International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 2 (2015), 130-143
http://www.etamaths.com

A NEW MONOTONE ITERATION PRINCIPLE IN THE THEORY OF

NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

BAPURAO C. DHAGE

Abstract. In this paper the author proves the algorithms for the existence as well as approximations

of the solutions for the initial value problems of nonlinear fractional differential equations using the

operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage
iteration principle embodied in the recent hybrid fixed point theorems of Dhage (2014) in a partially

ordered normed linear space and the existence and approximations of the solutions of the considered

nonlinear fractional differential equations are obtained under weak mixed partial continuity and
partial Lipschitz conditions. Our hypotheses and existence and approximation results are also well

illustrated by some numerical examples.

1. Introduction

The Dhage iteration principle or method (in short DIP or DIM) developed in [3, 4, 5, 6, 7] is
relatively new to the literature on nonlinear analysis, particularly in the theory of nonlinear differential
and integral equations. But it is is a powerful tool in the subject of nonlinear analysis because of
its utility of applications to nonlinear equations for different qualitative aspects of the solutions. See
Dhage and Dhage [8], Dhage et al. [13], Pathak and Lopez [16] and the references therein. Very
recently, the above method has been applied in Dhage [3, 4, 6, 7] and Dhage and Dhage [8, 9, 10, 11]
to nonlinear ordinary differential and integral equations for proving the existence and algorithms
of the solutions. Similarly, DIP has also some interesting applications to the theory of nonlinear
fractional differential and integral equations and in the present paper we prove the existence as well
as approximations of the solutions for the initial value problems of fractional differential equations.

Before stating the main differential problems of the paper, we recall the following basic definitions
of fractional calculus [15, 17] which are useful in what follows.

Definition 1.1. Let J = [t0, t0 + a] be a closed and bounded of the real line R for some t0,a ∈ R
with t0 ≥ 0 and a > 0. If x ∈ ACn(J,R), then the Caputo derivative CDqx of x of fractional order q
is defined as

CDqx(t) =
1

Γ(n−q)

∫ t
t0

(t−s)n−q−1x(n)(s)ds, n− 1 < q < n,n = [q] + 1,

where [q] denotes the integer part of the real number q, and Γ is the Euler’s gamma function. Here
ACn(J,R) denote the space of real valued functions x(t) which have continuous derivatives up to order
n− 1 on J such that the nth derivative x(n) ∈ AC(J,R).

Definition 1.2. If J∞ = [t0,∞) is a closed interval of the real line R for some t0 ∈ R with t0 ≥ 0,
then for any x ∈ C(J,R), the Riemann-Liouville fractional integral of order q > 0 is defined as

Iqx(t) =
1

Γ(q)

∫ t
t0

x(s)

(t−s)1−q
ds, t ∈ J∞,

2010 Mathematics Subject Classification. 34A12, 34H34, 47H07, 47H10.
Key words and phrases. Fractional differential equation; Dhage iteration principle; Hybrid fixed point theorem;

Existence and uniqueness theorems.

c©2015 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

130



NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 131

provided the right hand side is defined pointwise on (t0,∞), where Γ is the Euler’s gamma function.
Given a closed and bounded interval J = [t0, t0 + a] of the real line R for some t0,a ∈ R with

t0 ≥ 0 and a > 0, consider the initial value problem (in short IVP) of nonlinear fractional differential
equation (FDE),

(1.1)

CDqx(t) = f(t,x(t)), t ∈ J,

x(t0) = α0,




where CDq is the Caputo derivative of fractional order q, 0 < q < 1 and f : J × R → R is a
continuous function.

By a solution of the FDE (1.1) we mean a function x ∈ C1(J,R) that satisfies equation (1.1), where
C1(J,R) is the space of continuously differentiable real-valued functions defined on J.

The nonlinear FDE (1.1) is well-known and it has been discussed at length for the existence and the
uniqueness of the solutions under compactness and Lipschitz conditions which are considered to be
very strong in the theory of nonlinear differential and integral equations. In the present paper we prove
the existence and uniqueness of the solutions of FDE (1.1) under weaker partially compactness and
partially Lipschitz type conditions via Dhage iteration principle and also indicate some realizations.

The rest of the paper will be organized as follows. In Section 2 we give some preliminaries and key
fixed point theorem that will be used in subsequent part of the paper. In Section 3 we discuss the
existence result for initial value problems and in Section 4 we discuss the existence result for initial
value problems of hybrid differential differential equations with linear perturbation of first type.

