/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 4, 2016, 346-359

ISSN 1307-5543 – www.ejpam.com

Generalized Monotone Iterative Method for Caputo Fractional

Integro-differential Equations

J. Vasundhara Devi∗, Ch. V. Sreedhar

GVP-Prof. V.Lakshmikantham Institute for Advanced Studies, Department of Mathematics,

GVP College of Engineering, Visakhapatnam, AP, India.

Abstract. Using coupled lower and upper solutions we develop the generalized monotone iterative

technique to solve Caputo fractional integro-differential equation of order q with periodic boundary

condition, via initial value problem (IVP) where 0 < q < 1. We construct monotone iterates which are

solutions of initial value problems associated with linear integro-differential equations, that are easy

to obtain. We show that these iterates converge uniformly and monotonically to coupled minimal and

maximal solutions of the problem considered. We have obtained explicit solution of the linear IVP of

Caputo fractional integro-differential equation.

2010 Mathematics Subject Classifications: 34A08,34K10,45K99

Key Words and Phrases: Caputo fractional integro-differential equation, linear integro-differential

equations, monotone iterative technique, maximal and minimal solutions

1. Introduction

The theory of fractional calculus [3, 8] is more than three centuries old and but its study

has been restricted mainly to mathematicians till a few decades ago. The book of Oldham

and Spanier [6] attracted the attention of many researchers and study of various areas us-

ing fractional derivatives quickly gained impetus. With the monograph published by Prof. V.

Lakshmikantham et al. [4], there has been extensive work in this area of research.

As the monotone iterative technique MIT combined with method of lower and upper solu-

tions offers a flexible mechanism to obtain a solution of the considered mathematical model,

this technique was developed in various setups [5] over the years. The interest in fractional

differential equations led to developing MIT for IVPs and BVPs. There have been several papers

[1] dealing with iterative techniques for systems involving fractional derivatives.

It is quite obvious that the study of IVPs is relatively simpler than the study of BVPs. De-

veloping iterative techniques for BVPs is quite cumbersome. At this stage Pandit et al. [7], ob-

tained the solution of a BVP using the monotone iterates of the corresponding IVP introduced

∗Corresponding author.

Email addresses: jvdevi@igmail.com (J. Devi), chaduvulasreedhar@gmail.com (Ch. Sreedhar)

http://www.ejpam.com 346 c© 2016 EJPAM All rights reserved.



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 347

the development of MIT of a PBVP through the MIT of a IVP. This work has been extended by

J. D. Ramirez and A. S. Vatsala for Caputo fractional differential equations in [9].

Wen-Li Wang and Jing Feng Tian proposed a different approach to obtain a unique solution

for the BVP in [10].

In this paper, we consider the PBVP of Caputo fractional integro differential equation and

obtain its solution through a sequence of iterates developed for the corresponding IVP.

2. Preliminaries

In this section, we state a few definitions, some properties of fractional derivatives and

recall required results pertaining to Caputo fractional integro differential equations which are

useful in proving the main result.

Consider the Caputo fractional integro-differential equation of the type

c Dqu= f (t,u, Iq(u)), (1)

with

u(0) = u0, (2)

where f , G ∈ C[J ×R×R+,R], u ∈ C1[J ,R], J = [0, T],

c Dqu(t) =
1

Γ(1− q)

∫ t

0

(t − s)−qu′(s)ds, (3)

and

Iq(u(t)) =
1

Γq

∫ t

0

(t − s)q−1u(s)ds. (4)

We start by stating a couple of lemmas from [3] related to IVPs of Riemann Liouville and

Caputo fractional derivative of order q.

Lemma 1. Let m(t) ∈ C1([0, T]R). If there exists t1 ∈ [0, T] such that m(t1) = 0 and m(t)≤ 0

on [0, T] then Dqm(t1)≥ 0.

Lemma 2. Let m(t) ∈ C1([0, T],R). If there exists t1 ∈ [0, T] such that m(t1) = 0 and m(t)≤ 0

on [0, T] then c Dqm(t1)≥ 0.

The following theorem, which is a new result, gives the explicit solution of the linear IVP

of Caputo fractional integro-differential equation, this is the generalization of the result in [8].

Theorem 1. If λ ∈ C1([0, T],R). The solution of c Dqλ(t) = L1λ(t) +M Iq(λ(t)) is given by

λ(t) = Σ∞n=0Σ
∞
k=0

2n+kM n Lk
1

n+kCk t(2n+1)q

Γ[(2n+ 1)q+ 1]

where L1, M > 0.



