Ratio Mathematica Volume 43, 2022

Existence and Uniqueness of solution of
Volterra Integrodifferential Equation of

Fractional Order via S-Iteration

Haribhau L. Tidke ∗

Gajanan S. Patil †

Rupesh T. More ‡

Abstract

In this paper, we study the existence, uniqueness and other properties
of solutions of Volterra integrodifferential equation of fractional order
involving the Caputo fractional derivative. The tool employed in the
analysis is based on application of S− iteration method. Since the
study of qualitative properties in general required differential and in-
tegral inequalities, but here S−iteration method itself has equally im-
portant contribution to study various properties such as dependence
on initial data, closeness of solutions and dependence on parameters
and functions involved therein. An example in support of the all-
established results is given.
Keywords: Existence and uniqueness; Normal S−iterative method;
Fractional derivative; Continuous dependence; Closeness; Parame-
ters.
2020 AMS subject classifications: 35A01, 35A02, 34A12, 26A33,
35B30, 45D05, 47H10. 1

∗(Department of Mathematics, School of Mathematical Sciences, Kavayitri Bahinabai Chaud-
hari North Maharashtra University, Jalgaon, India); tharibhau@gmail.com.
†(Department of Mathematics, PSGVPM’s ASC College, Shahada, India); ga-

janan.umesh@rediffmail.com.
‡(Department of Mathematics, Arts, Commerce and Science College, Bodwad, India); rupesh-

more82@gmail.com.
1Received on May 28th, 2022. Accepted on December 25th, 2022. Published on December 30th,
2022. doi: 10.23755/rm.v41i0.791. ISSN: 1592-7415. eISSN: 2282-8214. c©The Authors. This
paper is published under the CC-BY licence agreement.



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

1 Introduction
We consider the following Volterra integrodifferential equation of fractional order
involving the Caputo fractional derivative of the type:

(
Dα∗a

)
y(t) = F

(
t,y(t),

∫ t
a

h
(
s,y(s)

)
ds
)
, (1)

for t ∈ I = [a,b], n− 1 < α ≤ n, n ∈ N, with the given initial conditions

y(j)(a) = cj, j = 0, 1, 2, · · · ,n− 1, (2)

where F : I × X × X → X, h : I × X → X are continuous functions and
cj (j = 0, 1, 2, . . . ,n− 1) are given elements in X.

Several researchers have introduced many iteration methods for certain classes of
operators in the sense of their convergence, equivalence of convergence and rate
of convergence etc. (see [1, 3, 4, 5, 6, 8, 9, 18, 19, 20, 21, 22, 23, 24, 31, 32]).
The most of iterations devoted for both analytical and numerical approaches. The
S− iteration method, due to simplicity and fastness, has attracted the attention
and hence, it is used in this paper.

The problems of existence, uniqueness and other properties of solutions of special
forms of IVP (1)-(2) and its variants have been studied by several researchers
under variety of hypotheses by using different techniques, [2, 7, 10, 11, 12, 13,
14, 15, 16, 26, 27, 29, 30] and some of references cited therein. In recently, Soltuz
and Grosan [33] have studied the special version of equation (1) for different
qualitative properties of solutions. Authors are motivated by the work of Sahu
[31] and influenced by [5,33].

The main objective of this paper is to use normal S−iteration method to establish
the existence and uniqueness of solution of the initial value problem (1)-(2) and
other qualitative properties of solutions.

2 Preliminaries
Before proceeding to the statement of our main results, we shall setforth some
preliminaries and hypotheses that will be used in our subsequent discussion.

Let X be a Banach space with norm ‖ · ‖ and I = [a,b] denotes an interval of
the real line R. For the fractional order α, n − 1 < α ≤ n, n ∈ N, we define
B = Cr(I,X), (where r = n for α ∈ N and r = n− 1 for α /∈ N), as a Banach
space of all r times continuously differentiable functions from I into X, endowed
with the norm



Existence and Uniqueness of solution via S-Iteration

‖y‖B = sup{‖y(t)‖ : y ∈ B}, t ∈ I.