2. Auxiliary Results

Unless otherwise mentioned, throughout this paper that follows, let E denote a partially ordered
real normed linear space with an order relation � and the norm ‖ · ‖ in which the addition and the
scalar multiplication by positive real numbers are preserved by � . A few details of such partially
ordered spaces appear in Dhage [2] and the references therein.

Two elements x and y in E are said to be comparable if either the relation x � 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.
It is known that E is regular if {xn} is a nondecreasing (resp. nonincreasing) sequence in E such
that xn → x∗ as n → ∞, then xn � x∗ (resp. xn � x∗) for all n ∈ N. The conditions guaranteeing
the regularity of E may be found in Heikkilä and Lakshmikantham [14] and the references therein.

We need the following definitions in the sequel.

Definition 2.1. A mapping T : E → E is called isotone or monotone nondecreasing if it
preserves the order relation �, that is, if x � y implies T x � T y for all x,y ∈ E. Similarly, T is
called monotone nonincreasing if x � y implies T x � T y for all x,y ∈ E. Finally, T is called
monotonic or simply monotone if it is either monotone nondecreasing or monotone nonincreasing
on E.

The following terminologies may be found in any book on nonlinear analysis and applications.

Definition 2.2. An operator T on a normed linear space E into itself is called compact if T (E)
is a relatively compact subset of E. T is called totally bounded if for any bounded subset S of E,
T (S) is a relatively compact subset of E. If T is continuous and totally bounded, then it is called a
completely continuous on E.

Definition 2.3 (Dhage [3]). A mapping T : E → E is called partially continuous at a point
a ∈ E if for � > 0 there exists a δ > 0 such that ‖T x−T a‖ < � whenever x is comparable to a and
‖x−a‖ < δ. T is called a partially continuous on E if it is partially continuous at every point of it.
It is clear that if T is a partially continuous on E, then it is continuous on every chain C contained
in E.



132 DHAGE

Definition 2.4 (Dhage [3, 4]). A non-empty subset S of the partially ordered Banach space E is
called partially bounded if every chain C in S is bounded. An operator T on a partially normed linear
space E into itself is called partially bounded if T (C) is bounded for every chain C in E. T is
called uniformly partially bounded if all chains T (C) in E are bounded by a unique constant.

Definition 2.5 (Dhage [3, 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. The operator T is called partially compact if
T (C) is a relatively compact subset of E for all totally ordered sets or chains C in E. T is called
uniformly partially compact if T is a uniformly partially bounded and partially compact operator
on E. T is called partially totally bounded if for any totally ordered and bounded subset C of E,
T (C) is a relatively compact subset of E. If T is partially continuous and partially totally bounded,
then it is called a partially completely continuous operator on E.

Remark 2.6. Note that every compact mapping on a partially normed linear space is partially com-
pact and every partially compact mapping is partially totally bounded, however the reverse implica-
tions do not hold. Again, every completely continuous mapping is partially completely continuous and
every partially completely continuous mapping is partially continuous and partially totally bounded,
but the converse may not be true.

Definition 2.7 (Dhage [3]). The order relation � and the metric d on a non-empty set E are said
to be compatible if {xn} is a monotone sequence, that is, monotone nondecreasing or monotone
nonincreasing sequence in E and if a subsequence {xnk} of {xn} converges to x

∗ implies that the
original sequence {xn} 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.

Clearly, the set R of real numbers with usual order relation ≤ and the norm defined by the absolute
value function has this property. Similarly, every finite dimensional euclidean space Rn possesses
compatibility property with respect to the usual component-wise order relation ≤ and the standard
norm ‖ · ‖ in Rn.

The Dhage iteration principle or method developed in Dhage [3, 4, 6] may be described as
“the monotonic convergence of the sequence of successive approximations to the solution
of a nonlinear equation beginning with a lower or an upper solution of the equation as
its initial or first approximation ” which forms a useful tool in the subject of existence theory of
nonlinear analysis. It is clear that Dhage iteration method is different from usual Picard’s successive
iterations and embodied in some of the following applicable hybrid fixed point theorems of Dhage [4]
which are the key tools for our work contained in the present paper. A few other hybrid fixed point
theorems containing the Dhage iteration method along with their applications also appear in Dhage
[3, 4].

Theorem 2.8 (Dhage [4]). Let
(
E,�,‖ · ‖

)
be a regular partially ordered complete normed linear

space such that the order relation � and the norm ‖ · ‖ in E are compatible in every compact chain
C of E. Let T : E → E be a partially continuous, nondecreasing and partially compact operator. If
there exists an element x0 ∈ E such that x0 �T x0 or T x0 � x0, then the operator equation T x = x
has a solution x∗ in E and the sequence {T nx0} of successive iterations converges monotonically to
x∗.

