/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 12, No. 2, 2019, 432-447
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Monotone iterative technique and Ulam-Hyers stability

analysis for nonlinear fractional order differential

equations with integral boundary value conditions

Sajjad Ali1, Kamal Shah2,∗, Hassan Khan1, Muhammad Arif1, Shahid Mahmood3

3 Sarhad University of Science and IT, Peshawar, Khyber Pakhtunkhwa, Pakistan
1 Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan
2 University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan

Abstract. In this manuscript, the monotone iterative scheme has been extended to the nature of
solution to boundary value problem of fractional differential equation that consist integral boundary
conditions. In this concern, some sufficient conditions are developed in this manuscript. On
the base of sufficient conditions, the monotone iterative scheme combined with lower and upper
solution method for the existence, uniqueness, error estimates and various view plots of the extremal
solutions to boundary value problem of nonlinear fractional differential equations have been studied.
The obtain results have clarified the nature of the extremal solutions. Further, the Ulam–Hyers and
Ulam–Hyers–Rassias stability have been investigated for the considered problem. Two illustrative
examples of the BVP of the nonlinear fractional differential equations have been provided to justify
our contribution.

2010 Mathematics Subject Classifications: 26A33, 34A08, 34A45

Key Words and Phrases: Nonlinear Fractional differential equations, Iterative technique, Max-
imal and minimal solutions, Uniqueness and existence

1. Introduction

In the last few decades, the class of nonlinear fractional differential equations (NFDEs)
has been attracted the attentions of researchers very well. The area devoted to study
NFDEs has attracted many researcher in almost every field of science. We refer some pa-
pers in [1–8] about the applications. Various aspects like existence theory for the aforesaid
equations has been considered in plenty of research papers, see [9–12, 17, 18] for existence
and uniqueness of solutions to NFDEs.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3407

Email addresses: sajjad ali@sbbu.edu.pk (S. Ali),
kamalshah408@gmail.com (K. Shah), hassanmath@awkum.edu.pk (H. Khan),
marifmaths@awkum.edu.pk (M. Arif), shahidmahmood757@gmail.com (S. Mahmood)

http://www.ejpam.com 432 c© 2019 EJPAM All rights reserved.



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 433

The iterative technique with the method of upper and lower solutions has gained much at-
tention of most of the researcher in last few decades. It is very useful tools for the existence
and approximation of solutions to the initial and boundary value problems. This scheme
is well studied for the intial value problems, we refer to see the work in [13–16, 19–24].
However, The study of the scheme is on intial stage for integral boundary value problem.
A few authors have its study for NFDEs with boundary conditions. for instance, Khan
[25] have considered it for the following class of FDEs

{

CDpz(t) + h(t, z(t)) = 0, 1 < p < 2,

z′(t)|t=0 = 0, z(1) = δz(t), 4

where δ, t ∈ (0, 1).
The aforementioned method has also been applied to the study of a coupled system of the
NFDEs with three point boundary conditions by kamal[26] and his co-author.

Recently, the investigation of stability is also a key research area for the the develop-
ment of the fractional calculus. It an important field of fractional calculus. In this concern
many researchers have introduced numerous type of stability including exponential sta-
bility, Lyapunov stability, Ulam type stability analysis etc. Among them, the Ulam type
stability has attracted the attention of most of the authors. The Ulam type stability has
been well investigated for classical differential equations. In last few years, for initial value
problems of FDEs, the Ulam type stability has been well investigated in many articles
for detail see [27–31]. But, as far we know, its study is very limited for BVPs of FDEs.
Therefore inspired from the aforesaid work, we study the following NFDEs with integral
boundary value conditions as

CDpz(s) +Q(s, z(s)) = 0, s ∈ [0, 1], 2 < p ≤ 3,

z(0) = z′′(s)|s=0 = 0, z(1) = δ

∫ 1

0
z(s)ds,

(1)

where CDp denotes Caputo derivative, Q : I × R → R, 2 < p ≤ 3 and 0 < δ < 2 ,
z ∈ AC2[0, 1].