Definition 2.1 (28). The Riemann Liouville fractional integral (left-sided) of a
function h ∈ C1[a,b] of order α ∈ R+ = (0,∞) is defined by

Iαa h(t) =
1

Γ(α)

∫ t
a

(t−s)α−1h(s) ds, t ∈ I

where Γ is the Euler gamma function.

Definition 2.2 (28). Let n− 1 < α ≤ n, n ∈ N. Then the expression

Dαah(t) =
dn

dtn
[
In−αa h(t)

]
, t ∈ [a,b]

is called the (left-sided) Riemann Liouville derivative of h of order α whenever
the expression on the right-hand side is defined.

Definition 2.3 (25). Let h ∈ Cn[a,b] and n − 1 < α ≤ n, n ∈ N. Then the
expression (

Dα∗a
)
h(t) = In−αa h

(n)(t), t ∈ [a,b]

is called the (left-sided) Caputo derivative of h of order α.

Lemma 2.1 (17). If the function f = (f1, · · · ,fn) ∈ C1[a,b], then the initial
value problems(
Dαi∗a

)
y(t) = fi(t,y1, · · · ,yn), y

(k)
i (0) = c

i
k, i = 1, 2, · · · ,n, k = 1, 2, · · · ,mi

where mi < αi ≤ mi + 1 is equivalent to Volterra integral equations:

yi(t) =

mi∑
k=0

cik
tk

k!
+ Iαia fi(t,y1, · · · ,yn), 1 ≤ i ≤ n.

As a consequence of the Lemma 2.1, it is easy to observe that if y ∈ B and
F ∈ C1[a,b], then y(t) satisfies the integral equation

y(t) =
n−1∑
j=0

cj
j!

(t−a)j +
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ
)
ds,

(3)

which is equivalent to (1)-(2).

We need the following pair of known results:



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

Theorem 2.1. ([31], p.194) Let C be a nonempty closed convex subset of a Ba-
nach space X and T : C → C a contraction operator with contractivity factor
m ∈ [0, 1) and fixed point x∗. Let αk and βk be two real sequences in [0, 1] such
that α ≤ αk ≤ 1 and β ≤ βk < 1 for all k ∈ N and for some α,β > 0. For given
u1 = v1 = w1 ∈ C, define sequences uk,vk and wk in C as follows:

S-iteration process:
{
uk+1 = (1 −αk)Tuk + αkTyk,
yk = (1 −βk)uk + βkTuk,k ∈ N.

Picard iteration: vk+1 = Tvk,k ∈ N.
Mann iteration process: wk+1 = (1 −βk)wk + βkTwk,k ∈ N.
Then we have the following:

(a) ‖uk+1 −x∗‖≤ mk
[
1 − (1 −m)αβ

]k
‖u1 −x∗‖, for all k ∈ N.

(b) ‖vk+1 −x∗‖≤ mk‖v1 −x∗‖, for all k ∈ N.

(c) ‖wk+1 −x∗‖≤
[
1 − (1 −m)β

]k
‖w1 −x∗‖, for all k ∈ N.

Moreover, the S-iteration process is faster than the Picard and Mann iteration
processes.

Definition 2.4. ([31], p.194) In particular, for αk = 1, k ∈ N ∪ {0} in the
S-iteration process, then it reduces to as follows:


u0 ∈ C,
uk+1 = Tyk,
yk = (1 − ξk)uk + ξkTuk, k ∈ N∪{0}.

(4)

This is called normal S−iteration method.

Note: For our convenience, we replaced βk in the S-iteration process by ξk.

Lemma 2.2. ([33], p.4) Let {βk}∞k=0 be a nonnegative sequence for which one as-
sumes there exists k0 ∈ N, such that for all k ≥ k0 one has satisfied the inequality

βk+1 ≤ (1 −µk)βk + µkγk, (5)

where µk ∈ (0, 1), for all k ∈ N ∪{0},
∞∑
k=0

µk = ∞ and γk ≥ 0, ∀k ∈ N ∪{0}.

