A common fixed point for generalized (ψ, φ)f,g-weak contractions
We extend the common fixed point theorem established by Zhang and Song in 2009 to generalized (ψ, φ)f,g weak contractions. Moreover, we give an example that illustrates the main result. Finally, some common fixed point results are obtained for mappings satisfying a contraction condition of the integ...
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irk-123456789-1663912020-02-20T01:27:13Z A common fixed point for generalized (ψ, φ)f,g-weak contractions Razani, A. Parvaneh, V. Abbas, M. Статті We extend the common fixed point theorem established by Zhang and Song in 2009 to generalized (ψ, φ)f,g weak contractions. Moreover, we give an example that illustrates the main result. Finally, some common fixed point results are obtained for mappings satisfying a contraction condition of the integral type in complete metric spaces. Теорему про спiльну нерухому точку, що була встановлена Чжаном i Суном у 2009 роцi, поширено на узагальненi (ψ, φ)f,g-слабкi стискуючi вiдображення. Наведено приклад, що iлюструє основний результат. Отримано деякi результати про спiльну нерухому точку для вiдображень, що задовольняють умову стиску iнтегрального типу у повних метричних просторах. 2011 Article A common fixed point for generalized (ψ, φ)f,g-weak contractions / A. Razani, V. Parvaneh, M. Abbas // Український математичний журнал. — 2011. — Т. 63, № 11. — С. 1544–1554. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166391 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Razani, A. Parvaneh, V. Abbas, M. A common fixed point for generalized (ψ, φ)f,g-weak contractions Український математичний журнал |
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We extend the common fixed point theorem established by Zhang and Song in 2009 to generalized (ψ, φ)f,g weak contractions. Moreover, we give an example that illustrates the main result. Finally, some common fixed point results are obtained for mappings satisfying a contraction condition of the integral type in complete metric spaces. |
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Razani, A. Parvaneh, V. Abbas, M. |
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Razani, A. Parvaneh, V. Abbas, M. |
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Razani, A. |
| title |
A common fixed point for generalized (ψ, φ)f,g-weak contractions |
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A common fixed point for generalized (ψ, φ)f,g-weak contractions |
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A common fixed point for generalized (ψ, φ)f,g-weak contractions |
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A common fixed point for generalized (ψ, φ)f,g-weak contractions |
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A common fixed point for generalized (ψ, φ)f,g-weak contractions |
| title_sort |
common fixed point for generalized (ψ, φ)f,g-weak contractions |
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Інститут математики НАН України |
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2011 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166391 |
| citation_txt |
A common fixed point for generalized (ψ, φ)f,g-weak contractions / A. Razani, V. Parvaneh, M. Abbas // Український математичний журнал. — 2011. — Т. 63, № 11. — С. 1544–1554. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
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2025-07-14T21:21:55Z |
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2025-07-14T21:21:55Z |
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1837658936935710720 |
| fulltext |
UDC 517.5
A. Razani, V. Parvaneh (Islamic Azad Univ., Karaj, Iran),
M. Abbas (Lahore Univ. Management Sci., Pakistan)
A COMMON FIXED POINT
FOR GENERALIZED (ψ,ϕ)f,g-WEAK CONTRACTIONS
СПIЛЬНА НЕРУХОМА ТОЧКА ДЛЯ УЗАГАЛЬНЕНИХ
(ψ,ϕ)f,g-СЛАБКИХ СТИСКУЮЧИХ ВIДОБРАЖЕНЬ
We extend the common fixed point theorem established by Zhang and Song in 2009 to generalized (ψ,ϕ)f,g-
weak contractions. Moreover, we give an example that illustrates the main result. Finally, some common fixed
point results are obtained for mappings satisfying a contraction condition of the integral type in complete
metric spaces.