Remark 2.9. The regularity of E in above Theorem 2.8 may be replaced with a stronger continuity
condition of the operator T on E which is a result proved in Dhage [4].

The following hybrid fixed point theorems are employed for proving the existence and uniqueness
of the solutions for the FDE considered in the subsequent section of the paper. Before stating these
results, we consider the following definition in what follows.

Definition 2.10. An upper semi-continuous and nondecreasing function ψ : R+ → R+ is called a
D-function provided ψ(0) = 0. An operator T : E → E is called partially nonlinear D-contraction if



NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 133

there exists a D-function ψ such that

(2.1) ‖T x−T y‖≤ ψ
(
‖x−y‖

)
for all comparable elements x,y ∈ E, where 0 < ψ(r) < r for r > 0. In particular if ψ(r) = k r, T is
a partially linear contraction on E with a contraction constant k.

Theorem 2.11 (Dhage [3]). Let (E,�,‖·‖) be a partially ordered Banach space and let T : E → E be
a nondecreasing and partially nonlinear D-contraction. Suppose that there exists an element x0 ∈ E
such that x0 �T x0 or x0 �T x0. If T is continuous or E is regular, then T has a fixed point x∗ and
the sequence {T nx0} of successive iterations converges monotonically to x∗. Moreover, the fixed point
x∗ is unique if every pair of elements in E has a lower and an upper bound.

Theorem 2.12 (Dhage [4]). Let
(
E,�,‖ · ‖

)
be a regular partially ordered complete normed linear

space such that the order relation � and the norm ‖·‖ in E are compatible in every compact chain C
of E. Let A,B : E → E be two nondecreasing operators such that

(a) A is partially bounded and partially nonlinear D-contraction,
(b) B is partially continuous and partially compact, and
(c) there exists an element x0 ∈ E such that x0 �Ax0 + Bx0 or x0 �Ax0 + Bx0.

Then the operator equation Ax + Bx = x has a solution x∗ in E and the sequence {xn} of successive
iterations defined by xn+1 = Axn + Bxn, n=0,1,. . . , converges monotonically to x∗.

Remark 2.13. The compatibility of the order relation � and the norm ‖ ·‖ in every compact chain
of E holds if every partially compact subset of E possesses the compatibility property with respect
to � and ‖ · ‖. This simple fact has been utilized in proving the main existence and approximations
results for the considered the FDE (3.1) on J.

Note that the Dhage iteration method presented in the above hybrid fixed point theorems have
been employed in Dhage and Dhage [9, 10, 11] for approximating the solutions of initial value prob-
lems of nonlinear first order ordinary differential equation under some natural hybrid conditions. In
the following section we approximate the solutions of certain IVPs of nonlinear integro-differential
equations via successive approximations beginning with the loser or upper solution.

3. Existence and Uniqueness Theorems

The equivalent integral form of the FDE (1.1) is considered in the function space C(J,R) of contin-
uous 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) 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 crucial lemma concerning the compatibility of the order relation
and the norm in C(J,R) 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 ‖ · ‖ and ≤ are compatible in every
partially compact subset of C(J,R).



134 DHAGE

Proof. Let S be a partially compact subset of C(J,R) and let {xn}n∈N be a monotone nondecreasing
sequence of points in S. Then we have

x1(t) ≤ x2(t) ≤ ···≤ xn(t) ≤ ··· , (ND)

for each t ∈ J.
Suppose that a subsequence {xnk}k∈N of {xn}n∈N is convergent and converges to a point x in

S. Then the subsequence {xnk (t)}k∈N of the monotone real sequence {xn(t)}n∈N is convergent. By
monotone characterization, the whole sequence {xn(t)}n∈N is convergent and converges to a point x(t)
in S for each t ∈ J. This shows that the sequence {xn}n∈N converges to x point-wise on J. To show the
convergence is uniform, it is enough to show that the sequence {xn(t)}n∈N is equicontinuous. Since S
is partially compact, every chain or totally ordered set and consequently {xn}n∈N is an equicontinuous
sequence by Arzelá-Ascoli theorem. Hence {xn}n∈N is convergent and converges uniformly to x. As a
result, ‖ ·‖ and ≤ are compatible in S. This completes the 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 FDE (1.1) if it satisfies
CDqu(t) ≤ f(t,u(t)), t ∈ J,

u(t0) ≤ α0,


 (∗)

Similarly, an upper solution v ∈ C1(J,R) to the FDE (3.1) is defined on J, by the above inequalities
with reverse sign.