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 348

Proof. By hypothesis we have c Dqλ(t) = L1λ(t)+M Iq(λ(t)). Now by applying the Laplace

transform on both sides we get

sq ¯λ(s)− sq−1λ(0) =L1
¯λ(s) +Ms−q

sq ¯λ(s)[sq −Ms−q − L1] =λ(0)s
q−1

¯λ(s) =
sq−1

[sq −Ms−q − L1]
λ(0)

¯λ(s) =
sq−1

[s2q −M − L1sq]
λ(0)

L−1( ¯λ(s)) =L−1(
sq−1

[s2q −M − L1sq]
)λ(0)

λ(t) =Σ∞n=0Σ
∞
k=0

2n+kM n Lk
1

n+kCk t(2n+1)q

Γ[(2n+ 1)q+ 1]
λ(0).

Next we shall establish the following comparison theorem.

Theorem 2. Let J = [0, T], f ∈ C[J×R×R+,R], v, w ∈ C1[J ,R] and suppose that the following

inequalities hold, for all t ∈ J.

c Dqv(t)≤ f (t, v(t), Iq(v(t))), v(0)≤ u0, (5)
c Dqw(t)≥ f (t, w(t), Iq(w(t))), w(0)≥ u0. (6)

Suppose further that f (t,u(t), Iq(u(t))) satisfies the following Lipschitz-like condition,

f (t, x , Iq(x))− f (t, y, Iq(y))≤ L(x − y) +M(Iq(x)− Iq(y)), (7)

for x ≥ y, L, M > 0. Then, v(0)≤ w(0) implies that

v(t)≤ w(t), 0≤ t ≤ T. (8)

Proof. Assume without loss of generality that one of the inequalities in (5), (6) is strict,

say c Dqv(t)< f (t, v(t), Iq(v(t))) and v(0)< w(0), where v(0) = v0 and w(0) = w0. We claim

that v(t)< w(t) for t ∈ J .

Suppose there exists t1 such that 0< t1 ≤ T for which

v(t1) = w(t1), v(t)≤ w(t), for t < t1. (9)

If we set m(t) = v(t)−w(t). Then m(t1) = 0 and m(t) = v(t)−w(t)≤ 0 for t < t1.

Then by Lemma 2 we have c Dqm(t1)≥ 0. Thus

f (t1, v(t1), Iq(v(t1)))>
c Dqv(t1)

≥c Dqw(t1)

≥ f (t1, w(t1), Iq(w(t1))),



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 349

which is a contradiction. So v(t) < w(t) for t ∈ J . By assuming that the inequalities in (5)

and (6) are non strict, we now prove that v(t)≤ w(t). Set wε(t) = w(t) + ελ(t) where ε > 0

and

λ(t) = Σ∞n=0Σ
∞
k=0

2n+kM n Lkn+kCk t(2n+1)q

Γ[(2n+ 1)q+ 1]

is a solution of the Caputo fractional integro differential equation

c Dqλ(t) = 2Lλ(t) + 2M Iqλ(t)withλ(0) = 1.

We have wε(0) = w(0) + ε > w0 and wε(t) > w(t) for t ∈ J . Using (5), (6), and (7), we find

that

c Dqwε(t) =
c Dqw(t) + εc Dqλ(t)

≥ f (t, w(t), Iq(w(t))) + 2Lλ(t) + 2M Iq(λ(t))

≥ f (t, wε(t), Iq(wε(t)))− Lλ(t)−M Iq(λ(t)) + 2Lλ(t) + 2M Iq(λ(t))

≥ f (t, wε(t), Iq(wε(t))) + Lλ(t) +M Iq(λ(t))

> f (t, wε(t), Iq(wε)(t)),

for 0≤ t ≤ T . Applying the result for strict inequalities to v(t), wε(t) we obtain v(t)< wε(t)

for t ∈ J , for every ε > 0 and consequently as ε→ 0, we get that v(t)≤ w(t) for t ∈ J .

Corollary 1. Let m ∈ C1[J ,R] be such that

c Dqm(t)≤ Lm(t) +M Iq(m(t)), m(0) = m0 ≤ 1,

then

m(t)≤ λ(t),

for 0≤ t ≤ T and L, M > 0, λ(0) = 1,λ(t) = Σ∞n=0Σ
∞
k=0

2n+k M n Lkn+kCk t(2n+1)q

Γ[(2n+1)q+1]
.

