International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 2 (2016), 101-111
http://www.etamaths.com

DHAGE ITERATION METHOD FOR APPROXIMATING POSITIVE SOLUTIONS

OF PBVPS OF NONLINEAR QUADRATIC DIFFERENTIAL EQUATIONS WITH

MAXIMA

SHYAM B. DHAGE, BAPURAO C. DHAGE∗

Abstract. In this paper authors prove the existence as well as approximation of the positive solu-

tions for a periodic boundary value problem of first order ordinary nonlinear quadratic differential
equations with maxima. An algorithm for the solutions is developed and it is shown that certain

sequence of successive approximations converges monotonically to the positive solution of considered

quadratic differential equations under some suitable mixed hybrid conditions. Our results rely on
the Dhage iteration principle embodied in a recent hybrid fixed point theorem of Dhage (2014). A

numerical example is also provided to illustrate the hypotheses and abstract theory developed in this

paper.

1. Introduction

Motivated by the linear differential equation with maxima that occurs in the automatic control
of some technical system, several mathematicians started working in the field of differential equations
with maxima for qualitative aspects of the solutions. See Magomedov [17, 18], Bainov and Hristova [1],
Otrocol and Rus [15] and the references therein for the details. Again, Mishkis [19] pointed out in his
survey the necessity of the study of differential equations with maxima and gave an impetus in the study
of such equations. Similarly, the study of periodic boundary value problems of nonlinear quadratic
differential equations is initiated in the works of Dhage et.al. [13] and Dhage [2]. Very recently, Dhage
et.al. [13] initiated the study of initial value problems of nonlinear quadratic differential equations
with maxima and proved the existence and approximation results for such equations. The purpose of
the present paper is to blend these two ideas together and discuss the PBVPs of first order quadratic
differential equations with maxima for existence and numerical aspects of the solutions. It is well-known
that the hybrid differential equations can be tackled using the Dhage iteration method embodied in
the hybrid fixed point theory initiated by Dhage [3, 5, 7]) which also yields the algorithms for the
solutions. Therefore, it is of interest to establish algorithms for the quadratic differential equations
with maxima (1.1) for existence and approximation of the solutions along similar lines. We claim that
our method as well as our results of this paper are new to the literature in the theory of nonlinear
differential equations with maxima.

Given a closed and bounded interval J = [0,T] in the real line R , consider the periodic boundary
value problem (in short PBVP) of nonlinear first order ordinary quadratic differential equation (QDE)
with maxima,

(1.1)

d

dt

[
x(t)

f(t,x(t))

]
+ λ

[
x(t)

f(t,x(t)))

]
= g(t,x(t),X(t))), t ∈ J,

x(0) = x(T),




for λ ∈ R, λ > 0, where f : J ×R×R → R\{0} and g : J ×R×R → R are continuous functions and
X(t) = max

t0≤ξ≤t
x(ξ) for t ∈ J.

2010 Mathematics Subject Classification. 34A12, 34A38.
Key words and phrases. quadratic differential equation with maxima; periodic boundary value problem; Dhage

iteration principle; approximate positive solution.

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.

101



102 S.B. DHAGE AND B.C. DHAGE

By a solution of the QDE (1.1) we mean a function x ∈ C1(J,R) that satisfies
(i) t 7→

x

f(t,x)
is a continuously differentiable function for each x,y ∈ R, and

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

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

The QDE (1.1) is a quadratic perturbation of second type of the nonlinear PBVP of first order
differential equations,

(1.2)
x′(t) = f(t,x(t)), t ∈ J,

x(0) = x(T),

}
The details of different types perturbations is given in Dhage [3]. The QDE (1.1) includes some

known classes of nonlinear differential equations studied earlier in the literature. In the following we
list a few special cases of (1.1) defined on J.

1. When f(t,x) = 1 for all t ∈ J and x,y ∈ R, the QDE (1.1) reduces to the following known
nonlinear differential equation with maxima

(1.3)
x′(t) + λx(t) = g(t,x(t),X(t)), t ∈ J,

x(0) = x(T).

}
The above nonlinear differential equation with maxima (3.10) has already been discussed in the

literature for existence and uniqueness of the solutions via classical methods of Schauder and Banach
fixed point principles. See Bainov and Hristova [1] and the references therein. Here our method is
different and constructive. Therefore, Theorem 3.1 includes the existence and approximation theorem
for the differential equation with maxima (3.10) as a special case under weak partial compactness type
conditions.

2. Again, when g(t,x,y) = g(t,x) for all t ∈ J and x,y ∈ R, the QDE (1.1) reduces to the following
QDE without maxima,

(1.4)

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).,




which has been discussed in Dhage and Dhage [11] via Dhage iteration method and established the
existence and approximation result for positive solutions.

