Common fixed-point theorems for nonlinear weakly contractive mappings
Some common fixed-point results for mappings satisfying a nonlinear weak contraction condition within the framework of ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known from the literature.
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irk-123456789-1660622020-02-19T01:26:13Z Common fixed-point theorems for nonlinear weakly contractive mappings Chandok, S. Khan, M.S. Abbas, M. Статті Some common fixed-point results for mappings satisfying a nonlinear weak contraction condition within the framework of ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known from the literature. Отримано дєякі спільні теореми про нерухому точку для відображень, що задовольняють нелінійну слабкостискальну умову в рамках упорядкованих метричних просторів. Отримані результати узагальнюють та розширюють декілька порівняльних результатів, відомих із літературних джерел. 2014 Article Common fixed-point theorems for nonlinear weakly contractive mappings / S. Chandok, M.S. Khan, M. Abbas // Український математичний журнал. — 2014. — Т. 66, № 4. — С. 531–537. — Бібліогр.: 17 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166062 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Chandok, S. Khan, M.S. Abbas, M. Common fixed-point theorems for nonlinear weakly contractive mappings Український математичний журнал |
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Some common fixed-point results for mappings satisfying a nonlinear weak contraction condition within the framework of ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known from the literature. |
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Article |
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Chandok, S. Khan, M.S. Abbas, M. |
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Chandok, S. Khan, M.S. Abbas, M. |
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Chandok, S. |
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Common fixed-point theorems for nonlinear weakly contractive mappings |
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Common fixed-point theorems for nonlinear weakly contractive mappings |
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Common fixed-point theorems for nonlinear weakly contractive mappings |
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Common fixed-point theorems for nonlinear weakly contractive mappings |
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Common fixed-point theorems for nonlinear weakly contractive mappings |
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common fixed-point theorems for nonlinear weakly contractive mappings |
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Інститут математики НАН України |
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Common fixed-point theorems for nonlinear weakly contractive mappings / S. Chandok, M.S. Khan, M. Abbas // Український математичний журнал. — 2014. — Т. 66, № 4. — С. 531–537. — Бібліогр.: 17 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT chandoks commonfixedpointtheoremsfornonlinearweaklycontractivemappings AT khanms commonfixedpointtheoremsfornonlinearweaklycontractivemappings AT abbasm commonfixedpointtheoremsfornonlinearweaklycontractivemappings |
| first_indexed |
2025-07-14T20:41:36Z |
| last_indexed |
2025-07-14T20:41:36Z |
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1837656393090334720 |
| fulltext |
UDC 517.9
S. Chandok ( Khalsa College Engineering and Technology (Punjab Techn. Univ.), India),
M. S. Khan (College Sci., Sultan Qaboos Univ. Al-Khod, Sultanate of Oman),
M. Abbas (Univ. Pretoria, South Africa)
COMMON FIXED POINT THEOREMS
FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS
СПIЛЬНI ТЕОРЕМИ ПРО НЕРУХОМУ ТОЧКУ
ДЛЯ НЕЛIНIЙНИХ СЛАБКОСТИСКАЛЬНИХ ВIДОБРАЖЕНЬ
Some common fixed point results for mappings satisfying a nonlinear weak contractive condition in the framework of
ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known
from the literature.
Отримано деякi спiльнi теореми про нерухому точку для вiдображень, що задовольняють нелiнiйну слабкостис-
кальну умову в рамках упорядкованих метричних просторiв. Отриманi результати узагальнюють та розширюють
декiлька порiвняльних результатiв, вiдомих iз лiтературних джерел.
Introduction and preliminaries. Banach contraction principle is one of the pivotal results of metric
fixed point theory. It is a popular tool for solving existence problems in different fields of mathe-
matics. There are several generalizations of Banach contraction principle in the related literature on
metric fixed point theory.
Ran and Reurings [15] extended Banach contraction principle in partially ordered metric spaces
with some applications to linear and nonlinear matrix equations. While Nieto and López [14] extended
the result of Ran and Reurings and applied their main result to obtain a unique solution for a first
order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham
[3] introduced a concept of mixed monotone mappings and obtained some coupled fixed point results.
Also, they applied their results to a first order differential equation with periodic boundary conditions.