Proof. We have c Dqm(t)≤ Lm(t) +M Iq(m(t)) and

c Dqλ(t) =2Lλ(t) + 2M Iq(λ(t))

≥Lλ(t) +M Iq(λ(t)),

for m(0) = m0 ≤ 1= λ(0). Hence from Theorem 2 we conclude that m(t)≤ λ(t) for t ∈ J .

The result of Corollary 1 is still true even if L = M = 0, which is given below.

Corollary 2. Let c Dqm(t)≤ 0 on [0, T]. If m(0)≤ 0 then m(t)≤ 0, t ≤ J .

Proof. By definition of c Dqm(t) and by hypothesis,

c Dqm(t) =
1

Γ(1− q)

∫ t

0

(t − s)−qm′(s)ds ≤ 0,

which implies that m′(t) ≤ 0 on [0, T]. Therefore m(t) ≤ m(0) ≤ 0 on [0, T]. The proof is

complete.



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 350

3. The Technique

In this section, we develop generalized monotone iterative technique to obtain a coupled

minimal and maximal solutions for the Caputo fractional integro-differential equation of the

form
c Dqu= F(t,u, Iq(u)) + G(t,u, Iq(u)), (10)

with the boundary condition

g(u(0),u(T )) = 0, (11)

where F, G ∈ C[J × R × R+,R], u ∈ C1[J ,R]. We begin with various definitions of coupled

lower and upper solutions of (10) and (11).

Definition 1. Let v0, w0 ∈ C1[J ,R]. Then v0 and w0 are said to be

(i) natural lower and upper solutions of (10), (11) if,

c Dqv0(t)≤F(t, v0(t), Iq(v0(t))) + G(t, v0(t), Iq(v0(t))), g(v0(0), v0(T ))≤ 0, (12)
c Dqw0(t)≥F(t, w0(t), Iq(w0(t))) + G(t, w0(t), Iq(w0(t))), g(w0(0), w0(T ))≥ 0, (13)

(ii) coupled lower and upper solutions of Type I of (10), (11) if

c Dqv0(t)≤F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq(w0(t))), g(v0(0), v0(T ))≤ 0, (14)
c Dqw0(t)≥F(t, w0(t), Iq(w0(t))) + G(t, v0(t), Iq(v0(t))), g(w0(0), w0(T ))≥ 0, (15)

(iii) coupled lower and upper solutions of Type II of (10), (11) if

c Dqv0(t)≤F(t, w0(t), Iq(w0(t))) + G(t, v0(t), Iq(v0(t))), g(v0(0), v0(T ))≤ 0, (16)
c Dqw0(t)≥F(t, v0(t), Iqv0(t)) + G(t, w0(t), Iqw0(t)), g(w0(0), w0(T ))≥ 0, (17)

(iv) coupled lower and upper solutions of Type III of (10), (11) if

c Dqv0(t)≤F(t, w0(t), Iq(w0(t))) + G(t, w0(t), Iq(w0(t))), g(v0(0), v0(T ))≤ 0, (18)
c Dqw0(t)≥F(t, v0(t), Iq(v0(t))) + G(t, v0(t), Iq(v0(t))), g(w0(0), w0(T ))≥ 0. (19)

We note that whenever v(t) ≤ w(t), t ∈ J , if F(t, x1, x2) is nondecreasing in x1 for each

(t, x2) ∈ J×R+ and is nondecreasing in x2 for each (t, x1) ∈ J×R, further, if G(t, y1, y2) is non

increasing in y1 for each (t, y2) ∈ J ×R+ and is non increasing in y2 for each (t, y1) ∈ J ×R,

then the lower and upper solutions defined by (12), (13) and those defined by (18) and (19)

reduce to (14), (15), and (16), (17) respectively. Hence it is sufficient to investigate the cases

(14), (15) and (16), (17).

Based on the concepts defined above, we now develop the monotone iterative technique for

the considered problem. To do so we use sequences of iterates which are solutions sequences

of IVP of linear Caputo fractional integro-differential equations. Since the solution of the

linear Caputo fractional differential equation is unique, the sequence of iterates is a unique



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 351

sequence converging to a solution of the considered problem defined by (10) and (11). In

this approach, we do not need to prove the existence of solution for BVP of nonlinear Caputo

fractional integro differential equation, as it follows from the construction of the monotone

sequences.

