Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type
We prove general fixed-point theorems (generalizing some recent results) in a complete G-metric space.
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irk-123456789-1655722020-02-15T01:27:00Z Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type Popa, V. Patriciu, A.-M. Статті We prove general fixed-point theorems (generalizing some recent results) in a complete G-metric space. Доведено загальні теореми про нерухому точку у повних G-метричних просторах, що узагальнюють дєякі результати, отримані нещодавно. 2013 Article Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type / V. Popa, A.-M. Patriciu // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 814–821. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165572 517.91 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Popa, V. Patriciu, A.-M. Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type Український математичний журнал |
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We prove general fixed-point theorems (generalizing some recent results) in a complete G-metric space. |
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Article |
| author |
Popa, V. Patriciu, A.-M. |
| author_facet |
Popa, V. Patriciu, A.-M. |
| author_sort |
Popa, V. |
| title |
Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type |
| title_short |
Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type |
| title_full |
Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type |
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Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type |
| title_full_unstemmed |
Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type |
| title_sort |
fixed-point results on complete g-metric spaces for mappings satisfying an implicit relation of new type |
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Інститут математики НАН України |
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2013 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165572 |
| citation_txt |
Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type / V. Popa, A.-M. Patriciu // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 814–821. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT popav fixedpointresultsoncompletegmetricspacesformappingssatisfyinganimplicitrelationofnewtype AT patriciuam fixedpointresultsoncompletegmetricspacesformappingssatisfyinganimplicitrelationofnewtype |
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2025-07-14T18:58:52Z |
| last_indexed |
2025-07-14T18:58:52Z |
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1837649923316645888 |
| fulltext |
UDC 517.91
V. Popa, A.-M. Patriciu (“Vasile Alecsandri” Univ. Bacău, Romania)
FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES
FOR MAPPINGS SATISFYING AN IMPLICIT RELATION OF A NEW TYPE
РЕЗУЛЬТАТИ ПРО НЕРУХОМУ ТОЧКУ
НА ПОВНИХ G-МЕТРИЧНИХ ПРОСТОРАХ ДЛЯ ВIДОБРАЖЕНЬ,
ЩО ЗАДОВОЛЬНЯЮТЬ НЕЯВНЕ СПIВВIДНОШЕННЯ НОВОГО ТИПУ
We prove some general fixed-point theorems in complete G-metric space that generalize some recent results.
Доведено загальнi теореми про нерухому точку у повних G-метричних просторах, що узагальнюють деякi резуль-
тати, отриманi нещодавно.
1. Introduction. In [3, 4] Dhage introduced a new class of generalized metric space, named D-metric
space. Mustafa and Sims [7, 8] proved that most of the claims concerning the fundamental topological
structures on D-metric spaces are incorrect and introduced appropriate notion of generalized metric
space, named G-metric space. In fact, Mustafa, Sims and other authors [2, 9 – 11] studied many
fixed-point results for self mappings in G-metric spaces under certain conditions.
Quite recently [12], Mustafa et al. obtained new results for mappings in G-metric spaces.
In [13, 14], Popa initiated the study of fixed points in metric spaces for mappings satisfying an
implicit relation.
Let T be a self mapping of a metric space (X, d). We denote by Fix (T ) the set of all fixed
points of T. T is said to satisfy property (P ) if Fix (T ) = Fix (Tn) for each n ∈ N. An interesting
fact about mappings satisfying property (P ) is that they have not nontrivial periodic points. Papers
dealing with property (P ) are, between others, [2, 13 – 15].
The purpose of this paper is to prove a general fixed-point theorem in complete G-metric space
which generalize the results from [1, 10 – 12] for mappings satisfying a new form of implicit relation.
In the last part of this paper is proved a general theorem for mappings in G-metric space satisfying
property (P ), which generalize some results from [1].