Alber and Guerre-Delabriere [1] introduced a concept of weakly contractive mappings and proved
the existence of fixed point of such mappings in Hilbert spaces. Thereafter, in 2001, Rhoades [17]
proved the fixed point theorem which is one of the generalizations of Banach’s contraction principle.
Weakly contractive mappings are closely related to the mappings of Boyd and Wong [4] and of Reich
types [16]. Recently, Doric [9] proved a common fixed point theorem for a generalized (ψ, φ)-weakly
contractive mappings. Fixed point problems involving weak contractions and mappings satisfying
weak contractive type inequalities have been studied by many authors (see [1, 5 – 10, 17] and ref-
erences cited therein). In this paper, we generalize Chatterjea type contraction mappings to (µ, ψ)-
generalized Chatterjea type contraction mappings and derive some common fixed point results for
single-valued mappings on ordered metric spaces.
First, we recall some basic definitions and notations.
Let (X, d) be a metric space. A mapping T : X → X is said to be:
(a) Kannan type (see [11]) if there exists a k ∈
(
0,
1
2
]
such that d(Tx, Ty) ≤ k[d(x, Tx) +
+ d(y, Ty)] for all x, y ∈ X;
(b) Chatterjea type [7] if there exists a k ∈
(
0,
1
2
]
such that d(Tx, Ty) ≤ k[d(x, Ty)+d(y, Tx)]
for all x, y ∈ X.
c© S. CHANDOK, M. S. KHAN, M. ABBAS, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4 531
532 S. CHANDOK, M. S. KHAN, M. ABBAS
Khan et al. [12] initiated the use of a control function that alters distance between two points in
a metric space. So they called it an altering distance function.
A function µ : [0,∞)→ [0,∞) is called an altering distance function if the following properties
are satisfied:
(i) µ is monotone increasing and continuous;
(ii) µ(t) = 0 if and only if t = 0.
Using the control function, we generalize the Chatterjea type contraction mappings as follows:
Suppose that T and f are self-mappings defined on a metric space X. A pair of mappings (T, f)
is said to satisfy (µ, ψ)-generalized Chatterjea type contractive condition if for all x, y ∈ X,
µ(d(Tx, fy)) ≤ µ
(
1
2
[d(x, fy) + d(y, Tx)]
)
− ψ(d(x, fy), d(y, Tx)), (1)
holds, where µ : [0,∞)→ [0,∞) is an altering distance function and ψ : [0,∞)2 → [0,∞) is a lower
semicontinuous mapping such that ψ(x, y) = 0 if and only if x = y = 0.
Let M be a nonempty subset of a metric space X, a point x ∈M is a common fixed (coincidence)
point of f and T if x = fx = Tx (fx = Tx). The set of fixed points (respectively, coincidence
points) of f and T is denoted by F (f, T ) (respectively, C(f, T )).
Definition 1. Let (X,≤) be a partially ordered set. Two mappings f, g : X → X are said to be
weakly increasing if fx ≤ gfx and gx ≤ fgx for all x ∈ X.
The following example shows that there exist discontinuous not nondecreasing mappings which
are weakly increasing.
Example 1. Let X = (0,∞), endowed with usual ordering. Let f, g : X → X be defined by
fx =
3x+ 2 if 0 < x < 1,
2x+ 1 if 1 ≤ x <∞
and
gx =
4x+ 1 if 0 < x < 1,
3x if 1 ≤ x <∞.
For 0 < x < 1, fx = 3x+2 ≤ 3(3x+2) = gfx and gx = 4x+1 ≤ 4x+3 = 2(2x+1)+1 = fgx
and for 1 ≤ x < ∞, fx = 2x + 1 ≤ 3(2x + 1) = gfx and gx = 3x ≤ 2(3x) + 1 = fgx. Thus f
and g are weakly increasing maps but not nondecreasing.
Common fixed point theorem in ordered metric spaces. Suppose that (X,�) is a partially
ordered set. A mapping T : X → X is said to be monotone increasing if for all x, y ∈ X,
x � y if and only if Tx � Ty. (2)
A subset W of a partially ordered set X is said to be well ordered if every two elements of W
are comparable.
Theorem 1. Let (X,�) be a partially ordered set such that there exists a complete metric d
on X. Suppose that T and f are weakly increasing self mappings on X, and satisfy (1) for all
comparable elements x, y ∈ X.
