@
Appl. Gen. Topol. 23, no. 1 (2022), 121-133

doi:10.4995/agt.2022.11368

© AGT, UPV, 2022

Fixed point results with respect to a

wt-distance in partially ordered b-metric spaces

and its application to nonlinear fourth-order

differential equation

Reza Babaei, Hamidreza Rahimi* and Ghasem Soleimani Rad

Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad Uni-

versity, Tehran, Iran

(rez.babaei.sci@iauctb.ac.ir, rahimi@iauctb.ac.ir, gha.soleimani.sci@iauctb.ac.ir)

Communicated by M. Abbas

Abstract

In this paper we study the existence of the fixed points for Hardy-Rogers
type mappings with respect to a wt-distance in partially ordered metric
spaces. Our results provide a more general statement, since we replace
a w-distance with a wt-distance and ordered metric spaces with ordered
b-metric spaces. Some examples are presented to validate our obtained
results and an application to nonlinear fourth-order differential equa-
tion are given to support the main results.

2020 MSC: 54H25; 47H10; 05C20.

Keywords: partially ordered set; b-metric space; wt-distance; fixed point.

1. Introduction and preliminaries

Fixed point theory is an important and useful tool for different branches
of mathematical analysis and it has many applications in mathematics and
sciences. In 1922, Banach proved the famous contraction mapping principle [3].
Afterward, other authors considered various definitions of contractive mappings

*Corresponding author

Received 7 February 2019 – Accepted 24 November 2021

http://dx.doi.org/10.4995/agt.2022.11368
https://orcid.org/0000-0003-4883-4250
https://orcid.org/0000-0002-0758-2758


R. Babaei, H. Rahimi and G. Soleimani Rad

and proved several fixed and common fixed point theorems (see Rhoades survey
[22] and references therein). On the other hand, the symmetric spaces as metric-
like spaces lacking the triangle inequality was introduced in 1931 by Wilson [24].
In the sequel, a new type of spaces which they called b-metric spaces (or metric
type spaces) are defined by Bakhtin [2] and Czerwik [6]. After that, several
papers have dealt with fixed point theory for single-valued and multi-valued
operators in b-metric spaces (for example, see [4, 5, 15, 17]).

Definition 1.1 ([2, 6]). Let X be a nonempty set and s ≥ 1 be a real number.
Suppose that the mapping d : X ×X → [0,∞) satisfies

(d1) d(x,y) = 0 if and only if x = y;
(d2) d(x,y) = d(y,x) for all x,y ∈ X;
(d3) d(x,z) ≤ s[d(x,y) + d(y,z)] for all x,y,z ∈ X.

Then d is called a b-metric and (X,d) is called a b-metric space (or metric type
space).

Obviously, for s = 1, a b-metric space is a metric space. Also, for notions
such as convergent and Cauchy sequences, completeness, continuity, etc in b-
metric spaces, we refer to [1, 5, 17].

In 1996, Kada et al. [16] introduced the concept of w-distance in metric
spaces, where nonconvex minimization problems were treated. Then some fixed
point results and common fixed point theorem with respect to w-distance in
metric spaces were proved by Ilić and Rakočević [12] and Shioji et al. [23].
In 2014, Hussain et al. [11] introduced the concept of wt-distance on a b-
metric space and proved some fixed point theorems under wt-distance in a
partially ordered b-metric space. Then Demma et al. [7] considered multi-
valued operators with respect to a wt-distance on b-metric spaces and proved
some results on fixed point theory.

Definition 1.2 ([11]). Let (X,d) be a b-metric space and s ≥ 1 be a given real
number. A function ρ : X ×X → [0, +∞) is called a wt-distance on X if the
following properties are satisfied:

(ρ1) ρ(x,z) ≤ s[ρ(x,y) + ρ(y,z)] for all x,y,z ∈ X;
(ρ2) ρ is b-lower semi-continuous in its second variable i.e., if x ∈ X and

yn → y in X, then ρ(x,y) ≤ s lim infn ρ(x,yn);
(ρ3) for each ε > 0 there exists δ > 0 such that ρ(z,x) ≤ δ and ρ(z,y) ≤ δ

imply d(x,y) ≤ ε.

