International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 2 (2014), 154-166
http://www.etamaths.com

PREŠIĆ-BOYD-WONG TYPE RESULTS IN ORDERED METRIC

SPACES

SATISH SHUKLA1,∗ AND STOJAN RADENOVIĆ2

Abstract. The purpose of this paper is to prove some Prešić-Boyd-Wong

type fixed point theorems in ordered metric spaces. The results of this paper

generalize the famous results of Prešić and Boyd-Wong in ordered metric s-
paces. We also initiate the homotopy result in product spaces. Some examples

are provided which illustrate the results proved herein.

1. Introduction and preliminaries

In 1922 Banach [26] proved the following theorem known as Banach contraction
mapping theorem.

Theorem 1. Let (X,d) be a complete metric space and f : X → X be a mapping
such that

(1) d(fx,fy) ≤ λd(x,y)

for all x,y ∈ X, where 0 ≤ λ < 1, then there exists a unique x ∈ X such that
fx = x. This point x is called the fixed point of mapping f.

Due to simplicity and usefulness, several authors generalized the Banach contrac-
tion mapping theorem. One such generalization is given by Prešić [24, 25]. Prešić
generalized the Banach contraction mapping theorem in product spaces and proved
the following theorem.

Theorem 2. Let (X,d) be a complete metric space, k a positive integer and f :
Xk → X be a mapping satisfying the following contractive type condition:

(2) d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤
k∑
i=1

qid(xi,xi+1),

for every x1,x2, . . . ,xk,xk+1 ∈ X, where q1,q2, . . . ,qk are nonnegative constants
such that q1 + q2 + · · · + qk < 1. Then there exists a unique point x ∈ X such
that f(x,x,. . . ,x) = x. Moreover if x1,x2, . . . ,xk are arbitrary points in X and for
n ∈ N, xn+k = f(xn,xn+1, . . . ,xn+k−1), then the sequence {xn} is convergent and
lim xn = f(lim xn, lim xn, . . . , lim xn).

2010 Mathematics Subject Classification. 54H25, 47H10.
Key words and phrases. Common fixed point; Prešić type mapping; Boyd-Wong fixed point

theorem; Partial order.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

154



PREŠIĆ-BOYD-WONG TYPE RESULTS 155

Condition (2) in the case k = 1 reduces to the condition (1). So, Theorem 1 is a
generalization of the Banach fixed point theorem. The results of Prešić is useful in
proving the convergence of some particular sequences and in proving the existence
of solutions of differences equations, for example, see [15,24,25,40]. For more on the
generalizations of Prešić type operators the reader is referred to [12,16–19,21,28–36].

On the other hand Boyd and Wong [4] generalized the Banach contraction map-
ping theorem and proved the following theorem.

Theorem 3. Let (X,d) be a complete metric space and f : X → X a mapping that
satisfies

(3) d(fx,fy) ≤ ψ(d(x,y)) for all x,y ∈ X,
where ψ : R+ → R+ is upper semi-continuous function from the right (i.e., λi ↓
λ ≥ 0 ⇒ lim sup

i→∞
ψ(λi) ≤ ψ(λ)) such that ψ(t) < t for each t > 0. Then f has a

unique fixed point u ∈ X. Moreover, for each x ∈ X, lim
n→∞

fnx = u.

Note that the condition (3) in the case ψ(t) = λt reduces to condition (1). So,
Theorem 3 is a generalization of the Banach fixed point theorem. Some generaliza-
tion of the Boyd-Wong theorem can be found in [9, 10, 13, 14, 20, 23, 38].

The existence of fixed point in partially ordered sets was investigated by Ran and
Reurings [1] and then by Nieto and Lopez [7, 8]. Some applications of fixed point
theorems in ordered metric spaces to differential equations can be seen in [7, 8].
Several authors generalized the results of these papers in different directions for
example, see [2, 3, 5, 6, 22, 27, 37, 39, 41]. The following version of the fixed point
theorem was proved, among others, in these papers.

Theorem 4. Let (X,�) be a partially ordered set and d be a metric on X such
that (X,d) is a complete metric space. Let f : X → X be a nondecreasing map
with respect to �. Suppose that the following conditions hold:

(i) there exists k ∈ (0, 1) such that d(fx,fy) ≤ kd(x,y) for all x,y ∈ X with
y � x;

(ii) there exists x0 ∈ X such that x0 � fx0;
(iii) if a nondecreasing sequence {xn} in X converges to x ∈ X, then xn ≤ x,

for all n ∈ N.
Then f has a fixed point x∗ ∈ X.

Recently, in [36] Malhotra et al. defined the ordered Prešić type contraction
mappings the setting of cone metric spaces (see also [19, 33]) and generalized the
result of Prešić in ordered case. In the present paper, we generalize the results of
Prešić, Boyd and Wong, Theorem 4 and several known results in and prove some
Prešić-Boyd-Wong type fixed point theorems in ordered metric spaces. A homotopy
result in the product spaces is also proved. Examples are included which illustrate
the results.

Following definitions will be needed in sequel.