In the following theorem, we use coupled lower and upper solutions of Type I and obtain

monotone sequences which converge uniformly and monotonically to coupled minimal and

maximal solutions of the problem defined by (10) and (11).

Theorem 3. Suppose that

(A1) v0, w0 are coupled lower and upper solutions of Type I for problem defined by (10), (11)

with v0(t)≤ w0(t) on J,

(A2) the function g(u, v) ∈ C[R2,R] is nonincreasing in v for each u and there exists a constant

M > 0 such that

g(u1, v)− g(u2, v)≤ M(u1 − u2), (20)

for v0(0)≤ u2 ≤ u1 ≤ w0(0), v0(T )≤ v ≤ w0(T ),

(A3) F, G ∈ C[J ×R×R+,R] and F(t, x1, x2) is non-decreasing in x1 for each (t, x2) ∈ J ×R+
and is nondecreasing in x2 for each (t, x1) ∈ J ×R. Further, G(t, y1, y2) is nonincreasing

in y1 for each (t, y2) ∈ J ×R+ and is nonincreasing in y2 for each (t, y1) ∈ J ×R.

Then the iterative scheme given by

c Dqvn+1 =F(t, vn, Iq(vn)) + G(t, wn, Iq(wn)), (21)

vn+1(0) =vn(0)−
1

M
g(vn(0), vn(T )), (22)

c Dqwn+1 =F(t, wn, Iq(wn)) + G(t, vn, Iq(vn)), (23)

wn+1(0) =wn(0)−
1

M
g(wn(0), wn(T )), (24)

yields two monotone sequences {vn(t)} and {wn(t)} such that

v0 ≤ v1 ≤ . . .≤ vn ≤ wn ≤ . . .≤ w1 ≤ w0.

Further, vn → ρ and wn → r in C1[J ,R] uniformly and monotonically, such that ρ and r are

respectively the coupled minimal and maximal solutions of the problem defined by (10) and (11),

that is, ρ and r satisfy the coupled system

c Dqρ =F(t,ρ, Iq(ρ)) + G(t, r, Iq(r)),

g(ρ(0),ρ(T )) = 0,
c Dqr =F(t, r, Iq(r)) + G(t,ρ, Iq(ρ)),

g(r(0), r(T )) = 0.



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 352

Proof. Putting n= 0 in (21), (22), we get

c Dqv1(t) =F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq(w0(t))),

v1(0) =v0(0)−
1

M
g(v0(0), v0(T )).

Clearly the above IVP has a unique solution denoted by v1(t), t ∈ J . We use induction on n

to establish the relation v0 ≤ v1 ≤ . . . ≤ vn ≤ wn ≤ . . . ≤ w1 ≤ w0. We start by showing

v0 ≤ v1 ≤ w1 ≤ w0. For this set p(t) = v0(t)− v1(t), then

c Dqp(t) =c Dqv0(t)−
c Dqv1(t),

≤F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq(w0(t)))

− [F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq)(w0(t)))] = 0

and p(0) = v0(0)−v0(0)+
1
M g(v0(0), v0(T ))≤ 0. Thus the hypothesis of Corollary 2 is satisfied

and we conclude that p(t) ≤ 0 on J , and obtain. Similarly we can show that w1 ≤ w0 on J .

Next we consider p(t) = v1(t) − w1(t), then by adding and subtracting suitable terms, and

using the fact that F is nondecreasing in second and third variables, G is nonincreasing in

second and third variables, and by taking Caputo fractional derivative we arrive at,

c Dqp(t) =c Dqv1(t)−
c Dqw1(t),

=F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq(w0(t)))

− [F(t, w0(t), Iq(w0(t))) + G(t, v0(t), Iq(v0(t)))]

≤0,

and

p(0) =v0(0)−w0(0)−
1

M
[g(v0(0), v0(T ))− g(w0(0), w0(T ))

≤v0(0)−w0(0)− [v0(0)−w0(0)] = 0.

Hence by Corollary 2 we have p(t) ≤ 0 on J that is, v1(t) ≤ w1(t), on J . Thus the claim

v0 ≤ v1 ≤ w1 ≤ w0 on J is proved.

