International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 1 (2016), 49-61
http://www.etamaths.com

DHAGE ITERATION METHOD FOR NONLINEAR FIRST ORDER

HYBRID DIFFERENTIAL EQUATIONS WITH A LINEAR

PERTURBATION OF SECOND TYPE

B.C. DHAGE∗

Abstract. In this paper the authors prove algorithms for the existence and approximation of the

solutions for an initial and a periodic boundary value problem of nonlinear first order ordinary hybrid
differential equations with a linear perturbation of second type via Dhage iteration method. Examples

are furnished to illustrate the hypotheses and main abstract results of this paper.

1. Introduction

Given a closed and bounded interval J = [0,T] in the real line R, consider the initial and periodic
boundary value problems of first order nonlinear hybrid differential equation (in short HDE),

(1.1)

d

dt

[
x(t) −f(t,x(t))

]
+ λ

[
x(t) −f(t,x(t))

]
= g(t,x(t)), t ∈ J,

x(0) = α0,




and

(1.2)

d

dt

[
x(t) −f(t,x(t))

]
+ λ

[
x(t) −f(t,x(t))

]
= g(t,x(t)), t ∈ J,

x(0) = x(T),




where λ ∈ R, λ > 0 and f,g : J ×R → R are continuous functions.

By a solution of the HDE (1.1) or (1.2) we mean a function x ∈ C(J,R) such that
(i) the function t 7→ x−f(t,x) is differentiable for each x ∈ R, and

(ii) x satisfies the equations in (1.1) or (1.2),

where C(J,R) is the space of continuous real-valued functions defined on J.

The HDEs (1.1) and (1.2) are linear perturbations of the second type of the nonlinear differential
equations

(1.3)
x′(t) = g(t,x(t)), t ∈ J,

x(t0) = α0,

}
and

(1.4)
x′(t) = g(t,x(t)), t ∈ J,

x(0) = x(T),

}
and a sharp classification of different types of perturbations of a differential equation appears in
Dhage [2] which can be treated with the hybrid fixed point theory (see Dhage [2, 3] and Dhage and
Lakshmikantham [14]). The HDE (1.1) with λ = 0 has been thoroughly discussed in the literature for
different basic aspects of the solutions such as existence theorem, differential inequalities, maximal and
minimal solutions, comparison principle under some mixed Lipschitz and compactness type conditions.

2010 Mathematics Subject Classification. 34A12, 34H34, 47H07, 47H10.
Key words and phrases. hybrid differential equation; Dhage iteration method; hybrid fixed point theorem; approxi-

mation theorem.

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.

49



50 DHAGE

See for example Dhage and Jadhav [13], Dhage [5, 6] and references therein. However, the HDE (1.2)
is new to the literature . In this paper we prove algorithms in terms of successive approximations for
proving the existence and approximate solutions of the considered hybrid differential equations. We
claim that the results of this paper are new basic and important contribution to the theory of nonlinear
ordinary differential equations.

2. Auxiliary Result

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 a partially ordered
normed linear space appear in Dhage [3], Heikkilä and Lakshmikantham [16] 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 the 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 [16] 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.

Definition 2.2 (Dhage [4]). 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 called partially continuous on E if it is partially continuous at every point of it. It is clear that if T
is partially continuous on E, then it is continuous on every chain C contained in E.

Definition 2.3 (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. A nondecreasing 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 C in E, T (C) are bounded by a unique constant.

Definition 2.4 (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. A nondecreasing mapping T : E → E 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 partially completely continuous on E.

Definition 2.5 (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. 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.

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 and so is a
Janhavi Banach space.



DHAGE ITERATION METHOD FOR HYBRID DIFFERENTIAL EQUATIONS 51

Definition 2.6. An upper semi-continuous and monotone 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 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,
k > 0, T is called a partial Lipschitz operator with a Lischitz constant k and moreover, if 0 < k < 1,
T is called a partial linear contraction on E with a contraction constant k.