Let us recall that a real-valued function f defined on a metric space X is
said to be b-lower semi-continuous at a point x ∈ X if either lim inf

xn→x
f(xn) = ∞

or f(x) ≤ lim inf
xn→x

sf(xn), whenever xn ∈ X and xn → x for each n ∈ N [12].
Note that each b-metric d is a wt-distance, but the converse is not hold. Thus,
wt-distance is a generalization of b-metric d. Obviously, for s = 1, every wt-
distance is a w-distance. But, a w-distance is not necessary a wt-distance.
Thus, each wt-distance is a generalization of w-distance.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 122



Fixed point results with respect to a wt-distance

Example 1.3 ([11]). Let X = R and define a mapping d : X × X → R by
d(x,y) = (x−y)2 for all x,y ∈ X. Then (X,d) is a b-metric with s = 2. Define
a mapping ρ : X × X → [0,∞) by ρ(x,y) = y2 or ρ(x,y) = x2 + y2 for all
x,y ∈ X. Then ρ is a wt-distance.

From Example 1.3, we have two important results:

(1) for any wt-distance ρ, ρ(x,y) = 0 is not necessarily equivalent to x = y
for all x,y ∈ X.

(2) for any wt-distance ρ, ρ(x,y) = ρ(y,x) does not necessarily hold for all
x,y ∈ X.

Lemma 1.4 ([11]). Let (X,d) be a b-metric space with parameter s ≥ 1 and ρ
be a wt-distance on X. Also, let {xn} and {yn} be sequences in X, and {αn}
and {βn} be a sequences in [0, +∞) converging to zero and x,y,z ∈ X. Then
the following conditions hold:

(i) if ρ(xn,y) ≤ αn and ρ(xn,z) ≤ βn for all n ∈ N, then y = z. In
particular, if ρ(x,y) = 0 and ρ(x,z) = 0, then y = z;

(ii) if ρ(xn,yn) ≤ αn and ρ(xn,z) ≤ βn for n ∈ N, then {yn} converges to
z;

(iii) if ρ(xn,xm) ≤ αn for all m,n ∈ N with m > n, then {xn} is a Cauchy
sequence in X;

(iv) if ρ(y,xn) ≤ αn for all n ∈ N, then {xn} is a Cauchy sequence in X.

Existence of fixed points in ordered metric spaces has been applied by Ran
and Reurings [21]. The key feature in this fixed point theorem is that the
contractivity condition on the nonlinear map is only assumed to hold on ele-
ments that are comparable in the partial order. However, the map is assumed
to be monotone. They showed that under such conditions the conclusions of
Banach’s fixed point theorem still hold. This fixed point theorem was extended
by Nieto and Rodŕıguez-López [18] and was applied to the periodic boundary
value problem (see, e.g., [8, 9, 13, 14, 19, 20]). A relation v on X is called

(i) reflexive if x v x for all x ∈ X;
(ii) transitive if x v y and y v z imply x v z for all x,y,z ∈ X;

(iii) antisymmetric if x v y and y v x imply x = y for all x,y ∈ X;
(iv) pre-order if it is reflexive and transitive.

A pre-order v is called partial order or an order relation if it is antisymmetric.
Given a partially ordered set (X,v); that is, the set X equipped with a partial
order v, the notation x < y stands for x v y and x 6= y. Also, let (X,v) be
a partially ordered set. A mapping f : X → X is said to be nondecreasing if
x v y implies that fx v fy for all x,y ∈ X.

2. Main results

Our main result is the following theorem for mappings satisfying Hardy-
Rogers type conditions [10] with respect to a given wt-distance in a complete
partially ordered b-metric space.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 123



R. Babaei, H. Rahimi and G. Soleimani Rad

Theorem 2.1. Let (X,v) be a partially ordered set, (X,d) be a complete b-
metric space with given real number s ≥ 1 and ρ be a wt-distance on X. Suppose
that there exist mappings αi : X → [0, 1) such that

αi(fx) ≤ αi(x)(2.1)

for all x ∈ X and i = 1, 2, · · · , 5, where f : X → X be a continuous and
nondecreasing mapping with respect to v satisfying the following conditions:

ρ(fx,fy) ≤ α1(x)ρ(x,y) + α2(x)ρ(x,fx) + α3(x)ρ(y,fy)
+ α4(x)ρ(x,fy) + α5(x)ρ(y,fx),(2.2)