Definition 1. Let X be any nonempty set, k a positive integer and f : Xk → X
be a mapping. An element x ∈ X is called a fixed point of f if f(x,x,. . . ,x) = x.



156 SHUKLA AND RADENOVIĆ

Definition 2. Let X be a nonempty set, k a positive integer, f : Xk → X and
g : X → X be mappings.

(i) An element x ∈ X said to be a coincidence point of f and g if gx =
f(x,x,. . . ,x).

(ii) If w = gx = f(x,x,. . . ,x), then w is called a point of coincidence of f and
g.

(iii) If x = gx = f(x,x,. . . ,x), then x is called a common fixed point of f and
g.

(iv) Mappings f and g are said to be commuting if g(f(x,x,. . . ,x)) = f(gx,gx,. . . ,gx)
for all x ∈ X.

(v) Mappings f and g are said to be weakly compatible if they commute at their
coincidence points.

Definition 3. Let X be a nonempty set equipped with a partial order relation “ � ”,
k a positive integer and f : Xk → X be a mapping. A sequence {xn} in X is said
to be nondecreasing with respect to “ � ”, if xn � xn+1 for all n ∈ N. The mapping
f is said to be nondecreasing with respect to “ � ” if for any finite nondecreasing
sequence {xn}k+1n=1 we have f(x1,x2, . . . ,xk) � f(x2,x3, . . . ,xk+1). Let g : X → X
be a mapping. f is said to be g-nondecreasing with respect to “ � ” if for any finite
nondecreasing sequence {gxn}k+1n=1 we have f(x1,x2, . . . ,xk) � f(x2,x3, . . . ,xk+1).

Note that for k = 1, above definitions reduce to usual definitions of nondecreasing
and g-nondecreasing mappings.

Definition 4. Let X be a nonempty set equipped with a partial order relation “ � ”,
and g : X → X be a mapping. A nonempty subset A of X is said to be well ordered
if every two elements of A are comparable. The elements a,b ∈ A are called g-
comparable if ga and gb are comparable. The set A is called g-well ordered if for
all a,b ∈A, a and b are g-comparable i.e. ga and gb are comparable.

Example 1. Let X = [0,∞), A = [0, 1] and define a relation “ � ” on X by

x � y ⇔{(x = y) or (x,y ∈ [0,
1

2
] with x ≤ y)}.

Then � is a partial order relation on X. Define g : X → X by gx = x
2

for all
x,y ∈ X.
Note that A is not well ordered. Indeed if x,y ∈ ( 1

2
,∞),x 6= y then neither x � y

nor y � x. But g(A) = [0, 1
2
], therefore A is g-well ordered.

Let (X,d) be a metric space equipped with partial order relation “ � ”, then
(X,�,d) is called an ordered metric space. Let k be a positive integer and f :
Xk → X be a mapping. f is called ordered Prešić contraction if

(4) d(f(x1,x2 . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤
k∑
i=1

αid(xi,xi+1),

for all x1,x2, . . . ,xk,xk+1 ∈ X with x1 � x2 � ··· � xk � xk+1, where αi are

nonnegative constants such that

k∑
i=1

αi < 1.

If (5) is satisfied for all x1,x2, . . . ,xk,xk+1 ∈ X, then f is called Prešić contraction.
Note that in ordered metric spaces a Prešić contraction is necessarily an ordered



PREŠIĆ-BOYD-WONG TYPE RESULTS 157

Prešić contraction, but converse may not be true (see examples 3.1 and 3.2 of [36]).

Let ψ : Rk+ → R+ be a function satisfying the following conditions:
(1) for tn ∈ R+, n ∈ N and tn ↓ t ≥ 0 implies lim sup

n→∞
ψ(tn, tn, . . . , tn) ≤

ψ(t,t, . . . , t);
(2) ψ(t,t, . . . , t) < t for each t > 0;
(3) ψ(t, 0, . . . , 0) + ψ(0, t, 0, . . . , 0) + · · · + ψ(0, . . . , 0, t) ≤ ψ(t,t, . . . , t) for each

t ∈ R+.
We denote the class of all such functions by Ψ, i.e., ψ ∈ Ψ if and only if ψ satisfies
the all above conditions.

Example 2. Let ψ : Rk+ → R+ be defined by ψ(t1, t2, . . . , tk) =
k∑
i=1

αiti, where αi

are nonnegative constants such that
k∑
i=1

αi < 1. Then ψ ∈ Ψ.

Mapping f : Xk → X is said to be an ordered Prešić-Boyd-Wong contraction if
(5)
d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ ψ(d(x1,x2),d(x2,x3), . . . ,d(xk.xk+1))

for all x1,x2, . . . ,xk,xk+1 ∈ X with x1 � x2 � ··· � xk � xk+1, where ψ ∈ Ψ. If
(5) is satisfied for all x1,x2, . . . ,xk,xk+1 ∈ X, then f is called Prešić-Boyd-Wong
contraction.

Now we can state our main results.