Also suppose that either
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
COMMON FIXED POINT THEOREMS FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS 533
(i) if {xn} ⊂ X is a nondecreasing sequence with xn → z in X, then xn � z, for every n ∈ N,
or
(ii) T or f is continuous.
Then T and f have a common fixed point. Moreover, the set of common fixed points of f and T
is well ordered if and only if f and T have one and only one common fixed point.
Proof. Let x0 ∈ X. We can choose x1, x2 ∈ X such that x1 = Tx0 and x2 = fx1. By induction,
we construct a sequence {xn} in X such that x2n+1 = Tx2n and x2n+2 = fx2n+1, for every n ≥ 0.
As T and f are weakly increasing mappings, so we obtain
x1 = Tx0 � fx1 = x2 � Tx2 = x3.
By induction on n, we conclude that
x1 � x2 � . . . � x2n+1 � x2n+2 � . . . .
Since x2n+1 and x2n+2 are comparable, by inequality (1) we have
µ(d(x2n+1, x2n+2)) = µ(d(Tx2n, fx2n+1)) ≤
≤ µ
(
1
2
[d(x2n, fx2n+1) + d(x2n+1, Tx2n)]
)
− ψ(d(x2n, fx2n+1), d(x2n+1, Tx2n)) =
= µ
(
1
2
d(x2n, x2n+2)
)
− ψ(d(x2n, x2n+2), 0) ≤
≤ µ
(
1
2
d(x2n, x2n+2)
)
.
Since µ is a monotone increasing function, for all n = 1, 2, . . . , we get
d(x2n+1, x2n+2) ≤
1
2
d(x2n, x2n+2) ≤
1
2
[
d(x2n, x2n+1) + d(x2n+1, x2n+2)
]
.
This implies that d(x2n+1, x2n+2) ≤ d(x2n, x2n+1). Following the similar arguments, we obtain
d(x2n+2, x2n+3) ≤ d(x2n+1, x2n+2). Hence, d(xn, xn+1) ≤ d(xn−1, xn). Thus {d(xn, xn+1)} is a
monotone decreasing sequence of nonnegative real numbers. Hence there exists r ≥ 0 such that
d(xn, xn+1)→ r. As
d(x2n+1, x2n+2) ≤
1
2
d(x2n, x2n+2) ≤
1
2
[
d(x2n, x2n+1) + d(x2n+1, x2n+2)
]
.
Taking limit as n → ∞, we have r ≤ lim
1
2
d(x2n, x2n+2) ≤
1
2
r +
1
2
r. Therefore limn→∞ d(x2n,
x2n+2) = 2r. Using the continuity of µ and lower semicontinuity of ψ, we have µ(r) ≤ µ(r) −
− ψ(2r, 0). This implies that ψ(2r, 0) = 0 and hence r = 0. Thus d(xn+1, xn)→ 0.
Now, we prove that {xn} is a Cauchy sequence. It is sufficient to show that {x2n} is a Cauchy
sequence. On contrary, suppose that {x2n} is not a Cauchy sequence. Then there exists ε > 0 for
which we can find subsequences {x2m(k)} and {x2n(k)} of {x2n} such that n(k) is the smallest index
for which n(k) > m(k) > k, d(x2m(k), x2n(k)) ≥ ε. This means that d(x2m(k), x2n(k)−2) < ε. So,
we have
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
534 S. CHANDOK, M. S. KHAN, M. ABBAS
ε ≤ d(x2m(k), x2n(k)) ≤
≤ d(x2m(k), x2n(k)−2) + d(x2n(k)−2, x2n(k)−1) + d(x2n(k)−1, x2n(k)) <
< ε+ d(x2n(k)−2, x2n(k)−1) + d(x2n(k)−1, x2n(k)).
Taking limit as k →∞, we get
lim
n→∞
d(x2m(k), x2n(k)) = ε. (3)
Also,
ε ≤ d(x2m(k), x2n(k)) ≤ d(x2m(k), x2m(k)−1) + d(x2m(k)−1, x2n(k)) ≤
≤ 2d(x2m(k), x2m(k)−1) + d(x2m(k), x2n(k)).
