@
Appl. Gen. Topol. 20, no. 2 (2019), 449-469

doi:10.4995/agt.2019.11683

c© AGT, UPV, 2019

Existence results of delay and fractional

differential equations via fuzzy weakly

contraction mapping principle

Rehana Tabassum a, Akbar Azam a and Mohammed Shehu Shagari b

a Department of Mathematics, COMSATS University, Chak Shahzad, Islamabad, 44000, Pak-

istan (reha alvi@yahoo.com,akbarazam@yahoo.com)
b Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University,Zaria,

Nigeria (ssmohammed@abu.edu.ng)

Communicated by S. Romaguera

Abstract

The purpose of this article is to extend the results derived through for-
mer articles with respect to the notion of weak contraction into intu-
itionistic fuzzy weak contraction in the context of (T ,N ,α)−cut set of
an intuitionistic fuzzy set. We intend to prove common fixed point the-
orem for a pair of intuitionistic fuzzy mappings satisfying weakly con-
tractive condition in a complete metric space which generalizes many
results existing in the literature. Moreover, concrete results on exis-
tence of the solution of a delay differential equation and a system of
Riemann-Liouville Cauchy type problems have been derived. In addi-
tion, we also present illustrative examples to substantiate the usability
of our main result.

2010 MSC: 46S40; 47H10; 54H25.

Keywords: common fixed point; intuitionistic fuzzy set-valued maps;
(T ,N ,α) − cut set; weakly contractive condition; delay differ-
ential equation; Riemann-Liouville fractional differential equa-
tions.

Received 15 April 2019 – Accepted 29 July 2019

http://dx.doi.org/10.4995/agt.2019.11683


R. Tabassum, A. Azam and M. S. Shagari

1. Introduction

Metric fixed point theory is generally based on the Banach Contraction
Principle, which has been used to study the existence and uniqueness of fixed
points. This principle has been extensively studied in different directions. In
1997, Alber and Guerr-Delabriere [3] proposed the notion of weak contractive
mappings on Hilbert spaces and studied the existence of fixed point results
in the context of weakly contractive single valued maps on Hilbert spaces as
a generalization of Banach Contraction Principle. However, in 2001, Rhoades
[30] presented some results of [3] to arbitrary Banach spaces. Later on, Bae [10]
established the fixed points of weakly contractive multivalued mappings and
Beg and Abbas [19] demonstrated the fixed point results for a pair of single
valued mappings one is weakly contractive relative to the other.

On the other hand, there are many complicated practical problems in the
domain of real world such as engineering, economics, social sciences, medical
science and many other fields that involve data which are not always precise.
To overcome these difficulties, classical mathematical notions may not be ap-
plied effectively, because there are numerous types of vagueness appear in these
domains. However, in response to this fact, Zadeh [39] developed the concept
of fuzzy set as an extension of conventional set theory. Over the years, sev-
eral mathematicians extended this notion in different directions, for instance,
L-fuzzy set, intuitionstic fuzzy set, fuzzy soft set and hesitant fuzzy soft set.
Consequently, in 1981, Heilpern [20] initiated the idea of fuzzy mapping and
proved a fixed point theorem for fuzzy contraction mappings as an extension
of multivalued mappings of Nadler’s contraction principle. Thus, this result
motivated several researchers to study and establish the fixed point results
satisfying a fuzzy contractive inequalities ( see, [1, 6, 7]).

One of the generalizations of fuzzy set theory [39] is the notion of intuition-
istic fuzzy set (IF-set) introduced by Atanassov [5]. Moreover, IF-sets cre-
ate a valuable mathematical structure to deal with inaccuracy and hesitancy
originating from insufficient decision information and as a consequence ,it has
remarkable applications in various fields like image proccessing [21], medical
diagnosis [18], drug selection [22], decision analysis [26], etc.

Until now, research on IF-set has been very active and many results have
been proved with different aspects. Recently, Azam et al. [8] developed new
approach to discuss the fixed point theorems using the idea of intuitionistic
fuzzy mappings [38] on a complete metric space. Later on, Azam and Rehana
[9] presented existence of common coincidence point for three intuitionistic
fuzzy set valued maps and they also studied existence results for a system of
integral equations.

In this manuscript, the idea of weakly contraction is used for intuitionistic
fuzzy mappings in association with (T ,N ,α)−cut set of an IF-set [27]. On the
basis of this concept, an existence result of common fixed point on complete
metric space is presented. From an application perspective, we apply our main

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 450



Existence results of delay and fractional differential equations

result to establish existence theorems of the solution of a delay differential
equation and a system of Riemann-Liouville Cauchy type problems.

2. Preliminaries

Throughout this paper, (U,ρ), (W,ρW ) and V (W) denote a metric space,
a metric linear space and a subcollection of all approximate quantities in W ,
respectively. Let

CB (U) = {A∗ : A∗ is nonempty, closed and bounded subset of U},
C(U) = { A∗ : A∗ is nonempty compact subset of U}.

For A∗, B∗ ∈ CB(U), define
ρ(u,A∗) = inf ρ

v∈A∗
(u,v),

ρ(A∗,B∗) = inf
u∈A∗, v∈B∗

ρ(u,v).

The Hausdorff metric ρH on CB(U) induced by ρ is defined as

ρH (A
∗,B∗) = max

{
sup
u∈A∗

ρ (u,B∗) , sup
v∈B∗

ρ (A∗,v)

}
.

Definition 2.1 ([3, 17]). Let (U,ρ) be a metric space and a mapping f : U → U
is called a weakly contractive mapping if for u, v ∈ U,

ρ (f (u) ,f (v)) ≤ ρ (u,v) −φ (ρ (u,v)) ,
where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) = 0
if and only if t = 0.