Теорему про спiльну нерухому точку, що була встановлена Чжаном i Суном у 2009 роцi, поширено
на узагальненi (ψ,ϕ)f,g-слабкi стискуючi вiдображення. Наведено приклад, що iлюструє основний
результат. Отримано деякi результати про спiльну нерухому точку для вiдображень, що задовольняють
умову стиску iнтегрального типу у повних метричних просторах.
1. Introduction. Let (X, d) be a metric space. A mapping T : X → X is said to be ϕ-
weak contraction, if there exists a map ϕ : [0,∞)→ [0,∞) with ϕ(0) = 0 and ϕ(t) > 0
for all t > 0 such that
d(Tx, Ty) ≤ d(x, y)− ϕ(d(x, y))
for all x, y ∈ X.
The above notion has been defined by Alber et al. [2] in 1997. They obtained some
fixed point results in Hilbert spaces. Then Rhoades [14] extended those results in Banach
spaces. In 2006, Beg and Abbas [5] studied some generalizations of Rhoades’s results
[14] for a pair of mappings such that one is weakly contractive with respect to the other.
In 2009, Zhang et al. [15] introduced the concept of generalized ϕ-weak contraction
as follows:
Definition 1.1. Two mappings T, S : X → X are called generalized ϕ-weak con-
tractions, if there exists a lower semicontinuous function ϕ : [0,∞) → [0,∞) with
ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 such that
d(Tx, Sy) ≤ N(x, y)− ϕ(N(x, y)),
for all x, y ∈ X, where
N(x, y) = max
{
d(x, y), d(x, Tx), d(y, Sy),
1
2
[
d(x, Sy) + d(y, Tx)
]}
.
Zhang et al. proved the following theorem.
Theorem 1.1. Let (X, d) be a complete metric space, and T, S : X → X are
generalized ϕ-weak contractions mappings where ϕ : [0,∞)→ [0,∞) is a lower semi-
continuous function with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0. Then there exists a
unique point u ∈ X such that u = Tu = Su.
So far, many authors extended Theorem 1.1 (see [1, 7, 12]). Moreover, Doric [7]
generalized it, by the definition of generalized (ψ,ϕ)-weak contractions.
c© A. RAZANI, V. PARVANEH, M. ABBAS, 2011
1544 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
A COMMON FIXED POINT FOR GENERALIZED (ψ,ϕ)f,g-WEAK CONTRACTIONS 1545
Definition 1.2. Two mappings T, S : X → X are called generalized (ψ,ϕ)-weak
contractive, if there exist two maps ϕ,ψ : [0,∞)→ [0,∞) such that
ψ(d(Tx, Sy)) ≤ ψ(N(x, y))− ϕ(N(x, y)),
for all x, y ∈ X, where N and ϕ are as in Definition 1.1 and ψ : [0,∞) → [0,∞) is a
continuous monotone nondecreasing function with ψ(0) = 0 and ψ(t) > 0 for all t > 0.
Theorem 1.2 [7]. Let (X, d) be a complete metric space, and T, S : X → X be
generalized (ψ,ϕ)-weak contractive self-mappings. Then there exists a unique point
u ∈ X such that u = Tu = Su.
Moradi et al. [12] extended the Zhang and Song’s result by introducing the notion
of ϕf -weak contractive mappings.
Definition 1.3. Two mappings T, S : X → X are called generalized ϕf -weak
contractive, if there exist two maps ϕ : [0,∞) → [0,∞) and f : X → X where ϕ is a
lower semicontinuous function with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 such that
d(Tx, Sy) ≤ P (x, y)− ϕ(P (x, y)),
for all x, y ∈ X, where
P (x, y) = max
{
d(fx, fy), d(fx, Tx), d(fy, Sy),
1
2
[
d(fy, Tx) + d(fx, Sy)
]}
.
Moradi et al. [12] proved the following theorem:
Theorem 1.3. Let (X, d) be a complete metric space and E be a nonempty closed
subset of X. Let T, S : E → E be two generalized ϕf -weak contractive.
Assume that f is a continuous function on E and
(I) TE ⊆ fE and SE ⊆ fE.