ρ(fy,fx) ≤ α1(x)ρ(y,x) + α2(x)ρ(fx,x) + α3(x)ρ(fy,y)
+ α4(x)ρ(fy,x) + α5(x)ρ(fx,y)(2.3)

for all x,y ∈ X with y v x such that

(s(α1 + α3 + 2α4) + α2 + (s
2 + s)α5)(x) < 1.(2.4)

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Proof. If fx0 = x0, then x0 is a fixed point of f and the proof is finished.
Now, suppose that fx0 6= x0. Since f is nondecreasing with respect to v and
x0 v fx0, we obtain by induction that

x0 v fx0 v f2x0 v ···v fnx0 v fn+1x0 v ··· ,

where xn = fxn−1 = f
nx0. First we shall prove that {xn} is a Cauchy se-

quence. Now, setting x = xn and y = xn−1 in (2.2) and applying (2.1) and
(ρ1), we have

ρ(xn+1,xn) = ρ(fxn,fxn−1)

≤ α1(xn)ρ(xn,xn−1) + α2(xn)ρ(xn,fxn) + α3(xn)ρ(xn−1,fxn−1)
+ α4(xn)ρ(xn,fxn−1) + α5(xn)ρ(xn−1,fxn)

= α1(fxn−1)ρ(xn,xn−1) + α2(fxn−1)ρ(xn,xn+1)

+ α3(fxn−1)ρ(xn−1,xn) + α4(fxn−1)ρ(xn,xn)

+ α5(fxn−1)ρ(xn−1,xn+1)

≤ α1(xn−1)ρ(xn,xn−1) + (α3 + sα5)(xn−1)ρ(xn−1,xn)
+ sα4(xn−1)ρ(xn+1,xn) + (α2 + sα4 + sα5)(xn−1)ρ(xn,xn+1)

...

≤ α1(x0)ρ(xn,xn−1) + (α3 + sα5)(x0)ρ(xn−1,xn)
+ sα4(x0)ρ(xn+1,xn) + (α2 + sα4 + sα5)(x0)ρ(xn,xn+1).(2.5)

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Fixed point results with respect to a wt-distance

Similarly, setting x = xn and y = xn−1 in (2.3) and applying (2.1) and (ρ1),
we have

ρ(xn,xn+1) ≤ α1(x0)ρ(xn−1,xn) + (α3 + sα5)(x0)ρ(xn,xn−1)
+ sα4(x0)ρ(xn,xn+1) + (α2 + sα4 + sα5)(x0)ρ(xn+1,xn).(2.6)

Now, adding up (2.5) and (2.6), we obtain

ρ(xn+1,xn) + ρ(xn,xn+1) ≤ (α1 + α3 + sα5)(x0)[ρ(xn,xn−1) + ρ(xn−1,xn)]
+ (α2 + 2sα4 + sα5)(x0)[ρ(xn+1,xn) + ρ(xn,xn+1)].

Let an = ρ(xn+1,xn) + ρ(xn,xn+1). Then we get

an ≤ (α1 + α3 + sα5)(x0)an−1 + (α2 + 2sα4 + sα5)(x0)an.

Therefore, an ≤ kan−1 for all n ∈ N, where

0 ≤ k =
(α1 + α3 + sα5)(x0)

1 − (α2 + 2sα4 + sα5)(x0)
<

1

s

by (2.4) and since (α1 + α3 + sα5)(x0) ≥ 0. By repeating the procedure, we
obtain an ≤ kna0 for all n ∈ N. It follows that

ρ(xn,xn+1) ≤ an ≤ kn[ρ(x1,x0) + ρ(x0,x1)].(2.7)

Let m > n. It follows from (2.7) and 0 ≤ sk < 1 that

ρ(xn,xm) ≤ s[ρ(xn,xn+1) + ρ(xn+1,xm)]
≤ sρ(xn,xn+1) + s[sρ(xn+1,xn+2) + ρ(xn+2,xm)]

...

≤ sρ(xn,xn+1) + s2ρ(xn+1,xn+2) + · · · + sm−nρ(xm−1,xm)]
≤ (skn + s2kn+1 · · · + sm−nkm−1)[ρ(x1,x0) + ρ(x0,x1)]

≤
skn

1 −sk
[ρ(x1,x0) + ρ(x0,x1)].