3.1. Existence theorem. We consider the following set of assumptions in what follows:

(H1) There exists a constant Mf > 0 such that |f(t,x)| ≤ Mf for all t ∈ J and x ∈ R.
(H2) The function f(t,x) is monotone nondecreasing in x for each t ∈ J.
(H3) The FDE (3.1) has a lower solution u ∈ C1(J,R).
The following lemma is useful in what follows and may be found in Kilbas et.al. [15] and Podlubny

[17].

Lemma 3.3. For a given continuous function h : J → R, a function x ∈ C1(J,R) is a solution of the
FDE

(3.3)

CDqx(t) = h(t), t ∈ J,

x(t0) = α0,

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

(3.4) x(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1 h(s) ds, t ∈ J.

Theorem 3.4. Assume that the hypotheses (H1) through (H3) hold. Then the FDE (3.1) has a
solution x∗ defined on J and the sequence {xn}∞n=0 of successive approximations defined by

(3.5) x0 = u, xn+1(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1f((s,xn(s)) ds,

for all t ∈ J, converges monotonically to x∗.

Proof. By Lemma 3.3, the FDE (1.1) is equivalent to the nonlinear integral equation

(3.6) x(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds, 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.



NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 135

Define the operator T on E by

(3.7) T x(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds, t ∈ J.

From the continuity of the integral, it follows that T defines the map T : E → E. Then, the FDE
(3.1) is equivalent to the operator equation

(3.8) T x(t) = x(t), t ∈ J.

We shall show that the operator T satisfies all the conditions of Theorem 2.8. This is achieved in
the series of following steps.

Step I: T is nondecreasing operator on E.
Let x,y ∈ E be such that x ≤ y. Then by hypothesis (H2), we obtain

T x(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds

≤ α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,y(s)) ds

= T y(t),

for all t ∈ J. This shows that T is nondecreasing operator on E into E.

Step II: T 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→∞

T xn(t) = lim
n→∞

[
α0 +

1

Γq

∫ t
t0

(t−s)q−1f(s,xn(s)) ds
]

= α0 +
1

Γq

∫ t
t0

(t−s)q−1
[

lim
n→∞

f(s,xn(s))
]
ds

= α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds

= T x(t),

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

arbitrary with t1 < t2. Then

|T xn(t2) −T xn(t1)|

≤
∣∣∣∣ 1Γq

∫ t2
t0

(t2 −s)q−1f(s,xn(s)) ds−
1

Γq

∫ t2
t0

(t1 −s)q−1f(s,xn(s)) ds
∣∣∣∣

+

∣∣∣∣ 1Γq
∫ t2
t0

(t1 −s)q−1f(s,xn(s)) ds−
1

Γq

∫ t1
t0

(t1 −s)q−1f(s,xn(s)) ds
∣∣∣∣

≤
1

Γq

∣∣∣∣
∫ t2
t0

|(t2 −s)q−1 − (t1 −s)q−1| |f(s,xn(s))|ds
∣∣∣∣

+
1

Γq

∣∣∣∣
∫ t2
t1

(t1 −s)q−1|f(s,xn(s))|ds
∣∣∣∣



136 DHAGE

≤
Mf
Γq

∣∣∣∣
∫ t2
t0

|(t2 −s)q−1 − (t1 −s)q−1|ds
∣∣∣∣ + MfΓq

∣∣∣∣
∫ t2
t1

(t1 −s)q−1 ds
∣∣∣∣

≤
Mf
Γq

∣∣∣∣
∫ t0+a
t0

|(t2 −s)q−1 − (t1 −s)q−1|ds
∣∣∣∣ + MfΓq |p(t1) −p(t2)|

where, p(t) =
Mf
Γq

∫ t
t0

(t0 + a−s)q−1 ds.

Since the functions t 7→ (t−s)q−1 and t 7→ p(t) are uniformly continuous on compact J = [t0, t0 +a],
we have that

|T xn(t2) −T xn(t1)|→ 0 as t2 → t1
uniformly for all n ∈ N. This shows that the convergence T xn → T x is uniformly and hence T is a
partially continuous on E.