(II) The pairs (T, f) and (S, f) are weakly compatible.
If for all x ∈ X
d(fTx, Tfx) ≤ d(fx, Tx) and d(fSx, Sfx) ≤ d(fx, Sx),
then f, T and S have a unique common fixed point.
2. Main results. In this paper, we establish common fixed point theorems for
mappings satisfying (ψ,ϕ)f,g-weakly contractive condition in a complete metric space.
Our result is an extension of Theorem 1.1 and Theorem 1.2. In fact, our generalization
is different from other generalization in [1, 7, 12].
From now as in [1], we assume:
Φ =
{
ϕ
∣∣ϕ : [0,∞)→ [0,∞) is a lower semicontinuous function,
ϕ(t) > 0 for all t > 0 and ϕ(0) = 0
}
,
and
Ψ =
{
ψ
∣∣ψ : [0,∞)→ [0,∞) is a continuous and nondecreasing function
and ψ(t) = 0⇐⇒ t = 0
}
.
We introduce the following definitions.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1546 A. RAZANI, V. PARVANEH, M. ABBAS
Definition 2.1. Two mappings T, S : X → X are called generalized (ψ,ϕ)f,g-
weak contractive, if there exists maps ϕ,ψ : [0,∞) → [0,∞) and f, g : X → X such
that
ψ(d(Tx, Sy)) ≤ ψ(M(x, y))− ϕ(M(x, y)), (2.1)
for all x, y ∈ X, where
M(x, y) = max
{
d(fx, gy), d(fx, Tx), d(gy, Sy),
1
2
[
d(gy, Tx) + d(fx, Sy)
]}
,
ψ ∈ Ψ and ϕ ∈ Φ.
Abbas et al. extended Zhang and Song’s theorem by the above concept [1]. We call
this class of mappings, generalized (ψ,ϕ)f,g-weak contractive mappings.
Definition 2.2. Let T and S be two self mappings of a metric space (X, d). T
and S are said to be weakly compatible, if for all x ∈ X the equality Tx = Sx implies
TSx = STx.
With respect to the above definition, we prove a common fixed point theorem as
follows:
Theorem 2.1. Let (X, d) be a complete metric space and E be a nonempty closed
subset of X. Suppose f and g are continuous functions of X. Let T, S : E → E be two
generalized (ψ,ϕ)f,g-weak contractive maps, such that
(A) TE ⊆ gE and SE ⊆ fE,
(B) T and f as well as S and g are weakly compatible.
In addition, for all x ∈ X
d(fTx, Tfx) ≤ d(fx, Tx) and d(gSx, Sgx) ≤ d(gx, Sx), (2.2)
and for all x, y ∈ X
d(fgx, gfy) ≤ d(gx, fy). (2.3)
Then T, f, S and g have a unique common fixed point.
Proof. Let x0 ∈ E be arbitrary. From (A), we can find two sequences {xn}∞n=0
and {yn}∞n=0 such that y1 = Tx0 = gx1, y2 = Sx1 = fx2, y3 = Tx2 = gx3, . . .
. . . , y2n+1 = Tx2n = gx2n+1, y2n+2 = Sx2n+1 = fx2n+2, . . . , respectively.
The rest of the proof is done in three steps as follows:
Step I. For all n = 0, 1, . . .
lim
n→∞
d(yn, yn+1) = 0.
Define dn = d(yn, yn+1). Suppose dn0
= 0 for some n0. Then yn0
= yn0+1. Conse-
quently, the sequence yn is constant for n ≥ n0. Indeed, let n0 = 2k. Then y2k = y2k+1
and we obtain from (2.1)
ψ(d(y2k+1, y2k+2)) = ψ(d(Tx2k, Sx2k+1)) ≤
≤ ψ(M(x2k, x2k+1))− ϕ(M(x2k, x2k+1)), (2.4)
where
M(x2k, x2k+1) = max
{
d(y2k, y2k+1), d(y2k, y2k+1), d(y2k+2, y2k+1),
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
A COMMON FIXED POINT FOR GENERALIZED (ψ,ϕ)f,g-WEAK CONTRACTIONS 1547
1
2
[
d(y2k, y2k+2) + d(y2k+1, y2k+1)
]}
=
= max
{
0, 0, d(y2k+1, y2k+2),
1
2
[d(y2k, y2k+2)]
}
= d(y2k+1, y2k+2).