Definition 2.2 ([10]). Let (U,ρ) be a metric space. A mapping f : U →
C (U) is said to be a weakly contractive multivalued mapping, if there exists
a continuous non-decreasing function φ : [0,∞) → [0,∞) with φ (0) = 0 and
φ (t) > 0 for all t > 0, such that

ρH (f (u) ,f (v)) ≤ ρ (u,v) −φ (ρ (u,v)) ,
for all u,v ∈ U.
Definition 2.3 ([20, 39]). Let Z be a universal set. A fuzzy set in U is an
object of the form

A∗ = {(z,A∗(z)) : z ∈ Z},
where A∗(z) denotes the membership values of z in A∗.

Definition 2.4 ([20, 39]). Let A∗ be a fuzzy set of universe Z. The α − cut
set of A∗ denoted by [A∗]α is a crisp subset of Z whose membership value in
A∗ is greater than or equal to some specific value of α, i.e.

[A∗]α = {z ∈ Z : A
∗(z) ≥ α} if α ∈ (0, 1] .

Definition 2.5 ([20]). A fuzzy set A∗ in a metric linear space W is said to be
an approximate quantity if and only if only if [A∗]α is compact and convex in
W for each α ∈ (0, 1] with sup A∗ (w)

w∈W
= 1.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 451



R. Tabassum, A. Azam and M. S. Shagari

Definition 2.6 ([20]). Let Z be an arbitrary set and U be a metric space. A
mapping from Z into IU is called a fuzzy mapping.

Definition 2.7 ([7]). An element u∗ ∈ U is called a fuzzy fixed point of a
fuzzy mapping S : U → IU if there exists α ∈ (0, 1] such that u∗ ∈ [S (u∗)]α.

Let IU be the collection of all fuzzy sets in U and

F (U) =
{
A∗ ∈ IU : [A∗]α ∈ C (U) for all α ∈ [0, 1]

}
.

For A∗, B∗ ∈ IU , if there exists an α ∈ [0, 1] such that [A∗]α, [B
∗]α ∈ C (U),

then define

Dα (A
∗,B∗) = ρH ([A

∗]α , [B
∗]α) ,

D (A∗,B∗) = sup
α
Dα ([A

∗]α , [A
∗]α) ,

where D is a metric on F (U) and the completness of (U,ρ) implies (C (U) ,Hρ)
and (F (U) ,D) are complete.

Lemma 2.8 ([28]). Let (U,ρ) be a metric space and A∗, B∗ ∈ C(U), then for
each u ∈ A∗, there exists an element v ∈ B∗ such that ρ(u,v) ≤ ρH(A∗,B∗).

Lemma 2.9 ([28]). Let (U,ρ) be a metric space and A∗, B∗ ∈ CB(U). If
u ∈ A∗, then ρ(u,B∗) ≤ ρH(A∗,B∗).

Definition 2.10 ([5]). Let Z be a fixed set. Then an IF-set E in Z is a set of
ordered triples given by

E = {〈z,µE (z) ,υE (z)〉 : z ∈ Z} ,

where µE : Z → [0, 1] and υE : Z → [0, 1] define the degree of membership
and the degree of non-membership respectively, of the elements z in E and
satisfying 0 ≤ µE (z) + υE (z) ≤ 1, for each element z ∈ Z.
In addition, the degree of hesitancy of z to E is defined by

πE (z) = 1 −µE (z) −υE (z) .

Particularly, If πE (z) = 0, for all z ∈ Z, then an IF-set E is reduced to a fuzzy
set A∗.

Example 2.11. Consider an IF-set E of high-experienced and low-experienced
employees of a company Z, whose degrees of membership µE(z) and non-
membership υE (z) are depicted in Fig. 1.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 452



Existence results of delay and fractional differential equations

Definition 2.12 ([5]). Let E be an IF-set of universe Z. The α − cut set of
E is a classical subset of elements of Z denoted by [E]α and is defined by

[E]α = {z ∈ Z : µE(z) ≥ α and υE (z) ≤ 1 −α} if α ∈ [0, 1] .

Definition 2.13 ([27]). A mapping T : [0, 1]2 → [0, 1] is called a triangular
norm (t-norm), if the following conditions are satisfied:
(i) . T (z1,T (z2,z3)) = T (T (z1,z2) ,z3) for all z1, z2, z3 ∈ Z.
(ii) . T (z1,z2) = T (z2,z1) for all z1, z2 ∈ Z.
(iii) . If z1, z2, z3 ∈ [0, 1] and z1 ≤ z2, then T (z1,z3) ≤T (z2,z3) .
(iv) . T (z1, 1) = z1 for all z1 ∈ Z.
Minium t-norm denoted by TM and is defined by
TM (z1,z2) = min (z1,z2) for all z1, z2 ∈ [0, 1] .

Definition 2.14 ([27]). Fuzzy negation is a non-increasing mapping N :
[0, 1] → [0, 1] such that N (0) = 1, N (1) = 0. If N is continuous and strictly
decreasing, then it is called strict. Fuzzy negations with N (N (z)) = z, for all

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 453



R. Tabassum, A. Azam and M. S. Shagari

z ∈ [0, 1] , are called strong fuzzy negations. The example of fuzzy negation is
a standard negation defined by NS (z) = 1 −z, for all z ∈ Z.

Definition 2.15 ([27]). Let E be an IF-set of U, T and N a triangular norm
and a fuzzy negation, respectively. Then (T ,N ,α) −cut set of E is a crisp set
denoted by [E](T ,N,α) and is defined by

[E](T ,N,α) = {z ∈ Z : T (µE(z),N (υE (z))) ≥ α} if α ∈ [0, 1] .

Remark 2.16. If we take T = TM and N = NS, then (T ,N ,α) − cut set is
reduced into original definition of a cut set by Atanassov [5].

Definition 2.17 ([38]). Let Z be an arbitrary set, U a metric space. A map-
ping S is called intuitionistic fuzzy mapping if S is a mapping from Z into

(IFS)
U

.