The Dhage iteration principle or method (in short DIP or DIM) developed in Dhage [3, 4, 7] may be
described as “ the sequence of successive approximations of a nonlinear equation beginning with a lower
or an upper solution as its first or initial approximation converges monotonically to the solution.” The
aforsaid convergence principle forms a basic and powerful tool in the study of nonlinear differential
and integral equations. See Dhage [7, 8], Dhage and Dhage [9, 10, 11, 12] and the references therein.
The following applicable hybrid fixed point theorem of Dhage [4] containing the DIP is used as a key
tool for the work contained in this paper.

Theorem 2.7 (Dhage [4]). Let
(
E,�,‖·‖

)
be a regular partially ordered complete normed linear space

such every compact chain C of E is Janhavi. 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 2.8. The condition that every compact chain of E is Janhavi holds if every partially compact
subset of E possesses the compatibility property with respect to the order relation � and the norm
‖ · ‖ in it. This simple fact is used to prove the desired characterization of the mild solution of the
problem (1.1) on J.

Remark 2.9. We remark that hypothesis (a) of Theorem 2.7 implies that the operator A is partially
continuous and consequently both the operators A and B in the theorem are partially continuous on
E. The regularity of E in above Theorem 2.7 may be replaced with a stronger continuity condition of
the operators A and B on E which is a result proved in Dhage [3, 4].

3. Main Results

In this section, we prove an existence and approximation result for the HDE (1.1) on a closed and
bounded interval J = [0,T] under mixed partial Lipschitz and partial compactness type conditions on
the nonlinearities involved in it. We place the HDE (1.1) 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) 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 lattice so that every pair of elements of E has a lower and an upper bound in
it. The following lemma concerning the Janhaviness of subsets of C(J,R) follows immediately form
the Arzelá-Ascoli theorem for compactness.

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 of C(J,R)
is Janhavi.

Proof. The proof of the lemma appears in Dhage and Dhage [9] and so we omit the details. �



52 DHAGE

We consider the following basic hypotheses in what follows.

(A0) The mapping x 7→ x−f(t,x) is increasing in R for each t ∈ J.
(A1) There exists a D-function ψ such that

0 ≤ f(t,x) −f(t,y) ≤ ψ(x−y)

for all t ∈ J and x,y ∈ R with x ≥ y. Moreover, 0 < ψ(r) < r for r > 0.
(A2) There exists aconstant Mf > 0 such that |f(t,x)| ≤ Mf for all t ∈ J and x ∈ R.
(B1) There exists a Mg > 0 such that |g(t,x)| ≤ Mg, for all t ∈ J and x ∈ R.
(B2) g(t,x) is nondecreasing in x for each t ∈ J.

Remark 3.2. If the hypothesis (A0) holds, then the function x 7→ x−f(t,x) is injective in R for each
t ∈ J.

Remark 3.3. The hypotheses (A1) and (A2) hold, in particular if f satisfies the inequality

0 ≤ f(t,x) −f(t,y) ≤
L(x−y)

K + (x−y)
for all x,y ∈ R with x ≥ y, where L > 0 and K > 0 are constants satisfying L ≤ K.

3.1. Initial value problem. The following useful lemma follows from the theory of calculus and
linear differential equations.

Lemma 3.4. Assume that hypothesis (A0) holds. Then for any continuous function h : J → R, the
function x ∈ C(J,R) is a solution of the HDE

(3.3)

d

dt
[x(t) −f(t,x(t))] + λ[x(t) −f(t,x(t))] = h(t), t ∈ J,

x(0) = α0,




if and only if x satisfies the hybrid integral equation (HIE)

(3.4) x(t) = ce−λt + f(t,x(t)) + e−λt
∫ t
0

eλsh(s) ds, t ∈ J,

where c = α0 −f(0,α0).

Proof. Let h ∈ C(J,R). Assume first that x is a solution of the HDE (3.3) defined on J. By definition,

the function t 7→ x(t) − f(t,x(t)) is continuous on J, and so, differentiable there, whence
d

dt

[
x(t) −

f(t,x(t))
]

is integrable on J. Applying integration to (3.3) from 0 to t, we obtain the HIE (3.4) on J.