Now from (2.1)
ψ(d(y2k+1, y2k+2)) = ψ(d(Sx2k+1, Tx2k)) ≤
≤ ψ(d(y2k+1, y2k+2))− ϕ(d(y2k+1, y2k+2)),
and so ϕ(d(y2k+1, y2k+2)) = 0, that is, y2k+1 = y2k+2.
Similarly, if n0 = 2k+ 1, one can easily obtain y2k+2 = y2k+3 and so the sequence
yn is constant (for n ≥ n0) and yn0
is a common fixed point of T, S, f and g. If we set
z = yn0 , then z is a unique common fixed point for T, S, f and g.
Suppose dn = d(yn, yn+1) > 0 for all n. We prove for each n = 1, 2, 3, . . .
d(yn+1, yn+2) ≤M(xn+1, xn+2) = d(yn, yn+1). (2.5)
Let n = 2k. Using condition (2.1), we obtain
ψ(d(y2k+1, y2k+2)) = ψ(d(Tx2k, Sx2k+1)) ≤
≤ ψ(M(x2k, x2k+1))− ϕ(M(x2k, x2k+1)) ≤
≤ ψ(M(x2k, x2k+1))
and since the function ψ is nondecreasing, it follows
d(y2k+1, y2k+2) ≤M(x2k, x2k+1). (2.6)
Here,
M(x2k, x2k+1) = max
{
d(fx2k, gx2k+1), d(fx2k, Tx2k), d(gx2k+1, Sx2k+1),
1
2
[
d(gx2k+1, Tx2k) + d(fx2k, Sx2k+1)
]}
=
= max
{
d(y2k, y2k+1), d(y2k, y2k+1), d(y2k+1, y2k+2),
1
2
[
d(y2k+1, y2k+1) + d(y2k, y2k+2)
]}
≤
≤ max
{
d(y2k, y2k+1), d(y2k+2, y2k+1),
1
2
[
d(y2k, y2k+1) + d(y2k+1, y2k+2)
]}
=
= max
{
d(y2k, y2k+1), d(y2k+1, y2k+2)
}
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1548 A. RAZANI, V. PARVANEH, M. ABBAS
If d(y2k+1, y2k+2) ≥ d(y2k, y2k+1) > 0, then
M(x2k+2, x2k+1) = d(y2k+2, y2k+1),
and this implies
ψ(d(y2k+2, y2k+1)) ≤ ψ(d(y2k+2, y2k+1))− ϕ(d(y2k+2, y2k+1))
which is only possible when d(y2k+2, y2k+1) = 0. This is a contradiction.
Hence, d(y2k+1, y2k+2) ≤ d(y2k+1, y2k) and
M(x2k+2, x2k+1) ≤ d(y2k+1, y2k).
Since, by definition of M(x, y),
M(x2k+2, x2k+1) ≥ d(y2k+1, y2k),
(2.5) is proved for d(y2k+1, y2k+2). Similarly, one can obtain
d(y2k+2, y2k+3) ≤M(x2k+1, x2k+2) = d(y2k+1, y2k+2).
So, (2.5) holds for all n.
Thus (2.5) shows that the sequence d(yn, yn+1) is a nonincreasing sequence of real
numbers and so there exists limn→∞ d(yn, yn+1) = limn→∞M(xn, xn+1) = r ≥ 0.
