there exists a δ0 ∈ N such that
hδ0p(xn0 , T (xn0)) < L. From (19), we get
L < p(xn0+δ, T (xn0+δ)) ≤ hδ0p(xn0 , T (xn0)) < L,
which is a contradiction. Hence, assumption K > L is wrong, so L = K. Now, we claim that K = 0.
Since K = L ≤ p(xn, T (xn)) ≤ p(xn, xn+1), then from (14) we have
lim inf
n−→∞
p(xn, xn+1) = K+.
Hence, we can choose a subsequence {p(xnδ , xnδ+1)} of {p(xn, xn+1)} such that
lim
δ−→∞
p(xnδ , xnδ+1) = K+.
By condition (i), we get
lim sup
p(xnδ ,xnδ+1)−→K+
β(p(xnδ , xnδ+1))
α(p(xnδ , xnδ+1))
< 1 (20)
and from (11), we obtain
p(xnδ+1, T (xnδ+1)) ≤
β(p(xnδ , xnδ+1))
α(p(xnδ , xnδ+1))
p(xnδ , T (xnδ)).
Taking the limit as δ −→∞ and have in mind that {p(xn, T (xn))} is convergent to L ≥ 0, thus, we
get
L = lim sup
δ−→∞
p(xnδ+1, T (xnδ+1)) ≤
≤
(
lim sup
δ−→∞
β(p(xnδ , xnδ+1))
α(p(xnδ , xnδ+1))
)
lim sup
δ−→∞
p(xnδ , T (xnδ)) =
=
(
lim sup
p(xnδ ,xnδ+1)−→K+
β(p(xnδ , xnδ+1))
α(p(xnδ , xnδ+1))
)
L.
If we suppose that L > 0, then from high inequality, we have
lim sup
p(xnδ ,xnδ+1)−→K+
β(p(xnδ , xnδ+1))
α(p(xnδ , xnδ+1))
≥ 1,
which contradicts (20). Thus L = 0. Obviously, {p(xn, T (xn))} is convergent to L ≥ 0, and since
p(xn, xn+1) ≤
1
b
p(xn, T (xn)), then we obtain
lim
n−→∞
p(xn, T (xn)) = 0+ (21)
and
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
12 A. AGHAJANI, R. ALLAHYARI
lim
n−→∞
p(xn, xn+1) = 0+.
Now, from condition (i), we can choose q such that
lim sup
p(xnδ ,xnδ+1)−→0+
β(p(xnδ , xnδ+1))
α(p(xnδ , xnδ+1))
< q < 1.
So
β(p(xn, xn+1))
α(p(xn, xn+1))
< q for all n ≥ n0, for some n0 ∈ N.
Again using (11), we get
p(xn+1, T (xn+1)) ≤ q p(xn, T (xn)) for all n ≥ n0.
Hence, by induction, we have
p(xn+1, T (xn+1) ≤ qn+1−n0p(xn0 , T (xn0)) for all n ≥ n0. (22)
From (13), (22) and the fact that xn+1 ∈ T (xn), we obtain
p(xn, xn+1) ≤
1
b
qn−n0p(xn0 , T (xn0)) for all n ≥ n0.
In this step of the proof, we show that {xn} is a Cauchy sequence.
For any m,n ∈ N,m > n ≥ n0, and using property (p4) of the partial metric p we have
p(xn, xm) ≤
m−1∑
δ=n
p(xδ, xδ+1) ≤
1
b
m−1∑
δ=n
qδ−n0f(xn0) ≤
1
b
(
qn−n0
1− q
)
f(xn0).
Since q < 1, we conclude that the sequence {xn} is a Cauchy sequence. Due to the completeness
of X, there exists v0 ∈ X such that limn−→∞ xn = v0, with respect to τp. Since f is lower
semicontinuous and from (21), we get
0 ≤ f(v0) ≤ lim inf
n−→∞
f(xn) = lim inf
n−→∞
p(xn, T (xn)) = 0,
and thus, f(v0) = p(v0, T (v0)) = 0. Now if p(v0, v0) = 0, since T (v0) is closed, it follows from
Lemma 1.1 that v0 ∈ T (v0).