In comparison to the work [10, 11, 25, 26], we contribute the improvement of itera-
tive scheme together with upper and lower solution to study existence, uniqueness, error
estimates and plotting of the iterative extremal solutions of nonlinear fractional differen-
tial equations with integral boundary conditions (1). In addition, we investigate stability
analysis for the solution of nonlinear fractional differential equations which not established
in the work of [10, 11, 25, 26] . Two illustrative examples of problem (1) have been con-
sidered to demonstrate our existence theory. Behavior of the upper and lower solutions of
proposed examples are plotted as Figure 1 and 2 via using Matlab which have not been
done in the work of cited above.

2. Preliminaries

We provide the following few preliminary results, we refer to see [2–5, 17–19].



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 434

Definition 1. The Riemann-Liouville integral of fractional order δ ∈ R+ of a function φ
is defined by

Ipφ(s) =
1

Γ(p)

∫ s

0
(s− γ)p−1φ(γ)dγ,

provided the right side exits.

Definition 2. [3] The Caputo fractional order derivative of order p of function φ is defined
by

CDpφ(s) =
1

Γ(n− p)

∫ s

0
(s− γ)n−p−1φ(n)(γ)dγ,

provided that integral on the right is pointwise defined on (0,∞), where n = ⌊p⌋+1 and
⌊p⌋ denotes the greatest integer which is less than or equal to the real number p.

Definition 3. [33] Let V = C[0, 1] be the Banach space endowed with ‖z‖ = maxs∈[0,1] |z(s)|
which satisfies the partially ordering and let U = [z1, z2] with z1 ≤ z2 be a set such that
U ⊂ V ,
Then the operator T : U → V is known as increasing function if for each z1, z2 ∈ U and
z1 ≤ z2 gives Tz1 ≤ Tz2. The operator T is known as decreasing function if for each
z1, z2 ∈ U and z1 ≤ z2 gives Tz1 ≥ Tz2.

Definition 4. [33] Let (I − T )z1 ≤ 0, then z1 ∈ U is known as a lower solution of
(I−T )z = 0 and (I−T )z2 ≥ 0, then z2 ∈ U is known as an upper solution of (I−T )z = 0.

Lemma 1. [18] Let W be Banach Space with satisfying partially order and zn, yn ∈ U so
that zn ≤ yn, n ∈ Z+. If zn → z and yn → y, then z ≤ y.

Assume that

(C1) Q : [0, 1]× R → R satisfies Caratheodory conditions ;

(C2) Q(s, z) is increasing with respect to z for each s ∈ [0, 1];

(C3) There is a constant B > 0 so that 0 ≤ Q(s, z1(s))−Q(s, z2(s) ≤ B(z1 − z2).

Lemma 2. [32] Let p > 0, then

z (s) =

⌊p⌋
∑

i=0

zi (0)

i!
si (2)

is the solution of fractional differential equation

CDpz (s) = 0. (3)

Lemma 3. [32] Let p > 0, then in view of Definition 24 and Lemma 2, the solution of
the solution of fractional differential equation

CDpz (s) = y(s) (4)

is given by

z (s) = Ipy(s) +

⌊p⌋
∑

i=0

zi (0)

i!
si. (5)



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 435

3. Iterative Solutions

This portion of the manuscript is devoted to existence and uniqueness of extremal
solutions to the considered BVP of the nonlinear FDE (1). Further, we study the iterative
technique coupled with method of the lower and upper solutions for the approximation
and error estimates to the extremal solution of (1). We also investigate the various type
stabilities like Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias
stability and generalized Ulam-Hyers-Rassias stability for the considered BVP. First, we
convert our considered problem into the integral equation.

Lemma 4. If Q ∈ C[0, 1], then the linear problem

CDpz(s) +Q(s) = 0, s ∈ [0, 1], 2 < p ≤ 3,

z(0) = z′′(s)|s=0 = 0, z(1) = δ

∫ 1

0
z(s)ds.