2. Preliminaries.
Definition 2.1 [8]. Let X be a nonempty set and G : X3 → R+ be a function satisfying the
following properties:
(G1) G(x, y, z) = 0 if x = y = z;
(G2) 0 < G(x, x, y) for all x, y ∈ X with x 6= y;
(G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with z 6= y;
(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = . . . (symmetry in all three variables);
(G5) G(x, y, z) ≤ G(x, a, a) +G(a, y, z) for all x, y, z, a ∈ X (rectangle inequality).
Then the function G is called a G-metric and the pair (X,G) is called a G-metric space.
Note that if G(x, y, z) = 0 then x = y = z [8].
Lemma 2.1 [8]. G(x, y, y) ≤ 2G(x, x, y) for all x, y ∈ X.
Definition 2.2 [8]. Let (X,G) be a metric space. A sequence (xn) in X is said to be:
a) G-convergent to x ∈ X if for any ε > 0 there exists k ∈ N such that G(x, xn, xm) < ε for all
m,n ≥ k;
c© V. POPA, A.-M. PATRICIU, 2013
814 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS . . . 815
b) G-Cauchy if for ε > 0, there exists k ∈ N such that for all n,m, p ≥ k, G(xn, xm, xp) < ε
that is G(xn, xm, xp)→ 0 as m,n, p→∞.
A G-metric space is said to be G-complete if every G-Cauchy sequence in X is G-convergent.
Lemma 2.2 [8]. Let (X,G) be a G-metric space. Then, the following properties are equivalent:
1) (xn) is G-convergent to x;
2) G(x, xn, xn)→ 0 as n→∞;
3) G(xn, x, x)→ 0 as n→∞.
Lemma 2.3 [8]. Let (X,G) be a G-metric space. Then the following properties are equivalent:
1) The sequence (xn) is G-Cauchy.
2) For every ε > 0, there exists k ∈ N such that G(xn, xm, xm) < ε for n,m > k.
Definition 2.3 [8]. Let (X,G) and (X ′, G′) be two G-metric spaces and f : (X,G)→ (X ′, G′).
Then, f is said to be G-continuous at x ∈ X if for ε > 0, there exists δ > 0 such that for all x, y ∈ X
and G(a, x, y) < δ, then G′(fa, fx, fy) < ε. f is G-continuous if it is G-continuous at each a ∈ X.
Lemma 2.4 [8]. Let (X,G) and (X ′, G′) be twoG-metric spaces. Then, a function f : (X,G)→
→ (X ′, G′) is G-continuous at a point x ∈ X if and only if f is sequentially continuous, that is,
whenever (xn) is G-convergent to x we have that f(xn) is G-convergent to fx.
Lemma 2.5 [8]. Let (X,G) be a G-metric space. Then, the function G(x, y, z) is continuous in
all three of its variables.
Quite recently, the following theorem is proved in [12].
Theorem 2.1. Let (X,G) be a complete G-metric space and T : X → X be a mapping which
satisfies the following condition, for all x, y ∈ X
G(Tx, Ty, Ty) ≤ max{aG(x, y, y), b[G(x, Tx, Tx) + 2G(y, Ty, Ty)],
b[G(x, Ty, Ty) +G(y, Ty, Ty) +G(y, Tx, Tx)]}, (2.1)
where a ∈ [0, 1) and b ∈
[
0,
1
3
)
. Then T has a unique fixed point.
The purpose of this paper is to prove a general fixed point theorem in G-metric space for map-
pings satisfying a new type of implicit relation which generalize Theorem 2.1 and other results from
[1, 2, 10 – 12].
3. Implicit relations.
Definition 3.1. Let Fu be the set of all continuous functions F (t1, . . . , t6) : R6
+ → R such that
(F1) F is nonincreasing in variables t5 and t6;
(F2) there exists h ∈ [0, 1) such that for each u, v ≥ 0 and F (u, v, v, u, u + v, 0) ≤ 0, then
u ≤ hv;
(F3) F (t, t, 0, 0, t, 2t) > 0 ∀t > 0.
