International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 2 (2016), 157-167
http://www.etamaths.com

EXISTENCE AND APPROXIMATE SOLUTIONS FOR NONLINEAR HYBRID

FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

B.C. DHAGE1,∗, G.T. KHURPE1, A.Y. SHETE2 AND J.N. SALUNKE2

Abstract. In this paper we prove existence and approximation of the solutions for initial value

problems of nonlinear hybrid fractional differential equations with maxima and with a linear as well

as quadratic perturbation of second type. The main results rely on Dhage iteration method embodied
in the recent hybrid fixed point theorem of Dhage (2014) in a partially ordered normed linear space.

The approximation of the solutions of the considered nonlinear fractional differential equations are

obtained under weaker mixed partial continuity and Lipschitz conditions. Our hypotheses and the
main results are also illustrated by a numerical example.

1. Introduction

In this paper we prove existence and approximations of the solutions for initial value problems
of nonlinear hybrid fractional differential equations. Consider the following initial value problem of
fractional differential equations,

(1.1)




cDα

(
x(t) − Iβh(t,x(t))

f(t,x(t))

)
= g

(
t,x(t),

∫ t
0

k(s,x(s)) ds

)
, t ∈ J := [0,T],

x(0) = x0 ∈ R+,

where cDα denotes the Caputo fractional derivative of order α, 0 < α < 1, Iβ is the Riemann-Liouville
fractional integral of order β, and f : J × R → R \{0}, h,k : J × R → R and g : J × R × R → R are
given continuous functions.

By a solution of the problem (1.1) we mean a function x ∈ C1(J,R) if

(i) the function t 7→
x(t) − Iβh(t,x(t))

f(t,x(t))
is Caputo differentiable, and

(ii) x satisfies the relations in ((1.1) on J,

where C1(J,R) is the space of continuously differentiable real-valued functions defined on J.

Fractional differential equations have aroused great interest, which is caused by both the intensive
development of the theory of fractional calculus and the applications to rheology, physics, mechanics
and chemistry engineering [16, 17]. For some recent development on the topic see [1] and the references
cited therein. For some recent results on hybrid fractional differential equations we refer to [1], [2],
[14], [18], [19] and the references cited therein.

The origin of the problem (1.1) lies in the initial value problems of first order quadratic differential
equations with ordinary derivative wherein only existence of the solutions is proved using classical
hybrid fixed point theorem of Dhage [3]. The problem (1.1) considered here is general in the sense
that it includes the following three well-known classes of initial value problems of fractional differential
equations.

2010 Mathematics Subject Classification. 34A08, 47H07, 47H10.
Key words and phrases. fractional differential equation; fixed point theorem; Dhage iteration method; existence and

uniqueness theorems.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

157



158 DHAGE, KHURPE, SHETE AND SALUNKE

Case I: Let f(t,x) = 1 and h(t,x) = 0 for all t ∈ J and x ∈ R. Then the problem (1.1) reduces to
standard initial value problem of fractional differential equation

(1.2)




cDαx(t) = g

(
t,x(t),

∫ t
0

k(s,x(s)) ds

)
, t ∈ J := [0,T],

xx(0) = x0 ∈ R.

Case II: If h(t,x) = 0 for all t ∈ J and x ∈ R in (1.1), we obtain the following quadratic fractional
differential equation,

(1.3)




cDα

(
x(t)

f(t,x(t))

)
= g

(
t,x(t),

∫ t
0

k(s,x(s)) ds

)
, t ∈ J := [0,T],

x(0) = x0 ∈ R.

Case III: If f(t,x) = 1 for all t ∈ J and x ∈ R in (1.1), we obtain the following interesting fractional
differential equation,

(1.4)




cDα
[
x(t) − Iβh(t,x(t))

]
= g

(
t,x(t),

∫ t
0

k(s,x(s)) ds

)
, t ∈ J := [0,T],

x(0) = x0 ∈ R.

Therefore, the main result of this paper also includes the existence as well as approximation results
for the solutions of above mentioned initial value problems of fractional differential equations as special
cases. Again our approach here in this paper is different than that employed in the related paper of
Dhage [3].