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

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

|T x(t)| ≤ |α0| +
1

Γq

∫ t
t0

(t−s)q−1|f(s,x(s))|ds

≤ |α0| +
1

Γq

∫ t
t0

(t−s)q−1|f(s,x(s))|ds

≤ |α0| +
aq Mf

Γ(q + 1)

= r,

for all t ∈ J. Taking the supremum over t, we obtain ‖T x‖ ≤ r for all x ∈ C. Hence T (C) is a
uniformly bounded subset of E. Next, we will show that T (C) is an equicontinuous set in E. Let
t1, t2 ∈ J be arbitrary with t1 < t2. Then

|T x(t2) −T x(t1)|

≤
∣∣∣∣ 1Γq

∫ t2
t0

(t2 −s)q−1f(s,x(s)) ds−
1

Γq

∫ t2
t0

(t1 −s)q−1f(s,x(s)) ds
∣∣∣∣

+

∣∣∣∣ 1Γq
∫ t2
t0

(t1 −s)q−1f(s,x(s)) ds−
1

Γq

∫ t1
t0

(t1 −s)q−1f(s,x(s)) ds
∣∣∣∣

≤
1

Γq

∣∣∣∣
∫ t2
t0

|(t2 −s)q−1 − (t1 −s)q−1| |f(s,x(s))|ds
∣∣∣∣

+
1

Γq

∣∣∣∣
∫ t2
t1

(t1 −s)q−1|f(s,x(s))|ds
∣∣∣∣

≤
Mf
Γq

∣∣∣∣
∫ t2
t0

|(t2 −s)q−1 − (t1 −s)q−1|ds
∣∣∣∣ + MfΓq

∣∣∣∣
∫ t2
t1

(t1 −s)q−1 ds
∣∣∣∣

≤
Mf
Γq

∣∣∣∣
∫ t0+a
t0

|(t2 −s)q−1 − (t1 −s)q−1|ds
∣∣∣∣ + MfΓq |p(t1) −p(t2)| .

Since the functions t 7→ (t−s)q−1 and t 7→ p(t) are uniformly continuous on compact J = [t0, t0 +a],
we have that

|T x(t2) −T x(t1)|→ 0 as t2 → t1



NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 137

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

Step IV: u satisfies the operator inequality u ≤T u.
Since the hypothesis (H3) holds, u is a lower solution of (3.1) defined on J. Then,

(3.9) CDqu(t) ≤ f(t,u(t)),
satisfying,

(3.10) u(t0) ≤ α0,
for all t ∈ J.

Integrating (3.9) from t0 to t, we obtain

(3.11) u(t) ≤ α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,u(s)) ds,

for all t ∈ J. This show that u is a lower solution of the operator equation x = T x.

Thus T satisfies all the conditions of Theorem 2.8 in view of Remark 2.9 and we apply to conclude
that the operator equation T x = x has a solution. Consequently the fractional integral equation (3.6)
and the FDE (1.1) has a solution x∗ defined on J. Furthermore, the sequence {xn} of successive
approximations defined by (3.5) converges monotonically to x∗. This completes the proof. �

Remark 3.5. The conclusion of Theorem 3.4 also remains true if we replace the hypothesis (H3)
with the following one:

(H′3) The FDE (3.1) has an upper solution v ∈ C1(J,R).

Example 3.6. Given a closed and bounded interval J = [0, 1], consider the FDE,

(3.12)

CD1/2x(t) = π + tanh x(t), t ∈ J,

x(0) = 1.




Here, f(t,x) = π + tanh x. Clearly, the function f is continuous on J ×R. The function f satisfies
the hypothesis (H1) with Mf = π + 2. Moreover, the function f(t,x) = π + tanh x is nondecreasing
in x for each t ∈ J and so the hypothesis (H2) is satisfied.

Finally, the FDE (3.12) has a lower solution u defined by

u(t) = 1 −
1

Γ(1/2)

∫ t
0

(t−s)−1/2 ds = 1 +
2 t1/2
√
π

defined on J. Thus all the hypotheses of Theorem 3.4 are satisfied. Hence we conclude that the FDE
(3.12) has a solution x∗ defined on J and the sequence {xn}∞n=0 defined by

x0 = u, xn+1(t) = 1 +
1
√
π

∫ t
0

(t−s)−1/2 tanh xn(s) ds,

for all t ∈ J, converges monotonically to x∗.

3.2. Uniqueness theorem. Next, we prove the uniqueness theorem for the FDE (3.1) under weak
Lipschitz condition. We need the following hypothesis in what follows.

(H4) There exists a D-function φ such that
0 ≤ f(t,x) −f(t,y) ≤ φ(x−y)

for all x,y ∈ R, x ≥ y. Moreover, 0 <
aq

Γ(q + 1)
φ(r) < r for each r > 0.

Theorem 3.7. Assume that hypotheses (H2) through (H4) hold. Then the FDE (3.1) has a unique
solution x∗ defined on J and the sequence {xn} of successive approximations defined by (3.5) converges
monotonically to x∗.