Conversely, assume that x satisfies the HIE (3.3). Then by direct differentiation we obtain the first
equation in (3.4). Again, substituting t = 0 in (3.4) yields

x(0) −f(0,x(0)) = α0 −f(0,α0).

Since the mapping x 7→ x−f(t,x) is increasing in R for all t ∈ J, the mapping x 7→ x−f(0,x) is
injective in R, whence x(0) = α0. Hence the proof of the lemma is complete. �

We need the following definition in what follows.

Definition 3.5. A function u ∈ C(J,R) is called a lower solution of the HDE (1.1) on J if the function
t 7→ x−f(t,x) is differentiable and satisfies the inequalities

d

dt

[
x(t) −f(t,x(t))

]
+ λ
[
x(t) −f(t,x(t))

]
≤ g(t,x(t)),

x(0) ≤ α0,




for all t ∈ J. Similarly, an upper solution of the HDE (1.1) on J is defined.

(B3) The HDE (1.1) has a lower solution u ∈ C(J,R).
Now we are in a position to prove the following existence and approximation theorem for the HDE

(1.1) on J.



DHAGE ITERATION METHOD FOR HYBRID DIFFERENTIAL EQUATIONS 53

Theorem 3.6. Assume that the hypotheses (A0) through (A2) and (B1) through (B3) hold. Then
the HDE (1.1) 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) = ce
−λt + f(t,xn(t)) + e

−λt
∫ t
0

eλsg(s,xn(s)) ds, t ∈ J,

converges monotonically to x∗, where c = α0 −f(0,α0).

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

Now, using the hypotheses (A0) and (A2), it can be shown by an application of Lemma 3.4 that
the HDE (1.1) is equivalent to the nonlinear HIE

(3.6) x(t) = ce−λt + f(t,x(t)) + e−λt
∫ t
0

eλsg(s,x(s)) ds

for t ∈ J.

Define two operators A,B : E → E by
(3.7) Ax(t) = f(t,x(t)), t ∈ J,
and

(3.8) Bx(t) = ce−λt + e−λt
∫ t
0

eλsg(s,x(s)) ds, t ∈ J.

Then, the HIE (3.6) is transformed into an operator equation as

(3.9) Ax(t) + Bx(t) = x(t), t ∈ J.
We shall show that the operators A and B satisfy all the conditions of Theorem 2.7. Firstly, we

show that the operators A and B are nondecreasing on E. Let x,y ∈ E be such that x ≥ y. Then, by
hypothesis (A1),

Ax(t) = f(t,x(t)) ≥ f(t,y(t)) = Ay(t)
for all t ∈ J. Similarly, by hypothesis (A3),

Bx(t) = ce−λt + e−λt
∫ t
0

eλsg(s,x(s)) ds

≥ ce−λt + e−λt
∫ t
0

eλsg(s,y(s)) ds

= By(t)
for all t ∈ J. This shows that A and B are nondecreasing operators on E into E.

From (A2) it follows that

‖Ax‖≤ sup
t∈J
|Ax(t)| ≤ sup

t∈J
|f(t,x)| ≤ Mf

for all x ∈ E. As a result, A is bounded and consequently partially bounded on E. Next, we show that
A is a partial nonlinear D-contraction on E with a D function ψ. Let x,y ∈ E be such that x ≥ y.
Then, by hypothesis (A1),

|Ax(t) −Ay(t)| = |f(t,x(t)) −f(t,y(t))|

= f(t,x(t)) −f(t,y(t))

≤ ψ(x(t) −y(t))

= ψ(|x(t) −y(t)|)

≤ ψ(‖x−y‖)



54 DHAGE

for all t ∈ J. Taking the supremum over t, we obtain

‖Ax−Ay‖≤ ψ(‖x−y‖)

for all x,y ∈ E with x ≥ y. This shows that A is a partial nonlinear D-contraction on E with the
D-function ψ.