Then the following inequality holds

0 ≤ lim sup
k→∞

βk ≤ lim sup
k→∞

γk. (6)



Existence and Uniqueness of solution via S-Iteration

3 Existence and Uniqueness of Solutions via S−iteration

Now, we are able to state and prove the following main theorem which deals with
the existence and uniqueness of solutions of the problem (1)-(2).

Theorem 3.1. Assume that there exist functions p, q ∈ C(I,R+) such that

‖F
(
t,u1,u2

)
−F

(
t,v1,v2

)
‖≤ p(t)

[
‖u1 −v1‖ + ‖u2 −v2‖

]
(7)

and

‖h(t,u1) −h(t,v1)‖≤ q(t)‖u1 −v1‖,

for t ∈ I. If Θ = Iaαp(t)
(
1 + (b − a)Q

)
< 1

(
where Q = sup

a≤t≤b
q(t)

)
, then

the iterative sequence {yk}∞k=0 generated by normal S− iteration method (4) with

the real control sequence {ξk}∞k=0 in [0, 1] satisfying
∞∑
k=0

ξk = ∞, converges to a

unique point y ∈ B, which is the required solution of the equations (1)-(2) with
the following estimate:

‖yk+1 −y‖B ≤
Θk+1

e

(
1−Θ
)∑k

i=0 ξi

‖y0 −y‖B. (8)

Proof. Let y(t) ∈ B and define the operator

(Ty)(t) =
n−1∑
j=0

cj
j!

(t−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ
)
ds, t ∈ I. (9)

Let {yk}∞k=0 be iterative sequence generated by normal S−iteration method (4)
for the operator given in (9).
We will show that yk → y as k →∞.
From (4), (9) and assumption, we obtain

‖yk+1(t) −y(t)‖
= ‖(Tzk)(t) − (Ty)(t)‖

= ‖
n−1∑
j=0

cj
j!

(t−a)j +
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
ds



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

−
n−1∑
j=0

cj
j!

(t−a)j −
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ
)
ds‖

≤
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)

−F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ
)
‖ds

≤
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk(s) −y(s)‖ +

∫ s
a

q(σ)‖zk(σ) −y(σ)‖dσ
]
ds.

(10)

Now, we estimate

‖zk(t) −y(t)‖ =
[
(1 − ξk)‖yk(t) −y(t)‖ + ξk‖(Tyk)(t) − (Ty)(t)‖

]
≤ (1 − ξk)‖yk(t) −y(t)‖ + ξk

1

Γ(α)

∫ t
a

(t−s)α−1p(s)

×
[
‖yk(s) −y(s)‖ +

∫ s
a

q(σ)‖yk(σ) −y(σ)‖dσ
]
ds. (11)

Now, by taking supremum in the inequalities (10) and (11), we obtain

‖yk+1 −y‖B ≤
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk −y‖B +

∫ s
a

q(σ)‖zk −y‖Bdσ
]
ds

≤
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk −y‖B + (b−a)Q‖zk −y‖B

]
ds

≤ Iaαp(t)
(

1 + (b−a)Q
)
‖zk −y‖B

= Θ‖zk −y‖B (12)

and

‖zk −y‖B ≤
[
(1 − ξk)‖yk −y‖B + ξkΘ‖yk −y‖B

]
=
[
1 − ξk

(
1 − Θ

)]
‖yk −y‖B, (13)

respectively.
Therefore, using (13) in (12), we have

‖yk+1 −y‖B ≤ Θ
[
1 − ξk

(
1 − Θ

)]
‖yk −y‖B. (14)

Thus, by induction, we get

‖yk+1 −y‖B ≤ Θk+1
k∏
j=0

[
1 − ξk

(
1 − Θ

)]
‖y0 −y‖B. (15)



Existence and Uniqueness of solution via S-Iteration

Since ξk ∈ [0, 1] for all k ∈ N∪{0}, the definition of Θ and ξk ≤ 1 yields,

⇒ ξkΘ < ξk

⇒ ξk
(

1 − Θ
)
< 1, ∀ k ∈ N∪{0}. (16)

From the classical analysis, we know that

1 −x ≤ e−x = 1 −x +
x2

2!
−
x3

3!
+ · · · , x ∈ [0, 1].