2. Main results

Theorem 5. Let (X,�,d) be a complete ordered metric space, k a positive integer.
Let f : Xk → X, g : X → X be two mappings such that f(Xk) ⊂ g(X) and g(X)
is a closed subspace of X. Suppose following conditions hold:

(I)
(6)
d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ ψ(d(gx1,gx2),d(gx2,gx3), . . . ,d(gxk,gxk+1)),

for all x1,x2, . . . ,xk,xk+1 ∈ X with gx1 � gx2 � ···� gxk � gxk+1, where
ψ ∈ Ψ;

(II) there exist x1 ∈ X such that gx1 � f(x1,x1, . . . ,x1);
(III) f is g-nondecreasing;
(IV) if a nondecreasing sequence {gxn} converges to gu ∈ X, then gxn � gu for

all n ∈ N and gu � ggu.
Then f and g have a point of coincidence. If in addition f and g are weakly
compatible, then f and g have a common fixed point v ∈ X. Moreover, the set of
common fixed points of f and g is g-well ordered if and only if f and g have a
unique common fixed point.

Proof. Starting with given x1 ∈ X, we define a sequence {yn} as follows: let
y1 = gx1, as f(X

k) ⊂ g(X) and gx1 � f(x1,x1, . . . ,x1), define yn+1 = gxn+1 =
f(xn,xn, . . . ,xn), n ∈ N. Then gx1 � gx2 i.e. y1 � y2 and f is g-nondecreasing, so

y2 = f(x1,x1, . . . ,x1) � f(x1,x1, . . . ,x1,x2) � f(x1,x1, . . . ,x1,x2,x2)
� ···� f(x1,x2, . . . ,x2) � f(x2,x2, . . . ,x2) = gx3 = y3



158 SHUKLA AND RADENOVIĆ

i.e. y2 = gx2 � y3 = gx3.
Continuing this procedure, we obtain

gx1 � gx2 � ···� gxn � gxn+1 � ··· ,

i.e.

y1 � y2 � ···� yn � yn+1 � ··· .

Thus {yn} = {gxn} is a nondecreasing sequence with respect to “ � ”.
For simplicity set dn = d(yn,yn+1),n ∈ N. We may assume that dn > 0 for all
n ∈ N, otherwise coincidence point and point of coincidence of f and g exist trivially.
We shall show that lim

n→∞
dn = 0.

Note that

dn+1 = d(yn+1,yn+2)

= d(f(xn,xn, . . . ,xn),f(xn+1,xn+1, . . . ,xn+1))

≤ d(f(xn,xn, . . . ,xn),f(xn, . . . ,xn,xn+1))
+d(f(xn, . . . ,xn,xn+1),f(xn, . . . ,xn,xn+1,xn+1))

+ · · · + d(f(xn,xn+1, . . . ,xn+1),f(xn+1, . . . ,xn+1))

and gxn � gxn+1, ψ ∈ Ψ, so it follows from (6) that

dn+1 ≤ ψ(0, . . . , 0,d(gxn,gxn+1)) + ψ(0, . . . , 0,d(gxn,gxn+1), 0)
+ · · · + ψ(d(gxn,gxn+1), 0, . . . , 0)

= ψ(0, . . . , 0,dn) + ψ(0, . . . , 0,dn, 0) + · · · + ψ(dn, 0, . . . , 0)
≤ ψ(dn,dn, . . . ,dn)
< dn,

for all n ∈ N. Therefore {dn} is a monotonic nondecreasing sequence and bounded
below, so lim

n→∞
dn exists. Let lim

n→∞
dn = δ ≥ 0. Assume δ > 0, then as ψ ∈ Ψ we

obtain

δ = lim
n→∞

dn+1 ≤ lim
n→∞

ψ(dn,dn, . . . ,dn) ≤ ψ(δ,δ, . . . ,δ) < δ,

a contradiction, so δ = 0. We shall show that {yn} is Cauchy sequence.
Assume that {yn} is not Cauchy, then there exists � > 0 and integers ml,nl, l ∈ N
such that ml > nl ≥ l and

d(ynl,yml ) ≥ � for l ∈ N.

Also, choosing ml as small as possible, it may be assumed that

d(yml−1,ynl ) < �.

So for each l ∈ N, we have

� ≤ d(yml,ynl ) ≤ d(yml,yml−1) + d(yml−1,ynl )
≤ dml−1 + �



PREŠIĆ-BOYD-WONG TYPE RESULTS 159

and it follows from the fact lim
n→∞

dn = 0 that lim
l→∞

d(yml,ynl ) = �. Observe that

� ≤ d(yml,ynl ) ≤ d(yml,yml+1) + d(yml+1,ynl+1) + d(ynl+1,ynl )
= dml + dnl + d(f(xnl, . . . ,xnl ),f(xml, . . . ,xml ))

≤ dml + dnl + d(f(xnl, . . . ,xnl ),f(xnl, . . . ,xnl,xml ))
+d(f(xnl, . . . ,xnl,xml ),f(xnl, . . . ,xnl,xml,xml ))

+ · · · + d(f(xnl,xml, . . . ,xml ),f(xml, . . . ,xml )).