Next, we show that B is a partially compact and partially continuous operator on E into E. First
we show that B is a partially continuous on E. Let {xn} be a sequence in a chain C of E converging
to a point x ∈ C. Then by the dominated convergence theorem for integration, we obtain

lim
n→∞

Bxn(t) = lim
n→∞

[
ce−λt + e−λt

∫ t
0

eλsg(s,xn(s)) ds

]
= ce−λt + lim

n→∞
e−λt

∫ t
0

eλsg(s,xn(s)) ds

= ce−λt + e−λt
∫ t
0

eλs
[

lim
n→∞

g(s,xn(s))
]
ds

= ce−λt + e−λt
∫ t
0

eλsg(s,x(s)) ds

= Bx(t)

for all t ∈ J. Moreover, it can be shown as below that {Bxn} is an equicontinuous sequence of functions
in E. Now, following the arguments similar to that given in Granas and Dugundji [15], it is proved
that B is a a partially continuous operator on E.

Next, we show that B is a partially compact operator on E. It is enough to show that B(C) is a
uniformly bounded and equi-continuous set in E for every chain C in E. Let x ∈ C be arbitrary. Then
by yhe hypothesis (A2),

|Bx(t)| ≤
∣∣ce−λt∣∣ + ∣∣∣∣e−λt

∫ t
0

eλsg(s,x(s)) ds

∣∣∣∣
≤ |α0 −f(t0,α0)| +

∫ T
t0

eλsMg ds

≤ |α0 −f(t0,α0)| +
(eλT − 1)Mg

λ

for all t ∈ J. Taking the supremum over t, we obtain

‖Bx‖≤ |α0 −f(t0,α0)| +
(eλT − 1)Mg

λ

for all x ∈ C. This shows that B is uniformly bounded on C.



DHAGE ITERATION METHOD FOR HYBRID DIFFERENTIAL EQUATIONS 55

Again, let t1, t2 ∈ J be arbitrary. Then for any x ∈ C, one has
|Bx(t1) −Bx(t2)| = |c|

∣∣e−λt1 −e−λt2∣∣
+

∣∣∣∣e−λt1
∫ t1
0

eλsg(s,x(s)) ds−e−λt2
∫ t2
0

eλsg(s,x(s)) ds

∣∣∣∣
≤ |c|

∣∣e−λt1 −e−λt2∣∣
+
∣∣e−λt1 −e−λt2∣∣ ∣∣∣∣

∫ t1
0

eλsg(s,x(s)) ds

∣∣∣∣
+
∣∣e−λt2∣∣ ∣∣∣∣

∫ t1
0

eλsg(s,x(s)) ds−
∫ t2
0

eλsg(s,x(s)) ds

∣∣∣∣
≤ |c|

∣∣e−λt1 −e−λt2∣∣ + [(eλT − 1)Mg
λ

]∣∣e−λt1 −e−λt2∣∣
+

∣∣∣∣
∫ t1
t2

eλs|g(s,x(s))|ds
∣∣∣∣

≤
[
|c| +

(eλT − 1)Mg
λ

]∣∣e−λt1 −e−λt2∣∣ + |p(t1) −p(t2)|
where, p(t) =

∫ t
0

eλsMg ds.

Since the functions t 7→ e−λt and t 7→ p(t) are continuous on compact J, they are uniformly
continuous there. Hence, for � > 0, there exists a δ > 0 such that

|t1 − t2| < δ =⇒ |Bx(t1) −Bx(t2)| < �
uniformly for all t1, t2 ∈ J and for all x ∈ S. This shows that B(C) is an equi-continuous set in E.
Now the set B(C) is uniformly bounded and equicontinuous in E, so it is compact by Arzelá-Ascoli
theorem. As a result, B is a partially continuous and partially compact operator on E.

Thus, all the conditions of Theorem 2.7 are satisfied and hence the operator equation Ax +Bx = x
has a solution x∗ in E and the sequence of successive approximations {xn} defined by xn = Axn−1 +
Bxn−1 converges monotonically to x∗. As a result, the HDE (1.1) has a solution x∗ defined on J and
the sequence of successive approximations {xn} defined by (3.5) converges monotonically to x∗. This
completes the proof. �

Remark 3.7. We remark that Theorem 3.6 also remains true if we replace the hypothesis (B3) with
the following one:

(B′3) The HDE (1.1) has an upper solution v ∈ C(J,R).