(6)

has a solution defined by

z(s) =

∫ 1

0
R(t, γ)Q(γ)dγ, (7)

where R(s, γ) is defined by

R (s, γ) =
1

(2− δ) Γ (p+ 1)

{

2s(1− γ)p−1 (p− δ + δγ)− (2− δ) p(s− γ)p−1, 0 ≤ γ ≤ s ≤ 1,

2s(1− γ)p−1 (p− δ + δγ) , 0 ≤ s ≤ γ ≤ 1.
(8)

Proof. We may apply a well known Lemma 3 to reduce the equation (6) to equivalent
integral form

z (s) = −IpQ (s) +
2

∑

i=0

z(i) (0)

i!
si = −

s
∫

0

(s− γ)p−1

Γ (p)
Q (γ) dγ +

2
∑

i=0

z(i) (0)

i!
si. (9)

By considering the given conditions z(0) = z
′′

(0) = 0, z (1) = δ
1
∫

0

z (s) ds , we can obtain

that

z (s) = −

s
∫

0

(s− γ)p−1

Γ (p)
Q (γ) dγ + s

1
∫

0

(1− γ)p−1

Γ (p)
Q (γ) dγ + tδ

1
∫

0

z (s)ds. (10)



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 436

Let say

A =

1
∫

0

z (s)ds

A = −

1
∫

0

s
∫

0

(s− γ)p−1

Γ (p)
Q (γ) dγds+

1
∫

0

1
∫

0

s
(1− γ)p−1

Γ (p)
Q (γ) dγds+ δA

1
∫

0

sds

A = −
2

(2− δ)

1
∫

0

(1− γ)p

pΓ (p)
Q (γ) dγ +

1

(2− δ)

1
∫

0

(1− γ)p−1

Γ (p)
Q (γ) dγ.

Then equation (3.5) implies

z (s) = −
s
∫

0

(s−γ)p−1

Γ(p) Q (γ) dγ +
1
∫

0

s(1−γ)p−1

Γ(p) Q (γ) dγ − 2δ
(2−δ)

1
∫

0

s(1−γ)p

pΓ(p) Q (γ) dγ

+ δ
(2−δ)

1
∫

0

t (1−γ)p−1

Γ(p) Q (γ) dγ.

z (s) =

1
∫

0

K (s, γ)Q (γ) dγ.

Clearly, for all s, γ ∈ [0, 1], we have R(s, γ) ≥ 0.

Lemma 5. If we consider the Lemma 4 and condition (C1) with Q ∈ C([0, 1],R), then we
see that the integral equation

z(s) =

∫ 1

0
R(s, γ)Q(γ)dγ, (11)

is the solution of problem (1), where R(s, γ) is provided in the equation (8).

Lemma 6. For Green’s function R(s, γ), we have that

max
s∈[0,1]

∫ 1

0
R(s, γ)dγ ≤

2

(2− δ)Γ(p)
, s ∈ [0, 1].

Proof. Since R(s, γ) ≤ 2t(1−γ)p−1(p−δ+δγ)
(2−δ)Γ(p+1) , it follows that

max
s∈[0,1]

∫ 1

0
R(s, γ)dγ ≤

2

(2− δ)Γ(p)
.

We define the operator T : U → U by

Tz(s) =

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ. (12)



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 437

Then, in view of (11) and (12), we obtain the operator equation

(Iz − Tz)(s) = 0, s ∈ [0, 1]. (13)

Let consider the condition (C2), then for z1, z2 ∈ U with z1 ≤ z2, we obtain

Tz1(s) =

∫ 1

0
R(s, γ)Q

(

γ, z1(γ)
)

dγ

≤

∫ 1

0
R(s, γ)Q

(

γ, z2(γ)
)

dγ = Tz2(s)

(14)

that is, T is nondecreasing.
We can assume another condition which is

(C4) For minimal and maximal solutions α, β ∈ U of the equation (13), the inequality
α ≤ β on [0, 1] is obvious .

Lemma 7. Under the conditions (C1)− (C4) the iterative solutions of operator equation
(13) is the monotonic sequence which converges to the solution of equation (11).

Proof. We consider that the conditions (C1) − (C4) hold, then we show that T is
continuous. For this, let z1, z2 ∈ U , then

|Tz1(s)− Tz2(s)|

=

∣

∣

∣

∣

∫ 1

0
R(s, γ)Q

(

γ, z1(γ)
)

dγ −

∫ 1

0
R(s, γ)Q

(

γ, z2(γ)
)

dγ

∣

∣

∣

∣

=

∣

∣

∣

∣

∫ 1

0

[

R(s, γ)Q
(

γ, x1(γ)
)

−R(s, γ)Q
(

γ, z2(γ)
)]

dγ

∣

∣

∣

∣

≤

∫ 1

0
R(s, γ)

∣

∣

∣

∣

Q
(

γ, z1(γ
)

−Q
(

γ, z2(γ)
)

∣

∣

∣

∣

dγ

≤
2B

(2− δ)Γ(p)
‖z1 − z2‖.