Now, Lemma 1.4 (iii) implies that {xn} is a Cauchy sequence in X. Since X
is complete, there exists a point x′ ∈ X such that xn → x′ as n → ∞. The
continuity of f implies that xn+1 = fxn → fx′ as n →∞, and since the limit
of a sequence is unique, we get that fx′ = x′. Thus, x′ is a fixed point of f.
Further, suppose that fz = z. Then, by using (2.2), we have

ρ(z,z) = ρ(fz,fz)

≤ α1(z)ρ(z,z) + α2(z)ρ(z,fz) + α3(z)ρ(z,fz)
+ α4(z)ρ(z,fz) + α5(z)ρ(z,fz)

≤ (α1 + α2 + α3 + α4 + α5)(z)ρ(z,z).

Since
5∑

i=1

αi(z) < s(α1 + α3 + 2α4)(z) + α2 + (s
2 + s)α5)(z) < 1, we obtain that

ρ(z,z) = 0 by using Lemma 1.4 (i). This completes the proof. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 125



R. Babaei, H. Rahimi and G. Soleimani Rad

Example 2.2. Let X = [0, 1] and define a mapping d : X × X → R by
d(x,y) = (x − y)2 for all x,y ∈ X. Then (X,d) is a complete b-metric space
with s = 2. Define a function ρ : X × X → [0,∞) by ρ(x,y) = d(x,y) for
all x,y ∈ X. Then ρ is a wt-distance. Let an order relation v be defined
by x v y if and only if x ≤ y Also, let a mapping f : X → X be defined by
fx = x

2

5
for all x ∈ X. Then f is a continuous and nondecreasing mapping with

respect to v and there exists a 0 ∈ X such that 0 v f0. Define the mappings
α1(x) =

(x+1)2

25
and αi(x) = 0 for all x ∈ X and i = 2, 3, 4, 5. Observe that

s(α1 + α3 + 2α4)(x) + α2 + (s
2 + s)α5)(x) = 2

(x + 1)2

25
< 1.

Also,

α1(fx) =
1

25

(x2
5

+ 1
)2 ≤ 1

25

(
x2 + 1

)2 ≤ (x + 1)2
25

= α1(x)

for all x ∈ X and αi(fx) = 0 = αi(x) for all x ∈ X and i = 2, 3, 4, 5. Moreover,
for all x,y ∈ X with y v x, we get

ρ(fx,fy) =

(
x2

5
−
y2

5

)2
=

(x + y)2(x−y)2

25

≤
(x + 1)2

25
(x−y)2

≤ α1(x)ρ(x,y) + α2(x)ρ(x,fx) + α3(x)ρ(y,fy)
+ α4(x)ρ(x,fy) + α5(x)ρ(y,fx).

Similarly, for all x,y ∈ X with y v x, we get
ρ(fy,fx) ≤ α1(x)ρ(y,x) + α2(x)ρ(fx,x) + α3(x)ρ(fy,y)

+ α4(x)ρ(fy,x) + α5(x)ρ(fx,y).

Therefore, all the conditions of Theorem 2.1 are satisfied. Hence, f has a fixed
point x = 0 with ρ(0, 0) = 0.

Several consequences of Theorem 2.1 follow now for particular choices of the
contractions.

Corollary 2.3. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exist mappings α,β,γ : X → [0, 1) such that

α(fx) ≤ α(x), β(fx) ≤ β(x), γ(fx) ≤ γ(x)
for all x ∈ X, where f : X → X be a continuous and nondecreasing mapping
with respect to v satisfying the following conditions:

ρ(fx,fy) ≤ α(x)ρ(x,y) + β(x)ρ(x,fy) + γ(x)ρ(y,fx),
ρ(fy,fx) ≤ α(x)ρ(y,x) + β(x)ρ(fy,x) + γ(x)ρ(fx,y)

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 126



Fixed point results with respect to a wt-distance

for all x,y ∈ X with y v x such that

(s(α + 2β) + (s2 + s)γ)(x) < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Proof. We obtain this result by applying Theorem 2.1 with α1(x) = α(x),
α2(x) = α3(x) = 0, α4(x) = β(x) and α5(x) = γ(x). �

In the process of proving Theorem 2.1, consider x = xn−1 and y = xn with
x v y (instead of x = xn and y = xn−1 with y v x). Then, we only need one
condition for some following types of the contractions.