Definition 2.18. A point u∗ ∈ U is said to be an intuitionistic fuzzy fixed point
of an intuitionistic fuzzy mapping S : U → (IFS)U if there exists α ∈ [0, 1]
such that u∗ ∈ [S (u∗)](T ,N,α) .
Let (IFS)

U
be the collection of all intuitionistic fuzzy subsets of U and define

FIF (U) =
{
E ∈ (IFS)U : [E](T ,N,α) ∈ C (U) for all α ∈ [0, 1]

}
.

For E1, E2 ∈ (IFS)
U

and α ∈ [0, 1] such that [E1](T ,N,α) , [E2](T ,N,α) ∈ C (U) ,
the following notations are defined by

Dα (E1,E2) = ρH

(
[E1](T ,N,α) , [E2](T ,N,α)

)
,

DIF (E1,E2) = sup
α
Dα

(
[E1](T ,N,α) , [E2](T ,N,α)

)
,

where DIF is a metric on FIF (U) .

3. Main results

In what follows hereafter, we present our main results.

Theorem 3.1. Let (U,ρ) be a complete metric space and F, G be a pair of

intuitionistic fuzzy mappings from U into (IFS)
U
. For u ∈ U, there exist

αF (ξ) , αG (ξ) ∈ [0, 1] such that [F (ξ)](T ,N,αF (ξ)) , [G (ξ)](T ,N,αG(ξ)) ∈ C (U) .
If for all u,v ∈ U,

(3.1) ρH

(
[F (u)](T ,N,αF (u)) , [G (v)](T ,N,αG(v))

)
≤ ρ (u,v) −φ (ρ (u,v)) ,

where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) =
0 if and only if t = 0 and lim

t→∞
φ (t) = ∞.

Thus, there exists ω ∈ U such that ω ∈ [F (ω)](T ,N,αF(ω)) ∩ [G (ω)](T ,N,αG(ω)).

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 454



Existence results of delay and fractional differential equations

Proof. Let u0 be an arbitrary but fixed element of U, then by assumptions,
there exists αF(u0) ∈ [0, 1] such that [F (u0)](T ,N,αF (u0)) ∈ C (U). Choose u1 ∈
[F (u0)](T ,N,αF (u0)). It follows from Lemma 2.8, there exists u2 ∈ [G (u1)](T ,N,αG(u1))
such that

ρ (u1,u2) ≤ ρH
(

[F (u0)](T ,N,αF (u0)) , [G (u1)](T ,N,αG(u1))

)
≤ ρ (u0,u1) −φ (ρ (u0,u1)) .

Again by Lemma 2.8, for u2 ∈ [G (u1)](T ,N,αG(u1)) there exists u3 ∈ [F (u2)](T ,N,αF (u2))
such that

ρ (u2,u3) ≤ ρH
(

[G (u1)](T ,N,αG(u1)) , [F (u2)](T ,N,αF (u2))

)
≤ ρ (u1,u2) −φ (ρ (u1,u2)) .

Continuing this process, for un ∈ U we obtain un+1 ∈ U such that

un+1 ∈ [F (un)](T ,N,αF(un)) , n = 0, 1, 2, · · ·,

un+2 ∈ [G (un+1)](T ,N,αG(un+1)) , n = 0, 1, 2, · · ·,
where,

ρ (un+1,un+2) ≤ ρH
(

[F (un)](T ,N,αF (un)) , [G (un+1)](T ,N,αG(un+1))

)
≤ ρ (un,un+1) −φ (ρ (un,un+1))
≤ ρ (un,un+1) , n = 0, 1, 2, · · ·.

It follows that {ρ (un,un+1)} is a non-increasing sequence of positive real num-
bers and hence tends to limit r ≥ 0. If r > 0, then we obtain

(3.2) ρ (un+1,un+2) ≤ ρ (un,un+1) −φ (r) .

Therefore,

(3.3) ρ (un+N,un+N+1) ≤ ρ (un,un+1) −Nφ (r) ,

which is a contradiction for large enough N. Hence, ρ (un,un+1) → 0.
Therefore, by a similar argument of [11], it follows {un} is a Cauchy sequence
in U. As U is complete, therefore there exists ω ∈ U such that un → ω.
Then by Lemma 2.9, we get

ρ
(

[F (ω)](T ,N,αF (ω)) ,un+2

)
≤ ρH

(
[F (ω)](T ,N,αF (ω)) , [G (un+1)](T ,N,αG(un+1))

)
≤ ρ (ω,un+1) −φ (ρ (ω,un+1)) .(3.4)

Letting n →∞ and using the fact that φ (0) = 0, we obtain

ρ
(

[F (ω)](T ,N,αF (ω)) ,ω
)
≤ 0.

This implies

ω ∈ [F (ω)](T ,N,αF (ω)) .

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 455



R. Tabassum, A. Azam and M. S. Shagari

Similarly,
ω ∈ [G (ω)](T ,N,αG(ω)) .

Hence, there exists ω ∈ U such that ω ∈ [F (ω)](T ,N,αF (ω))∩[G (ω)](T ,N,αG(ω)) .
�

Corollary 3.2. Let (U,ρ) be a complete metric space and F : U → (IFS)U
be a intuitionistic fuzzy mapping. For u ∈ U, there exists αF (u) ∈ [0, 1] such
that [F (u)](T ,N,αF (u)) ∈ C (U) . If for all u, v ∈ U,

ρH

(
[F (u)](T ,N,αF (u)) , [F (v)](T ,N,αF (v))

)
≤ ρ (u,v) −φ (ρ (u,v)) ,

where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) =
0 if and only if t = 0 and lim

t→∞
φ (t) = ∞.

Thus, there exists ω ∈ U such that ω ∈ [F (ω)](T ,N,αF (ω)).

If we take φ (t) = (1 −q) (t) , where 0 < q < 1, then corollary 3.2 reduces to
the following result.