In the present paper we prove the existence and approximations of the solutions of problem (1.1)
under weaker partial compactness and partial Lipschitz type conditions via Dhage iteration method
[7]. Very recently, Dhage iteration method has been applied in [7, 8, 9, 11, 12, 13] to nonlinear ordinary
differential equations for proving the existence and algorithms of the solutions.

We recall the basic definitions of fractional calculus [16, 17] which are useful in what follows.

Definition 1.1. The Riemann-Liouville fractional integral of order q with the lower limit zero for a
function f is defined as

Iqf(t) =
1

Γ(q)

∫ t
0

f(s)

(t−s)1−q
ds, t > 0, q > 0,

provided the right hand-side is point-wise defined on [0,∞), where Γ(·) is the gamma function, which

is defined by Γ(q) =

∫ ∞
0

tq−1e−tdt.

Definition 1.2. The Riemann-Liouville fractional derivative of order q > 0, n− 1 < q < n, n ∈ N,
is defined as

D
q
0+f(t) =

1

Γ(n−q)

(
d

dt

)n ∫ t
0

(t−s)n−q−1f(s)ds,

where the function f(t) has absolutely continuous derivative up to order (n− 1).

Definition 1.3. The Caputo derivative of order q for a function f ∈ Cn(J,R) can be written as

cDqf(t) = Dq

(
f(t) −

n−1∑
k=0

tk

k!
f(k)(0)

)
, t > 0, n− 1 < q < n.

Remark 1.4. If f ∈ Cn(J,R), then

cDqf(t) =
1

Γ(n−q)

∫ t
0

f(n)(s)

(t−s)q+1−n
ds = In−qf(n)(t), t > 0, n− 1 < q < n.



APPROXIMATE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS 159

Lemma 1.5. For q > 0, the general solution of the fractional differential equation cDqx(t) = 0 is
given by

x(t) = c0 + c1t + . . . + cn−1t
n−1,

where ci ∈ R, i = 1, 2, . . . ,n− 1 (n = [q] + 1).

Remark 1.6. In view of Lemma 1.5, it follows that

(1.5) Iq cDqx(t) = x(t) + c0 + c1t + . . . + cn−1t
n−1,

for some ci ∈ R, i = 1, 2, . . . ,n− 1 (n = [q] + 1).

The rest of the paper will be organized as follows. In Section 2 we give some preliminaries and key
fixed point theorems that will be used in subsequent part of the paper. In Section 3 we discuss the
main existence and approximation result for initial value problems of fractional differential equations
(1.1). An illustrative example is also discussed.

2. Auxiliary Results

Unless otherwise mentioned, throughout this paper we let E denote a partially ordered real normed
linear space with the order relation � and the norm ‖ · ‖ in which addition and scalar multiplication
by positive real numbers are preserved by �. A few details on such partially ordered normed linear
spaces appear in Dhage [5] and the references therein.

Two elements x and y in E are said to be comparable if either the relation x � y or y � x holds.
A non-empty subset C of E is called a chain or totally ordered if all elements of C are comparable.
We say that E is regular if for any nondecreasing (resp. nonincreasing) sequence {xn} in E such that
xn → x∗ as n →∞, we have that xn � x∗ (resp. xn � x∗) for all n ∈ N. Conditions guaranteeing the
regularity of E may be found in Heikkilä and Lakshmikantham [15] and the references therein.

We need the following definitions (see Dhage [5, 6, 6] and the references therein) in what follows.

Definition 2.1. A mapping B : E → E is called isotone or nondecreasing if it preserves the order
relation �, that is, if x � y implies Bx �By for all x,y ∈ E.

Definition 2.2. A mapping B : E → E is called partially continuous at a point a ∈ E if for � > 0
there exists a δ > 0 such that ‖Bx−Ba‖ < � whenever x is comparable to a and ‖x−a‖ < δ. B called
a partially continuous on E if it is partially continuous at every point of it. It is clear that if B is a
partially continuous on E, then it is continuous on every chain C contained in E.

Definition 2.3. A non-empty subset S of the partially ordered Banach space E is called partially
bounded if every chain C in S is bounded. A nondecreasing mapping B : E → E is called partially
bounded if B(C) is bounded for every chain C in E. B is called uniformly partially bounded if
all chains B(C) in E are bounded by a unique constant. B is called bounded if B(E) is a bounded
subset of E.