Remark 3.8. We note that if the HDE (1.1) has a lower solution u as well as an upper solution v
such that u ≤ v, then under the given conditions of Theorem 3.6 it has corresponding solutions x∗
and x∗ and these solutions satisfy x∗ ≤ x∗. Hence they are the minimal and maximal solutions of the
HDE (1.1) in the vector segment [u,v] of the Banach space E = C(J,R), where the vector segment
[u,v] is a set of elements in C(J,R) defined by

[u,v] = {x ∈ C(J,R) | u ≤ x ≤ v}.
This is because the order relation ≤ defined by (3.2) is equivalent to the order relation defined by the
order cone K = {x ∈ C(J,R) | x ≥ θ} which is a closed set in C(J,R).

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

(3.10)

d

dt

[
x(t) − tan−1 x(t)

]
+
[
x(t) − tan−1 x(t)

]
= tanh x(t), t ∈ J,

x(0) = 1.




Here, λ = 1 and the functions f and g are given by

f(t,x) = tan−1 x and g(t,x) = tanh x



56 DHAGE

for all t ∈ J and x ∈ R. We show that the functions f and g satisfy all the hypotheses of Theorem
3.6. First we show that f satisfies the hypotheses (A0)-(A2). Now,

∂

∂x

[
x−f(t,x)

]
=

d

dx

[
x− tan−1 x

]
= 1 −

1

1 + x2
> 0,

for all x ∈ R and t ∈ J, so that the function x 7→ x−f(t,x) is increasing in R for each t ∈ J. Therefore,
hypothesis (A0) holds. Next, let x,y ∈ R be such that x ≥ y. Then,

0 ≤ f(t,x) −f(t,y) ≤ tan−1 x− tan−1 y =
1

1 + ξ2
(x−y)

for all x > ξ > y, showing that f satisfies the hypothesis (A1) with D-function ψ given by

ψ(r) =
r

1 + ξ2
< r, r > 0,

where ξ 6= 0. Again,
|f(t,x)| = |tan−1 x| ≤

π

2
,

for all t ∈ J and x ∈ R. This shows that f satisfies hypothesis (A2) with Mf =
π

2
.

Furthermore,

|g(t,x)| = |tanh x| ≤ 1,
for all t ∈ J and x ∈ R, so that the hypothesis (B1) holds with Mg = 1. Again, since the function
x 7→ tanh x is nondecreasing in R and so the hypothesis (B2) is satisfied. Finally, the function
u(t) = −(t + 3) is a lower solution of the HDE (3.10) defined on J = [0, 1].

Thus the functions f and g satisfy all the conditions of Theorem 3.6. Hence we apply and conclude
that the HDE (3.10) has a solution x∗ defined on J and the sequence {xn} of successive approximations
defined by

x0 = u, xn+1(t) = 1 −
π

2
+ tan−1 xn(t) + e

−t
∫ t
0

es tanh xn(s) ds,

for each t ∈ J, converges monotonically to x∗. A similar conclusion also remains true if we replace the
lower solution u with the upper solution v(t) = t + 3, t ∈ J.

3.2. Periodic boundary value problem. The following useful lemma is obvious and may be found
in Dhage [1] and Nieto [17].

Lemma 3.10. For any function σ ∈ L1(J,R), x is a solution to the differential equation

(3.11)
x′(t) + λx(t) = σ(t), t ∈ J,

x(0) = x(T),

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

(3.12) x(t) =

∫ T
0

Gλ(t,s) σ(s) ds

where,

(3.13) Gλ(t,s) =



eλs−λt+λT

eλT − 1
, if 0 ≤ s ≤ t ≤ T,

eλs−λt

eλT − 1
, if 0 ≤ t < s ≤ T.

Notice that the Green’s function Gλ is continuous and nonnegative on J × J and therefore, the
number

Kλ := max{|Gλ(t,s)| : t,s ∈ [0,T]}
exists for all λ ∈ R+. For the sake of convenience, we write Gλ(t,s) = G(t,s) and Kλ = K.