On letting k →∞, we obtain
lim
n→∞
d(x2m(k)−1, x2n(k)) = ε. (4)
On the other hand, we have
d(x2m(k), x2n(k)) ≤ d(x2m(k), x2n(k)+1) + d(x2n(k)+1, x2n(k)) ≤
≤ d(x2m(k), x2n(k)) + 2d(x2n(k)+1, x2n(k)).
On taking limit as k →∞, we get
lim
n→∞
d(x2m(k), x2n(k)+1) = ε.
Also,
d(x2m(k)−1, x2n(k)) ≤ d(x2m(k)−1, x2n(k)+1) + d(x2n(k)+1, x2n(k)) ≤
≤ d(x2m(k)−1, x2n(k)) + 2d(x2n(k)+1, x2n(k)).
On taking limit as k →∞, we obtain
lim
n→∞
d(x2m(k)−1, x2n(k)+1) = ε.
Consider
µ(ε) ≤ µ(d(x2m(k), x2n(k))) = µ(d(Tx2m(k)−1, fx2n(k)−1)) ≤
≤ µ
(
1
2
[
d(x2m(k)−1, fx2n(k)−1) + d(x2n(k)−1, Tx2m(k)−1)
])
−
−ψ(d(x2m(k)−1, fx2n(k)−1), d(x2n(k)−1, Tx2m(k)−1)) =
= µ
(
1
2
[d(x2m(k)−1, x2n(k)) + d(x2n(k)−1, x2m(k))]
)
−
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
COMMON FIXED POINT THEOREMS FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS 535
−ψ(d(x2m(k)−1, x2n(k)), d(x2n(k)−1, x2m(k))).
Taking limit as k → ∞, and using the continuity of µ and lower semicontinuity of ψ, we have
µ(ε) ≤ µ
(
1
2
[ε+ ε]
)
− ψ(ε, ε) and consequently ψ(ε, ε) ≤ 0, a contradiction as ε > 0. Thus {x2n}
is a Cauchy sequence and hence {xn} is a Cauchy sequence. As X is a complete metric space, there
exists t ∈ X such that limn→∞ xn = t. Since {xn} is a nondecreasing sequence, by (i), we have
xn � t. Consider
µ(d(x2n+1, ft)) = µ(d(Tx2n, ft)) ≤
≤ µ
(
1
2
[d(x2n, ft) + d(t, Tx2n)]
)
− ψ(d(x2n, ft), d(t, Tx2n)) =
= µ
(
1
2
[d(x2n, ft) + d(t, x2n+1)]
)
− ψ(d(x2n, ft), d(t, x2n+1)).
Taking limit as n → ∞, we have µ(d(t, ft)) ≤ µ
(
1
2
d(t, ft)
)
− ψ(d(t, ft), 0)) ≤ µ
(
1
2
d(t, ft)
)
.
This implies that d(t, ft) = 0 and hence t = ft.
Again, consider
µ(d(Tt, t)) = µ(d(Tt, ft)) ≤ µ
(
1
2
[d(t, ft) + d(t, T t)]
)
− ψ(d(t, ft), d(t, T t)) =
= µ
(
1
2
d(t, T t)
)
− ψ(0, d(t, T t)) ≤ µ
(
1
2
d(t, T t)
)
.
This implies that d(Tt, t) = 0, T t = t. Therefore, t = Tt = ft, i.e., t is a common fixed point of T
and f.
If condition (ii) holds: Assume that T is continuous. Then t = limn→∞ Txn = x2n+1 = Tt.
Now
µ(d(t, ft)) = µ(d(Tt, ft)) ≤ µ
(
1
2
[d(t, ft) + d(t, T t)]
)
− ψ(d(t, ft), d(t, T t)) =
= µ
(
1
2
d(t, ft)
)
− ψ(d(t, ft), 0) ≤ µ
(
1
2
d(t, ft)
)
implies that d(t, ft) = 0, ft = t. Therefore, t = Tt = ft, i.e., t is a common fixed point of T and f.
If f is continuous, then following arguments similar to those given above, the result follows.