Definition 2.4. A non-empty subset S of the partially ordered Banach space E is called partially
compact if every chain C in S is compact. A nondecreasing mapping B : E → E is called partially
compact if B(C) is a relatively compact subset of E for all totally ordered sets or chains C in E. B is
called uniformly partially compact if B(C) is a uniformly partially bounded and partially compact
on E. B is called partially totally bounded if for any totally ordered and bounded subset C of E,
B(C) is a relatively compact subset of E. If B is partially continuous and partially totally bounded,
then it is called partially completely continuous on E.

Definition 2.5. The order relation � and the metric d on a non-empty set E are said to be compatible
if {xn}n∈N is a monotone, that is, monotone nondecreasing or monotone nonincreasing sequence in E
and if a subsequence {xnk}n∈N of {xn}n∈N converges to x

∗ implies that the whole sequence {xn}n∈N
converges to x∗. Similarly, given a partially ordered normed linear space (E,�,‖·‖), the order relation
� and the norm ‖·‖ are said to be compatible if � and the metric d defined through the norm ‖·‖ are
compatible. A subset S of E is called Janhavi if the order relation � and the metric d or the norm
‖ · ‖ are compatible in it. In particular, if S = E, then E is called a Janhavi metric or Janhavi
Banach space.



160 DHAGE, KHURPE, SHETE AND SALUNKE

Clearly, the set R of real numbers with usual order relation ≤ and the norm defined by the absolute
value function | · | has this property. Similarly, the finite dimensional Euclidean space Rn with usual
componentwise order relation and the standard norm possesses the compatibility property.

Definition 2.6. An upper semi-continuous and nondecreasing function ψ : R+ → R+ is called a D-
function provided ψ(0) = 0. Let (E,�,‖ · ‖) be a partially ordered normed linear space. A mapping
T : E → E is called partially nonlinear D-Lipschitz if there exists a D-function ψ : R+ → R+
such that

(2.1) ‖T x−T y‖≤ ψ(‖x−y‖)

for all comparable elements x,y ∈ E. If ψ(r) = k r, k > 0, then T is called a partially Lipschitz with a
Lipschitz constant k. Furthermore, if ψ(r) < r, r > 0, T is called a partially nonlinear D-contraction
on E.

Let (E,�,‖ ·‖) be a partially ordered normed linear algebra. Denote

E+ =
{
x ∈ E | x � θ, where θ is the zero element of E

}
and

(2.2) K = {E+ ⊂ E | uv ∈ E+ for all u,v ∈ E+}.

The elements of the set K are called the positive vectors in E. Then following lemma is immediate.

Lemma 2.7 (Dhage [3]). If u1,u2,v1,v2 ∈K are such that u1 � v1 and u2 � v2, then u1u2 � v1v2.

Definition 2.8. An operator B : E → E is said to be positive if the range R(B) of B is such that
R (B) ⊆K.

The Dhage iteration method is embodied in the following hybrid fixed point theorem proved in
Dhage [6] which are useful tools in what follows. A few other such hybrid fixed point theorems appear
in Dhage [5, 6].

Theorem 2.9 (Dhage [7]). Let
(
E,�,‖ · ‖

)
be a regular partially ordered complete normed linear

algebra such that every compact chain C in E is Janhavi. Let A,B : E → K and C : E → E be three
nondecreasing operators such that

(a) A and C are partially bounded and partially nonlinear D-Lipschitz with D-functions ψA and
ψC respectively.

(b) B is partially continuous and uniformly partially compact,
(c) 0 < MψA(r) + ψC(r) < r, r > 0, where M = sup{‖B(C)‖ : C is a chain in E}, and
(d) there exists an element x0 ∈ E such that x0 �Ax0Bx0 + Cx0 or

x0 �Ax0 Bx0 + Cx0.
Then the operator equation AxBx+Cx = x has a solution x∗ in E and the sequence {xn} of successive
iterations defined by xn+1 = AxnBxn + Cxn, n = 0, 1, . . . converges monotonically to x∗.

Remark 2.10. The compatibility of the order relation � and the norm ‖ · ‖ in every compact chain
of E is held if every partially compact subset S of E possesses the compatibility property with respect
to � and ‖ · ‖. This simple fact is used to prove the desired characterization of the mild solution of
the problem (1.1) on J.