DHAGE ITERATION METHOD FOR HYBRID DIFFERENTIAL EQUATIONS 57

Lemma 3.11. If there exists a differentiable function u ∈ C(J,R) such that

(3.14)
u′(t) + λu(t) ≤ σ(t), t ∈ J,

u(0) ≤ u(T),

}
then

(3.15) u(t) ≤
∫ T
0

Gλ(t,s) σ(s) ds,

for all t ∈ J, where G(t,s) is a Green’s function given by (3.5).

Proof. Suppose that the function u ∈ C(J,R) satisfies the inequalities given in (3.14). Multiplying the
first inequality in (3.14) by eλt, (

eλtu(t)
)′
≤ eλtσ(t).

A direct integration of above inequality from 0 to t yields

(3.16) eλtu(t) ≤ u(0) +
∫ t
0

eλsσ(s) ds,

for all t ∈ J. Therefore, in particular,

(3.17) eλTu(T) ≤ u(0) +
∫ T
0

eλsσ(s) ds.

Now u(0) ≤ u(T), so one has

(3.18) u(0)eλT ≤ u(T)eλT .

From (3.17) and 3.18) it follows that

(3.19) eλTu(0) ≤ u(0) +
∫ T
0

eλsσ(s) ds

which further yields

(3.20) u(0) ≤
∫ T
0

eλs

(eλT − 1)
σ(s) ds.

Substituting (3.20) in (3.16) we obtain

u(t) ≤
∫ T
0

G(t,s)σ(s) ds,

for all t ∈ J. This completes the proof. �

We need the following definition in what follows.

Definition 3.12. A function u ∈ C(J,R) is called a lower solution of the HDE (1.1) if the function
t 7→ x−f(t,x) is differentiable and satisfies the inequalities

d

dt

[
u(t) −f(t,u(t))

]
+ λ
[
u(t) −f(t,u(t))

]
≤ g(t,u(t)),

u(0) ≤ u(T),




for all t ∈ J. Similarly, an upper solution v ∈ C(J,R) of the HDE (1.2) is defined.

We need the following hypotheses in what follows.

(B4) The function f(t,x) is periodic in t with period T for all x ∈ R, i.e., f(0,x) = f(T,x) for all
x ∈ R.

(B5) The HDE (1.2) has a lower solution u ∈ C(J,R).



58 DHAGE

Lemma 3.13. Assume that hypothesis (A0) holds. Then for any λ ∈ R+, the function x ∈ C(J,R) is
a solution of the HDE

(3.21)

d

dt
[x(t) −f(t,x(t))] + λ[x(t) −f(t,x(t))] = g(t,x(t)), t ∈ J,

x(0) = x(T),




if and only if x satisfies the hybrid integral equation (HIE)

(3.22) x(t) = f(t,x(t)) +

∫ T
0

G(t,s)g(s,x(s)) ds, t ∈ J.

Proof. Assume first that x is a solution of the HDE (3.22) defined on J. By definition, the function

t 7→ x(t) − f(t,x(t)) is continuous on J, and so, differentiable there, whence
d

dt

[
x(t) − f(t,x(t))

]
is

integrable on J. Again by hypothesis (B2), x(0) − f(0,x(0)) = x(T) − f(T,x(T)). So by a direct
application of Lemma 3.10, we obtain the HIE (3.22) on J.

Conversely, assume that x satisfies the HIE (3.21) on J. Then by a direct differentiation of (3.22),
we obtain the first equation in (3.21). Again, substituting t = 0 in (3.2) yields

x(0) −f(0,x(0)) = x(T) −f(0,x(T)).

Since the mapping x 7→ x−f(t,x) is increasing in R for all t ∈ J, the mapping x 7→ x−f(0,x) is
injective in R, and so x(0) = x(T). Hence the proof of the lemma is complete. �

Now we are in a position to prove the following existence and approximation theorem for the HDE
(1.2) on J.

Theorem 3.14. Assume that the hypotheses (A0) - (A2), (B1) - (B2) and (B4) - (B5) hold. Then
the HDE (1.2) has a solution x∗ defined on J and the sequence {xn}∞n=0 of successive approximations
defined by

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

∫ T
0

G(t,s)g(s,xn(s)) ds, t ∈ J,

converges monotonically to x∗.