As ml > nl and {yn} is nondecreasing with respect to “ � ”, so ynl � yml i.e.,
gxnl � gxml, therefore it follows from (6) and the above inequality that

� ≤ d(yml,ynl ) ≤ dml + dnl + ψ(0, . . . , 0,d(ynl,yml )) + ψ(0, . . . , 0,d(ynl,yml ), 0)
+ · · · + ψ(d(ynl,yml ), 0, . . . , 0)

≤ dml + dnl + ψ(d(ynl,yml ), . . . ,d(ynl,yml )).

Letting l →∞ and using the facts that lim
n→∞

dn = 0 and ψ ∈ Ψ, we have

� = lim
l→∞

d(yml,ynl ) ≤ lim
l→∞

ψ(d(yml,ynl ), . . . ,d(yml,ynl )) ≤ ψ(�, . . . ,�) < �,

which is a contradiction. Therefore {yn} = {gxn} is a Cauchy sequence in g(X).
As g(X) is closed, there exist u,v ∈ X such that v = gu and

(7) lim
n→∞

yn = lim
n→∞

gxn = gu = v.

We shall show that u is a coincidence point and v is a point of coincidence of f and
g. Note that

d(v,f(u,u,. . . ,u)) ≤ d(v,yn+1) + d(yn+1,f(u,u,. . . ,u))
= d(v,yn+1) + d(f(xn,xn, . . . ,xn),f(u,u,. . . ,u))

≤ d(v,yn+1) + d(f(xn,xn, . . . ,xn),f(xn, . . . ,xn,u))
+d(f(xn, . . . ,xn,u)),f(xn, . . . ,xn,u,u))

+ · · · + d(f(xn,u, . . . ,u)),f(u,. . . ,u)).

If v 6= f(u,u,. . . ,u), then by (IV) we have gxn � gu,gu � ggu, so using (6) it
follows from the above inequality that

d(v,f(u,u,. . . ,u)) ≤ d(v,yn+1) + ψ(0, . . . , 0,d(gxn,gu)) + ψ(0, . . . , 0,d(gxn,gu), 0)
+ · · · + ψ(d(gxn,gu), 0, . . . , 0)

≤ d(v,yn+1) + ψ(d(gxn,gu), . . . ,d(gxn,gu))
< d(v,yn+1) + d(gxn,gu),

letting n →∞ and using (7) we obtain

d(v,f(u,u,. . . ,u)) = 0 i.e. f(u,u,. . . ,u) = gu = v.

Thus u is a coincidence point and v is a point of coincidence of f and g.
Suppose f and g are weakly compatible, so

f(v,v, . . . ,v) = f(gu,gu,. . . ,gu) = g(f(u,u,. . . ,u)) = gv.



160 SHUKLA AND RADENOVIĆ

Again if d(gu,gv) > 0 then as gu � ggu = gv we obtain from (6) that
d(v,f(v,v, . . . ,v)) = d(f(u,u,. . . ,u),f(v,v, . . . ,v))

≤ d(f(u,u,. . . ,u),f(u,. . . ,u,v)) + d(f(u,. . . ,u,v),f(u,. . . ,u,v,v))
+ · · · + d(f(u,v, . . . ,v),f(v, . . . ,v))

≤ ψ(0, . . . , 0,d(gu,gv)) + ψ(0, . . . , 0,d(gu,gv), 0)
+ · · · + ψ(d(gu,gv), 0, . . . , 0)

≤ ψ(d(gu,gv), . . . ,d(gu,gv)) < d(gu,gv) = d(v,f(v,v, . . . ,v)),
a contradiction, therefore v = gv = f(v,v, . . . ,v). Thus v is the common fixed point
of f and g.
Suppose the set of common fixed points of f and g is g-well ordered. We shall show
that common fixed point is unique. Assume on contrary that v′ is another common
fixed point of f and g i.e. v′ = gv′ = f(v′,v′, . . . ,v′) and v 6= v′. As v and v′ are
g-comparable, let e.g. gv � gv′. From (6), it follows that
d(v,v′) = d(f(v,v, . . . ,v),f(v′,v′, . . . ,v′))

≤ d(f(v,v, . . . ,v),f(v, . . . ,v,v′)) + d(f(v, . . . ,v,v′),f(v, . . . ,v,v′,v′))
+ · · · + d(f(v,v′, . . . ,v′),f(v′,v′, . . . ,v′))

≤ ψ(0, . . . , 0,d(gv,gv′)) + ψ(0, . . . , 0,d(gv,gv′), 0)
+ · · · + ψ(d(gv,gv′), 0, . . . , 0)

≤ ψ(d(gv,gv′), . . . ,d(gv,gv′)) < d(v,v′)
a contradiction. Therefore, v = v′, i.e., the common fixed point is unique. For
converse, if common fixed point of f and g is unique then the set of common fixed
points of f and g being singleton therefore g-well ordered. �

Remark 1. For k = 1 the above theorem is a generalization and extension of result
of Boyd and Wong in ordered metric spaces.

Following is a simple example which illustrate the above result.

Example 3. Let X = [0,∞) with the usual metric and partial order �= {(x,y) :
x,y ∈ X,y ≤ x}. For k = 2, define f : X2 → X and g : X → X by

f(x1,x2) =
x1 + x2

3 + x1 + x2
for all x1,x2 ∈ X and gx = x for all x ∈ X.