Theorem 2.4. Suppose that all the hypotheses of Theorem 2.3 except (ii) hold. Assume that
inf{p(x, v) + p(x, T (x)) : x ∈ X} > 0, for every v ∈ X with v 6∈ T (v).
Then Fix(T ) 6= ∅.
Proof. Since the proof of this theorem can be completed essentially on the line of the proof of
Theorem 2.2, hence details are omitted.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED NONLINEAR CONTRACTIVE MAPS . . . 13
3. Examples.
Example 3.1. Let X = [0, 1] and p(x, y) = max{x, y}, then it is clear that (X, p) is a complete
partial metric space. Note that function p is not a metric. Let T : X −→ Cl(X) be defined by the
following formula:
T (x) =
{
1
6
x2
}
for x ∈
[
0,
1
2
)⋃(1
2
, 1
]
,
{
9
120
,
9
20
}
for x =
1
2
.
Define α : [0,∞) −→ (0,∞) by
α(t) =
1
2
for t ∈ [0, 1],
1
2
t for t ∈ (1,∞),
and, β : [0,∞) −→ (0,∞) by β(t) =
1
3
α(t).
Since T (x) ≤ x, we have f(x) = max{x, T (x)} = x for all x ∈ [0, 1] and f is lower semicon-
tinuous. Moreover, for each x ∈
[
0,
1
2
)⋃(1
2
, 1
]
, we have T (x) =
{
1
6
x2
}
, so y ∈ T (x) implies
y =
1
6
x2. Hence
p(x, y) = p
(
x,
1
6
x2
)
= max
{
x,
1
6
x2
}
= x ∈ [0, 1].
Now, for all x ∈ X −
{
1
2
}
and y =
1
6
x2
α(p(x, y))p(x, y) =
1
2
x ≤ x = p(x, T (x))
and
p(y, T (y)) =
(
1
6
x2
)
≤ 1
6
x = β(p(x, y))p(x, y).
If x =
1
2
then we have T (x) =
{
9
120
,
9
20
}
, and p(x, T (x)) =
1
2
. Thus, for x =
1
2
we can choose
y =
9
120
∈ T (x) such that
α(p(x, y))p(x, y) = α
(
1
2
)
1
2
=
1
4
< p(x, T (x))
and
p(y, T (y)) =
9
120
< β
(
1
2
)
1
2
=
1
12
= β(p(x, y))p(x, y).
Hence, T satisfies all the conditions of Theorem 2.3 and note that Fix(T ) = {0}.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
14 A. AGHAJANI, R. ALLAHYARI
Example 3.2. Consider xn = 1 − 1
n
for n ∈ N and x0 = 1. Suppose N, Ne, No denote
the sets of all positive integers, even positive integers and odd positive integers, respectively. Let
X = {x0, x1, x2, x3, x4, . . .} that is a bounded complete subset of R. Let p(x, y) = d(x, y), for all
x, y ∈ X. Define a mapping T from X into Cl(X) by
T (xn) =
x0 if n = 0,
{xn−1, xn2} if n ∈ {n ∈ Ne : n ≥ 2},
x1 if n ∈ No.
It is easy to verify that
f(xn) = p(xn, T (xn)) =
0 if n ∈ {0, 1},
1
n− 1
− 1
n
if n ∈ {n ∈ Ne : n ≥ 2},
1− 1
n
if n ∈ {n ∈ No : n > 2}.
So, f is lower semicontinuous in X.
Suppose α(t) : [0,∞) −→ (0, 1) and β(t) : [0,∞) −→ (0, 1) are defined by
α(t) =
1
4
if t ∈ {0} ∪ [1,∞),
√
t if t ∈ (0, 1),
and
β(t) =
t if t ∈ [0, 1),
1
8
if t ∈ [1,∞).
Since
β(t)
α(t)
=
√
t if t ∈ (0, 1),
0 if t ∈ {0},
1
2
if t ∈ [1,∞).
Then, we have for each t ∈ [0,∞), β(t) ≤ α(t) and
lim sup
r−→t+
β(r)
α(r)
< 1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
FIXED POINT THEOREMS FOR MULTIVALUED GENERALIZED NONLINEAR CONTRACTIVE MAPS . . . 15
For x = x1, x0 let y = x ∈ T (x), so
α(f(x))p(x, y) = 0 = f(x), f(y) = 0 = β(f(x))p(x, y).