138 DHAGE

Proof. Set E = C(J,R). Clearly, E is a lattice w.r.t. the order relation ≤ and so the lower and the
upper bound for every pair of elements in E exist. Define the operator T by (3.7). Then, the FDE
(1.1) is equivalent to the operator equation (3.8). We shall show that T satisfies all the conditions of
Theorem 2.11 in E.

Clearly, T is a nondecreasing operator on E into itself. We shall simply show that the operator
T is a nonlinear D-contraction on E. Let x,y ∈ E be any two elements such that x ≥ y. Then, by
hypothesis (H4),

|T x(t) −T y(t)| ≤
∣∣∣∣ 1Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds−
1

Γq

∫ t
t0

(t−s)q−1f(s,y(s)) ds
∣∣∣∣

≤
1

Γq

∣∣∣∣
∫ t
t0

(t−s)q−1|f(s,x(s)) −f(s,y(s))|ds
∣∣∣∣

≤
1

Γq

∣∣∣∣
∫ t
t0

(t−s)q−1φ(|x(s) −y(s)|) ds
∣∣∣∣

≤
1

Γq

∣∣∣∣
∫ t
t0

(t−s)q−1 ds
∣∣∣∣φ(‖x−y‖)

≤
aq

Γ(q + 1)
φ(‖x−y‖)

= ψ(‖x−y‖)(3.13)

for all t ∈ J, where ψ(r) =
aq

Γ(q + 1)
φ(r) < r, r > 0.

Taking the supremum over t, we obtain

‖T x−T y‖≤ ψ(‖x−y‖)

for all x,y ∈ E, x ≥ y. As a result T is a partially nonlinear D-contraction on E. Furthermore, it
can be shown as in the proof of Theorem 3.4 that the function u given in hypothesis (H3) satisfies
the operator inequality u ≤T u on J. Now a direct application of Theorem 2.11 yields that the FDE
(1.1) has a unique solution x∗ and the sequence {xn}∞n=0 of successive approximations defined by (3.5)
converges monotonically to x∗. �

Remark 3.8. The conclusion of Theorem 3.7 also remains true if we replace the hypothesis (H3)
with the following one:

(H′3) The FDE (3.1) has an upper solution v ∈ C1(J,R).

Example 3.9. Given a closed and bounded interval J = [0, 1], consider the FDE,

(3.14)

CD1/2x(t) = π +
1

2
arctan x(t), t ∈ J,

x(0) = 1.




Here, f(t,x) = π + 1
2

arctan x. Clearly, the function f is continuous on J × R. The function f
satisfies the hypothesis (H1) with Mf = π +

π

4
. Moreover, the function f(t,x) is nondecreasing in x

for each t ∈ J and so the hypothesis (H2) is satisfied. We show that f satisfies the hypothesis (H4)
on J ×R. Let x,y ∈ R be such that x ≥ y. Then, we have

0 ≤ f(t,x) −f(t,y) ≤
1

2

[
arctan x− arctan y

]
≤ φ(x−y)



NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 139

for all t ∈ J, where ψ is D-function defined by φ(r) =
1

2
·

r

1 + ξ2
and x > ξ > y. Furthermore,

aq

Γ(q + 1)
φ(r) =

1
√
π
·

r

1 + ξ2
< r for each 0 < ξ < r. Finally, the FDE (3.1) has a lower solution

u(t) = 1 −
π

2
·

1

Γ(1/2)
·
∫ t
0

(t−s)−1/2 ds = 1 +
√
π t1/2

defined on J. Thus all the hypotheses of Theorem 3.7 are satisfied and hence we conclude that the
FDE (3.14) has a unique solution x∗ defined on J and the sequence {xn}∞n=0 defined by

x0 = u, xn+1(t) = 1 +
1

2
√
π

∫ t
0

(t−s)−1/2 arctan xn(s) ds,

for all t ∈ J, converges monotonically to x∗.

4. Linear Perturbations of First Type

It is possible sometimes that the nonlinearity f involved in the FDE (1.1) neither satisfies the
hypothesis of Theorem 3.4 nor satisfies the hypothesis of Theorem 3.7, however the splitting functions
f1 and f2 of f in the form f = f1 + f2 satisfy the conditions of Theorems 3.4 and 3.7 respectively.
In the terminology of Dhage [2] it is called a hybrid fractional differential equation with a linear
perturbation of the FDE (1.1) of first type. Then in this case it of interest to obtain the conclusion
of Theorem 3.4 with stated algorithm for the solutions which is a problem of this section.

Given the notations of previous section, we consider the following nonlinear hybrid fractional dif-
ferential equation (in short HFDE),

(4.1)

CDqx(t) = f(t,x(t)) + g(t,x(t)), t ∈ J,

x(t0) = α0,




where f,g : J ×R → R are continuous functions.