3. Main Existence Result

The equivalent integral form of the problem (1.1) is considered in the function space C(J,R) of
continuous real-valued functions defined on J. We define a norm ‖ · ‖ and the order relation ≤ in
C(J,R) by

(3.1) ‖x‖ = sup
t∈J
|x(t)|

and

(3.2) x ≤ y ⇐⇒ x(t) ≤ y(t)



APPROXIMATE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS 161

for all t ∈ J. Clearly, C(J,R) is a Banach space with respect to above supremum norm and also
partially ordered w.r.t. the above partially order relation ≤. It is known that the partially ordered
Banach space C(J,R) is regular and a lattice so that every pair of elements of E has a lower and
an upper bound in it. It is known that the partially ordered Banach space C(J,R) has some nice
properties w.r.t. the above order relation in it. The following lemma follows by an application of
Arzellá-Ascoli theorem.

Lemma 3.1. Let
(
C(J,R),≤,‖ · ‖

)
be a partially ordered Banach space with the norm ‖ · ‖ and the

order relation ≤ defined by (3.1) and (3.2) respectively. Then every partially compact subset S of
C(J,R) is Janhavi.

Proof. The proof of the lemma is given in Dhage and Dhage [11]. Since the proof is well-known, we
omit the details of proof. �

We need the following definition in what follows.

Definition 3.2. A function u ∈ C1(J,R) is said to be a lower solution of the problem (1.1) if the

function t 7→
u(t) − Iβh(t,u(t))

f(t,u(t))
is continuously differentiable and satisfies

cDα

(
u(t) − Iβh(t,u(t))

f(t,u(t))

)
≤ g

(
t,u(t),

∫ t
0

k(s,u(s)) ds

)
, t ∈ J,

u(0) ≤ x0.


 (∗)

Similarly, an upper solution v ∈ C1(J,R) to the problem (1.1) is defined on J, by the above inequalities
with reverse sign.

We consider the following set of assumptions in what follows:

(A0) The map x 7→
x

f(t,x)
is injective for each t ∈ J.

(A1) There exists a constant Mf > 0 such that 0 < f(t,x) ≤ Mf for all t ∈ J and x ∈ R.
(A2) There exists a D-function ϕ such that

0 ≤ f(t,x) −f(t,y) ≤ ϕ(x−y)

for all t ∈ J and x,y ∈ R, x ≥ y.
(B1) There exists a constant Mg > 0 such that 0 < g(t,x,y) ≤ Mg for all t ∈ J and x,y ∈ R.
(B2) The function g(t,x,y) is monotone nondecreasing in x and y for each t ∈ J.
(B3) The function k(t,x) is monotone nondecreasing in x for each t ∈ J.
(C1) There exists a constant Mh > 0 such that 0 ≤ h(t,x) ≤ Mh for all t ∈ J and x ∈ R.
(C2) There exists a D-function ω such that

0 ≤ h(t,x) −h(t,y) ≤ ω(x−y)

for all t ∈ J and x,y ∈ R, x ≥ y.
(D1) The problem (1.1) has a lower solution u ∈ C1(J,R).
(D2) The problem (1.1) has an upper solution v ∈ C1(J,R).

Remark 3.3. Notice that Hypothesis (A0) holds in particular if the function x 7→
x

f(t,x)
is increasing

in R for each t ∈ J.

The following lemma is useful in what follows and may be found in Kilbas et.al. [16] and Podlubny
[17].

Lemma 3.4. For a given continuous function h : J → R, a function u ∈ C1(J,R) is a solution of the
QFDE

(3.3)

cDqx(t) = h(t), t ∈ J, 0 < q < 1,
x(0) = α0,

}



162 DHAGE, KHURPE, SHETE AND SALUNKE

if and only if it is a solution of the nonlinear integral equation,

(3.4) x(t) = α0 +
1

Γ(q)

∫ t
0

(t−s)q−1 h(s) ds, t ∈ J.