Corollary 2.4. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exist mappings αi : X → [0, 1) such that αi(fx) ≤ αi(x)
for all x ∈ X and i = 1, 2, 3, 4, where f : X → X be a continuous and
nondecreasing mapping with respect to v satisfying the following condition:

ρ(fx,fy) ≤ α1(x)ρ(x,y) + α2(x)ρ(x,fx) + α3(x)ρ(y,fy) + α4(x)ρ(x,fy)

for all x,y ∈ X with x v y such that

(s(α1 + α2) + α3 + (s
2 + s)α4)(x) < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Corollary 2.5. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exist mappings α,β,γ : X → [0, 1) such that

α(fx) ≤ α(x), β(fx) ≤ β(x), γ(fx) ≤ γ(x)

for all x ∈ X, where f : X → X be a continuous and nondecreasing mapping
with respect to v satisfying the following condition:

ρ(fx,fy) ≤ α(x)ρ(x,y) + β(x)ρ(x,fx) + γ(x)ρ(y,fy)

for all x,y ∈ X with x v y such that

(s(α + β) + γ)(x) < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Theorem 2.6. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exists a continuous and nondecreasing mapping f : X → X
with respect to v such that the following conditions hold:

ρ(fx,fy) ≤ α1ρ(x,y) + α2ρ(x,fx) + α3ρ(y,fy) + α4ρ(x,fy) + α5ρ(y,fx),
ρ(fy,fx) ≤ α1ρ(y,x) + α2ρ(fx,x) + α3ρ(fy,y) + α4ρ(fy,x) + α5ρ(fx,y)

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 127



R. Babaei, H. Rahimi and G. Soleimani Rad

for all x,y ∈ X with y v x, where αi are nonnegative coefficients for i =
1, 2, · · · , 5 with

s(α1 + α3 + 2α4) + α2 + (s
2 + s)α5 < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Proof. We can prove this result by applying Theorem 2.1 with αi(x) = αi for
i = 1, 2, · · · , 5. �

Several consequences of Theorem 2.6 follow now for particular choices of the
contractions.

Corollary 2.7. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exists a continuous and nondecreasing mapping f : X → X
with respect to v such that the following conditions hold:

ρ(fx,fy) ≤ αρ(x,y) + βρ(x,fy) + γρ(y,fx),
ρ(fy,fx) ≤ αρ(y,x) + βρ(fy,x) + γρ(fx,y)

for all x,y ∈ X with y v x, where α,β,γ are nonnegative coefficients with

s(α + 2β) + (s2 + s)γ < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Proof. We obtain this result by applying Theorem 2.6 with α1 = α, α2 = α3 =
0, α4 = β and α5 = γ. �

In the process of proving Theorem 2.6, consider x = xn−1 and y = xn with
x v y (instead of x = xn and y = xn−1 with y v x). Then, we only need one
condition for some types of the contractions.

Corollary 2.8. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exists a continuous and nondecreasing mapping f : X → X
with respect to v such that the following condition hold:

ρ(fx,fy) ≤ α1ρ(x,y) + α2ρ(x,fx) + α3ρ(y,fy) + α4ρ(x,fy)

for all x,y ∈ X with x v y, where αi for i = 1, 2, 3, 4 are nonnegative coeffi-
cients with

s(α1 + α2) + α3 + (s
2 + s)α4 < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Corollary 2.9. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 128



Fixed point results with respect to a wt-distance

Suppose that there exists a continuous and nondecreasing mapping f : X → X
with respect to v such that the following condition hold:

ρ(fx,fy) ≤ αρ(x,y) + βρ(x,fx) + γρ(y,fy)

for all x,y ∈ X with x v y, where α,β,γ are nonnegative coefficients with

s(α + β) + γ < 1.

If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover,
if fz = z, then ρ(z,z) = 0.