Define ψ : R2+ → R by

ψ(t1, t2) =
t1 + t2

3 + |t1 − t2|
for all t1, t2 ∈ R+.

Then it easy to see that all the conditions of Theorem 5 are satisfied and 0 is the
unique common fixed point of f and g in X.

Taking g = IX i.e. identity mapping of X in Theorem 5, we get the following
fixed point result for ordered Prešić-Boyd-Wong contraction.

Corollary 6. Let (X,�,d) be a complete ordered metric space, k a positive integer.
Let f : Xk → X be a mapping such that the following conditions hold:

(I) f is ordered Prešić-Boyd-Wong contraction;
(II) there exist x1 ∈ X such that x1 � f(x1,x1, . . . ,x1);

(III) f is nondecreasing;



PREŠIĆ-BOYD-WONG TYPE RESULTS 161

(IV) if a nondecreasing sequence {xn} converges to u ∈ X, then xn � u for all
n ∈ N.

Then f has a fixed point v ∈ X. Moreover, the set of fixed points of f is well
ordered if and only if f has a unique fixed point.

The following example illustrate the case when the known results are not appli-
cable but the Corollary 6 of this paper is applicable.

Example 4. Let X = [0, 2] and d is the usual metric on X, then (X,d) is a
complete metric space. For k = 2, define a mapping f : X2 → X by

f(x,y) =




x

1 + x
, if (x,y) ∈ [0, 1) × [0, 1) ∪ [1, 2] × [0, 1);

y

1 + y
, if (x,y) ∈ [0, 1) × [1, 2];

0, otherwise,

and a function ψ : Rk+ → R+ by

ψ(t1, t2) =
t1

1 + |t1/2 − t2|
for all t1, t2 ∈ R+.

Let � be a partial order define on X by

� = {(x,y) : (x,y) ∈ [0, 1) × [0, 1) with y ≤ x}∪{(x,y) : (x,y) ∈ [1, 2] × (0, 1)}
∪{(x,x) : x ∈ X},

then ψ ∈ Ψ. Now by careful calculations one can see that all the conditions of
Corollary 6 are satisfied and 0 is the unique fixed point of f. Note that, f is not an
ordered Prešić type contraction, therefore it is not a Prešić type contraction. To see
this, take arbitrary values x,y = z ∈ [0, 1) and then condition (5) is not satisfied.

Following theorem is a generalization of the result of Prešić and Boyd and Wong
in metric spaces.

Theorem 7. Let (X,d) be a complete metric space, k a positive integer. Let
f : Xk → X, g : X → X be two mappings such that f(Xk) ⊂ g(X) and g(X) is a
closed subspace of X. Suppose following conditions hold:
(8)
d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ ψ(d(gx1,gx2),d(gx2,gx3), . . . ,d(gxk,gxk+1)),

for all x1,x2, . . . ,xk,xk+1 ∈ X, where ψ ∈ Ψ. Then f and g have a point of
coincidence. If in addition f and g are weakly compatible, then f and g have a
unique common fixed point v ∈ X.

Proof. We note that the inequality (8) is true for all x1,x2, . . . ,xk,xk+1 ∈ X,
therefore the proof of theorem follows from similar process as used in the proof of
Theorem 5. �

Taking g = IX i.e. identity mapping of X in Theorem 7, we get the following
fixed point result for Prešić-Boyd-Wong contraction.

Corollary 8. Let (X,d) be a complete metric space, k a positive integer. Let
f : Xk → X be a Prešić-Boyd-Wong contraction. Then f has a unique fixed point
v ∈ X.



162 SHUKLA AND RADENOVIĆ

Remark 2. Note that, for ψ(t1, t2, . . . , tk) =
k∑
i=1

αiti, where αi are nonnegative

constants such that
k∑
i=1

αi < 1, Corollary 8 reduces to the Prešić theorem.

3. A homotopy result

In this section we prove a homotopy result for Prešić type mapping on product
space.

Theorem 9. Let (X,d) be any complete metric space, U an open subset of X.
Suppose H : (U)k × [0, 1] → X be a function such that the following conditions
hold:

(i) for every x ∈ ∂U(here ∂U is the boundary of U) and λ ∈ [0, 1], x 6=
H(x,x,. . . ,x,λ);

(ii) for all x1,x2, . . . ,xk,xk+1 ∈ U and λ ∈ [0, 1]

(9) d(H(x1,x2, . . . ,xk,λ),H(x2,x3, . . . ,xk+1,λ)) ≤
k∑
i=1

αid(xi,xi+1),

where αi are nonnegative constants such that
k∑
i=1

αi <
1
k

;

(iii) for all x1,x2, . . . ,xk ∈ U and λ,µ ∈ [0, 1] there exists M ≥ 0 such that
(10) d(H(x1,x2, . . . ,xk,λ),H(x1,x2, . . . ,xk,µ)) ≤ M|λ−µ|.
If Hλ=λ′ has a fixed point in U for at least one λ

′ ∈ [0, 1], then Hλ has a fixed point
in U for all λ ∈ [0, 1]. Furthermore, for any fixed λ ∈ [0, 1], the fixed point of Hλ is
unique.