Now suppose that the set of common fixed points of T and f is well ordered. Now, we claim
the uniqueness of common fixed points of T and f. Assume on contrary that Tu = fu = u and
Tv = fv = v but u 6= v. Consider
µ(d(u, v)) = µ(d(Tu, fv)) ≤
≤ µ
(
1
2
[d(u, fv) + d(v, Tu)]
)
− ψ(d(u, fv), d(v, Tu)) =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
536 S. CHANDOK, M. S. KHAN, M. ABBAS
= µ
(
1
2
[d(u, v) + d(v, u)]
)
− ψ(d(u, v), d(v, u)) =
= µ(d(u, v))− ψ(d(u, v), d(u, v)).
This implies that d(u, v) = 0, by the property of ψ. Hence u = v. Conversely, if T and f have
only one common fixed point then the set of common fixed point of f and T being singleton is well
ordered.
Theorem 1 is proved.
If T = f, we have the following result.
Corollary 1. Let (X,�) be a partially ordered set such that there exists a complete metric d on
X. Suppose that T is a monotone nondecreasing self mapping on X such that
µ(d(Tx, Ty)) ≤ µ
(
1
2
[d(x, Ty) + d(y, Tx)]
)
− ψ(d(x, Ty), d(y, Tx)),
is satisfied for all x, y ∈ X with x and y comparable.
Also suppose that either
(i) if {xn} ⊂ X is a nondecreasing sequence with xn → z in X, then xn � z, for every n ∈ N,
or
(ii) T is continuous.
Then T has a fixed point.
If µ(t) = t, we have the following result.
Corollary 2 (see [5, 10]). Let (X,�) be a partially ordered set such that there exists a complete
metric d on X. Suppose that T is a monotone nondecreasing self mapping on X such that
µ(d(Tx, Ty)) ≤ µ
(
1
2
[d(x, Ty) + d(y, Tx)]
)
− ψ(d(x, Ty), d(y, Tx)),
is satisfied for all comparable elements x, y ∈ X.
Also suppose that either
(i) if {xn} ⊂ X is a nondecreasing sequence with xn → z in X, then xn � z, for every n ∈ N,
or
(ii) T is continuous.
Then T has a fixed point.
Example 2. Let M = [0, 1] be endowed with partial order x � y if and only if x ≥ y. Let d be
defined by d(x, y) = |x− y|. We set Tx = 0 and fx =
x2
8
for all x ∈M. It is easy to see that f and
g are weakly increasing maps. We define µ : [0,∞)→ [0,∞) and ψ : [0,∞)× [0,∞)→ [0,∞) by
µ(t) =
t
2
and ψ(t, s) =
t+ s
16
.
Then for x, y ∈M, we have
µ(d(Tx, fy)) = µ
(
d
(
0,
y2
8
))
= µ
(
y2
8
)
=
y2
16
,
and
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
COMMON FIXED POINT THEOREMS FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS 537
µ
(
1
2
[d(x, fy) + d(y, Tx)]
)
− ψ(d(x, fy), d(y, Tx)) =
= µ
(
1
2
[
d
(
x,
y2
8
)
+ d(y, 0)
])
− ψ
(
d
(
x,
y2
8
)
, d(y, 0)
)
=
= µ
(
1
2
[ ∣∣∣∣x− y2
8
∣∣∣∣+ y
])
− ψ
(∣∣∣∣x− y2
8
∣∣∣∣, y) =
=
1
4
[ ∣∣∣∣x− y2
8
∣∣∣∣+ y
]
−
∣∣∣∣x− y2
8
∣∣∣∣+ y
16
=
3
16
[ ∣∣∣∣x− y2
8
∣∣∣∣+ y
]
≥ 3y
16
≥ y2
16
.
Hence
µ(d(Tx, fy)) ≤ µ
(
1
2
[d(x, fy) + d(y, Tx)]
)
− ψ(d(x, fy), d(y, Tx)).
Thus all conditions of Theorem 1 are satisfied. Moreover, T and f have a unique common fixed
point 0.
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Theory, Adv. Appl. / Eds I. Gohberg and Yu. Lyubich. – Basel: Birkhäuser, 1997. – 8. – P. 7 – 22.
2. Altun I., Damjanović B., Djorić D. Fixed point and common fixed point theorems on ordered cone metric spaces //
Appl. Math. Lett. – 2010. – 23. – P. 310 – 316.
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Received 11.06.12,
after revision — 04.04.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
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