Corollary 3.3. Let (U,ρ) be a complete metric space and F : U → (IFS)U be
intuitionistic fuzzy mapping. For u ∈ U there exists αF (u) ∈ [0, 1] such that
[F (u)](T ,N,αF (u)) ∈ C (U) . If 0 < q < 1 and for all u, v ∈ U,

ρH

(
[F (u)](T ,N,αF (u)) , [F (v)](T ,N,αF (v))

)
≤ qρ (u,v) .

Thus, there exists ω ∈ U such that ω ∈ [F (ω)](T ,N,αF (ω)).

Corollary 3.4. Let (U,ρ) be a complete metric space and F, G : U → IU be
a pair of fuzzy mappings. For u ∈ U there exists αF (u) , αG (u) ∈ (0, 1] such
that [F (u)]αF (ξ) , [G (u)]αG(ξ) ∈ C (U) . If for all u,v ∈ U,

ρH

(
[F (u)]αF (u) , [G (v)]αG(v)

)
≤ ρ (u,v) −φ (ρ (u,v)) ,

where φ : [0,∞) → [0,∞) is a continuous non-decreasing function with φ (t) =
0 if and only if t = 0 and lim

t→∞
φ (t) = ∞.

Thus, there exists ω ∈ U such that ω ∈ [F (ω)]αF (ω) ∩ [G (ω)]αG(ω).

Corollary 3.5. Let (W,ρW ) be a complete metric linear space and F, G : W →
V (W) be a pair of fuzzy mappings satisfying the following condition

D (F (u) ,G (v)) ≤ αρ (u,v) ,
for each u,v ∈ W, where φ : [0,∞) → [0,∞) is a continuous non-decreasing
function with φ (t) = 0 if and only if t = 0 and lim

t→∞
φ (t) = ∞. Thus, there

exists a ω ∈ W such that {ω}⊂ F (ω) and {ω}⊂ G (ω) .

Corollary 3.6. Let (W,ρW ) be a complete metric linear space and F : W →
V (W) be a fuzzy mapping satisfying

D (F (u) ,F (v)) ≤ ρ (u,v) −φ (ρ (u,v)) ,
for each u,v ∈ W. Thus, there exists ω ∈ W such that {ω}⊂ F (ω) .

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 456



Existence results of delay and fractional differential equations

Example 3.7. Let U = R+, ρ (u,v) = |u−v| , whenever u, v ∈ U and γ,
δ ∈ [0, 1]. Consider a pair of intuitionistic fuzzy mappings F = 〈µF ,υF〉 ,
G = 〈µG,υG〉 : U → (IFS)

U
as follow:

Case (i) : If u = v = 0, then we have

µF(0) (t) = µG(0) (t) =




1 if t = 0
2
5

if 0 < t ≤ 1002
0 if t > 1002

and

υF(0) (t) = υG(0) (t) =




0 if t = 0
3
5

if 0 < t ≤ 1003
1 if t > 1003 .

If we take αF(0) = 1 = αG(0), then we obtain

[F (0)](T ,N,1) = {0} = [G (0)](T ,N,1) .

Moreover,

ρH

(
[F (0)](T ,N,α

F (0)
) , [G (0)](T ,N,αG(0))

)
= ρ (u,v) −φ (ρ (u,v)) .

Case (ii) : If u 6= 0, v 6= 0 then we have

µF(u) (t) =




γ if 0 ≤ t ≤ u− u
2

2
γ
3

if u− u
2

2
< t ≤ u− u

2

4
γ

7
if u− u

2

4
< t < u

0 if u ≤ t < ∞ ,

υF(u) (t) =




0 if 0 ≤ t ≤ u− u
2

2
γ
4

if u− u
2

4
< t ≤ u− u

2

6
γ
2

if u− u
2

6
< t < u

1 if u ≤ t < ∞ .
and

µG(u) (t) =




δ if 0 ≤ t ≤ u− u
2

2
δ
2

if u− u
2

2
< t ≤ ξ − u

2

5
δ
3

if u− u
2

5
< t < u

0 if u ≤ t < ∞ ,

υG(u) (t) =




0 if 0 ≤ t ≤ u− u
2

2
δ
8

if u− u
2

2
< t ≤ u− u

2

8
δ
6

if u− u
2

8
< t < u

1 if u ≤ t < ∞ .
If αF(u) = γ and αG(u) = δ, then we have

[F (u)]
(T ,N,γ)

=
{
t ∈ U : T

(
µF(u) (t) ,N

(
υF(u) (t)

))
= γ

}
=

[
0,u−

u2

2

]

and

[G (u)](T ,N,δ) =
{
t ∈ U : T

(
µG(u) (t) ,N

(
υG(u) (t)

))
= δ
}

=

[
0,u−

u2

2

]
.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 457



R. Tabassum, A. Azam and M. S. Shagari

However,

ρH

(
[F (u)](T ,N,γ) , [G (u)](T ,N,δ)

)
=

∣∣∣∣u− u22 −v + v
2

2

∣∣∣∣
=

∣∣∣∣(u−v)
(

1 −
u + v

2

)∣∣∣∣
≤ |u−v|

∣∣∣∣1 − |u−v|2
∣∣∣∣

≤ |u−v|−
|u−v|2

2
= |u−v|−ϕ (|u−v|)
≤ ρ (u,v) −ϕ (ρ (u,v)) .

Thus, in both the cases, for ϕ (t) = 1
2
t2, all the assumptions of theorem 3.1 are

satisfied to obtain ω ∈ [F (ω)](T ,N,α
F (ω)

) ∩ [G (ω)](T ,N,α
G(ω)

) .