Example 3.1. F (t1, . . . , t6) = t1−max{at2, b(t3+2t4), b(t4+ t5+ t6)}, where a ∈ [0, 1) and
b ∈
[
0,
1
3
)
.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u − max{av, b(v + 2u)} ≤ 0. If u > v,
then u[1 − max{a, 3b}] ≤ 0, a contradiction. Hence u ≤ v, which implies u ≤ hv, where h =
= max{a, 3b} < 1.
(F3) F (t, t, 0, 0, t, 2t) = t(1−max{a, 3b}) > 0 ∀t > 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
816 V. POPA, A.-M. PATRICIU
Example 3.2. F (t1, . . . , t6) = t1−at2−b(t3+2t4)−c(t5+t6), where a, b, c ≥ 0, a+3b+2c < 1
and a+ 3c < 1.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u − av − b(v + 2u) − c(u + v) ≤ 0. Then
u ≤ hv, where h =
a+ b+ c
1− 2b− c
< 1.
(F3) F (t, t, 0, 0, t, 2t) = t[1− (a+ 3c)] > 0 ∀t > 0.
Example 3.3. F (t1, . . . , t6) = t1 − at2 − bmax{t3, t4} − cmax{t5, t6}, where a, b, c ≥ 0,
a+ b+ 2c < 1.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u − av − bmax{u, v} − c(u + v) ≤ 0. If
u > v, then u[1 − (a + b + 2c)] ≤ 0, a contradiction. Hence, u ≤ v which implies u ≤ hv, where
h =
a+ b+ c
1− c
< 1.
(F3) F (t, t, 0, 0, t, 2t) = t[1− (a+ 2c)] > 0 ∀t > 0.
Example 3.4. F (t1, . . . , t6) = t1 − kmax{t2, t3, . . . , t6}, where k ∈
[
0,
1
2
)
.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u − k(u + v) ≤ 0 which implies u ≤ hv,
where h =
k
k − 1
< 1.
(F3) F (t, t, 0, 0, t, 2t) = t(1− 2k) > 0 ∀t > 0.
Example 3.5. F (t1, . . . , t6) = t1−at2−bt3−cmax{t4+t5, 2t6}, where a, b, c ≥ 0, a+b+3c <
< 1, a+ 4c < 1.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u+ v, 0) = u− av − bv − c(2u+ v) ≤ 0. Then u ≤ hv,
where h =
a+ b+ c
1− 2c
< 1.
(F3) F (t, t, 0, 0, t, 2t) = t[1− (a+ 4c)] > 0 ∀t > 0.
Example 3.6. F (t1, . . . , t6) = t1 − kmax
{
t2, t3, t4,
2t4 + t6
3
,
2t4 + t3
3
,
t5 + t6
3
}
≤ 0, where
k ∈ [0, 1).
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u+ v, 0) = u− kmax
{
u, v,
2u
3
,
2u+ v
3
,
u+ v
3
≤ 0
}
. If
u > v, then u(1− k) ≤ 0, a contradiction. Hence, u ≤ v which implies u ≤ hv, where h = k < 1.
(F3) F (t, t, 0, 0, t, 2t) = t(1− k) > 0 ∀t > 0.
Example 3.7. F (t1, . . . , t6) = t1 − kmax
{
t2, t3, t4,
t5 + t6
2
}
, where k ∈
[
0,
2
3
)
.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u+ v, 0) = u− kmax
{
u, v,
u+ v
2
}
≤ 0. If u > v, then
u(1− k) ≤ 0, a contradiction. Hence, u ≤ v which implies u ≤ hv, where h = k < 1.
(F3) F (t, t, 0, 0, t, 2t) = t− kmax
{
t,
3t
2
}
= t
[
1− 3k
2
]
> 0 ∀t > 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS . . . 817
Example 3.8. F (t1, . . . , t6) = t21− t1(at2+ bt3+ ct4)−dt5t6, where a, b, c ≥ 0, a+ b+ c < 1,
a+ 2d < 1.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u2 − u(av + bv + cu) ≤ 0. If u > 0, then
u− av − bv − cu ≤ 0 which implies u ≤ hv, where h =
a+ b
1− c
< 1. If u = 0, then u ≤ hv.