We now show that v1, w1 are the coupled lower and upper solutions of Type I for (10), (11),

Using the fact that v0 ≤ v1, w1 ≤ w0 and from the assumption (A3), proceeding as earlier we

obtain c Dqv1(t)≤ 0. Also,

g(v1(0), v1(T )) =g(v1(0), v1(T ))− g(v0(0), v0(T ))−M v1(0) +M v0(0)

≤M(v1(0)− v0(0))−M(v1(0)− v0(0)) = 0.

Similarly we can show that w1 satisfies reverse inequalities. Hence v1, w1 are coupled

lower and upper solutions of (10), (11).

Assume that vk−1 ≤ vk ≤ wk ≤ wk−1 on J for k > 1, where vk−1, vk are the solutions of

the IVP (21), (22) and wk−1, wk are the solutions of the IVP (23), (24) for n = k − 1, n = k

respectively. We claim that the following relation holds.

vk ≤ vk+1 ≤ wk+1 ≤ wk



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 353

on J . To prove this we take p(t) = vk(t) − vk+1(t), then by adding and subtracting suitable

terms and working as earlier we arrive at,

c Dqp(t) =c Dqvk(t)−
c Dqvk+1(t)

≤[F(t, vk−1(t), Iq(vk−1(t))) + G(t, wk−1(t), Iq(wk−1)(t))]

− [F(t, vk(t), Iq(vk(t))) + G(t, wk(t), Iq(wk(t)))]≤ 0.

Further, p(0) = vk(0) − vk+1(0) = vk(0) − vk(0) +
1
M g(vk(0), vk(T )) ≤ 0. An application of

Corollary 2 yields that p(t) ≤ 0 and consequently, vk(t) ≤ vk+1(t), on J . In a similar manner

we can prove that wk+1(t)≤ wk(t).

Next to prove vk+1(t) ≤ wk+1(t), on J , consider p(t) = vk+1(t)− wk+1(t) and again fol-

lowing the earlier approach we deduce that

c Dqp(t) =c Dqvk+1(t)−
c Dqwk+1(t)≤ 0

and

p(0) =vk+1(0)−wk+1(0)

=vk(0)−
1

M
g(vk(0), vk(T ))−wk(0) +

1

M
g(wk(0), wk(T ))

≤vk(0)−wk(0) +
1

M
[g(wk(0), wk(T ))− g(vk(0), vk(T ))≤ 0.

which yields p(t)≤ 0 on using Corollary 2. Thus we obtain two monotone sequences {vn} and

{wn} satisfying

v0 ≤ v1 ≤ . . .≤ vn ≤ wn ≤ . . .≤ w1 ≤ w0.

Now we claim that these sequences are equicontinuous and uniformly bounded. By hypothesis

both v0(t), w0(t) are bounded on [0, T] and the sequences {vn} and {wn} are such that

v0 ≤ v1 ≤ . . .≤ vn ≤ wn ≤ . . .≤ w1 ≤ w0.

Therefore {vn} and {wn} are uniformly bounded. Next we prove that {vn} is equicontinous.

To do so for given ε > 0 choose δ = (
(εΓ(q+1))

2M2
)

1
q . Next for t1, t2 ∈ J such that t2 > t1 consider

|vn(t1)− vn(t2)|

=|vn(0) +
1

Γq

∫ t1

0

(t1 − s)q−1[F(s, vn−1(s), Iq(vn−1(s)) + G(s, wn−1(s), Iq(wn−1(s)))]ds

− vn(0) +
1

Γq

∫ t2

0

(t2 − s)q−1[F(s, vn−1(s), Iq(vn−1(s))) + G(s, wn−1(s), Iq(wn−1(s)))]ds|

≤
1

Γq

∫ t1

0

[(t1 − s)q−1 − (t2 − s)q−1]|F(s, vn−1(s), Iq(vn−1(s))) + G(s, wn−1(s), Iq(wn−1(s)))|ds

+

∫ t2

t1

(t2 − s)q−1|F(s, vn−1(s), Iq(vn−1(s))) + G(s, wn−1(s), Iq(wn−1(s)))|ds



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 354

≤
M2

Γq
{

∫ t1

0

[(t1 − s)q−1 − (t2 − s)q−1]ds+

∫ t2

t1

(t2 − s)q−1ds}

≤
M2

Γq+ 1
[(t1)

q + (t2 − t1)
q − (t2)

q + (t2 − t1)
q]

≤
M2

Γq+ 1
[t

q

1
− t

q

2
] +

2M2

Γq+ 1
(t2 − t1)

q

≤
2M2

Γq+ 1
(t2 − t1)

q,

here we have used the fact that {vn}, {I
qvn}, {wn}, {I

qvn} are uniformly bounded and F(t, x1, x2),

G(t, y1, y2) are continuous on [0, T].