Example 2.10. Consider X, d, s and order relation v as in Example 2.2.
Define a function ρ : X ×X → [0,∞) by ρ(x,y) = y2 for all x,y ∈ X. Then
ρ is a wt-distance. Also, let a mapping f : X → X be defined by fx = x

2

3
for

all x ∈ X. Then f is a continuous and nondecreasing mapping with respect to
v and there exists a 0 ∈ X such that 0 v f0. Take α = 1

9
, β = 1

8
and γ = 1

4
.

Then we obtain

ρ(fx,fy) = (fy)2 = (
y2

3
)2 =

y4

9

≤
1

9
y2

=
1

9
ρ(x,y) ≤ αρ(x,y) + βρ(x,fx) + γρ(y,fy).

Also, we have

s(α + β) + γ = 2(
1

9
+

1

8
) +

1

4
=

26

36
< 1.

Hence, all the conditions of Corollary 2.9 are satisfied. Therefore, f has a fixed
point x = 0. Moreover, ρ(0, 0) = 0.

Corollary 2.11. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exists a continuous and nondecreasing mapping f : X → X
with respect to v such that

(2.8) ρ(fx,fy) ≤ λρ(x,y)

for all x,y ∈ X with x v y, where λ ∈ [0,
1

s
). If there exists x0 ∈ X such that

x0 v fx0, then f has a fixed point. Moreover, if fz = z, then ρ(z,z) = 0.

Example 2.12. Let X = [0, 2] and consider d, s and order relation v as
in Example 2.2. Also, let a mapping f : X → X be defined by fx = x

2

2

for all x ∈ X. Since d(f0,f2) = d(0, 2), there is not 0 ≤ α < 1
s

such that
d(fx,fy) ≤ αd(x,y) for all x,y ∈ X. Hence, Banach-type result on b-metric
space cannot be applied for this example. Now, let X = [0, 1] and define a
function ρ : X ×X → [0,∞) by ρ(x,y) = y2 + x2 for all x,y ∈ X. Then ρ is a

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R. Babaei, H. Rahimi and G. Soleimani Rad

wt-distance.

ρ(fx,fy) = (fy)2 + (fx)2 = (
y2

2
)2 + (

x2

2
)2

=
y4 + x4

4
≤
y2 + x2

4
=

1

4
ρ(x,y).

Thus, (2.8) is hold with λ = 1
4
∈ [0, 1

2
). Hence, all conditions of Banach-type

fixed point results (or same Corollary 2.11) with respect to the wt-distance on
b-metric spaces are satisfied. Note that f has a (trivial) fixed point 0 ∈ [0, 1] ⊆
[0, 2] and ρ(0, 0) = 0.

Corollary 2.13. Let (X,v) be a partially ordered set, (X,d) be a complete
b-metric space with given real number s ≥ 1 and ρ be a wt-distance on X.
Suppose that there exists a continuous and nondecreasing mapping f : X → X
with respect to v such that

ρ(fx,fy) ≤ δ(ρ(x,fx) + ρ(y,fy))

for all x,y ∈ X with x v y, where δ ∈ [0, 1
s+1

). If there exists x0 ∈ X such
that x0 v fx0, then f has a fixed point. Moreover, if fz = z, then ρ(z,z) = 0.

3. An application

It is well-known that fourth-order differential equations are important and
useful tools for modeling the elastic beam deformation. Precisely, we refer
to beams in equilibrium state, whose two ends are simply supported. Conse-
quently, this study has many applications in engineering and physical science.
Now, we establish the existence of solutions of fourth-order boundary value
problems as a consequence of Theorem 2.1. In particular, the focus is on the
equivalent integral formulation of the boundary value problem below and the
use of Green’s functions. At the first, we introduce the mathematical back-
ground as follows (also, see [14]).

Let X = C([0, 1],R) be the set of all non-negative real-valued continuous
functions on the interval [0, 1]. Also, let X be endowed with the supremum
norm ‖x‖∞ = supt∈[0,1] |x(t)| and define a mapping d : X×X → R by d(x,y) =
supt∈[0,1](x(t) −y(t))2 for all x,y ∈ X. Also, consider the partial order

(x,y) ∈ X ×X, x v y ⇐⇒ x(t) ≤ y(t) for all t ∈ [0, 1].