By a solution of the HFDE (4.1) we mean a function x ∈ C1(J,R) that satisfies equation (4.1) on
J.

The HFDE (4.1) is a hybrid fractional differential equation with a linear perturbation of first type.
See Dhage [1, 2] and the references therein. The HFDE (4.1) is well-known in the literature and
discussed at length for existence as well as other aspects of the solutions. In the present discussion,
it is proved that the existence of the solutions may be proved under mixed partially Lipschitz and
partially compactness type conditions.

We need the following definition in what follows.

Definition 4.1. A function u ∈ C1(J,R) is said to be a lower solution of the HFDE (4.1) if it satisfies
CDqu(t) ≤ f(t,u(t)) + g(t,u(t)), t ∈ J,

u(t0) ≤ α0.


 (∗∗)

Similarly, an upper solution v ∈ C1(J,R) to the HFDE (4.1) is defined on J by the above inequalities
with reverse sign.

We consider the following set of assumptions in what follows:

(H5) There exists a constant Mg > 0 such that |g(t,x)| ≤ Mg for all t ∈ J and x ∈ R.
(H6) The mapping g(t,x) is monotone nondecreasing in x for each t ∈ J.
(H7) The HFDE (4.1) has a lower solution u ∈ C1(J,R).



140 DHAGE

Theorem 4.2. Assume that the hypotheses (H1) and (H4) through (H7) hold. Then the HFDE (4.1)
has a solution x∗ defined on J and the sequence {xn}∞n=0 of successive approximations defined by

x0 = u, xn+1(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,xn(s)) ds

+
1

Γq

∫ t
t0

(t−s)q−1g(s,xn(s)),(4.2)

for all t ∈ J, converges monotonically to x∗.

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

By Lemma 3.3, the HFDE (4.1) is equivalent to the nonlinear integral equation

x(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds

+
1

Γq

∫ t
t0

(t−s)q−1g(s,x(s)) ds, t ∈ J.(4.3)

Set E = C(J,R) and define the operators A and B on E by

(4.4) Ax(t) =
1

Γq

∫ t
t0

(t−s)q−1f(s,x(s)) ds, t ∈ J,

and

(4.5) Bx(t) = α0 +
1

Γq

∫ t
t0

(t−s)q−1g(s,x(s)) ds, t ∈ J.

From the continuity of the integrals, it follows that A and B define the operators A,B : E → E.
Now, the FDE (4.1) is equivalent to the operator equation

(4.6) Ax(t) + Bx(t) = x(t), t ∈ J.

Then following the arguments similar to those given in Theorems 3.4 and 3.7, it can be shown
that the operator A is partially bounded and nonlinear D-contraction and B is partially continuous
and partially compact operator on E into itself. Finally, by hypothesis (H7), we have an element
u ∈ C1(J,R) such that u ≤Au + Bu. Now by a direct application of Theorem 2.12 we conclude that
the operator equation Ax + Bx = x has a solution x∗. Consequently the FDE (4.1) has a solution
x∗ and the sequence {xn}∞n=1 defined by (4.2) converges monotonically to x∗. This completes the
proof. �

The conclusion of Theorems 4.2 also remains true if we replace the hypothesis (H7) with the
following one:

(H′7) The HFDE (4.1) has an upper solution v ∈ C1(J,R).

Example 4.3. Given a closed and bounded interval J = [0, 1], consider the HFDE,

(4.7)

CD1/2x(t) = π +
1

2
arctan x(t) + tanh x(t), t ∈ J,

x(0) = 1.




Here, f(t,x) = π + 1
2

arctan x and g(t,x) = tanh y. Then the function f satisfies the hypothesis

(H1) with Mf = π +
π

4
. Again, f satisfies (H4) with ψ(r) =

1

2
·

r

1 + ξ2
, 0 < ξ < r. Next, g satisfies



NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 141

(H5) with Mg = 1. Similarly, g satisfies the hypothesis (H6). Finally, the function

u(t) = 1 −
1

Γ(1/2)

∫ t
0

(t−s)−1/2 ds−
1

Γ(1/2)
·
∫ t
0

(t−s)−1/2 ds

= 1 +
4
√
π
t1/2,

for all t ∈ J is a lower solution of the HFDE (4.7) defined on J. As a result, we apply Theorem 4.2
and conclude that the FDE (4.7) has a solution x∗ on J and the sequence {xn}∞n=0 defined by

x0 = u, xn+1(t) = 1 +
1

2
√
π

∫ t
0

(t−s)−1/2 arctan x(s)

+
1
√
π

∫ t
0

(t−s)−1/2 tanh xn(s) ds

for each t ∈ J, converges monotonically x∗.