Hence by utilizing this fact with (16) in (15), we obtain

‖yk+1 −y‖B ≤ Θk+1e−
(

1−Θ
)∑k

j=0 ξj‖y0 −y‖B

=
Θk+1

e

(
1−Θ
)∑k

i=0 ξi

‖y0 −y‖B. (17)

Since
∞∑
k=0

ξk = ∞,

e
−
(

1−Θ
)∑k

j=0 ξj → 0 as k →∞. (18)

Hence, using this, the inequality (17) implies lim
k→∞
‖yk+1 −y‖B = 0 and therefore,

we have yk → y as k →∞.

Remark: It is an interesting to note that the inequality (17) gives the bounds
in terms of known functions, which majorizes the iterations for solutions of the
problem (1)-(2) for t ∈ I.

4 Continuous dependence via S−iteration
In this section, we shall deal with continuous dependence of solution of the prob-
lem (1) on the initial data, functions involved therein and also on parameters.

4.1 Dependence on initial data

Suppose y(t) and y(t) are solutions of (1) with initial data

y(j)(a) = cj, j = 0, 1, 2, · · · ,n− 1, (19)



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

and

y(j)(a) = dj, j = 0, 1, 2, · · · ,n− 1, (20)

respectively, where cj,dj are elements of the space X.
Then looking at the steps as in the proof of Theorem 3.1, we define the operator
for the equation (1) with the initial conditions (20):

(Ty)(t) =
n−1∑
j=0

dj
j!

(t−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ
)
ds, t ∈ I. (21)

We shall deal with the continuous dependence of solutions of equations (1) on
initial data.

Theorem 4.1. Suppose the function F in equation (1) satisfies the condition
(7). Consider the sequences {yk}∞k=0 and {yk}

∞
k=0 generated normal S− itera-

tive method associated with operators T in (9) and T in (21), respectively with
the real sequence {ξk}∞k=0 in [0, 1] satisfying

1
2
≤ ξk for all k ∈ N ∪{0}. If the

sequence {yk}
∞
k=0 converges to y, then we have

‖y −y‖B ≤
3M(

1 − Θ
), (22)

where

M =
n−1∑
j=0

‖cj −dj‖
j!

(b−a)j.

Proof. From iteration (4) and equations (9); (21) and assumptions, we obtain

‖yk+1(t) −yk+1(t)‖
= ‖(Tzk)(t) − (Tzk)(t)‖

= ‖
n−1∑
j=0

cj
j!

(t−a)j +
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
ds

−
n−1∑
j=0

dj
j!

(t−a)j −
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
ds‖

≤
n−1∑
j=0

‖cj −dj‖
j!

(b−a)j



Existence and Uniqueness of solution via S-Iteration

+
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)

−F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
‖ds

≤ M +
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk(s) −zk(s)‖ +

∫ s
a

q(σ)‖zk(σ) −zk(σ)‖dσ
]
ds.

(23)

Recalling the equations (12) and (13), the above inequality becomes

‖yk+1 −yk+1‖B ≤ M + Θ‖zk −zk‖B, (24)

and similarly, it is seen that

‖zk −zk‖B ≤ ξkM +
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B. (25)

Therefore, using (25) in (24) and using hypothesis Θ < 1, and 1
2
≤ ξk for all

k ∈ N∪{0}, the resulting inequality becomes

‖yk+1 −yk+1‖B ≤ M + ‖zk −zk‖B
≤ M + ξkM +

[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B

≤ 2ξkM + ξkM +
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B

≤
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B + ξk

(
1 − Θ

) 3M(
1 − Θ

). (26)
We denote

βk = ‖yk −yk‖B ≥ 0,

µk = ξk

(
1 − Θ

)
∈ (0, 1),

γk =
3M(

1 − Θ
) ≥ 0.