As an application of Lemma 3.4, we obtain

Lemma 3.5. Assume that the hypothesis (A0) holds. If a function x ∈ C1(J,R) is a solution of the
QFDE

(3.5)
cDq

[
x(t) − Iβh(t,x(t))

f(t,x(t))

]
= g

(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
, t ∈ J, 0 < q < 1,

x(t0) = α0,




then it satisfies the nonlinear integral equation,

x(t) =
[
f(t,x(t))

][ α0
f(t0,α0)

+
1

Γ(q)

∫ t
0

(t−s)q−1 g
(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

]
+ Iβh(t,x(t)), t ∈ J.

(3.6)

Proof. Assume first that x ∈ C1(J,R) is a solution to the QFDE (1.1) defined on J. By Lemma 1.5,
we have

(3.7)
x(t) − Iβh(t,x(t))

f(t,x(t))
=

∫ t
0

(t−s)α−1

Γ(α)
g

(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds + c0,

where c0 ∈ R. Since x(0) = α0,f(0,α0) 6= 0, it follows c0 =
α0

f(0,α0)
. Thus (3.6) holds. �

Definition 3.6. A function x ∈ C1(J,R) which satisfies the QFIE (3.7) is called a mild solution of
the QFDE (1.1) defined on J.

Theorem 3.7. Assume that the hypotheses (A0)-(A2), (B1)-(B2), (C1)-(C7) and (D1) hold. Further-
more, if

(3.8)

[∣∣∣ α0
f(t0,α0)

∣∣∣ + MgTα
Γ(α + 1)

]
ϕ(r) +

Tβ

Γ(β + 1)
ω(r) < r,

then the problem (1.1) has a mild solution x∗ defined on J and the sequence {xn}∞n=1 of successive
approximations defined by

xn+1(t) =

∫ t
0

(t−s)β−1

Γ(β)
h(s,xn(s))ds

+ f(t,xn(t))

[
α0

f(t0,α0)
+

∫ t
0

(t−s)α−1

Γ(α)
g

(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

](3.9)
for all t ∈ J, where x1 = u, converges monotonically to x∗.

Proof. By Lemma 3.5, the mild solution x of the problem (1.1) satisfies the nonlinear integral equation

x(t) =

∫ t
0

(t−s)β−1

Γ(β)
h(s,x(s))ds

+ f(t,x(t))

[
α0

f(t0,α0)
+

∫ t
0

(t−s)α−1

Γ(α)
g

(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

](3.10)
for all t ∈ J.

Set E = C(J,R). Then, from Lemma 3.1 it follows that every compact chain in E possesses the
compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in E.

Define the operators A, B, and C on E by

(3.11) Ax(t) = f(t,x(t)), t ∈ J,



APPROXIMATE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS 163

(3.12) Bx(t) =
α0

f(t0,α0)
+

1

Γ(α)

∫ t
0

(t−s)α−1g
(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds, t ∈ J,

and

(3.13) Cx(t) =
1

Γ(β)

∫ t
0

(t−s)β−1h(s,x(s)) ds, t ∈ J.

From the continuity of the integrals, it follows that A,B and C define the maps A,B : E →K and
C : E → E. Then, the problem (1.1) is equivalent to the operator equation

(3.14) Ax(t)Bx(t) + Cx(t) = x(t), t ∈ J.

We shall show that the operators A,B and C satisfy all the conditions of Theorem 2.9. This is
achieved in the series of following steps.

Step I: A,B and C are nondecreasing operators on E.
Let x,y ∈ E be such that x ≥ y. Then by hypothesis (A2), we obtain

Ax(t) = f(t,x(t)) ≥ f(t,y(t)) = Ay(t),

for all t ∈ J. This shows that A is nondecreasing operator on E into E. Similarly, we have by (A5),

Bx(t) =
α0

f(t0,α0)
+

1

Γ(α)

∫ t
0

(t−s)α−1g
(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

≥
α0

f(t0,α0)
+

1

Γ(α)

∫ t
0

(t−s)α−1g
(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

= By(t),

for all t ∈ J. This shows that B is nondecreasing operator on E into itself. The proof that C is
nondecreasing operator on E into itself is similar.

Step II: A and C are partially bounded and partially D-contraction on E.
Let x ∈ E be arbitrary. Then by (A1),

|Ax(t)| ≤ |f(t,x(t))| ≤ Mf,

for all t ∈ J. Taking supremum over t, we obtain ‖Ax‖ ≤ Mf and so, A is bounded. This further
implies that A is partially bounded on E.