Clearly, (X,v) is a partially ordered set and (X,d) is a complete b-metric
space with s = 2. Finally, consider the wt-distance ρ : X × X → R given by
ρ(x,y) = d(x,y) for all x,y ∈ X. Thus, we study the following fourth-order
two-point boundary value problem


xiv(t) = k(t,x(t)), 0 < t < 1,

x(0) = x′(0) = x′′(1) = x′′′(1) = 0,
(3.1)

with k ∈ C([0, 1] ×R,R).

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Fixed point results with respect to a wt-distance

It is well-known that the problem (3.1) may be equivalently expressed in
integral form: find x∗ ∈ X solution of

(3.2) x(t) =

∫ 1
0

G(t,τ)k(τ,x(τ)) dτ, t ∈ [0, 1],

where the Green function G(t,τ) is given by

G(t,τ) =
1

6




τ2(3t− τ), 0 ≤ τ ≤ t ≤ 1,

t2(3τ − t), 0 ≤ t ≤ τ ≤ 1.

Also, it is immediate to show that

(3.3) 0 ≤ G(t,τ) ≤
1

2
t2τ for all t,τ ∈ [0, 1].

Next, we consider the following hypotheses:

(I) There exists α1 : X → [0, 12 ) such that

(3.4) 0 ≤ k(t,y(t)) −k(t,x(t)) ≤ 4
√
α1(x)ρ(x,y)

for all x,y ∈ X with y v x and for all t ∈ [0, 1] and

(3.5) α1

(∫ 1
0

G(t,τ)k(τ,x(τ)) dτ

)
≤ α1(x) for all x ∈ X.

(II) There exists x0 ∈ X such that

x0(t) ≤
∫ 1
0

G(t,τ)k(τ,x0(τ)) dτ, t ∈ [0, 1];

that is, the integral equation (3.2) admits a lower solution in X.

Now, we prove the existence of at least a solution of (3.1) in X.

Theorem 3.1. The existence of at least a solution of problem (3.1) in X is es-
tablished, provided that the function k ∈ C([0, 1]×R,R) satisfies the hypotheses
(I) and (II).

Proof. The problem in study is equivalent to the fixed point problem obtained
by introducing the continuous integral operator f : X → X given as

(fx)(t) =

∫ 1
0

G(t,τ)k(τ,x(τ)) dτ, t ∈ [0, 1] and x ∈ X.

Now, we show that the operator f satisfies all the conditions in Theorem 2.1 to
conclude that there exists a fixed point of f in X. By using the inequality (3.4)
in hypothesis (I), we deduce that f is a nondecreasing mapping with respect
to v. Also, by using (3.4), for all t ∈ [0, 1] and for all x,y ∈ X with y v x, we

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 131



R. Babaei, H. Rahimi and G. Soleimani Rad

get

|(fy)(t) − (fx)(t)| =
∫ 1
0

G(t,τ)[k(τ,y(τ)) −k(τ,x(τ))] dτ

≤
∫ 1
0

G(t,τ)4
√
α1(x)ρ(x,y) dτ

≤
(∫ 1

0

G(t,τ) dτ

)
4
√
α1(x)ρ(x,y)

≤
√
α1(x)ρ(x,y) (from (3.3)).

Since ρ(x,y) = d(x,y) for all x,y ∈ X, by passing to square and taking the
supremum with respect to t, we get

ρ(fx,fy) = d(fx,fy)

= sup
t∈[0,1]

((fy)(t) − (fx)(t))2

≤ α1(x)ρ(x,y)

for all x,y ∈ X with y v x. It follows that the conditions (2.2) and (2.3) of
Theorem 2.1 hold αi(x) = 0 for all x ∈ X and i = 2, 3, 4, 5. By hypothesis (II),
we get that there exists x0 ∈ X such that x0 v fx0. Also, from (3.5) and the
fact that the function α1 assumes values in the interval [0,

1
2
), we have

α1(fx) ≤ α1(x) <
1

2
for all x ∈ X;

that is, the condition (2.1) of Theorem 2.1 hold with αi(x) = 0 for all x ∈ X
and i = 2, 3, 4, 5. We conclude that all the conditions of Theorem 2.1 hold and
so we deduce the existence of a fixed point of f; that is, there exists a solution
of problem (3.1) in X. �

Acknowledgements. The authors thank the Editorial Board and the referees
for their valuable comments.

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