Thus T is continuous. It is easy to prove that T is also uniformly bounded and equi-
continuous . Thank to the Arzela Ascoli theorem’s statement: “If M be a family (finite or
infinite) of an equi-continuous, uniformly bounded real valued functions z on an interval
[0, 1] . Then M contains a uniformly convergent sequence of functions zn , converging
to a function z in U as n → ∞, where U denotes the space of all continuous bounded
functions on [0, 1]. Thus any sequence in M contains a uniformly bounded convergent
subsequence on [0, 1] and consequently M has a compact closure in U”. We obtain T is
compact. Further, we may choose z0 = α. then, in the view of C4, we obtain

α ≤ β

z0 ≤ β



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 438

z0 ≤ Tz0 ≤ Tβ ≤ β, T is increasing

z0 ≤ z1 ≤ β on [0, 1], where z1 = Tz0.

Similarly,

Tz0 ≤ Tz1 ≤ Tβ ≤ β, that is, z1 ≤ z2 ≤ β on [0, 1], where z2 = Tz1.

In same way, we obtain the bounded monotone sequence {zn}, that is

z0 ≤ z1 ≤ z2 ≤ ...zn−1 ≤ zn ≤ β on [0, 1], (15)

where zn = Tzn−1. Then, there will be z ∈ U such that zn → z as n → ∞. Hence, we
obtain that z = Tz. But z is solution of the equation (11) given by

z(s) =

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ, s ∈ [0, 1].

Let consider the condition (C3), then for z, y ∈ U with z ≤ y, we have the inequality

‖Ty − Tz‖ ≤ ∆‖y − z‖,where ∆ =
2B

(2− δ)Γ(p)
. (16)

Hence in view of (15) and (16), we get

‖z2 − z1‖ = ‖Tz − Tz0‖ ≤ ∆e1,

‖z3 − z2‖ = ‖Tz2 − Tz0‖ ≤ ∆2e1,

‖z4 − z3‖ = ‖Tz3 − Tz2‖ ≤ ∆3e1,

...

‖zn+1 − zn‖ = ‖Tzn − Tz0‖ ≤ ∆ne1.

Therfore, we can obtain

‖zm+n−zn‖ ≤ ‖zn+p−zn+p−1‖ + ‖zn+m−1−zn+m−2‖ + . . . ‖zn+1−zn‖ ≤ ∆n 1−∆m

1−∆
e1.

(17)
where m,n are positive integers and ∆ < 1. Now if n→ ∞, then we obtain that ‖zm+n −
zn‖ → 0. Hence {zn} is Cauchy sequence in U . Thus z∗(s) = limn→∞ zn(s). Therefore,
we obtain that Tz∗ = z∗. Let m→ ∞ in the equation (17), then estimate of error for the
minimal solution is provided by

en = ‖z∗ − zn‖ ≤
∆n

1−∆
e1,where e1 = ‖z1 − z0‖.

Remark. If we consider that z0 = β, then we have that {zn} is Cauchy sequence such
that z0 ≥ z1 ≥ z2 ≥ ...zn−1 ≥ zn ≥ α on [0, 1],



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 439

which converges to the solution of the equation (11). Hence, we obtain corresponding
estimate of error for the maximal solutions which is provided by e∗n = ‖z∗n − z̄∗‖ ≤
∆n

1−∆e
∗
1,where e

∗
1 = ‖z∗0 − z

∗
1‖. In the view of Lemma 7, we can fix the iterative schemes for

NFDE (1) as

zm(s) =

∫ 1

0
R(s, γ)Q (γ, zm−1) dγ, m ≥ 1,

z∗m(s) =

∫ 1

0
R(s, γ)Q

(

γ, z∗m−1

)

dγ, m ≥ 1,

from which, we obtain z∗(s) = lim
m→∞

zm(s)

z̄∗(s) = lim
m→∞

z∗m(s).

Theorem 1. Under the assumptions (C1), (C2) and (C3) with ∆ < 1, there exist unique
lower and upper solutions to the problem (1).

Proof. Proof is similar as given in [33].