Proof. Set E = C(J,R). Then, in view of Lemma 3.1, every compact chain C in E possesses the
compatibility property with respect to the norm ‖ · ‖ and the order relation ≤ so that every compact
chain C is Janhavi set in E.

Now, using the hypotheses (A0), it can be shown by an application of Lemma 3.4 that the HDE
(1.1) is equivalent to the nonlinear HIE

(3.24) x(t) = f(t,x(t)) +

∫ T
0

G(t,s)g(s,x(s)) ds

for t ∈ J.

Define two operators A,B : E → E by

(3.25) Ax(t) = f(t,x(t)), t ∈ J,

and

(3.26) Bx(t) =
∫ T
0

G(t,s)g(s,x(s)) ds, t ∈ J.

Then, the HIE (3.24) is transformed into an operator equation as

(3.27) Ax(t) + Bx(t) = x(t), t ∈ J.

We shall show that the operators A and B satisfy all the conditions of Theorem 3.6. Now, it can
be shown as in the proof of Theorem 3.6 that A is a nondecreasing partially bounded and partially



DHAGE ITERATION METHOD FOR HYBRID DIFFERENTIAL EQUATIONS 59

nonlinear D-contraction on E with a D-function ψ. Next, we show that the operator B is a nonde-
creasing, partially continuous and partially compact operator on E. Let x,y ∈ E be such that x ≥ y.
Then, by hypothesis (B3),

Bx(t) =
∫ T
0

G(t,s)g(s,x(s)) ds ≥
∫ T
0

G(t,s)g(s,y(s)) ds = By(t)

for all t ∈ J. This shows that A and B are nondecreasing operators on E into E.

Next, let {xn} be a sequence in a chain C of E converging to a point x ∈ C. Then by dominated
convergence theorem for integration, we obtain

lim
n→∞

Bxn(t) = lim
n→∞

[∫ T
0

G(t,s)g(s,xn(s)) ds

]

=

∫ T
0

G(t,s)
[

lim
n→∞

g(s,xn(s))
]
ds

=

∫ T
0

G(t,s)g(s,x(s)) ds

= Bx(t)

for all t ∈ J. Moreover, it can be shown as below that {Bxn} is an equicontinuous sequence of functions
in E. Now, following the arguments similar to that given in Granas and Dugundji [15], it is proved
that B is a a partially continuous operator on E.

Next, we show that B is a partially compact operator on E. It is enough to show that B(C) is a
uniformly bounded and equi-continuous set in E for every chain C in E. Let x ∈ C be arbitrary. Then
by hypothesis (A2),

|Bx(t)| ≤

∣∣∣∣∣
∫ T
0

G(t,s)g(s,x(s)) ds

∣∣∣∣∣ ≤
∫ T
0

G(t,s)|g(s,x(s))|ds ≤ KTMg

for all t ∈ J. Taking the supremum over t, ‖Bx‖ ≤ KTMg for all x ∈ C. This shows that B is
uniformly bounded on C.

Again, let t1, t2 ∈ J be arbitrary. Then for any x ∈ C, one has

|Bxn(t2) −Bxn(t1)| =

∣∣∣∣∣
∫ T
0

G(t1,s)f̃(s,xn(s)) ds−
∫ T
0

G(t2,s)g(s,xn(s)) ds

∣∣∣∣∣
≤

∫ T
0

|G(t1,s) −G(t2,s)||g(s,xn(s))|ds

≤ Mg
∫ T
0

|G(t1,s) −G(t2,s)|ds

→ 0 as t2 − t1 → 0.

Since the functions t 7→ G(t,s) is continuous on compact J, it is uniformly continuous there. Hence,
for � > 0, there exists a δ > 0 such that

|t1 − t2| < δ =⇒ |Bx(t1) −Bx(t2)| < �
uniformly for all t1, t2 ∈ J and for all x ∈ S. This shows that B(C) is an equi-continuous set in E.
Now the set B(C) is uniformly bounded and equicontinuous set in E, so it is compact by Arzelá-Ascoli
theorem. As a result, B is a partially continuous and partially compact operator on E.