Suppose r > 0. Then from
ψ(d(yn+1, yn+2)) ≤ ψ(M(xn, xn+1))− ϕ(M(xn, xn+1)),
if n→∞, it follows that
ψ(r) ≤ ψ(r)− lim inf
n→∞
ϕ(M(xn, xn+1)) ≤ ψ(r)− ϕ(r),
i.e., ϕ(r) ≤ 0. But, ϕ ∈ Φ, so r = 0, which is a contradiction. We conclude that
lim
n→∞
d(yn, yn+1) = lim
n→∞
M(xn, xn+1) = 0. (2.7)
Step II. {yn} is a Cauchy sequence.
It is sufficient to show the subsequence {y2n} is a Cauchy sequence. If not, there
exists ε > 0 for which one can find subsequences {y2m(k)} and {y2n(k)} of {y2n} such
that
n(k) > m(k) > k and d(y2m(k), y2n(k)) ≥ ε
and n(k) is the least index with this property, that is,
d(y2m(k), y2n(k)−2) < ε. (2.8)
From (2.8) and triangle inequality
ε ≤ d(y2m(k), y2n(k)) ≤
≤ d(y2m(k), y2n(k)−2) + d(y2n(k)−2, y2n(k)−1) + d(y2n(k)−1, y2n(k)) ≤
≤ ε+ d(y2n(k)−2, y2n(k)−1) + d(y2n(k)−1, y2n(k)).
If k →∞ and using (2.7) we have
lim
k→∞
d(y2m(k), y2n(k)) = ε. (2.9)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
A COMMON FIXED POINT FOR GENERALIZED (ψ,ϕ)f,g-WEAK CONTRACTIONS 1549
In addition, from the known relation |d(x, z)− d(x, y)| ≤ d(y, z), we obtain∣∣d(y2m(k), y2n(k)+1)− d(y2m(k), y2n(k))
∣∣ ≤ d(y2n(k), y2n(k)+1), (2.10)∣∣d(y2m(k), y2n(k)+2)− d(y2m(k), y2n(k)+1)
∣∣ ≤ d(y2n(k)+2, y2n(k)+1), (2.11)∣∣d(y2n(k)+1, y2m(k)+1)− d(y2n(k)+1, y2m(k))
∣∣ ≤ d(y2m(k), y2m(k)+1), (2.12)∣∣d(y2n(k)+2, y2m(k)+1)− d(y2n(k)+1, y2m(k)+1)
∣∣ ≤ d(y2n(k)+1, y2n(k)+2), (2.13)
and using (2.7), (2.9), (2.10), (2.11), (2.12) and (2.13) we get
lim
k→∞
d(y2m(k), y2n(k)+1) = lim
k→∞
d(y2m(k), y2n(k)+2) =
= lim
k→∞
d(y2n(k)+1, y2m(k)+1) = lim
k→∞
d(y2n(k)+2, y2m(k)+1) = ε. (2.14)
From the definition of M(x, y) and the above limits,
lim
k→∞
M(x2m(k), x2n(k+1)) = ε.
Because,
M(x2m(k), x2n(k)+1) = max
{
d(fx2m(k), gx2n(k)+1), d(fx2m(k), Tx2m(k)),
d(gx2n(k)+1, Sx2n(k)+1),
1
2
[
d(gx2n(k)+1, Tx2m(k)) + d(fx2m(k), Sx2n(k)+1)
]}
=
= max
{
d(y2m(k), y2n(k)+1), d(y2m(k), y2m(k)+1),
d(y2n(k)+1, y2n(k)+2),
1
2
[
d(y2n(k)+1, y2m(k)+1) + d(y2m(k), y2n(k)+2)
]}
,
and if k →∞, we have
M(x2m(k), x2n(k)+1)→ max
{
ε, 0, 0,
1
2
[ε+ ε]
}
= ε.
Now, we apply condition (2.1), to obtain
ψ(d(y2m(k)+1, y2n(k)+2)) ≤ ψ(M(x2m(k), x2n(k)+1))− ϕ(M(x2m(k), x2n(k)+1)).
Again, if k →∞, we obtain ψ(ε) ≤ ψ(ε) − ϕ(ε) which is a contradiction with ε > 0.