4. Ulam type stability

In this section, we develop sufficient conditions for the Ulam stability analysis of the
solutions to the considered BVP of FDEs (1). The required definitions and lemmas can
be found in [27–31, 34–36].

Definition 5. If, we have cK ∈ R
+ and for every ε > 0 such that for any solution

z ∈ AC2[0, 1] of
|CDpz(s) +Q(s, z(s))| ≤ ε, s ∈ [0, 1], (18)

there exists v ∈ AC2[0, 1] is a unique solution of equation (1) such that

|z(s)− v(s)| ≤ cKε, s ∈ [0, 1].

Then BVP (1) is said to be Ulam–Hyers stable.

Definition 6. If, we have ϕ ∈ C(R+,R+), ϕ(0) = 0 and for every ε > 0 so that for any
solution z ∈ AC2[0, 1] of

|CDpz(s) +Q(s, z(s))| ≤ ε, s ∈ [0, 1], (19)

there exists v ∈ AC2[0, 1] is a unique solution of the considered problem (1) such that

|z(s)− v(s)| ≤ φ(s)ε, s ∈ [0, 1].

Then BVP of FDEs (1) is said to be generalized Ulam–Hyers stable.



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 440

Definition 7. If, we have φ ∈ C([0, 1],R+), cK > 0 and for every ε > 0 such that for any
solution z ∈ AC2[0, 1] of inequality

|CDpz(s) +Q(s, z(s))| ≤ εφ(s), s ∈ [0, 1], (20)

there exists v ∈ AC2[0, 1] is a unique solution of considered problem (1) such that

|z(s)− v(s)| ≤ cKεφ(s), s ∈ [0, 1].

Then BVP(1)is said to be the Ulam–Hyers–Rassias stable with respect to φ ∈ C([0, 1],R+).

Definition 8. If, we have φ ∈ C([0, 1],R) and cφ ∈ R
+ so that for any solution z ∈

AC2[0, 1] of inequality

|CDpz(s) +Q(s, z(s))| ≤ φ(s), s ∈ [0, 1], (21)

there exists v ∈ AC2[0, 1] is a unique solution of considered problem (1) so that

|z(s)− v(s)| ≤ cφφ(s), s ∈ [0, 1].

Then BVP(1) is said to be generalized Ulam–Hyers–Rassias stable with respect to φ ∈
C([0, 1],R).

Remark 1. Let z ∈ AC2[0, 1] is the solution of inequality (19) whenever, there exists the
function ψ ∈ C([0, 1],R) (dependent on z), so that

(i) CDpz(s) +Q(s, z(s)) = ψ(s), s ∈ [0, 1];

(ii) |ψ(s)| ≤ ε, for all s ∈ [0, 1].

Theorem 2. If consider the conditions (C2), (C3) coupled with δ < 1, then the Ulam-
Hyers stability results for the solutions of BVP (1) are obtained which further implies that
the solutions of BVP (1) are generalized Ulam-Hyers stable.

Proof. Let consider the conditions (C2),(C3) coupled with δ < 1. Let z ∈ AC2[0, 1] be
any solution of BVP (1). Then Using the condition (i) of the Remark 1 for z ∈ AC2[0, 1],
which is

CDpz(s) +Q(s, z(s)) = ψ(s), for all s ∈ [0, 1], 2 < p ≤ 3.

Thank to Lemma 4 , we obtain

z(s) =

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ +

∫ 1

0
R(s, γ)ψ(γ)dγ, where ψ ∈ C([0, 1],R).

Which yields

|z(s)−

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ| ≤

2ε

(2− δ)Γ(p)
, s ∈ [0, 1]. (22)



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 441

Let v ∈ AC2[0, 1] be a unique solution of BVP (1).Then

|z(s)− v(s)|

=

∣

∣

∣

∣

z(s)−

∫ 1

0
R(s, γ)Q(γ, v(γ))dγ

∣

∣

∣

∣

=

∣

∣

∣

∣

z(s)−

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ +

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ −

∫ 1

0
R(s, γ)Q(γ, v(γ))dγ

∣

∣

∣

∣

≤

∣

∣

∣

∣

z(s)−

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ

∣

∣

∣

∣

+

∫ 1

0
|R(s, γ)|Q(γ, z(γ))−Q(γ, v(γ))|dγ.