(F3) F (t, t, 0, 0, t, 2t) = t2[1− (a+ 2d)] > 0 ∀t > 0.
Example 3.9. F (t1, . . . , t6) = t1 − kmax
{
t2,
t3 + t4
2
,
t5 + t6
2
}
, where k ∈
[
0,
2
3
)
.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u − kmax
{
v,
u+ v
2
}
≤ 0. If u > 0, then
u(1− k) ≤ 0, a contradiction. Hence u ≤ v which implies u ≤ hv, where h = k < 1.
(F3) F (t, t, 0, 0, t, 2t) = t
[
1− 3k
2
]
> 0 ∀t > 0.
Example 3.10. F (t1, . . . , t6) = t1 − kmax
{
t2,
√
t3t4,
√
t5t6
}
, where k ∈
[
0,
2
3
)
.
(F1) Obviously.
(F2) Let u, v ≥ 0 be and F (u, v, v, u, u + v, 0) = u − kmax {v,
√
uv} ≤ 0. If u > v, then
u(1− k) ≤ 0, a contradiction. Hence, u ≤ v which implies u ≤ hv, where 0 ≤ h = k < 1.
(F3) F (t, t, 0, 0, t, 2t) = t(1−
√
2k) > 0 ∀t > 0.
4. Main results.
Theorem 4.1. Let (X,G) be a G-metric space and T : (X,G) → (X,G) be a mapping such
that
F (G(Tx, Ty, Ty), G(x, y, y), G(x, Tx, Tx), G(y, Ty, Ty), G(x, Ty, Ty), G(y, Tx, Tx)) ≤ 0
(4.1)
for all x, y ∈ X, where F satisfies property (F3). Then T has at most a fixed point.
Proof. Suppose that T has two distinct fixed points u and v. Then by (4.1) we have successively
F (G(Tu, Tv, Tv), G(u, v, v), G(u, Tu, Tu), G(v, Tv, Tv), G(u, Tv, Tv), G(v, Tu, Tu)) ≤ 0,
F (G(u, v, v), G(u, v, v), 0, 0, G(u, v, v), G(v, u, u)) ≤ 0.
By Lemma 2.1 G(v, u, u) ≤ 2G(u, v, v). Since F is nonincreasing in variable t6 we obtain
F (G(u, v, v), G(u, v, v), 0, 0, G(u, v, v), 2G(u, v, v)) ≤ 0,
a contradiction of (F3). Hence u = v.
Theorem 4.1 is proved.
Theorem 4.2. Let (X,G) be a complete G-metric space and T : (X,G) → (X,G) satisfying
inequality (4.1) for all x, y ∈ X, where F ∈ Fu. Then T has a unique fixed point.
Proof. Let x0 ∈ X be an arbitrary point in X. We define xn = Txn−1, n = 1, 2, . . . . Then by
(4.1) we have successively
F (G(Txn−1, Txn, Txn), G(xn−1, xn, xn), G(xn−1, Txn−1, Txn−1),
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
818 V. POPA, A.-M. PATRICIU
G(xn, Txn, Txn), G(xn−1, Txn, Txn), G(xn, Txn−1, Txn−1)) ≤ 0,
F (G(xn, xn+1, xn+1), G(xn−1, xn, xn), G(xn−1, xn, xn),
G(xn, xn+1, xn+1), G(xn−1, xn+1, xn+1), 0) ≤ 0.
By (G5), G(xn−1, xn+1, xn+1) ≤ G(xn−1, xn, xn)+G(xn, xn+1, xn+1). Since F is nonincreas-
ing in variable t5 we obtain
F (G(xn, xn+1, xn+1), G(xn−1, xn, xn), G(xn−1, xn, xn),
G(xn, xn+1, xn+1), G(xn−1, xn, xn) +G(xn, xn+1, xn+1, 0) ≤ 0
which implies by (F2) that
G(xn, xn+1, xn+1) ≤ hG(xn−1, xn, xn).