Thus, {y2n} is a Cauchy sequence and hence {yn} is a Cauchy sequence.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1550 A. RAZANI, V. PARVANEH, M. ABBAS
Step III. There exists t such that gt = ft = St = Tt = t.
Since (X, d) is complete and {yn} is Cauchy, there exists z ∈ X such that
limn→∞ yn = z. Since E is closed and {yn} ⊆ E, we have z ∈ E. We know that
z = lim
n→∞
y2n = lim
n→∞
fx2n = lim
n→∞
Sx2n−1 =
= lim
n→∞
y2n+1 = lim
n→∞
gx2n+1 = lim
n→∞
Tx2n.
Since f and g are continuous, we have fyn → fz and gyn → gz.
On the other hand, from (2.2) and (2.3)
d(Ty2n, gz) ≤ d(Ty2n, fy2n+1) + d(fy2n+1, gy2n) + d(gy2n, gz) =
= d(Tfx2n, fTx2n) + d(fgx2n+1, gfx2n) + d(gy2n, gz) ≤
≤ d(Tx2n, fx2n) + d(gx2n+1, fx2n) + d(gy2n, gz) =
= d(y2n+1, y2n) + d(y2n+1, y2n) + d(gy2n, gz).
Therefore, from (2.7) and continuity of g,
lim
n→∞
d(Ty2n, gz) = 0.
Also, from (2.3) we have
d(Ty2n, fz) ≤ d(Ty2n, fy2n+1) + d(fy2n+1, fz) =
= d(Tfx2n, fTx2n) + d(fy2n+1, fz) ≤
≤ d(Tx2n, fx2n) + d(fy2n+1, fz) =
= d(y2n+1, y2n) + d(fy2n+1, fz).
Therefore, from (2.7)
lim
n→∞
d(Ty2n, fz) = 0.
From (2.1)
ψ(d(Ty2n, Sz)) ≤ ψ(M(y2n, z))− ϕ(M(y2n, z)),
where
M(y2n, z) = max
{
d(fy2n, gz), d(fy2n, T y2n), d(gz, Sz),
1
2
[
d(gz, Ty2n) + d(fy2n, Sz)
]}
.
Also,
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A COMMON FIXED POINT FOR GENERALIZED (ψ,ϕ)f,g-WEAK CONTRACTIONS 1551
lim
n→∞
d(Ty2n, gz) = lim
n→∞
d(Ty2n, fz) = 0.
Consequently, fz = gz.
If n→∞, we have
lim
n→∞
M(y2n, z) = max
{
d(fz, gz), d(fz, fz), d(gz, Sz),
1
2
[
d(gz, fz) + d(fz, Sz)
]}
.
So, we have
lim
n→∞
M(y2n, z) = d(fz, Sz).
Therefore,
ψ(d(fz, Sz)) ≤ ψ(d(fz, Sz))− ϕ(d(fz, Sz))).
This implies ϕ(d(fz, Sz)) = 0, and hence Sz = fz. We can analogously prove Tz =
= gz. Therefore, Tz = gz = fz = Sz = t.
Using weak compatibility of the pairs (T, f) and (S, g), we have Tt = ft and
gt = St. So,
ψ(d(Tt, t)) = ψ(d(Tt, Sz)) ≤ ψ(M(t, z))− ϕ(M(t, z)) =
= ψ
(
max
{
d(ft, gz), d(ft, T t), d(gz, Sz),
1
2
[d(gz, T t) + d(ft, Sz)]
})
−
−ϕ
(
max
{
d(ft, gz), d(ft, T t), d(gz, Sz),
1
2
[d(gz, T t) + d(ft, Sz)]
})
=
= ψ
(
max
{
d(Tt, t), d(Tt, T t), d(t, t),
1
2
[d(t, T t) + d(Tt, t)]
})
−
−ϕ
(
max
{
d(Tt, t), d(Tt, T t), d(t, t),
1
2
[d(t, T t) + d(Tt, t)]
})
=
= ψ(d(Tt, t))− ϕ(d(Tt, t)).