The assumption 1
2
≤ ξk for all k ∈ N ∪{0} implies

∞∑
k=0

ξk = ∞. Now, it can be

easily seen that (26) satisfies all the conditions of Lemma 2.2 and hence, we have

0 ≤ lim sup
k→∞

βk ≤ lim sup
k→∞

γk



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

⇒ 0 ≤ lim sup
k→∞
‖yk −yk‖B ≤ lim sup

k→∞

3M(
1 − Θ

)
⇒ 0 ≤ lim sup

k→∞
‖yk −yk‖B ≤

3M(
1 − Θ

). (27)
Using the assumptions, lim

k→∞
yk = y, lim

k→∞
yk = y, we get from (27) that

‖y −y‖B ≤
3M(

1 − Θ
), (28)

which shows that the dependency of solutions of the equations (1)-(2) and (1) with
the initial conditions (20) on given initial data.

4.2 Closeness of solution via S−iteration
Consider the problem (1)-(2) and the corresponding problem

(
Dα∗a

)
y(t) = F

(
t,y(t),

∫ t
a

h
(
s,y(s)

)
ds
)
, (29)

for t ∈ I = [a,b], n− 1 < α ≤ n, n ∈ N, with the given initial conditions

y(j)(a) = dj, j = 0, 1, 2, · · · ,n− 1, (30)

where F is defined as F and dj (j = 0, 1, 2, . . . ,n−1) are given elements in X.

Then looking at the steps as in the proof of Theorem 3.1, we define the operator
for the equations (29)- (30)

(Ty)(t) =
n−1∑
j=0

dj
j!

(t−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ
)
ds, t ∈ I. (31)

The next theorem deals with the closeness of solutions of the problems (1)-(2) and
(29)-(30).

Theorem 4.2. Consider the sequences {yk}∞k=0 and {yk}
∞
k=0 generated normal

S− iterative method associated with operators T in (9) and T in (31), respectively
with the real sequence {ξk}∞k=0 in [0, 1] satisfying

1
2
≤ ξk for all k ∈ N ∪{0}.

Assume that



Existence and Uniqueness of solution via S-Iteration

(i) all conditions of Theorem 3.1 hold, and y(t) and y(t) are solutions of (1)-
(2) and (29)-(30) respectively,

(ii) there exist non negative constant � such that

‖F
(
t,u1,u2

)
−F

(
t,u1,u2

)
‖≤ �, ∀ t ∈ I. (32)

If the sequence {yk}
∞
k=0 converges to y, then we have

‖y −y‖B ≤
3
[
M +

�(b−a)α
Γ(α+1)

]
(

1 − Θ
) . (33)

Proof. From iteration (4) and equations (9); (31) and hypotheses, we obtain

‖yk+1(t) −yk+1(t)‖
= ‖(Tzk)(t) − (Tzk)(t)‖

= ‖
n−1∑
j=0

cj
j!

(t−a)j +
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
ds

−
n−1∑
j=0

dj
j!

(t−a)j −
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
ds‖

≤
n−1∑
j=0

‖cj −dj‖
j!

(b−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)

−F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
‖ds

≤ M +
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)

−F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
‖ds

+
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)

−F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ
)
‖ds

≤ M +
1

Γ(α)

∫ t
a

(t−s)α−1�ds



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

+
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk(s) −zk(s)‖ +

∫ s
a

q(σ)‖zk(σ) −zk(σ)‖dσ
]
ds

≤ M +
�(b−a)α

Γ(α + 1)

+
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk(s) −zk(s)‖ +

∫ s
a

q(σ)‖zk(σ) −zk(σ)‖dσ
]
ds.

(34)

Recalling the derivations obtained in equations (12) and (13), the above inequality
becomes

‖yk+1 −yk+1‖B ≤ M +
�(b−a)α

Γ(α + 1)
+ Θ‖zk −zk‖B, (35)

and similarly, it is seen that

‖zk −zk‖B ≤ ξk
[
M +

�(b−a)α

Γ(α + 1)

]
+
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B. (36)

Therefore, using (36) in (35) and using hypothesis Θ < 1, and 1
2
≤ ξk for all

k ∈ N, the resulting inequality becomes

‖yk+1 −yk+1‖B

≤
[
M +

�(b−a)α

Γ(α + 1)

]
+ ‖zk −zk‖B

≤
[
M +

�(b−a)α

Γ(α + 1)