Next, let x,y ∈ E be such that x ≥ y. Then,

|Ax(t) −Ay(t)| = |f(t,x(t)) −f(t,y(t))| ≤ ϕ(|x(t) −y(t)|) ≤ ϕ(‖x−y‖).

Then, ‖Ax−Ay‖≤ ϕ(‖x−y‖) for all x,y ∈ E with x ≥ y and hence A is a partially D-Lipschitz on
E with D-functions ϕ(r), which further implies that A is also a partially continuous on E.

Again, we have

|Cx(t)| ≤
∫ t
0

(t−s)β−1

Γ(β)
|h(s,x(s))|ds

≤ Mh
∫ t
0

(t−s)β−1

Γ(β)
ds

≤
Mh t

β

Γ(β + 1)

≤
Mh T

β

Γ(β + 1)
,

which means that C is bounded and consequently partially bounded on E.



164 DHAGE, KHURPE, SHETE AND SALUNKE

Next, let x,y ∈ E be such that x ≥ y. Then,

|Cx(t) −Cy(t)| =
∫ t
0

(t−s)β−1

Γ(β)
|h(s,x(s)) −h(s,y(s))|ds

≤
Tβ

Γ(β + 1)
ω(‖x−y‖).

Hence C is a partially D-Lipschitz on E with D-functions
Tβ

Γ(β + 1)
ω(r), which further implies that C

is a partially continuous on E.

Step III: B is a partially continuous operator on E.
Let {xn} be a sequence of points of a chain C in E such that xn → x for all n ∈ N. Then, by

dominated convergence theorem, we have

lim
n→∞

Bxn(t) = lim
n→∞

[
α0

f(t0,α0)
+

1

Γ(α)

∫ t
0

(t−s)α−1g
(
s,xn(s),

∫ s
0

k(τ,xn(τ)) dτ

)
ds

]
=

α0
f(t0,α0)

+
1

Γ(α)

∫ t
0

(t−s)α−1
[

lim
n→∞

g

(
s,xn(s),

∫ s
0

k(τ,xn(τ)) dτ

)]
ds

=
α0

f(t0,α0)
+

1

Γ(α)

∫ t
0

(t−s)α−1g
(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

= Bx(t),

for all t ∈ J. This shows that {Bxn} converges to Bx pointwise on J.
Next, we will show that {Bxn} is an equicontinuous sequence of functions in E. Let t1, t2 ∈ J be

arbitrary with t1 < t2. Then

|Bxn(t2) −Bxn(t1)|

≤
1

Γ(α)

∣∣∣∣
∫ t2
0

|(t2 −s)α−1 − (t1 −s)α−1|
∣∣∣g(s,x(s),∫ s

0

k(τ,x(τ)) dτ

)∣∣∣ds∣∣∣∣
+

1

Γ(α)

∣∣∣∣
∫ t2
t1

(t1 −s)α−1
∣∣∣g(s,x(s),∫ s

0

k(τ,x(τ)) dτ

)∣∣∣ds∣∣∣∣
≤

Mg
Γ(α + 1)

(tα2 − t
α
1 ).

Consequently, we obtain

|Bxn(t2) −Bxn(t1)|→ 0 as t2 → t1

uniformly for all n ∈ N. This shows that the convergence Bxn → Bx is uniformly and hence B is a
partially continuous on E.

Step IV: B is a partially compact operator on E.
Let C be an arbitrary chain in E. We show that B(C) is a uniformly bounded and equicontinuous

set in E. First we show that B(C) is uniformly bounded. Let x ∈ C be arbitrary. Then,

|Bx(t)| ≤
∣∣∣ α0
f(t0,α0)

∣∣∣ + 1
Γ(α)

∫ t
0

(t−s)α−1
∣∣∣∣g
(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)∣∣∣∣ ds
≤

∣∣∣ α0
f(t0,α0)

+
∣∣∣ + MgTα

Γ(α + 1)
= r,

for all t ∈ J. Taking the supremum over t, we obtain ‖Bx‖ ≤ r for all x ∈ C. Hence B(C) is a
uniformly bounded subset of E. Next, we will show that B(C) is an equicontinuous set in E. Let