Proof. Define

F = {λ ∈ [0, 1] : x = H(x,x,. . . ,x,λ) for some x ∈ U}.
As Hλ=λ′ for at least one λ

′ ∈ [0, 1], has a fixed point in U, i.e., there exists x ∈ U
such that H(x,x,. . . ,x,λ′) = x, so λ′ ∈ F and Fx 6= ∅. We shall show that F is
both open and closed in [0, 1] and therefore by connectedness F = [0, 1].
(I) F is closed: Let {λn} be any sequence in F and lim

n→∞
λn = λ ∈ [0, 1]. As

λn ∈ F for all n ∈ N so there exists xn ∈ U such that xn = H(xn,xn, . . . ,xn,λn)
for all n ∈ N.
Note that, for all n,m ∈ N with m > n we have

d(xn,xm) = d(H(xn,xn, . . . ,xn,λn),H(xm,xm, . . . ,xm,λm))

≤ d(H(xn, . . . ,xn,λn),H(xn, . . . ,xn,xm,λn))
+d(H(xn, . . . ,xn,xm,λn),H(xn, . . . ,xn,xm,xm,λn))

+ · · · + d(H(xn,xm, . . . ,xm,λn),H(xm, . . . ,xm,λn))
+d(H(xm, . . . ,xm,λn),H(xm, . . . ,xm,λm)).

Using (9) and (10) it follows that

d(xn,xm) ≤ αkd(xn,xm) + αk−1d(xn,xm) + · · · + α1d(xn,xm) + M|λn −λm|

= [

k∑
i=1

αi]d(xn,xm) + M|λn −λm|



PREŠIĆ-BOYD-WONG TYPE RESULTS 163

i.e.

d(xn,xm) ≤
M

1 −
k∑
i=1

αi

|λn −λm|.

Letting n → ∞ and using the fact that lim
n→∞

λn = λ it follows from the above

inequality that lim
n→∞

d(xn,xm) = 0, therefore {xn} is a Cauchy sequence. As X is
complete, there exists u ∈ U such that

lim
n→∞

xn = u.

Now for any n ∈ N we obtain

d(xn,H(u,u,. . . ,u,λ)) = d(H(xn,xn, . . . ,xn,λn),H(u,u,. . . ,u,λ))

≤ d(H(xn, . . . ,xn,λn),H(xn, . . . ,xn,u,λn))
+d(H(xn, . . . ,xn,u,λn),H(xn, . . . ,xn,u,u,λn))

+ · · · + d(H(xn,u, . . . ,u,λn),H(u,. . . ,u,λn))
+d(H(u,u,. . . ,u,λn),H(u,u,. . . ,u,λ)),

using (9) and (10) it follows that

d(xn,H(u,u,. . . ,u,λ)) ≤ αkd(xn,u) + αk−1d(xn,u) + · · · + α1d(xn,u) + M|λn −λ|

= [

k∑
i=1

αi]d(xn,u) + M|λn −λ|.

As lim
n→∞

λn = λ and lim
n→∞

xn = u, we obtain

lim
n→∞

d(xn,H(u,u,. . . ,u,λ)) = d(u,H(u,u,. . . ,u,λ)) = 0

i.e. u = H(u,u,. . . ,u,λ) and u ∈ U. As (i) holds, therefore u ∈ U so λ ∈F. Thus
F is closed.

(II) F is open: Let λ0 ∈F, then there exists u0 ∈ U such that u0 = H(u0, . . . ,u0,λ0).
As U is open, there exists δ > 0 such that B(u0,δ) = {x ∈ X : d(x,u0) < δ} ⊂ U.
Fix � > 0 with

(11) � <

1 −k
k∑
i=1

αi

M
δ.

Let λ ∈ (λ0 −�,λ0 + �), then for all x1,x2, . . . ,xk ∈ B(u0,δ) = {x ∈ X : d(x,u0) ≤
δ}, we have

d(u0,H(x1,x2, . . . ,xk,λ)) = d(H(u0,u0, . . . ,u0,λ0),H(x1,x1, . . . ,xk,λ))

≤ d(H(u0,u0, . . . ,u0,λ0),H(u0, . . . ,u0,x1,λ0))
+d(H(u0, . . . ,u0,x1,λ0),H(u0, . . . ,u0,x1,x2,λ0))

+ · · · + d(H(u0,x1, . . . ,xk−1,λ0),H(x1, . . . ,xk,λ0))
+d(H(x1,x2, . . . ,xk,λ0),H(x1,x2, . . . ,xk,λ)).



164 SHUKLA AND RADENOVIĆ

It follows from (9), (10) and the above inequality that

d(u0,H(x1,x2, . . . ,xk,λ)) ≤ [
k∑
i=1

αi]d(u0,x1) + [

k∑
i=2

αi]d(x1,x2) + · · ·

+[

k∑
i=k−1

αi]d(xk−2,xk−1) + αkd(xk−1,xk) + M|λ0 −λ|

< [k

k∑
i=1

αi]δ + M�.