For x = xn, n ≥ 2, and n ∈ Ne if we let y = xn2 ∈ T (x) then
α(f(x))p(x, y) = α
(
1
n− 1
− 1
n
)(
1
n
− 1
n2
)
=
=
(√
1
n− 1
− 1
n
)(
1
n
− 1
n2
)
<
(
1
n− 1
− 1
n
)
= f(x),
f(y) =
(
1
n2 − 1
− 1
n2
)
<
(
1
n− 1
− 1
n
)(
1
n
− 1
n2
)
= β(f(x))p(x, y).
For x = xn, n > 2, and n ∈ No, let y = x1 ∈ T (x), so
α(f(x))p(x, y) = α
(
1− 1
n
)(
1− 1
n
)
=
(√
1− 1
n
)(
1− 1
n
)
<
(
1− 1
n
)
= f(x),
f(y) = 0 <
(
1− 1
n
)(
1− 1
n
)
= β(f(x))p(x, y).
Therefore, all the assumptions of Theorem 2.1 are satisfied and x1, x0 are two fixed points of T. Let
us observe that T does not satisfy the assumptions of Theorem 1.2 provided p(x, y) = d(x, y), for
all x, y ∈ X. Indeed, for any function ϕ : [0,∞) −→ [b, 1), b ∈ (0, 1), there exists n > 2, n ∈ Ne,
such that for x = xn, if y = xn2 ∈ T (x), we have
f(x) =
1
n− 1
− 1
n
<
√
b
(
1
n
− 1
n2
)
≤
√
ϕ(f(x))d(x, y),
and if y = xn−1 ∈ T (x), we have
f(y) = d(y, T (y)) = 1− 1
n− 1
>
1
n(n− 1)
≥ ϕ(f(x))d(x, y),
which does not match the assumptions of Theorem 1.2.
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Received 25.04.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
|
| id | umjimathkievua-article-2106 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:49Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9c/9d60d37cb93c1770afd57bec27e0469c.pdf |
| spelling | umjimathkievua-article-21062019-12-05T10:24:29Z Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces Теореми про нерухому точку для багатозначних узагальнених нелінійних стискаючих відображень в частково метричних просторах Aghajani, A. Allahyari, R. Агажані, А. Аллахярі, Р. We prove some fixed-point results for multivalued generalized nonlinear contractive mappings in partial metric spaces, which generalize and improve the corresponding recent fixed-point results due to Ćirić [L. B. Ćirić, “Multivalued nonlinear contraction mappings,” Nonlin. Anal., 71, 2716–2723 (2009)] and Klim and Wardowski [D. Klim and D. Wardowski, “Fixed-point theorems for set-valued contractions in complete metric spaces,” J. Math. Anal. Appl., 334, 132–139 (2007)]. Доведено ДЄЯКІ теореми про нерухому точку в частково метричних просторах, що узагальнюють та покращують відповідні нові результати про нерухому точку, отримані Чірічем (CiriC L. B. Multivalued nonlinear contraction mappings // Nonlinear Anal. - 2009. - 71. - P. 2716-2723) та Клімом i Вардовським (Klim D., Wardowski D. Fixed point theorems for set-valued contractions in complete metric spaces // J. Math. Anal. and Appl. - 2007. - 334. - P. 132-139). Institute of Mathematics, NAS of Ukraine 2014-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2106 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 1 (2014); 3–16 Український математичний журнал; Том 66 № 1 (2014); 3–16 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2106/1214 https://umj.imath.kiev.ua/index.php/umj/article/view/2106/1215 Copyright (c) 2014 Aghajani A.; Allahyari R. |
| spellingShingle | Aghajani, A. Allahyari, R. Агажані, А. Аллахярі, Р. Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces |
| title | Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces |
| title_alt | Теореми про нерухому точку для багатозначних узагальнених нелінійних стискаючих відображень в частково метричних просторах |
| title_full | Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces |
| title_fullStr | Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces |
| title_full_unstemmed | Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces |
| title_short | Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces |
| title_sort | fixed-point theorems for multivalued generalized nonlinear contractive maps in partial metric spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2106 |
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