APPROXIMATE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS 165

t1, t2 ∈ J be arbitrary with t1 < t2. Then,

|Bx(t2) −Bx(t1)|

≤
1

Γ(α)

∫ t2
0

|(t2 −s)α−1 − (t1 −s)α−1|
∣∣∣g(s,x(s),∫ s

0

k(τ,x(τ)) dτ

)∣∣∣ds
+

1

Γ(α)

∣∣∣∣
∫ t2
t1

(t1 −s)α−1
∣∣∣g(s,x(s),∫ s

0

k(τ,x(τ)) dτ

)∣∣∣ds∣∣∣∣
≤

Mg
Γ(α + 1)

(tα2 − t
α
1 ).

Thus, we have that

|Bx(t2) −Bx(t1)|→ 0 as t2 → t1

uniformly for all x ∈ C. This shows that B(C) is an equicontinuous set in E. Hence B(C) is compact
subset of E and consequently B is a partially compact operator on E into itself.

Step V: D-functions ϕ and ω satisfy the inequality 0 < MψA(r) + ψC(r) < r, r > 0.
We have

MψA(r) + ψC(r) =

[∣∣∣ α0
f(t0,α0)

∣∣∣ + MgTα
Γ(α + 1)

]
ϕ(r) +

Tβ

Γ(β + 1)
ω(r) < r,

by assumption.

Step VI: u satisfies the operator inequality u ≤AuBu + Cu.
Since the hypothesis (A6) holds, u is a lower solution of (1.1) defined on J. Then,

(3.15) cDα

(
u(t) − Iβh(t,u(t))

f(t,u(t))

)
≤ g(t,u(t)),

satisfying,

(3.16) u(0) ≤ α0,

for all t ∈ J.

Taking the Riemann-Livoulle Integration of fractional order α from 0 to t on both sides of the above
inequality (3.15), we obtain

u(t) ≤
∫ t
0

(t−s)β−1

Γ(β)
h(s,u(s))ds

+
[
f(t,u(t))

][ α0
f(0,α0)

+

∫ t
0

(t−s)α−1

Γ(α)
g

(
s,x(s),

∫ s
0

k(τ,x(τ)) dτ

)
ds

]
,

(3.17)

for all t ∈ J. This show that u is a solution of the operator inequality u ≤AuBu + Cu.

Thus, the operators A,B and C satisfy all the conditions of Theorem 2.9 in view of Remark 2.9
and we apply it to conclude that the operator equation AxBx + Cx = x has a solution defined on
J. Consequently the integral equation has a solution x∗ defined on J which is also a mild solution
of the QFDE (1.1) and the sequence {xn} of successive approximations defined by (3.9) converges
monotonically to x∗. This completes the proof. �

Remark 3.8. The conclusion of Theorem 3.7 alsi ramains true if we replace the hypothesis (D1) with
(D2). The oroof under this new hypothesis is similar with obvious modifications.



166 DHAGE, KHURPE, SHETE AND SALUNKE

Example 3.9. Given a closed and bounded interval J = [0, 1] in R, consider the initial value problem
of quadratic fractional nonlinear integro-differential equation,

(3.18)



cD1/2

[
x(t) − I3/2(arctan x(t))

f(t,x(t))

]
=

2 + tanh x(t) + tanh

(∫ t
0

k(s,x(s)) ds

)
16

,

x(0) = 0,

for all t ∈ J := [0, 1], where cD1/2 denotes the Caputo fractional derivative of order 1/2, k : J×R → R
and f : J ×R → R\{0} are two continuous functions defined by

f(t,x) =




1, if x ≤ 0,

1 +
x

1 + x
, if x > 0.

and

k(t,x) =

{
0 if x ≤ 0,

log(1 + x) if x > 0

for t ∈ J and x ∈ R.

If we take h(t,x) = arctan x, g(t,x,y) =
2 + tanh x + tanh y

16
and then it is easy to check that the

conditions of Theorem 3.7 are satisfied with the lower solution u defined by u(t) = −
4t3/2

3
√
π

+
t1/2

6
√
π

,

t ∈ J. Therefore, the problem (3.17) has a mild solution defined on [0, 1].

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1Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur, Maharashtra, India

2School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, nanded, Maharash-

tra, India

∗Corresponding author: bcdhage@email.email