]
+ ξk

[
M +

�(b−a)α

Γ(α + 1)

]
+
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B

≤ 2ξk
[
M +

�(b−a)α

Γ(α + 1)

]
+ ξk

[
M +

�(b−a)α

Γ(α + 1)

]
+
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B

≤
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B + ξk

(
1 − Θ

)3[M + �(b−a)α
Γ(α+1)

]
(

1 − Θ
) . (37)

We denote

βk = ‖yk −yk‖B ≥ 0,

µk = ξk

(
1 − Θ

)
∈ (0, 1),

γk =
3
[
M +

�(b−a)α
Γ(α+1)

]
(

1 − Θ
) ≥ 0.



Existence and Uniqueness of solution via S-Iteration

The assumption 1
2
≤ ξk for all k ∈ N ∪{0} implies

∞∑
k=0

ξk = ∞. Now, it can be

easily seen that (37) satisfies all the conditions of Lemma 2.2 and hence, we have

0 ≤ lim sup
k→∞

βk ≤ lim sup
k→∞

γk

⇒ 0 ≤ lim sup
k→∞
‖yk −yk‖B ≤ lim sup

k→∞

3
[
M +

�(b−a)α
Γ(α+1)

]
(

1 − Θ
)

⇒ 0 ≤ lim sup
k→∞
‖yk −yk‖B ≤

3
[
M +

�(b−a)α
Γ(α+1)

]
(

1 − Θ
) . (38)

Using the assumptions, lim
k→∞

yk = y, lim
k→∞

yk = y, we get from (38) that

‖y −y‖B ≤
3
[
M +

�(b−a)α
Γ(α+1)

]
(

1 − Θ
) , (39)

which shows that the dependency of solutions of IVP (1)-(2) on the function in-
volved on the right hand side of the given equation.

Remark: The inequality (39) relates the solutions of the problems (1)-(2) and
(29)-(30) in the sense that, if F and F are close as � → 0, then not only the solu-
tions of the problems (1)-(2) and (29)-(30) are close to each other (i.e. ‖y−y‖B →
0), but also depends continuously on the functions involved therein and initial
data.

4.3 Dependence on Parameters
We next consider the following problems

(
Dα∗a

)
y(t) = F

(
t,y(t),

∫ t
a

h
(
s,y(s)

)
ds,µ1

)
, (40)

for t ∈ I = [a,b], n− 1 < α ≤ n, n ∈ N, with the given initial conditions

y(j)(a) = cj, j = 0, 1, 2, · · · ,n− 1, (41)

and (
Dα∗a

)
y(t) = F

(
t,y(t),

∫ t
a

h
(
s,y(s)

)
ds,µ2

)
, (42)



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

for t ∈ I = [a,b], n− 1 < α ≤ n, n ∈ N, with the given initial conditions

y(j)(a) = dj, j = 0, 1, 2, · · · ,n− 1, (43)

where F : I × X × X × R → X is continuous function, cj, dj (j =
0, 1, 2, . . . ,n− 1) are given elements in X and constants µ1, µ2 are real parame-
ters.

Let y(t), y(t) ∈ B and following steps from the proof of Theorem 3.1, define the
operators for the equations (40) and (42), respectively

(Ty)(t) =
n−1∑
j=0

cj
j!

(t−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ,µ1

)
ds, t ∈ I;

(44)

and

(Ty)(t) =
n−1∑
j=0

dj
j!

(t−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,y(s),

∫ s
a

h
(
σ,y(σ)

)
dσ,µ2

)
ds, t ∈ I.

(45)

The following theorem states the continuous dependency of solutions on parame-
ters.

Theorem 4.3. Consider the sequences {yk}∞k=0 and {yk}
∞
k=0 generated normal

S− iterative method associated with operators T in (44) and T in (45), respec-
tively with the real sequence {ξk}∞k=0 in [0, 1] satisfying

1
2
≤ ξk for all k ∈ N∪{0}.