3.1. Application to Delay Differential Equations. In this section, we will
establish an existence result of delay differential equation with constant delay,
where the only independent variable is the time variable.
Delay differential equations appear naturally in modelling the numerous bio-
logical systems. For instance, primary infection [14], drug therapy [29] and
immune response [15]. They have also been used in the study of epidemiology
[16], the respiratory system [36] and tumor growth [37]. Moreover, statistical
analysis of ecological data [34], indicates the delay effects in many classes of
population dynamics.
General Form of Delay Differential Equation

Consider the general form of equation with delay

u· (t) = g (t,ut) ,

where ut : [−τ, 0] → Rn is a function such that ut (λ) = u (t + λ) for λ ∈
[−τ, 0] , as shown in Fig. 2.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 458



Existence results of delay and fractional differential equations

For an ordinary differential system, a unique solution is obtained using an
initial point in Euclidean space at an initial time t0. On the other hand, one
needs information on the entire interval [t0 − τ,t0], for a delay differential equa-
tions.
Delay differential equations are solved by considering previous values of depen-
dent variable u at every time step. For this, one requires initial function or
initial history, the value of u (t) for the interval [−τ, 0] is used to demonstrate
the behavior of the system prior to the starting time.

Theorem 3.8. Let U = C ([a,b] ,R) be the space of all continuous real valued
functions on [a,b] with a metric ρ : X ×X → R defined by

ρ (u,v) = max
t∈[a,b]

|u (t) −v (t)| ,

for all u, v ∈ C [a,b] .
Assume that g : [t0,b] × R2 → R and ψ : [t0 − τ,b] → R are continuous
mappings, where t0, b ∈ R and τ > 0. If there exists λg > 0 such that λg <

1
2(b−t0)

and

(3.5) |g (t,u1,u2) −g (t,v1,v2)| ≤ λg
2∑
i=1

|ui −vi|,

for all ui, vi ∈ R, i = 1, 2, t ∈ [t0,b] .
Thus, the delay differential equation

(3.6) u′ (t) = g (t,u (t) ,u (t− τ)) , t ∈ [t0,b]

with initial condition

(3.7) u (t) = ψ (t) , t ∈ [t0 − τ,t0]

has a solution u ∈ C ([t0 − τ,b] ,R) ∩C1 ([t0,b] ,R) .

Proof. Let F : U → (IFS)U be intuitionistic fuzzy mapping and define an
arbitrary mapping h from U into (0, 1].
The integral reformulation of problem (3.5)-(3.7) is given by

(3.8) u (t) =




ψ (t) , t ∈ [t0 − τ,t0]

ψ (t0) +
t∫
t0

g (s,u (s) ,u (s− τ)) ds, t ∈ [t0,b] .

Define an intuitionistic fuzzy mapping F = 〈µF ,υF〉 : U → (IFS)
U

as follows:

µF(u) (e) =

{
h (u) if e (t) = u (t) for all t ∈ [t0,b]
0 otherwise ,

υF(u) (e) =

{
0 if e (t) = u (t) for all t ∈ [t0,b]
h (u) otherwise .

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 459



R. Tabassum, A. Azam and M. S. Shagari

If αF(u) = h (u) , then we have

[F (u)](T ,N,αF (u)) =
{
e ∈ U : T

(
µF(u) (e) ,N

(
υF(u) (e)

))
= h (u)

}
= {u} .

However,

ρH

(
[F (u)](T ,N,αF (u)) , [F (v)](T ,N,αF (v))

)
= max
t∈[t0−τ,b]

|u (t) −v (t)| .

Therefore, by assumptions, we obtain

max
t∈[t0−τ,b]

|u (t) −v (t)| = max
t∈[t0−τ,b]

∣∣∣∣∣∣
t∫
t0

g (s,u (s) ,u (s− τ)) ds−
t∫
t0

g (s,v (s) ,v (s− τ)) ds

∣∣∣∣∣∣
≤ max

t∈[t0−τ,b]

t∫
t0

|g (s,u (s) ,u (s− τ)) −g (s,v (s) ,v (s− τ))|ds

≤ max
t∈[t0−τ,b]

t∫
t0

λg (|u (s) −v (s)| + |u (s− τ) −v (s− τ)|) ds

≤
t∫
t0

λg

(
max

s∈[t0−τ,b]
|u (s) −v (s)| + max

s∈[t0−τ,b]
|u (s− τ) −v (s− τ)|

)
ds

≤
t∫
t0

λg (ρ (u,v) + ρ (u,v)) ds

≤ 2λgd (u,v)
t∫
t0

ds

≤ 2λg (b− t0) ρ (u,v)
≤ ρ (u,v) − (1 −q) ρ (u,v)
≤ ρ (u,v) −ϕ (ρ (u,v)) .

Where, q = 2λg (b− t0) and ϕ (u) = (1 −q) (u) . Thus, all the assumptions of
Theorem 3.1 are satisfied for F = G to obtain ω ∈ U such that

ω ∈
[
[F (ω)](T ,N,αF (ω))

]
.

Hence, ω is a solution of (3.5) and (3.6). �

Example 3.9. Consider the delay differential equation

(3.9) u′ (t) = t3 +
1

10
u5 (t) +

1

10
u5
(
t−

1

2

)
, t ∈ [0, 1]

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 460



Existence results of delay and fractional differential equations

with initial condition

(3.10) u (t) = t + 1, t ∈
[
−

1

2
, 0

]
,

where τ = 1
2

and ψ (t) = t + 1,

g (t,u (t) ,u (t− τ)) = t3 +
1

10
u5 (t) +

1

10
u5
(
t−

1

2

)
.

The associated integral equation of problem (3.9)-(3.10) is given by

u (t) =




t + 1, t ∈
[
−1

2
, 0
]

1 +
t∫
0

(
s3 + 1

10
u5 (s) + 1

10
u5
(
s− 1

2

))
ds, t ∈ [0, 1] .

If ui, vi ∈ R, i = 1, 2, and t ∈ [0, 1] then we obtain

|g (t,u1,u2) −g (t,v1,v2)| =
∣∣∣∣t3 + 110u51 + 110u52 − t3 − 110v51 − 110v52

∣∣∣∣
=

∣∣∣∣ 110 (u51 −v51) + 110 (u52 −v52)
∣∣∣∣

≤
1

10

∣∣(u51 −v51)∣∣ + 110 ∣∣(u52 −v52)∣∣
≤

1

10

2∑
i=1

∣∣u5i −v5i ∣∣ .
Hence, for λg =

1
10
, all the conditions of Theorem 3.8 are satisfied to obtain a

solution of the given delay differential equation.