Using (11) in the above inequality we obtain

d(u0,H(x1,x2, . . . ,xk,λ)) < [k

k∑
i=1

αi]δ + [1 −k
k∑
i=1

αi]δ

= δ.

Therefore

d(u0,H(x1,x2, . . . ,xk,λ)) < δ i.e. H(x1,x2, . . . ,xk,λ) ∈ B(u0,δ).

Thus, for each fixed λ ∈ (λ0 − �,λ0 + �), Hλ is a self map of B(u0,δ), so we can
apply Corollary 8 and Remark 2, to deduce that Hλ has a fixed point in U, and
as (i) holds, this fixed point must be in U. Thus λ ∈F for all λ ∈ (λ0 − �,λ0 + �).
Therefore F is open in [0, 1] i.e. F = [0, 1]. Thus Hλ has a fixed point in U for all
λ ∈ [0, 1].
For uniqueness, let λ ∈ [0, 1] be fixed and for this fixed λ, u and v be two fixed
points of Hλ in U i.e. u = H(u,u,. . . ,u,λ) and v = H(v,v, . . . ,v,λ) and u 6= v.
Then it follows from (9) that

d(u,v) = d(H(u,u,. . . ,u,λ),H(v,v, . . . ,v,λ))

≤ d(H(u,. . . ,u,λ),H(u,. . . ,u,v,λ)) + d(H(u,. . . ,u,v,λ),H(u,. . . ,u,v,v,λ))
+ · · · + d(H(u,v, . . . ,v,λ),H(v,v, . . . ,v,λ))

≤ αkd(u,v) + αk−1d(u,v) + · · · + α1d(u,v)

= [

k∑
i=1

αi]d(u,v)

< d(u,v),

a contradiction. Thus fixed point is unique. �

For k = 1 in the above theorem, we obtain following Homotopy result.

Corollary 10. Let (X,d) be any complete metric space, U an open subset of X.
Suppose H : U × [0, 1] → X be a function such that the following conditions hold:

(i) for every x ∈ ∂U(here ∂U is the boundary of U) and λ ∈ [0, 1], x 6= H(x,λ);
(ii) for all x,y ∈ U and λ ∈ [0, 1]

d(H(x,λ),H(y,λ)) ≤ αd(x,y),
where 0 ≤ α < 1;

(iii) for all x ∈ U and λ,µ ∈ [0, 1] there exists M ≥ 0 such that
d(H(x,λ),H(x,µ)) ≤ M|λ−µ|.



PREŠIĆ-BOYD-WONG TYPE RESULTS 165

If Hλ=λ′ has a fixed point in U for at least one λ
′ ∈ [0, 1], then Hλ has a fixed point

in U for all λ ∈ [0, 1]. Furthermore, for any fixed λ ∈ [0, 1], the fixed point of Hλ is
unique.

Conflict of Interests. The authors declare that there is no conflict of interests
regarding the publication of this paper.

References

[1] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some

application to matrix equations, Proc. Am. Math. Soc. 132(2004), 1435-1443.

[2] C.M. Chen, Fixed Point Theorems for ψ-Contractive Mappings in Ordered Metric Spaces, J.
Appl. Math. Volume 2012(2012), Article ID 756453, 10 pages

[3] D. O’Regan, A. Petrusel, Fixed point theorems for generalized contractions in ordered metric

spaces, J. Math. Anal. Appl. 341(2008), 1241-1252.
[4] D.W. Boyd, J.S. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20(1969), 458C464.

[5] H. Aydi, Some fixed point results in ordered partial metric spaces, J. Nonlinear Sci. Appl.
4(2011):210-217.

[6] I. Altun A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces,

Fixed Point Theory Appl., 2011(2011), Article ID 508730.
[7] J.J. Nieto, R.R. Lopez, Contractive mapping theorems in partially ordered sets and applica-

tions to ordinary differential equations, Order 22(2005), 223-239.

[8] J.J. Nieto., R.R. Lopez, Existence and uniqueness of fixed point in partially ordered set-
s and applications to ordinary differential equations, Acta. Math. Sinica, English Ser.

23(12)(2007), 2205-2212.

[9] J.R. Morales, Generalizations of some fixed point theorems, Notas de Mathemática, No. 199
Mérida, 1999

[10] J. Maŕın, S. Romaguera and P. Tirado, Weakly contractive multivalued maps and w-distances

on complete quasi-metric spaces, Fixed Point Theory Appl. 2011 (2011),Article ID 2.
[11] K.P.R. Rao, G. N. V. Kishore, M. Md. Ali, A generalization of the Banach contraction

principle of Prešić type for three maps, Math. Sci. 3(3)(2009), 273-280.

[12] L.B. Ćirić and S. B. Prešić, On Prešić type generalisation of Banach contraction principle,

Acta. Math. Univ. Com. 76, No. 2(2007), 143-147
[13] M. Akkouchi, On a fixed point theorem of D. W. Boyd and J. S. Wong, Acta Math. Vietnam.

27, Number 2 (2002), 231-237.

[14] M.R. Tasković, A generalization of Banach’s contraction principle, Publ. Inst. Math.
Beograd, 23(37)(1978), pp. 179-191.