By using the inequality (22), we obtain that

‖z − v‖ ≤
2ε

(2− δ)Γ(p)
+

2B

(2− δ)Γ(p)
‖z − v‖,

hence we have ‖z − v‖ ≤ cKε,where cK =
2

(2− δ)Γ(p)− 2B
> 0.

Hence the solutions of BVP (1) are Ulam-Hyers stable. When, we substitute φ(ε) = cKε,

φ(0) = 0. Then consequently generalized Ulam-Hyers stability for the solutions of BVP
(1) are obtained.

Remark 2. Let z ∈ AC2[0, 1] is the solution of inequality (20) whenever there exists the
function ψ ∈ C([0, 1],R) (dependent on z) so that

(i) CDpz(s) +Q(s, z(s)) = ψ(s), s ∈ [0, 1];

(ii) |ψ(s)| ≤ εφ(s), for all s ∈ [0, 1], where φ ∈ C([0, 1],R).

By Using the Remark 2, Let z ∈ AC2[0, 1] be any solution of BVP (1) such that

CDpz(s) +Q(s, z(s)) = ψ(s), s ∈ [0, 1]. (23)

Then, thank to Lemma 4, we have

z(s) =

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ +

∫ 1

0
R(s, γ)ψ(γ)dγ, where ψ ∈ C([0, 1],R)

which yields

|z(s)−

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ| ≤

2ε

(2− δ)Γ(p)
φ(s), s ∈ [0, 1]. (24)

Theorem 3. If consider the conditions (C2), (C3) coupled with δ < 1, then the Ulam-
Hyers-Rassias stability results for the solutions of BVP (1) are obtained which further
implies that the solutions of BVP (1) are generalized Ulam-Hyers-Rassias stable.



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 442

Proof. Let consider the conditions (C2),(C3) coupled with δ < 1. If v ∈ AC2[0, 1] is
the unique solution of BVP (1) and z ∈ AC2[0, 1] be any solution of BVP (1) which is
also the solution (24). Then, we have

|v(s)− z(s)|

=

∣

∣

∣

∣

∫ 1

0
R(s, γ)Q(γ, v(γ))dγ − z(s)

∣

∣

∣

∣

=

∣

∣

∣

∣

∫ 1

0
R(s, γ)Q(γ, v(γ))dγ −

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ +

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ − z(s)

∣

∣

∣

∣

≤

∫ 1

0
|R(s, γ)|Q(γ, v(γ))−Q(γ, z(γ))|dγ +

∣

∣

∣

∣

z(s)−

∫ 1

0
R(s, γ)Q(γ, z(γ))dγ

∣

∣

∣

∣

.

Which implies that

‖v − z‖ ≤
2B

(2− δ)Γ(p)
‖v − z‖+

2εφ(s)

(2− δ)Γ(p)
, (25)

from which we get ‖v − z‖ ≤ εcKφ(s),where cK =
2

(2− δ)Γ(p)− 2B
> 0. (26)

Therefore the solutions of BVP (1) are Ulam-Hyers-Rassias stable. When, we substitute
ϕ(ε) = cKεφ(s), ϕ(0) = 0. Then consequently generalized Ulam-Hyers-Rassias stability
for the solutions of BVP (1) are obtained.

5. Examples

In the concern section, we provide two examples of concern problem (1). We find iter-
ative solution for the extremal solution of the considered examples. We also provide error
estimates for the extremal solutions. Furthermore, stability analysis and error estimates
for the solution of examples of the problem (1) is also our concern. In addition, we provide
various plots view of every example of problem (1).

Example 1.










CD
5

2 z(s) +Q (s, z(s)) = 0, s ∈ [0, 1],

z(0) = z′′(s)
∣

∣

s=0
= 0, z(1) =

1

2

∫ 1

0
z(s)ds.