Then
G(xn, xn+1, xn+1) ≤ hG(xn−1, xn, xn) ≤ . . . ≤ hnG(x0, x1, x1).
Moreover, for all m,n ∈ N, m > n, we have repeated use the rectangle inequality
G(xn, xm, xm) ≤ G(xn, xn+1, xn+1) +G(xn+1, xn+2, xn+2) + . . .+G(xm−1, xm, xm) ≤
≤ (hn + hn+1 + . . .+ hm−1)G(x0, x1, x1) ≤
hn
1− h
G(x0, x1, x1),
which implies limn,m→∞G(xn, xm, xm) = 0. Hence, (xn) is a G-Cauchy sequence. Since (X,G)
is G-complete, there exists u ∈ X such that limn→∞ xn = u.
We prove that u = Tu. By (F1) we have successively
F (G(Txn−1, Tu, Tu), G(xn−1, u, u), G(xn−1, Txn−1, Txn−1),
G(u, Tu, Tu), G(xn−1, Tu, Tu), G(u, Txn−1, Txn−1)) ≤ 0,
F (G(xn, Tu, Tu), G(xn−1, u, u), G(xn−1, xn, xn),
G(u, Tu, Tu), G(xn−1, Tu, Tu), G(u, xn, xn)) ≤ 0.
By continuity of F and G, letting n tend to infinity, we obtain
F (G(u, Tu, Tu), 0, 0, G(u, Tu, Tu), G(u, Tu, Tu), 0) ≤ 0.
By (F2) we obtain G(u, Tu, Tu) = 0, hence u = Tu and u is a fixed point of T. By Theorem
4.1 u is the unique fixed point of T.
Theorem 4.2 is proved.
Corollary 4.1. Theorem 2.1.
Proof. The proof follows from Theorem 4.2 and Example 3.1.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS . . . 819
Corollary 4.2 (Theorem 2.2 [11]). Let (X,G) be a G-complete metric space and T : (X,G)→
→ (X,G) be a mapping satisfying the following condition:
G(Tx, Ty, Tz) ≤ αG(x, y, z) + β[G(x, Tx, Tx) +G(y, Ty, Ty) +G(z, Tz, Tz)], (4.2)
for all x, y, z ∈ X and 0 ≤ α+ 3β < 1. Then T has a unique fixed point.
Proof. By (4.2) for z = y we obtain
G(Tx, Ty, Ty) ≤ αG(x, y, y) + β[G(x, Tx, Tx) + 2G(y, Ty, Ty)],
for all x, y ∈ X. By Theorem 4.2 and Example 3.2 for α = a, β = b and c = 0 it follows that T has
a unique fixed point.
Corollary 4.3 (Theorem 2.3 [11]). Let (X,G) be a G-complete metric space and T : (X,G)→
→ (X,G) be a mapping satisfying the condition
G(Tx, Ty, Tz) ≤ αG(x, y, z) + βmax{G(x, Tx, Tx), G(y, Ty, Ty), G(z, Tz, Tz)}, (4.3)
for all x, y, z ∈ X and 0 ≤ α+ β < 1. Then T has a unique fixed point.
Proof. By (4.3) for z = y we obtain
G(Tx, Ty, Ty) ≤ αG(x, y, y) + βmax{G(x, Tx, Tx), G(y, Ty, Ty)},
for all x, y ∈ X. By Theorem 4.2 and Example 3.3 for α = a, β = b and c = 0 it follows that T has
a unique fixed point.
Corollary 4.4 (Theorem 2.1 [10]). Let (X,G) be a G-complete metric space and T : (X,G)→
→ (X,G) be a mapping satisfying the condition
G(Tx, Ty, Tz) ≤ kmax{G(x, y, z), G(x, Tx, Tx), G(y, Ty, Ty),
G(y, Tz, Tz), G(x, Ty, Ty), G(y, Tz, Tz), G(z, Tx, Tx)},
(4.4)
for all x, y, z ∈ X, where k ∈
[
0,
1
2
)
. Then T has a unique fixed point.