Thus for any given ε > 0 there exists δ > 0 independent of n such that for each n,

|vn(t1)− vn(t2)| < ε whenever δ = (
(εΓ(q+1))

2M2
)

1
q . Therefore {vn} is equicontinuous. Similarly

we can prove that {wn} is equicontinuous. Hence by Arzela-Ascoli’s theorem there exist sub-

sequences {vnk
} and {wnk

} which converge uniformly to ρ(t) and r(t) respectively. Since the

sequences are monotone, the entire sequences converge uniformly to ρ and r respectively on

J .

To prove that ρ and r are coupled minimal and maximal solutions of (10) and (11) re-

spectively, we need to show that if u is any solution of (10) and (11), such that v0 ≤ u ≤ w0,

then ρ ≤ u1, u2 ≤ r. Assume that there exists a positive integer n such that vn ≤ u≤ wn on J .

Then using the monotone nature of F , G we have

c Dqp(t) =c Dqvn+1(t)−
c Dqu(t),

≤[F(t, vn(t), Iq(vn(t))) + G(t, wn(t), Iq(wn(t)))]

− [F(t,u(t), Iq(u(t))) + G(t,u(t), Iq(u(t)))]
c Dqp(t)≤0

and

p(0) =vn+1(0)− u(0)

=vn(0)−
1

M
g(vn(0), vn(T ))− u(0)

≤vn(0)− u(0)−
1

M
[g(vn(0), vn(T ))− g(u(0),u(T ))]≤ 0

so vn+1(t)≤ u1(t), on J follows from Corollary 2. Similarly we can show that u(t)≤ wn+1(t),

on J . By applying induction on n we conclude that vn+1 ≤ u ≤ wn+1 on J . Taking limit as

n→∞, we get ρ ≤ u≤ r, t ∈ J . Hence

v0 ≤ v1 ≤ v2 ≤ . . .≤ vn ≤ . . .≤ ρ ≤ u≤ r ≤ . . .≤ wn ≤ . . . w1 ≤ w0,

on J , where ρ and r are coupled minimal and maximal solutions of (10) and (11). Thus the

proof is complete.



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 355

Remark 1.

(i) In Theorem 3, if G(t,u, Iq(u)) = 0, then we get a result when F is nondecreasing in first

and second variables,

(ii) If F(t,u, Iqu) = 0, in Theorem 3 then we obtain the results for G nonincreasing in first and

second variables.

Theorem 4. Assume that conditions (A1), (A2), and (A3) of Theorem 3 are true. Then for any

solution u(t) of (10), (11) with v0 ≤ u ≤ w0. on J, we have the iterates {v2n, w2n+1} and

{v2n+1, w2n} satisfying

v0 ≤ w1 ≤ . . .≤ v2n ≤ w2n+1 ≤ u≤ v2n+1 ≤ w2n ≤ . . .≤ v1 ≤ w0. (25)

for each n≥ 1 on J, Further more {v2n, w2n+1} → ρ and {v2n+1, w2n} → r in C1[J ,R] uniformly

and monotonically, such that ρ and r are coupled minimal and maximal solutions of (10) and

(11), respectively, that is, ρ ≤ u≤ r, ρ and r satisfy the coupled system

c Dqρ =F(t,ρ, Iq(ρ)) + G(t, r, Iq(r)),

g(ρ(0),ρ(T )) = 0,
c Dqr =F(t, r, Iq(r)) + G(t,ρ, Iq(ρ))

g(r(0), r(T )) = 0.

Proof. Consider the following IVP

c Dqvn+1 =F(t, wn, Iq(wn)) + G(t, vn, Iq(vn)), (26)

vn+1(0) =wn(0)−
1

M
g(wn(0), wn(T )), (27)

c Dqwn+1 =F(t, vn, Iq(vn)) + G(t, wn, Iq(wn)), (28)

wn+1(0) =vn(0)−
1

M
g(vn(0), vn(T )), (29)

where v0 ≤ w0. Our aim is to show that the solutions vn+1, wn+1 of (26), (27), and (28), (29)

satisfy

v0 ≤ w1 ≤ . . .≤ v2n ≤ w2n+1 ≤ u≤ v2n+1 ≤ w2n ≤ . . .≤ v1 ≤ w0.