4. Application to a System of Riemann-Liouville Fractional
Differential Equations

In recent time, fractional calculus has drawn the interests of researchers due
to its wide range of applications in solving problems in diverse areas such as
viscoelasticity, biological science, aerodynamics, statistical physics, etc. For
some noted applications and developmental history of fractional calculus, the
interested reader may see [24, 35]. Undoubtably, the first problem of every
fractional differential equation is the conditions for the existence of its solution.
Thus, this section is devoted to providing existence conditions of solutions to
Riemann-Liouville Cauchy type problem on a finite interval of the real line in a
space of summable and continuous functions. Our investigations are based on
reformulating the problem to Volterra integral equation of the second kind and
using intuitionistic fuzzy maps. The nonlinear Riemann-Liouville fractional

derivative (D
ξ
a+
v)(u) of order ξ, defined for Re(ξ) > 0 on a finite interval [a,b]

is given by

(4.1) (D
ξ
a+
v)(u) = g(u,v(u)),

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 461



R. Tabassum, A. Azam and M. S. Shagari

with initial conditions

(4.2) (D
ξ−i
a+

v)(a+) = di, di ∈ C (i = 1, 2, 3, · · ·n),

where n = Re(ξ) + 1 for ξ /∈ N and ξ = n for ξ ∈ N. Notice that for ξ = n ∈ N,
problems (4.1)-(4.2) are reduced to classical Cauchy problem for the ordinary
differential equation. The Cauchy type problem (4.1)-(4.2) with complex ξ ∈ C
was first studied by Kilbas [23] in the space of summable functions L(a,b).
Al-Bassam [12] studied problem (4.1)-(4.2) for a real 0 < ξ ≤ 1 in the space of
continuous functions C[a,b], provided that g(u,v) is a real-valued continuous
function in a domain H ⊂ R2. Most likely, he was the first to show that
the method of contraction mapping could be employed to prove the existence
of solution to (4.1)-(4.2). It was however observed by Kilbas [23] that the
condition given by Al-Bassam [12] was not suitable for solving the problem.
Afterwards, Delbosco and Rodino [19] studied the nonlinear Riemann-Liouville
Cauchy problem:

(4.3) (D
ξ
a+
v)(u) = g(u,v(u)), v(i)(0) = vi ∈ R (i = 1, 2, 3, · · ·n)

with 0 ≤ u ≤ 1, λ > 0 and g(u,v) is a continuous function on [0, 1] × R.
They showed the equivalence to the corresponding Volterra integral equation
and applied Schauder’s fixed point theorem to prove that problem (4.3) has at
least one solution v(u) defined on [0,τ] provided that uκg(u,v) is continuous
on [0, 1] × R for some κ ∈ [0, 1). Later on, problems (4.1)-(4.2) and (4.3) were
studied by several authors (see, [2, 4, 32]). But the above investigations were
not complete due to the missing of some techniques of nonlinear functional
analysis [23]. For details in this observation, the interested readers may go
through the survey paper by Kilbas and Trujillo [33].

As far as we know, no contribution exists in the literature concerning with
the study of existence conditions of the Riemann-Liouville Cauchy type prob-
lem (4.1)-(4.2) in the setting of intuitionistic fuzzy mappings and even then for
fuzzy and multivalued mappings. Thus, in this section, we establish existence
conditions for the solution of problem (4.1)-(4.2) in the space L1(a,b) = L(a,b)
of summable functions on a finite interval [a,b] of R by appealing to intuition-
istic fuzzy mappings defined on a complete metric space.

For our convenience, we recall the definitions of Riemann-Liouville frac-
tional integrals and fractional derivatives on a finite interval of the real line
and present specific results. For these basic concepts and notations, we follow
the books of Kilbas et al. [23] and Samko et al. [31].

The Riemann-Liouville fractional integrals I
ξ
a+
g and I

ξ
b−
g of order ξ ∈ C

where Re(ξ) > 0 are defined by

(4.4) (I
ξ
a+
g)(u) =

1

Γ(ξ)

∫ u
a

g(t)

(u− t)1−ξ
dt

and

(4.5) (I
ξ
b−
g)(u) =

1

Γ(ξ)

∫ b
u

g(t)

(u− t)1−ξ
dt,

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 462



Existence results of delay and fractional differential equations

where Γ(.) is the Gamma function. The integrals (4.4) and (4.5) are called the
left-sided and right-sided fractional integrals, respectively.

The Riemann-Liouville fractional derivatives D
ξ
a+
v, D

ξ
b−
v of order ξ ∈ C are

defined by

(D
ξ
a+
v)(u) =

(
d

du

)n (
I
n−ξ
a+

)
(u)

=
1

Γ(n− ξ)

(
d

du

)n ∫ u
a

v(t)

(u− t)ξ−n+1
dt (n = [Re(ξ)] + 1)

and

(D
ξ
b−
v)(u) =

(
−
d

du

)n (
I
n−ξ
b−

)
(u)

=
1

Γ(n− ξ)

(
−
d

du

)n ∫ b
u

v(t)

(u− t)ξ−n+1
dt (n = [Re(ξ)] + 1),

respectively, where [Re(ξ)] means the integral part of Re(ξ).
we denote by Lp(a,b), where 1 ≤ p ≤∞, the set of all Lebesgue complex-valued
measurable functions g on Ω for which ‖g‖p < ∞ with

‖g‖p =
(∫

Ω

|f(t)|p dt
)1
p

and

‖g‖∞ = esssup
a≤u≤b

|g(u)|.

The following result shows that fractional differentiation is an operator in-
verse to the fractional integral operator from the left.

Lemma 4.1 ([31]). If Re(ξ) > 0 and g(u) ∈ Lp(a,b), where 1 ≤ p ≤∞, then
the following equalities(

D
ξ
a+
I
ξ
a+
g
)

(u) = g(u) and
(
D
ξ
b−
I
ξ
b−
g
)

(u) = g(u),

hold almost everywhere on [a,b].