Finally, we show that u is a lower solution of the operator equation Ax +Bx = x. Since u is a lower
solution of the HDE (1.2) on J, we have

d

dt

[
u(t) −f(t,u(t))

]
+ λ
[
u(t) −f(t,u(t))

]
≤ g(t,u(t)),

u(0) ≤ u(T),






60 DHAGE

for all t ∈ J. Again, since the hypotheses (A0) and (B4) hold, one has
d

dt

[
u(t) −f(t,u(t))

]
+ λ
[
u(t) −f(t,u(t))

]
≤ g(t,u(t)),[

u(0) −f(0,u(0))
]
≤
[
u(T) −f(T,u(T))

]
,




for all t ∈ J. Now an application Lemma 3.11 yields that

(3.28) u(t) ≤ f(t,u(t)) +
∫ T
0

G(t,s)g(s,u(s)) ds

for t ∈ J. This further in view of definitions of the operators A and B implies that u ≤Au + Bu and
that u is a lower solution of the operator equation Ax + Bx = x.

Thus, all the conditions of Theorem 3.6 are satisfied by the operators A and B and hence the
operator equation Ax + Bx = x has a solution x∗ in E and the sequence of successive approximations
{xn} defined by xn = Axn−1 + Bxn−1 converges monotonically to x∗. Consequently, the HDE (1.1)
has a solution x∗ defined on J and the sequence of successive approximations {xn} defined by (3.3)
converges monotonically to x∗. This completes the proof. �

Remark 3.15. We remark that Theorem 3.14 also remains true if we replace the hypothesis (B5)
with the following one:

(B′5) The HDE (1.2) has an upper solution v ∈ C(J,R).

Remark 3.16. We note that if the HDE (1.2) has a lower solution u as well as an upper solution v
such that u ≤ v, then under the given conditions of Theorem 3.14 it has corresponding solutions x∗
and x∗ and these solutions satisfy x∗ ≤ x∗. Hence they are the minimal and maximal solutions of the
HDE (1.2) in the vector segment [u,v] of the Banach space E = C(J,R), where the vector segment
[u,v] is a set of elements in C(J,R) defined by

[u,v] = {x ∈ C(J,R) | u ≤ x ≤ v}.

This is because the order relation ≤ defined by (3.2) is equivalent to the order relation defined by the
order cone K = {x ∈ C(J,R) | x ≥ θ} which is a closed set in C(J,R).

Example 3.17. Given a closed and bounded interval J = [0, 1] in R, consider the HDE

(3.29)

d

dt

[
x(t) − tan−1 x(t)

]
+
[
x(t) − tan−1 x(t)

]
= tanh x(t), t ∈ J,

x(0) = x(1).




Here, λ = 1 and the functions f and g are given by

f(t,x) = tan−1 x and g(t,x) = tanh x

for all t ∈ J and x ∈ R. Now, it can be shown that the functions f and g satisfy all the hypotheses of
Theorem 3.14 with u(t) = −4et, t ∈ [0, 1]. Hence we conclude that the HDE (3.29) has a solution x∗
defined on J and the sequence {xn} of successive approximations defined by

x0 = u, xn+1(t) = tan
−1 xn(t) +

∫ 1
0

G(t,s) tanh xn(s) ds,

for each t ∈ J, converges monotonically to x∗, where G(t,s) is a Green’s function associated with the
homogeneous PBVP

(3.30)
x′(t) + x(t) = 0, t ∈ J,

x(0) = x(1),

}
given by

G(t,s) =



es−t+1

e− 1
, if 0 ≤ s ≤ t ≤ 1,

es−t

e− 1
, if 0 ≤ t < s ≤ 1.



DHAGE ITERATION METHOD FOR HYBRID DIFFERENTIAL EQUATIONS 61

Again, a similar conclusion holds if we replace the lower solution u with the upper solution v(t) = 4et,
t ∈ [0, 1] in view of Remark 3.14.

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∗Corresponding author: bcdhage@email.email