[15] M.S. Khan, M. Berzig and B. Samet, Some convergence results for iterative sequences of

Prešić type and applications, Adv. Difference Equ. 2012, 2012:38 doi:10.1186/1687-1847-2012-
38

[16] M. Pǎcurar, A multi-step iterative method for approximating common fixed points of Prešić-

Rus type operators on metric spaces, Studia Univ. “Babeş-Bolyai”, Mathematica, Volume
LV, Number 1, March 2010.

[17] M. Pǎcurar, Approximating common fixed points of Prešić-Kannan type operators by a multi-
step iterative method, An. Şt. Univ. Ovidius Constanţa 17(1)(2009), 153-168.

[18] M. Pǎcurar, Common fixed points for almost Prešić type operators, Carpathian J. Math. 28
No. 1 (2012), 117-126.

[19] N.V. Luong, N.X. Thuan, Some fixed point theorems of Prešić-Ćirić type, Acta Univ. Apu-

lensis Math. Inform. 2012(2012), no. 30, 237-249.
[20] P. Das, L.K. Dey, Fixed point of contractive mappings in generalized metric spaces, Math.

Slovaca 59 no. 4 (2009), 499-504.
[21] R. George, K. P. Reshma and R. Rajagopalan, A generalised fixed point theorem of Prešić

type in cone metric spaces and application to Morkov process, Fixed Point Theory Appl.

2011(2011):85, doi:10.1186/1687-1812-2011-85.

[22] R.P. Agarwal, M. A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered
metric spaces, Appl. Anal. 87(2008), 109-116.

[23] R.P. Pant, V. Pant, V.P. Pandey, Generalization of Meir-Keeler type fixed point theorems,
Tamkang J. Math. 35, no. 3(2004), 179-187.



166 SHUKLA AND RADENOVIĆ

[24] S.B. Prešić, Sur une classe dinequations aux differences finite et sur la convergence de cer-
taines suites, Publ. de lInst. Math. Belgrade 5(19)(1965), 75-78.

[25] S.B. Prešić, Sur la convergence des suites, Comptes. Rendus. de l’Acad. de Paris 260(1965),

3828-3830.
[26] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations

intégrales, Fund Math., 3(1922), 133-181.

[27] S. Radenović, Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces,
Comput. Math. Appl. 60(2010), 1776- 1783. doi:10.1016/j.camwa.2010.07.008

[28] S. Shukla, Prešić type results in 2-Banach spaces, Afrika Matematika, (2013) DOI
10.1007/s13370-013-0174-2

[29] S. Shukla, Set-valued Prešić-Ćirić type contraction in 0-complete partial metric spaces,
Matematićki Vesnik, In press, (2013).

[30] S. Shukla, B. Fisher, A generalization of Prešić type mappings in metric-like spaces, Journal

of Operators, 2013 (2013), Article ID 368501, 5 pages.
[31] S. Shukla, R. Sen, Set-valued Prešić-Reich type mappings in metric spaces, Revista de la

Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas (2012). DOI

10.1007/s13398-012-0114-2
[32] S. Shukla, R. Sen, S. Radenović, Set-valued Prešić type contraction in metric spaces, An.

Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) (Accepted)(2012).

[33] S. Shukla, S. Radenović, A generalization of Prešić type mappings in 0-Complete ordered
partial metric spaces, Chinese Journal of Mathematics 2013 Article ID 859531, 8 pages.

[34] S. Shukla, S. Radojević, Z.A. Veljković, S. Radenović, Some coincidence and common fixed
point theorems for ordered Prešić-Reich type contractions, Journal of Inequalities and Appli-

cations, 2013 2013:520. doi: 10.1186/1029-242X-2013-520

[35] S. Shukla, M. Abbas, Fixed point results of cyclic contractions in product spaces, Carpathian
J. Math., In press (2014).

[36] S.K. Malhotra, S. Shukla, R. Sen, A generalization of Banach contraction principle in ordered

cone metric spaces, J. Adv. Math. Stud., 5(2) (2012), 59–67.
[37] S. K. Malhotra, S. Shukla, and R. Sen, Some fixed point theorems for ordered Reich type

contractions in cone rectangular metric spaces. Acta Mathematica Universitatis Comenianae,

LXXXII 2 (2013), 165C175.
[38] W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277(2003), 645-650.

[39] W. Shatanawi, B. Samet, M. Abbas, Coupled fixed point theorems for mixed monotone map-

pings in ordered partial metric spaces, Math. Comput. Model. 55(2012), 680-687.
[40] Y.Z. Chen, A Prešić type contractive condition and its applications, Nonlinear Anal. 71,

2012-2017 (2009). doi:10.1016/j.na.2009.03.006
[41] Z. Kadelburg, M. Pavlović, S. Radenović, Common fixed point theorems for ordered contrac-

tions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl. 59(2010),

3148-3159.

1Department of Applied Mathematics, Shri Vaishnav Institute of Technology & Sci-
ence, Gram Baroli, Sanwer Road, Indore (M.P.) 453331, India

2Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16,
11120 Beograd, Serbia

∗Corresponding author