Lemma 4.2 ([31]). The fractional integral operator I
ξ
a+

with ξ > 0 is bounded
in L(a,b) satisfying

‖Iξ
a+
z‖1 ≤

(b−a)ξ

Γ(ξ + 1)
‖z‖1.

Lemma 4.3 ([23]). Let ξ ∈ C and n− 1 < Re(ξ) < n (n ∈ N). Let H be an
open set in C and g : [a,b] ×H −→ C be a function such that g(u,v) ∈ L(a,b)
for any v ∈ H. If v(u) ∈ L(a,b), then v(u) satisfies almost every where the
Riemann-Liouville Cauchy type problem (4.1)-(4.2) if and only if v(u) satisfies
the integral equation

(4.6) v(u) =

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)ξ−i +
1

Γ(ξ)

∫ u
a

g(t,v(t))

(u− t)1−ξ
dt

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 463



R. Tabassum, A. Azam and M. S. Shagari

Our main result of this section runs as follows:

Theorem 4.4. Let H be an open set in C and g1,g2 : [a,b] × H −→ C be
functions such that g1(u,v), g2 (u,v) ∈ L(a,b) = U, where L(a,b) is the set
of all Lebesgue complex-valued measurable functions on [a,b] endowed with the
metric ρ : U ×U −→ R defined as

ρ(v1,v2) = ‖v1 −v2‖ =
∫ u
a

|v1(u) −v2(u)|du,

for all v1,v2 ∈ U and a < u < b. Assume that for all g1,g2 ∈ L(a,b), there
exists % > 0 such that

‖g1(u,v1) −g2(u,v2)‖≤ %‖v1 −v2‖.

Thus, the system of Riemann-Liouville Cauchy type problems (SRLCTPs) given
by

(4.7) (D
ξ
a+
v)(u) = g1(u,v(u)),

with initial conditions

(4.8) (D
ξ−i
a+

v)(a+) = di, di ∈ C (i = 1, 2, 3, · · ·n)

and

(4.9) (D
ξ
a+
v)(u) = g2(u,v(u)),

with initial conditions

(4.10) (D
ξ−i
a+

v)(a+) = di, di ∈ C (i = 1, 2, 3, · · ·n),

have a common solution in L(a,b).

Proof. By Lemma 4.3, the common solution of (4.7)-(4.10) is also the common
solution of their integral reformulation, respectively given as:

(4.11) v(u) =

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)λ−i +
1

Γ(ξ)

∫ u
a

g1(t,v(t))

(u− t)1−ξ
dt

(4.12) v(u) =

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)λ−i +
1

Γ(ξ)

∫ u
a

g2(t,v(t))

(u− t)1−ξ
dt

Clearly, the set U equipped with the given metric ρ is a complete metric space.
Let r,s : U −→ (0, 1] be any two arbitrary mappings and φ : [0,∞) −→ [0,∞)
be a continuous non-decreasing function. Choose u1 ∈ (a,b) such that

%
(u1 −a)ξ

Γ(ξ + 1)
≤
ρ(v1,v2) −ϕ(ρ(v1,v2))

1 + ρ(v1,v2)
.

For v ∈ U, we have

ωv(t) = v0(t) +
1

Γ(ξ)

∫ u
a

g1(t,v(t))

(u− t)1−ξ
dt

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 464



Existence results of delay and fractional differential equations

and

τv(t) = v0(t) +
1

Γ(ξ)

∫ u
a

g2(t,v(t))

(u− t)1−ξ
dt,

where

v0(t) =

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)ξ−i.

Consider a pair of intuitionistic fuzzy mappings F,G : U −→ (IFS)U defined
as follows:

µF(v)(q) =

{
r(v), q(t) = ωv(t), t ∈ [a,b]
0, q(t) 6= ωv(t),

υF(v)(q) =

{
0, q(t) = ωv(t), t ∈ [a,b]
r(v), q(t) 6= ωv(t)

and

µG(v)(q) =

{
s(v), q(t) = τv(t), t ∈ [a,b]
0, q(t) 6= τv(t),

υG(v)(q) =

{
0, q(t) = τv(t), t ∈ [a,b]
s(v), q(t) 6= τv(t).

If we take αF(v) = r(v) and αG(v) = s(v), then we have

[F(v)](T ,N,αF (v)) =
{
q ∈ U : T

(
µF(v)(q),N

(
υF(v)(q)

))
= r(v)

}
= {ωv}

and

[G(v)](T ,N,αG(v)) =
{
q ∈ U : T

(
µG(v)(q),N

(
υG(v)(q)

))
= s(v)

}
= {τv}.

Therefore, for v1,v2 ∈ U, we obtain

[F(v1)](T ,N,αF (v1)) = {ωv1}

and

[G(v2)](T ,N,αG(v2)) = {τv2}.
Consequently,

ρH

(
[F(v1)](T ,N,αF (v1)) , [G(v2)](T ,N,αG(v2))

)
= ‖ωv1 − τv2‖1.