Assume that

(i) y(t) and y(t) are solutions of (40)-(41) and (42)-(43) respectively,

(ii) there exist functions p, r ∈ C(I,R+) such that

‖F
(
t,u1,u2,µ1

)
−F

(
t,v1,v2,µ1

)
‖≤ p(t)

[
‖u1 −v1‖ + ‖u2 −v2‖

]
,

and

‖F
(
t,u1,u2,µ1

)
−F

(
t,u1,u2,µ2

)
‖≤ r(t)

∣∣∣µ1 −µ2∣∣∣.



Existence and Uniqueness of solution via S-Iteration

If the sequence {yk}
∞
k=0 converges to y, then we have

‖y −y‖B ≤
3
[
M + |µ1 −µ2|Iaαr(t)

]
(

1 − Θ
) , (46)

where Θ = Ia
αp(t)

(
1 + (b−a)Q

)
< 1, t ∈ I.

Proof. From iteration (4) and equations (44); (45) and hypotheses, we obtain

‖yk+1(t) −yk+1(t)‖
= ‖(Tzk)(t) − (Tzk)(t)‖

= ‖
n−1∑
j=0

cj
j!

(t−a)j +
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ1

)
ds

−
n−1∑
j=0

dj
j!

(t−a)j −
1

Γ(α)

∫ t
a

(t−s)α−1F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ2

)
ds‖

≤
n−1∑
j=0

‖cj −dj‖
j!

(b−a)j

+
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ1

)
−F

(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ2

)
‖ds

≤ M +
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ1

)
−F

(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ1

)
‖ds

+
1

Γ(α)

∫ t
a

(t−s)α−1‖F
(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ1

)
−F

(
s,zk(s),

∫ s
a

h
(
σ,zk(σ)

)
dσ,µ2

)
‖ds

≤ M +
1

Γ(α)

∫ t
a

(t−s)α−1r(s)|µ1 −µ2|ds

+
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk(s) −zk(s)‖ +

∫ s
a

q(σ)‖zk(σ) −zk(σ)‖dσ
]
ds

≤ M + |µ1 −µ2|Iaαr(t)



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

+
1

Γ(α)

∫ t
a

(t−s)α−1p(s)
[
‖zk(s) −zk(s)‖ +

∫ s
a

q(σ)‖zk(σ) −zk(σ)‖dσ
]
ds.

(47)

Recalling the derivations obtained in equations (12) and (13), the above inequality
becomes

‖yk+1 −yk+1‖B ≤ M + |µ1 −µ2|Ia
αr(t) + Θ‖zk −zk‖B, (48)

and similarly, it is seen that

‖zk −zk‖B ≤ ξk
[
M + |µ1 −µ2|Iaαr(t)

]
+
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B.

(49)

Therefore, using (49) in (48) and using hypothesis Θ < 1, and 1
2
≤ ξk for all

k ∈ N∪{0}, the resulting inequality becomes

‖yk+1 −yk+1‖B
≤
[
M + |µ1 −µ2|Iaαr(t)

]
+ ‖zk −zk‖B

≤
[
M + |µ1 −µ2|Iaαr(t)

]
+ ξk

[
M + |µ1 −µ2|Iaαr(t)

]
+
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B

≤ 2ξk
[
M + |µ1 −µ2|Iaαr(t)

]
+ ξk

[
M + |µ1 −µ2|Iaαr(t)

]
+
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B

≤
[
1 − ξk

(
1 − Θ

)]
‖yk −yk‖B + ξk

(
1 − Θ

)3[M + |µ1 −µ2|Iaαr(t)](
1 − Θ

) . (50)
We denote

βk = ‖yk −yk‖B ≥ 0,

µk = ξk

(
1 − Θ

)
∈ (0, 1),

γk =
3
[
M + |µ1 −µ2|Iaαr(t)

]
(

1 − Θ
) ≥ 0.