Remark 4.4. In this paper we have proved only the existence and uniqueness results for the FDE
(1.1) and its linear perturbations of first type (4.1). However, other aspects of the solutions of these
FDEs such as the existence of minimal and maximal solutions and comparison theorems could also be
proved using the same Dhage iteration method with appropriate modifications. Furthermore, if the
FDE (1.1) has a lower solution u and an upper solution v such that u ≤ v, then the corresponding
solutions x∗ and x

∗ of the FDE (1.1) satisfy x∗ ≤ x∗ and are the minimal and maximal solutions in
the vector segment [u,v] of the Banach space E = C(J,R). Because the order relation ≤ defined by
(3.1) is equivalent to the order relation defined by the order cone

(4.8) K =
{
x ∈ C(J,R) | x(t) ≥ 0 for all t ∈ J

}
,

which is a closed set in C(J,R).

5. Dhage iteration and Other principles

We remark that the existence theorem for the FDE (1.1) can also be proved via classical Schauder
fixed point principle under the compactness type arguments. The procedure involves the construction
of a closed convex and bounded subset S of the Banach space C(J,R) and a continuous and compact
operator T which maps S into itself. This is somewhat a troublesome job and moreover, it does not
yield any computational scheme for the solutions. In the present approach of the Dhage iteration prin-
ciple, we do not need any convexity argument and along with the existence we obtain an algorithm for
the numerical solutions via construction of a sequence of successive approximations. Again, the exis-
tence and uniqueness theorem for the FDE (1.1) is generally proved via classical Banach contraction
mapping principle which involves a strong Lipschitz condition on the nonlinearity f, but here in the
present approach, the Lipschitz condition is replaced by a weaker partial nonlinear Lipschitz condition
and obtained the same conclusion with some additional information about the characterizations of
convergence of the sequence of successive approximations. Similarly, the existence theorem for the
HFDE (4.1) may be proved via classical Krasnoselskii type fixed point principle under the usual mixed
Lipschitz and compactness type conditions which again does not yield any numerical or approximate
solution. See Dhage [1] and the references therein. But in the case of Dhage iteration principle for
such nonlinear hybrid equations, we need the weaker partial Lipschitz and partial compactness type
conditions and we obtain an algorithm in terms of a sequence of successive approximations for the
numerical solutions. Thus in all cases, the Dhage iteration principle has some advantages over the
classical existing methods involving the basic Schauder, Banach and Krasnoselskii fixed point princi-
ples. The limitation of the Dhage iteration principle is that one needs an extra monotone feature of
the nonlinearity in the unknown variable, but as a result we obtain an additional information about
the monotonic characterization of the convergence of the sequence of successive approximations to the
solutions of the problems in question.



142 DHAGE

Another approach of classic upper and lower solution method in the existence theory for the FDEs
(1.1), (4.1) and similar nonlinear problems is well-known in the literature in which one needs both
the upper and lower solutions to exist in order to prove the existence of the solutions of the related
differential equation. Moreover, the lower solution u and an upper solution v of the related problem
should satisfy the inequality u ≤ v. The work along this line may be found in Heikkilä and Laksh-
mikatham [14] and the references therein. The novelty of our present approach with respect to the
existing ones lies in the fact that we do not assume both lower and upper solutions in our existence
theorems even if they exist, but assume only one of them and prove the existence of solutions in a
constructive way. Furthermore, even if the lower and an upper solution exist, they may not satisfy
the inequality mentioned above. Therefore, in such a case the usual classic upper and lower solution
method does not work. This is the main advantage of the present approach over the previous classic
upper and lower solution method for the nonlinear differential and other different types of equations.

6. Conclusion

From the foregoing discussion it is clear that the Dhage iteration principle forms an interesting
powerful tool for discussing the existence results of certain nonlinear hybrid fractional differential
equations. However, it has some limitations that unlike Picard’s method, the new method does not
give any idea about the rate of convergence of the sequence of successive approximations. But in a way
we have been able to prove the numerical solutions of the considered nonlinear fractional differential
equation (1.1) and its perturbation (4.1). Finally, while concluding this paper we mention that the
fractional differential equation considered here is of very simple nature for which we have illustrated the
Dhage iteration principle to obtain the algorithms for the solutions under weaker partially Lipschitz
and compactness conditions. However, an analogous study could also be made for other complex
fractional differential equations using similar method with appropriate modifications. Some of the
results along this line will be reported elsewhere.

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NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 143

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