That is, ϕ(d(Tt, t)) = 0 and this implies Tt = t. Therefore, ft = Tt = t. Analogously,
gt = St = t. Hence gt = St = t = ft = Tt.
Theorem 2.1 is proved.
Note that the proof of Steps I and II is approximately analogous to what which has
been done in the other papers such as [1, 7, 12, 13], specially.
Example 2.1. Let X = R be endowed with the Euclidean metric and E = [0, 1].
Suppose T, S : E → E is defined by Tx =
1
2
= Sx, for all x ∈ E. We define functions
f, g : E → X by
f(x) =
x, 0 ≤ x ≤ 1
2
,
1
2
,
1
2
≤ x ≤ 1,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1552 A. RAZANI, V. PARVANEH, M. ABBAS
g(x) =
1
2
, 0 ≤ x ≤ 1
2
,
x,
1
2
≤ x ≤ 1,
and function ψ,ϕ : [0,∞)→ [0,∞) by ϕ(t) = t3 and ψ(t) = t2.
Thus for all x ∈ X
d(fTx, Tfx) ≤ d(fx, Tx), d(gSx, Sgx) ≤ d(gx, Sx),
and
d(fgx, gfy) ≤ d(gx, fy) for all x, y ∈ X.
Since 0 ≤M(x, y) ≤ 1 and d(Tx, Sy) = 0, we have
ψ(d(Tx, Sy)) ≤ ψ(M(x, y))− ϕ(M(x, y)).
So mappings T and S satisfy relation (2.1). This example cannot be studied by the
Theorem 1.3 ( Theorem 2.1 of [12]). But, all conditions of Theorem 2.1 are hold, and
T, S, f and g have a unique common fixed point
(
x =
1
2
)
.
3. Applications. In this section, we obtain some common fixed point theorems for
mappings satisfying a contraction condition of integral type in a complete metric space.
In [6], Branciari obtained a fixed point result for a single mapping satisfying an
integral type inequality. Then Altun et al. [3] established a fixed point theorem for
weakly compatible maps satisfying a general contractive inequality of integral type.
As in [13], we denote by Υ the set of all functions φ : [0,+∞)→ [0,+∞) verifying
the following conditions:
(I) φ is a positive Lebesgue integrable mapping on each compact subset of [0,+∞).
(II) For all ε > 0,
∫ ε
0
φ(t)dt > 0.
Corollary 3.1. Replace the generalized (ψ,ϕ)f,g-weak contractive condition of
Theorem 2.1 by the following condition:
There exists a φ ∈ Υ such that
ψ(d(Tx,Sy))∫
0
φ(t)dt ≤
ψ(M(x,y))∫
0
φ(t)dt−
ϕ(M(x,y))∫
0
φ(t)dt. (3.1)
If other conditions of Theorem 2.1 satisfy, then T, S, f and g have a unique common
fixed point.
Proof. Consider the function Γ(x) =
∫ x
0
φ(t)dt. Then (3.1) becomes
Γ(ψ(d(Tx, Sy))) ≤ Γ(ψ(M(x, y)))− Γ(ϕ(M(x, y)),
and taking ψ1 = Γoψ and ϕ1 = Γoϕ and applying Theorem 2.1, we obtain the proof (it
is easy to verify that ψ1 ∈ Ψ and ϕ1 ∈ Φ).
Corollary 3.2. Replace the generalized (ψ,ϕ)f,g-weak contractive condition of
Theorem 2.1 by the following condition:
There exists a φ ∈ Υ such that
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A COMMON FIXED POINT FOR GENERALIZED (ψ,ϕ)f,g-WEAK CONTRACTIONS 1553
ψ
d(Tx,Sy)∫
0
φ(t)dt
≤ ψ
M(x,y)∫
0
φ(t)dt
− ϕ
M(x,y)∫
0
φ(t)dt
. (3.2)
If other conditions of Theorem 2.1 satisfy, then T, S, f and g have a unique common
fixed point.