Clearly the IVPs (26), (27), and (28), (29) have unique solutions for each n = 0,1,2, . . .

denoted by vn+1, wn+1. First we show that v0 ≤ v1 ≤ w1 ≤ w0. Since v0 is a coupled lower

solution of Type I for (10), (11) we have

c Dqv0(t)≤ F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq(w0(t))), g(v0(0), v0(T ))≤ 0.

Setting n= 0 in (26), (27), we get that v1 is a solution of the boundary value problem,

c Dqv1(t) =F(t, w0(t), Iq(w0(t))) + G(t, v0(t), Iq(v0(t))),

v1(0) =w0(0)−
1

M
g(w0(0), w0(T )).



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 356

Set p(t) = v0(t)− v1(t), then by taking the Caputo fractional derivative on both sides and due

to the fact that F and G are in monotonic in the second and third variable we get that

c Dqp(t) =c Dqv0(t)−
c Dqv1(t)

≤[F(t, v0(t), Iq(v0(t))) + G(t, w0(t), Iq(w0(t)))]

− [F(t, w0(t), Iq(w0(t))) + G(t, v0(t), Iq(v0(t)))]

which means that
c Dqp(t)≤ 0.

Now p(0) = v0(0)− w0(0) +
1
M g(w0(0), w0(T )) ≤

1
M g(v0(0), v0(T )) ≤ 0. On applying Corol-

lary 2 we arrive at v0(t) ≤ v1(t), on J . In a similar fashion we get w1(t) ≤ w0(t), on J . Next

we proceed to show that

v0 ≤ w1 ≤ v2 ≤ w3 ≤ u≤ v3 ≤ w2 ≤ v1 ≤ w0. (30)

Writing p = u − v1, and working as earlier, we get c Dqp(t) ≤ 0 and p(t) ≤ 0 Again an ap-

plication of Corollary 2 gives u(t) ≤ v1(t), on J . A similar argument yields w1 ≤ u, v2 ≤ u,

u ≤ w2, u ≤ v3 and w3 ≤ u. Our next claim is that v0 ≤ w1 ≤ v2 ≤ w3 and v3 ≤ w2 ≤ v1 ≤ w0.

For this, let p(t) = v0(t)− w1(t), then c Dqp(t) ≤ 0 due to the fact that v0 ≤ w0, also p0 ≤ 0.

By applying Corollary 2 we get p(t) ≤ 0. Thus v0 ≤ w1. Proceeding in the same way we can

obtain w1 ≤ v2, v1 ≤ w0, v2 ≤ w3, v3 ≤ w2, w2 ≤ v1 on J . Thus we arrive at relation (30).

Suppose there exists an integer k ≥ 2 such that

w2k−1 ≤ v2k ≤ w2k+1 ≤ u≤ v2k+1 ≤ w2k ≤ v2k−1

holds, then we claim that

w2k+1 ≤ v2k+2 ≤ w2k+3 ≤ u≤ v2k+3 ≤ w2k+2 ≤ v2k+1.

Setting p(t) = w2k+1(t)− v2k+2(t).

c Dqp(t) =c Dqw2k+1(t)−
c Dqv2k+2(t)

≤[F(t, v2k(t), Iq(v2k(t))) + G(t, w2k(t), Iq(w2k(t)))]

− [F(t, w2k+1(t), Iq(w2k+1(t))) + G(t, v2k+1(t), Iq(v2k+1(t)))]

≤0,

which is obtained by adding and subtracting suitable terms and on using the monotone nature

of F , G. Next

p(0) =w2k+1(0)− v2k+2(0)

=v2k(0)−w2k+1(0) +
1

M
[g(w2k+1(0), w2k+1(T ))− g(v2k(0), v2k(T ))

≤0.