Proof. By (4.4) for z = y we obtain
G(Tx, Ty, Ty) ≤ kmax{G(x, y, y), G(x, Tx, Tx), G(y, Ty, Ty), G(x, Ty, Ty), G(y, Tx, Tx)}.
By Theorem 4.2 and Example 3.4, T has a unique fixed point.
Corollary 4.5. Let (X,G) be a G-complete metric space and T : (X,G) → (X,G) be a map-
ping which satisfy the following inequality for all x, y ∈ X,
G(Tx, Ty, Ty) ≤ kmax{G(y, Ty, Ty) +G(x, Ty, Ty), 2G(y, Tx, Tx)}, (4.5)
where k ∈
[
0,
1
3
)
. Then T has a unique fixed point.
Proof. By Theorem 4.2 and Example 3.5 for a = b = 0 and c = k, T has a unique fixed point.
Remark 4.1. In Theorem 2.8 [10], k ∈
[
0,
1
2
)
.
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820 V. POPA, A.-M. PATRICIU
Corollary 4.6. Let (X,G) be a G-metric space and T : (X,G) → (X,G) be a mapping satis-
fying the following inequality for all x, y, z ∈ X,
G(Tx, Ty, Tz) ≤ hmax
{
G(x, y, z), G(x, Tx, Tx), G(y, Ty, Ty), G(z, Tz, Tz),
G(y, Tx, Tx) +G(y, Ty, Ty) +G(y, Tz, Tz)
3
,
G(x, Tx, Tx) +G(y, Ty, Ty) +G(z, Tz, Tz)
3
}
,
(4.6)
where k ∈ [0, 1) . Then T has a unique fixed point.
Proof. If y = z, by (4.6) we obtain that
G(Tx, Ty, Ty) ≤ hmax
{
G(x, y, y), G(x, Tx, Tx), G(y, Ty, Ty),
G(y, Tx, Tx) + 2G(y, Ty, Ty)
3
,
G(x, Tx, Tx) + 2G(y, Ty, Ty)
3
}
≤
≤ hmax
{
G(x, y, y), G(x, Tx, Tx), G(y, Ty, Ty),
G(y, Tx, Tx) + 2G(y, Ty, Ty)
3
,
G(x, Tx, Tx) + 2G(y, Ty, Ty)
3
,
G(x, Ty, Ty) +G(y, Tx, Tx)
3
}
,
for all x, y ∈ X.
By Theorem 4.2 and Example 3.6, T has a unique fixed point.
Remark 4.2. Corollary 4.6 is a generalization of Theorem 2.6 [1], where k ∈
[
0,
1
2
)
.
Remark 4.3. By Theorem 4.2 and Examples 3.7 – 3.10 we obtain new results.
5. Property (P ) in G-metric spaces.
Theorem 5.1. Under the conditions of Theorem 4.2, T has property (P ).
Proof. By Theorem 4.2, T has a fixed point. Therefore, Fix (Tn) 6= ∅ for each n ∈ N. Fix
n > 1 and assume that p ∈ Fix (Tn). We prove that p ∈ Fix (T ). Using (4.1) we have
F (G(Tnp, Tn+1p, Tn+1p), G(Tn−1p, Tnp, Tnp), G(Tn−1p, Tnp, Tnp), G(Tnp, Tn+1p, Tn+1p),
G(Tn−1p, Tn+1p, Tn+1p), G(Tnp, Tnp, Tnp)) ≤ 0.
By rectangle inequality
G(Tn−1p, Tn+1p, Tn+1p) ≤ G(Tn−1p, Tnp, Tnp) +G(Tnp, Tn+1p, Tn+1p).
By (F1) we obtain
F (G(Tnp, Tn+1p, Tn+1p), G(Tn−1p, Tnp, Tnp), G(Tn−1p, Tnp, Tnp), G(Tnp, Tn+1p, Tn+1p),
G(Tn−1p, Tnp, Tnp) +G(Tnp, Tn+1p, Tn+1p), 0) ≤ 0.