Proof. Again, as in Corollary 3.1, define the function Γ(x) =
∫ x
0
φ(t)dt. Then
(3.2) changes to
ψ(Γ(d(Tx, Sy))) ≤ ψ(Γ(M(x, y)))− ϕ(Γ(M(x, y))).
Now, if we define ψ1 = ψoΓ and ϕ1 = ϕoΓ and applying Theorem 2.1, then the proof
is complete (it is easy to verify ψ1 ∈ Ψ and ϕ1 ∈ Φ).
Now, we recall the definition of altering distance function as follows [10]:
Definition 3.1. The function ϕ : [0,+∞)→ [0,+∞) is called an altering distance
function if the following properties are satisfied:
(a) ϕ is continuous and nondecreasing,
(b) ϕ(t) = 0⇐⇒ t = 0.
Remark 3.1. In Theorem 2.1, assume ψ and ϕ are altering distance functions, then
theorem is hold.
Corollary 3.3. Replace the generalized (ψ,ϕ)f,g-weak contractive condition of
Theorem 2.1 by the following condition:
There exists a φ ∈ Υ such that
ψ1
ψ2(d(Tx,Sy))∫
0
φ(t)dt
≤ ψ1
ψ2(M(x,y))∫
0
φ(t)dt
− ϕ1
ϕ2(M(x,y))∫
0
φ(t)dt
,
(3.3)
for altering distance functions ψ1, ψ2, ϕ1 and ϕ2. If other conditions of Theorem 2.1
satisfy, then T, S, f and g have a unique common fixed point.
Proof. Consider the function Γ(x) =
∫ x
0
φ(t)dt. Then (3.3) will be
ψ1(Γ(ψ2(d(Tx, Sy)))) ≤ ψ1(Γ(ψ2(M(x, y))))− ϕ1(Γ(ϕ2(M(x, y)))),
and taking Ψ̂ = ψ1oΓoψ2 and Φ̂ = ϕ1oΓoϕ2 and applying Theorem 2.1, we obtain the
proof (it is easy to verify that Ψ̂ and Φ̂ are altering distance functions).
As in [13], let N ∈ N∗ be fixed. Let {φi}1≤i≤N be a family of N functions which
belong to Υ. For all t ≥ 0, we define
I1(t) =
t∫
0
φ1(s)ds,
I2(t) =
I1t∫
0
φ2(s)ds =
∫ ∫ t
0
φ1(s)ds
0
φ2(s)ds,
I3(t) =
I2t∫
0
φ3(s)ds =
∫ ∫ ∫ t
0 φ1(s)ds
0 φ2(s)ds
0
φ3(s)ds,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1554 A. RAZANI, V. PARVANEH, M. ABBAS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IN (t) =
I(N−1)t∫
0
φN (s)ds.
We have the following result.
Corollary 3.4. Replace the generalized (ψ,ϕ)f,g-weak contractive condition of
Theorem 2.1 by the following condition:
ψ
I(N−1)(d(Tx,Sy))∫
0
φN (s)ds
≤
≤ ψ
I(N−1)(M(x,y))∫
0
φN (s)ds
− ϕ
I(N−1)(M(x,y))∫
0
φN (s)ds
, (3.4)
where ψ ∈ Ψ and ϕ ∈ Φ. If other conditions of Theorem 2.1 satisfy, then T, S, f and g
have a unique common fixed point.
Proof. Consider Ψ̂ = ψoIN and Φ̂ = ϕoIN . Then the above inequality becomes
Ψ̂(d(Tx, Sy)) ≤ Ψ̂(M(x, y))− Φ̂(M(x, y))).
Applying Theorem 2.1, we obtain the desired result (it is easy to verify that Ψ̂ ∈ Ψ and
Φ̂ ∈ Φ).
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Received 24.05.11
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