For the remaining steps, we employ a standard method for nonlinear Volterra
integral equations of the proof of result on a subinterval of [a,b], (see, [25, 23]).
Notice that equations (4.11)-(4.12) are valid in any interval [a,u1] ⊂ [a,b] for
a < u1 < b. Thus, for an interval [a,u1], a metric ρ : L(a,u1) ×L(a,u1) −→ R
is defined by

ρ(v1,v2) = ‖v1 −v2‖1 =
∫ u1
a

|v1(u) −v2(u)|du.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 465



R. Tabassum, A. Azam and M. S. Shagari

Since f, g ∈ L(a,b), therefore, by Lemma 4.2, u1 = b and z = g1(u,v1) −
g2(u,v2), we have

ρH

(
[F(v1)](T ,N,αF (v1)) , [G(v2)](T ,N,αG(v2))

)
= ‖ωv1 − τv2‖1

=

∥∥∥∥ 1Γ(ξ)
∫ u1
a

g1(t,v1(t))

(u− t)1−ξ
dt−

∫ u1
a

g2(t,v2(t))

(u− t)1−ξ
dt

∥∥∥∥
1

≤
∥∥∥∥ 1Γ(ξ)

∫ u1
a

[g1(t,v1) −g2(t,v2)]
(u− t)1−ξ

dt

∥∥∥∥
1

≤
∥∥∥Iξa+ [g1(t,v1) −g2(t,v2)]∥∥∥

1

≤
(u1 −a)ξ

Γ(ξ + 1)
‖g1(t,v1) −g2(t,v2)‖1

≤ %
(u1 −a)ξ

Γ(ξ + 1)
ρ(v1,v2)

≤ %
(u1 −a)ξ

Γ(ξ + 1)
(1 + ρ(v1,v2))

≤ ρ(v1,v2) −ϕ(ρ(v1,v2)).

Hence, by Theorem 3.1, there exists a common solution v∗ ∈ L(a,u1) to the
Volterra integral equations (4.11)-(4.12) in the interval [a,u1].
Next, consider the interval [u1,u2], where u2 = u1 + ζ1 and ζ1 > 0 are such
that u2 < b. Rewrite equations (4.11)-(4.12) as follows:

v(u) =
1

Γ(ξ)

∫ u
u1

g1(t,v(t))

(u− t)1−ξ
dt +

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)ξ−i

+
1

Γ(ξ)

∫ u1
a

g1(t,v(t))

(u− t)1−ξ
dt.(4.13)

v(u) =
1

Γ(ξ)

∫ u
u1

g2(t,v(t))

(u− t)1−ξ
dt +

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)ξ−i

+
1

Γ(ξ)

∫ u1
a

g2(t,v(t))

(u− t)1−ξ
dt.(4.14)

Again, equations (4.13)-(4.14) can be rewrriten as

v(u) = v01(u) +
1

Γ(ξ)

∫ u
u1

g1(t,v(t))

(u− t)1−ξ
dt,

v(u) = v01(u) +
1

Γ(ξ)

∫ u
u1

g2(t,v(t))

(u− t)1−ξ
dt,

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 466



Existence results of delay and fractional differential equations

where

v01(u) =

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)ξ−i +
1

Γ(ξ)

∫ u1
a

g1(t,v(t))

(u− t)1−ξ

=

n∑
i=1

di
Γ(ξ − i + 1)

(u−a)ξ−i +
1

Γ(ξ)

∫ u1
a

g2(t,v(t))

(u− t)1−ξ
(4.15)

are the known functions. The idea of (4.15) is to ignore the previous interval
[a,u1] for which a solution is known. Next, by re-considering any two arbitrary
mappings r, s : U −→ (0, 1], a pair of intuitionistic fuzzy mappings F,G :
U −→ (IFS)U and a nondecreasing continuous function φ : [0,∞) −→ [0,∞)
such that

%
(u2 −a)ξ

Γ(ξ + 1)
≤
ρ(v2,v3) −φ(ρ(v2,v3))

1 + ρ(v2,v3)
(v2,v3 ∈ L(a,b), a < u2 < b) .

So, one can obtain

ρH

(
[F(v2)](T ,N,αF (v2)) , [G(v3)](T ,N,αG(v3))

)
≤ ρ(v2,v3) −ϕ(ρ(v2,v3)).

Again, Theorem 3.1 can be applied to find a solution v∗(u) ∈ L(u1,u2) to
the integral equations (4.11)-(4.12) on the interval [u1,u2]. By repeating this
procedure inductively on the intervals [u2,u3], · · · , [un,un+1], where un+1 =
un+ζn and ζn > 0 are such that un+1 < b, therefore, we can conclude according
to Theorem 3.1 that there exists a common solution v(u) = v∗(u) ∈ L(a,b)
to the Riemann-Liouville Cauchy type problems (4.7)-(4.10) on the interval
[a,b]. �

Remark 4.5. The result of Theorem 4.4 only gives existence conditions for the
Riemann-Liouville Cauchy type problem (4.7)-(4.8) and its equivalent integral
equation (4.11) in the space L(a,b) for ξ ∈ C and n− 1 < Re(ξ) < n (n ∈ N).
The case of the problem (4.7)-(4.8) for order ξ = n+im, (n ∈ N,m ∈ R,m 6= 0)
may be considered in due course.

Conclusion

In the framework of IF-sets, we have established a common fixed point the-
orem using weakly contractive condition for a pair of intuitionistic fuzzy map-
pings in the context of (T ,N ,α)−cut set of an IF-set in a complete metric
space. Moreover, in our research work, we have constructed the iterations to
establish the fixed point of intuitionistic fuzzy mappings. By building on the
constructive approach, one will be able to define a procedure for obtaining
the solution of certain functional equations arising in dynamical systems. On
other hand, there is a rich variety of dynamics with multifaceted mathematical
structures such as industrial control devices and systems handling imprecise
information. Therefore, the knowledge of cut sets of an IF-set is beneficial to
handle such uncertain and imprecise informations and processes, because these
sets can transform an IF-set into a crisp set.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 467



R. Tabassum, A. Azam and M. S. Shagari

As an application, we have investigated the existence of solution of time de-
pendent delay differential equations with constant delay and Riemann-Liouville
Cauchy type fractional differential equations, which involve completeness prop-
erty of function spaces. Moreover, an example has been given to support the
validity of existence theorem of the considered delay differential equation.

In future, the presented results will be useful to handle several realistic
uncertain situations. On one hand, as an application, one can implement these
results for the existence of delay differential equations with variable delays and
n-systems of Cauchy problems of Riemann Liouville type.

Acknowledgements. The authors are grateful to the editors and the anony-
mous referee(s) for careful checking of the details and for their helpful comments
to improve this paper.

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