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FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS . . . 821
By (F2) we obtain
G(Tnp, Tn+1p, Tn+1p) ≤ hG(Tn−1p, Tnp, Tnp) ≤ . . . ≤ hnG(p, Tp, Tp).
Since p ∈ Tnp, then
G(p, Tp, Tp) = G(Tnp, Tn+1p, Tn+1p).
Therefore
G(p, Tp, Tp) ≤ hnG(p, Tp, Tp)
which implies G(p, Tp, Tp) = 0, i.e., p = Tp and T has property (P ).
Theorem 5.1 is proved.
Corollary 5.1. In the condition of Corollary 4.6, T has property (P ).
Remark 5.1. Corollary 5.1 is a generalization of the results from Theorem 2.6 [1].
Corollary 5.2. In the condition of Corollary 4.4 with k ∈
[
0,
1
2
)
, instead k ∈ [0, 1), T has
property (P ).
Remark 5.2. We obtain other new results from Examples 3.1 – 3.10.
1. Abbas M., Nazir T., Radanović S. Some periodic point results in generalized metric spaces // Appl. Math. and Comput. –
2010. – 217. – P. 4094 – 4099.
2. Chung R., Kasian T., Rasie A., Rhoades B. E. Property (P ) in G-metric spaces // Fixed Point Theory and Appl. –
2010. – Art. ID 401684. – 12 p.
3. Dhage B. C. Generalized metric spaces and mappings with fixed point // Bull. Calcutta Math. Soc. – 1992. – 84. –
P. 329 – 336.
4. Dhage B. C. Generalized metric spaces and topological structures I // An. şti. Univ. Iaşi. Ser. mat. – 2000. – 46, № 1. –
P. 3 – 24.
5. Jeong G. S. More maps for which F (T ) = F (Tn) // Demonstr. math. – 2007. – 40, № 3. – P. 671 – 680.
6. Jeong G. S., Rhoades B. E. Maps for which F (T ) = F (Tn) // Fixed Point Theory and Appl. – Nova Sci. Publ.,
2007. – 6.
7. Mustafa Z., Sims B. Some remarks concerning D-metric spaces // Int. Conf. Fixed Point. Theory and Appl. – 2004. –
P. 184 – 198.
8. Mustafa Z., Sims B. A new approach to generalized metric spaces // J. Nonlinear Convex Analysis. – 2006. – 7. –
P. 289 – 297.
9. Mustafa Z., Obiedat H., Awawdeh F. Some fixed point theorems for mappings on G-complete metric spaces // Fixed
Point Theory and Appl. – 2008. – Article ID 189870. – 10 p.
10. Mustafa Z., Sims B. Fixed point theorems for contractive mappings in complete G-metric spaces // Fixed Point Theory
and Appl. – 2009. – Article ID 917175. – 10 p.
11. Mustafa Z., Obiedat H. A fixed point theorem of Reich in G-metric spaces // Cubo A. Math. J. – 2010. – 12. –
P. 83 – 93.
12. Mustafa Z., Khandagji M., Shatanawi W. Fixed point results on complete G-metric spaces // Stud. Sci. Math. hung. –
2011. – 48, № 3. – P. 304 – 319.
13. Popa V. Fixed point theorems for implicit contractive mappings // Stud. cerc. St. Ser. Mat. Univ. Bacău. – 1997. – 7. –
P. 129 – 133.
14. Popa V. Some fixed point theorems for compatible mappings satisfying implicit relations // Demonstr. math. – 1999. –
32. – P. 157 – 163.
15. Rhoades B. E., Abbas M. Maps satisfying contractive conditions of integral type for which F (T ) = F (Tn) // Int.
Pure and Appl. Math. – 2008. – 45, № 2. – P. 225 – 231.
Received 14.02.12,
after revision — 19.11.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
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