The assumption 1
2
≤ ξk for all k ∈ N ∪{0} implies

∞∑
k=0

ξk = ∞. Now, it can be

easily seen that (50) satisfies all the conditions of Lemma 2.2 and hence we have

0 ≤ lim sup
k→∞

βk ≤ lim sup
k→∞

γk



Existence and Uniqueness of solution via S-Iteration

⇒ 0 ≤ lim sup
k→∞
‖yk −yk‖B ≤ lim sup

k→∞

3
[
M + |µ1 −µ2|Iaαr(t)

]
(

1 − Θ
)

⇒ 0 ≤ lim sup
k→∞
‖yk −yk‖B ≤

3
[
M + |µ1 −µ2|Iaαr(t)

]
(

1 − Θ
) . (51)

Using the assumption lim
k→∞

yk = y, lim
k→∞

yk = y, we get from (51) that

‖y −y‖B ≤
3
[
M + |µ1 −µ2|Iaαr(t)

]
(

1 − Θ
) , (52)

which shows the dependence of solutions of the problem (1)-(2) is on parameters
µ1 and µ2.

Remark: The result dealing with the property of a solution called “dependence
of solutions on parameters”. Here the parameters are scalars. Notice that the
initial conditions do not involve parameters. The dependence on parameters is an
important aspect in various physical problems.

5 Example
We consider the following problem:

(
Dα∗
)
y(t) =

3t

5

[t− sin(y(t))
2

+
1

9

∫ t
0

e−s

(2 + s)2
y(s)ds

]
, (53)

for t ∈ [0, 1], n− 1 < α ≤ n, n ∈ N, with the given initial conditions

y(j)(0) = cj, j = 0, 1, 2, · · · ,n− 1. (54)

Comparing this equation with the equation (1), we get F ∈ C(I ×R2,R), with

F
(
t,y(t),

∫ t
0

h(s,y(s))ds
)

=
3t

5

[t− sin(y(t))
2

+
1

9

∫ t
0

e−s

(2 + s)2
y(s)ds

]
and

h(t,y(t)) =
3t

45

e−t

(2 + t)2
y(t).



Haribhau L. Tidke, Gajanan S. Patil and Rupesh T. More

Now, one can easily show that∣∣∣F(t,y(t),z(t)) −F(t,y(t),z(t))∣∣∣
≤

3t

5

[1
2

∣∣∣ sin(y(t)) − sin(y(t))∣∣∣ + 1
9

∣∣∣z(t) −z(t)∣∣∣]
≤

3t

10

[∣∣∣y −y∣∣∣ + ∣∣∣z −z∣∣∣], (55)
and ∣∣∣h(t,y(t)) −h(t,z(t))∣∣∣ ≤ 3t

45

e−t

(2 + t)2

∣∣∣y −z∣∣∣, (56)
where p(t) =

3t

10
, and q(t) =

3t

45

e−t

(2 + t)2
. Therefore, we have

Q = sup
t∈[0,1]

{q(t)} =
3

180
=

1

60
.

Thus, we the estimate

Θ = Ia
αp(t)

(
1 + (b−a)Q

)
= Ia

α 3t

10

(
1 +

1

60

)
=

3

10

(
1 +

1

60

)
(Ia

α)(t)

=
61

200
(Ia

α)(t)

=
61

200

tα+1

Γ(α + 2)

≤
1

Γ(α + 2)
, (t ≤ 1). (57)

Therefore, the condition Θ < 1 is satisfied only if
1

Γ(α + 2)
< 1.

We define the operator T : B → B by

(Ty)(t) =
n−1∑
j=0

cj
j!
tj

+
1

Γ(α)

∫ t
0

(t−s)α−1
3s

5

[s− sin(y(s))
2

+
1

9

∫ s
0

e−σ

(2 + σ)2
y(σ)dσ

]
ds,

(58)

for t ∈ I. Since all conditions of Theorem 3.1 are satisfied and so by its con-
clusion, the sequence {yn} associated with the normal S−iterative method (4) for
the operator T in (58) converges to a unique solution y ∈ B.



Existence and Uniqueness of solution via S-Iteration

6 Conclusions

Firstly, we proved the main result, which address the existence and uniqueness
of the solution to the IVP (1)-(2) by the method of normal S−iteration. Next,
we discussed various properties of solutions like continuous dependence on the
initial data, closeness of solutions, and dependence on parameters and functions
involve therein. Finally, we provided an appropriate example to support all of the
findings.

Acknowledgement: The authors are very grateful to the referees for their com-
ments and remarks.

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