(27)

Where Q (s, z(s)) =
√

z(s) + sin2[t exp(z(s))] + log[1 + z(s)]. Let say n = 4 is large
enough.Then, approximate minimal and maximal solutions are given by

z(s) = z4(s) =

∫ 1

0
R(s, γ)Q (γ, z3(γ)) dγ,

z∗(s) = z∗4(s) =

∫ 1

0
R(s, γ)Q (γ, z∗3(γ)) dγ.

(28)



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 443

Let B = 0.00001 and the lower and upper solutions of BVP (1) are z0 = −0.2, and z∗0 =
0.2 respectively. Then we have ∆ = 2B

(2−δ)Γ(p) = 0.000010030 < 1. Therefore, the error
estimates are:

e3 = ‖z(s)− z3(s)‖ ≤
∆3

1−∆
× e1 ≤ 1.00904832686× 10−15 max

s∈[0,1]
|z1(s) + 0.2| ≃ 2.22668549047× 10−16,

e∗3 = ‖z(s)− z∗3(s)‖ ≤
∆3

1−∆
× e∗1 ≤ 1.00904832686× 10−15 max

s∈[0,1]
|0.2− z∗1(s)| ≃ 2.01809665372× 10−16.

The maximal and minimal solutions of the Example (1) are plotted in the Figure 1.
Clearly, the condition ∆ < 1 is sufficient for various kinds of Ulam stabilities for the

extremal solution of the Example (1). The stability of extremal solutions is also obvious
from plots in the given Figure 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.05

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.05

0.1

0.15

z
*

1
(s)=Maximal solution

z
1

(s)=Minimal solution

z
*

2
(s)=Maximal solution

z
2

(s)=Minimal solution

Figure 1 A line graph of the extremal solutions of the Example 1.

Example 2.










CD
11

4 z(s) +Q (s, z(s)) = 0, s ∈ [0, 1],

z(0) = z′′(s)
∣

∣

s=0
= 0, z(1) =

1

10

∫ 1

0
z(s)ds.

(29)

Where Q (s, z(s)) = t
√

z(s)+ exp(z(s). Let say n = 4 is large enough. Then, approximate
minimal and maximal solutions are given by

z(s) = z4(s) =

∫ 1

0
R(s, γ)Q (γ, z3(γ)) dγ,

z∗(s) = z∗4(s) =

∫ 1

0
R(s, γ)Q (γ, z∗3(γ)) dγ.

(30)



K. Shah et al. / Eur. J. Pure Appl. Math, 12 (2) (2019), 432-447 444

Let B = 0.00001 and the lower and upper solutions of BVP (2) are z0 = −0.5, and z∗0 = 0.5
respectively. Then, we have ∆ = 2B

(2−δ)Γ(p) = 6.5448e−06 < 1. Therefore, the corresponding
estimates of error are:

e3 = ‖z(s)− z3(s)‖ ≤
∆3

1−∆
× e1 ≤ 2.8034× 10−16 max

s∈[0,1]
|z1(s) + 0.5| ≃ 1.5469× 10−16,

e∗3 = ‖z(s)− z∗3(s)‖ ≤
∆3

1−∆
× e∗1 ≤ 2.8034× 10−16 max

s∈[0,1]
|0.5− z∗1(s)| ≃ 1.4017× 10−16.

The maximal and minimal solutions of the Example (2) are plotted in the Figure 2.
Clearly, the condition ∆ < 1 is sufficient for various kinds of Ulam stabilities for the

extremal solution of the Example (2). The stability of extremal solutions is also obvious
from plots in the given Figure 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.05

0.1

0.15

z
*

1
(s)=Maximal solution

z
1

(s)=Minimal solution

z
*

2
(s)=Maximal solution

z
2

(s)=Minimal solution

Figure. 2 A line graph of the extremal solutions of the Example 2.

6. Conclusion

We have successfully investigated sufficient conditions for maximal and minimal solu-
tions using monotone iterative schemes together with upper and lower solutions method.
We also provided maximum error estimates for the solutions. The obtained results were
verified by plotting the graph of the illustrative problem.

Acknowledgements

The authors would like to thank the reviewers of this paper for their valuable comments
on the earlier version of the paper. They would also like to acknowledge Prof. Dr. Salim
ur Rehman, V.C. Sarhad University of Science and I. T, for providing excellent research



REFERENCES 445

and academic environment.
Funding Source: We remark that this work has been supported by the HEC of Pakistan,
Grant No: 10039/KPK/ NRPU/R&D/HEC/ 2017 and HED of KPK, Government of
Pakistan at Grant No: HEREF-46
Authors contributions: All authors jointly worked on the results and they read and
approved the final manuscript.
Conflict of interest: It is declared that no conflict of interest exist regarding this paper.

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