J. Devi, Ch. Sreedhar / Eur. J. Pure Appl. Math, 9 (2016), 346-359 357

Corollary 2 yields that p(t) ≤ 0 and consequently, w2k+1 ≤ v2k+2, on J . Similarly, we obtain

w2k+2 ≤ v2k+1, v2k+2 ≤ w2k+3, v2k+3 ≤ w2k+2. Finally consider p(t) = v2k+2(t) − u(t), and

working in a similar fashion we arrive at

c Dqp(t) =c Dqv2k+2(t)−
c Dqu(t)

≤F(t, w2k+1(t), Iq(w2k+1(t))) + G(t, v2k+1(t), Iq(v2k+1(t)))

− [F(t,u(t), Iq(u(t))) + G(t,u(t), Iq(u(t)))]

≤0,

and p(0) ≤ 0. So by applying Corollary 2 we get u(t) ≤ v2k+1(t). The relations u ≤ v2k+2,

w2k+3 ≤ u, w2k+2 ≤ u, u≤ v2k+3 can be proved by working as in the previous case.

Now by induction we have

v0 ≤ w1 ≤ . . .≤ v2n ≤ w2n+1 ≤ u≤ v2n+1 ≤ w2n ≤ . . .≤ v1 ≤ w0.

By arguing as in Theorem 3, we get the sequences {v2n, w2n+1} → ρ and {v2n+1, w2n} → r in

C1[J ,R] uniformly and monotonically, such that ρ and r are coupled minimal and maximal

solutions of Type I for (10), (11). Hence the proof of the theorem.

To avoid repetition, we will state next two theorems without proof since it follows the same

of pattern as that for Theorem 3 and Theorem 4.

Theorem 5. Assume that the hypothesis (A1), (A2), (A3) of Theorem 3 hold and v0, w0 are

coupled lower and upper solutions of Type II for (10), (11) with v0(t) ≤ w0(t) on J. Then the

iterative scheme given by

c Dqvn+1 =F(t, vn, Iq(vn)) + G(t, wn, Iq(wn)),

vn+1(0) =vn(0)−
1

M
g(vn(0), vn(T )),

c Dqwn+1 =F(t, wn, Iq(wn)) + G(t, vn, Iq(vn)),

wn+1(0) =wn(0)−
1

M
g(wn(0), wn(T )),

result in two monotone sequences {vn(t)}, {wn(t)} satisfying

v0 ≤ v1 ≤ . . .≤ vn ≤ wn ≤ . . .≤ w1 ≤ w0.

Further more vn → ρ and wn → r in C1[J ,R] uniformly and monotonically, such that ρ and r

are coupled minimal and maximal solutions of Type II for (10), (11), respectively, provided that

v0 ≤ w0. Thus ρ and r satisfy the coupled system

c Dqρ =F(t,ρ, Iq(ρ)) + G(t, r, Iq(r)),

g(ρ(0),ρ(T )) = 0,
c Dqr =F(t,ρ, Iq(ρ)) + G(t, r, Iq(r)),

g(r(0), r(T )) = 0.



REFERENCES 358

Theorem 6. Let (A2), (A3) of Theroem 3 hold and v0, w0 are coupled lower and upper solutions

of Type II for (10), (11) with v0(t)≤ w0(t) on J. Then the iterative scheme given by

c Dqvn+1 =F(t, wn, Iq(wn)) + G(t, vn, Iq(vn)),

vn+1(0) =wn(0)−
1

M
g(wn(0), wn(T )),

c Dqwn+1 =F(t, vn, Iq(vn)) + G(t, wn, Iq(wn)),

wn+1(0) =vn(0)−
1

M
g(vn(0), vn(T )),

yields alternating monotone sequences {v2n, w2n+1} and {v2n+1, w2n} satisfying

v0 ≤ w1 ≤ . . .≤ v2n ≤ w2n+1 ≤ u≤ v2n+1 ≤ w2n ≤ . . .≤ v1 ≤ w0,

for each n≥ 1 on J, provided that v0 ≤ u≤ w0. Furthermore {v2n, w2n+1} → ρ and

{v2n+1, w2n} → r in C1[J ,R] uniformly and monotonically, such that ρ and r are coupled min-

imal and maximal solutions of (10) , (11), respectively, that is, if v0 ≤ u ≤ w0 then ρ ≤ u ≤ r,

and ρ and r satisfy the coupled system.

c Dqρ =F(t,ρ, Iq(ρ)) + G(t, r, Iq(r)),

g(ρ(0),ρ(T )) = 0,
c Dqr =F(t,ρ, Iq(ρ)) + G(t, r, Iq(r)),

g(r(0), r(T )) = 0.

4. conclusion

We consider periodic boundary value problem of Caputo fractional integro differential

equation and obtained its maximal and minimal solutions. We obtained this by using mono-

tone iterative technique of initial value problems.

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