3. If we take g(t,x,y) = py +F(t) for all t ∈ J and x,y ∈ R in (1.3), then it reduces to the standard
linear differential equation of automatic regulation,

(1.5)
x′(t) + λx(t) = pX(t) + F(t)

x(0) = x(T),

}
for all t ∈ J, where λ > 0, p > 0 are constants and F : J → R is a continuous perturbation
function. The differential equation with maxima (1.5) is the motivation for development of the subject
of differential equations with maxima. Therefore, our QDE (1.1) is more general and the existence
and approximation result of this problem includes the existence and approximation results for all the
above differential equations with maxima as special cases.

2. Auxiliary Results

In this section we give all the preliminaries and key tool that is used in subsequent part of the paper.
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 ‖ · ‖. It is known that E is regular if
{xn}n∈N is a nondecreasing (resp. nonincreasing) sequence in E such that xn → x∗ as n → ∞, then
xn � x∗ (resp. xn � x∗) for all n ∈ N. Clearly, the partially ordered Banach space C(J,R) is regular
and the conditions guaranteeing the regularity of any partially ordered normed linear space E may be
found in Heikkilä and Lakshmikantham [16] and the references therein.



POSITIVE SOLUTIONS OF QUADRATIC DIFFERENTIAL EQUATIONS WITH MAXIMA 103

We need the following preliminary definitions given in Dhage [4, 5, 6] in what follows.

A mapping T : E → E is called isotone or nondecreasing if it preserves the order relation �, that
is, if x � y implies T x �T y for all x,y ∈ E. 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 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. 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 : E → E is called partially bounded if every
chain C in T(E) is bounded. T is called uniformly partially bounded if all chains C in T (E) are
bounded by a unique constant. T is called bounded if T(E) is a bounded subset of E. A non-empty
subset S of the partially ordered Banach space E is called partially compact if every chain C in S
is compact. An operator T : E → E is called partially compact if every chain or totally ordered set
C in T (E) is a relatively compact subset of E. T is called uniformly partially compact if T (E) is
a uniformly partially bounded and partially compact on E. T is called partially totally bounded if
for any bounded subset S of E, T (S) 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.

Remark 2.1. Suppose that T is a nondecreasing operator on E into itself. Then T is a partially
bounded or partially compact if T (C) is a bounded or relatively compact subset of E for each chain C
in E.

Definition 2.1 (Dhage [4]). 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.

Definition 2.2 (Dhage [5]). 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.

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. The following lemma follows imme-

diately from the definition of the set K which is often times used in the hybrid fixed point theory of
Banach algebras and applications to nonlinear differential and integral equations.

Lemma 2.1 (Dhage [4]). If u1,u2,v1,v2 ∈K are such that u1 � v1 and u2 � v2, then u1u2 � v1v2.

Definition 2.3. An operator T : E → E is said to be positive if the range R(T ) of T is such that
R (T ) ⊆K.



104 S.B. DHAGE AND B.C. DHAGE

The Dhage iteration principle (in short DIP) developed in Dhage [5, 7] may be described as “the
monotonic convergence of the sequence of successive approximations to the solutions of
a nonlinear equation beginning with a lower or an upper solution of the equation as its
initial or first approximation.” The aforesaid convergence principle is a useful tool in nonlinear
analysis and is called Dhage iteration method (in short DIM) for nonlinear equations. Dhage iteration
method embodied in the following applicable hybrid fixed point theorem of Dhage [7] is used as a key
tool for our work contained in the present paper. A few other hybrid fixed point theorems containing
the Dhage iteration method along with applications appear in Dhage [5, 7].

Theorem 2.1 (Dhage [5, 7]). Let
(
E,�,‖ · ‖

)
be a regular partially ordered complete normed linear

algebra such that every compact chain of E is Janhavi. Let A,B : E → K be two nondecreasing
operators such that

(a) A is partially bounded and partially nonlinear D-Lipschitz with D-function ψA,
(b) B is partially continuous and uniformly partially compact,
(c) 0 < MψA(r) < r, r > 0, where M = sup{‖B(C)‖ : C is a chain in E}, and
(d) there exists an element x0 ∈ X such that x0 �Ax0 Bx0 or x0 �Ax0 Bx0.

Then the operator equation

(2.3) AxBx = x

has a positive solution x∗ in E and the sequence {xn} of successive iterations defined by xn+1 =
AxnBxn, n = 0, 1, . . . ; converges monotonically to x∗.

Remark 2.2. 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.

3. Main Results

The QDE (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)

for all t ∈ J respectively. Clearly, C(J,R) is a Banach algebra with respect to above supremum norm
and is also partially ordered w.r.t. the above partially order relation ≤. It is known that the partially
ordered Banach algebra 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-Ascolli 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 [9, 10, 11] and so we omit the details of
it. �

We need the following definition in what follows.

Definition 3.1. A function u ∈ C(J,R) is said to be a lower solution of the QDE (1.1) if the function

t 7→
u(t)

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

d

dt

[
u(t)

f(t,u(t))

]
+ λ

[
u(t)

f(t,u(t))

]
≤ g(t,u(t),U(t)),

u(0) ≤ u(T),






POSITIVE SOLUTIONS OF QUADRATIC DIFFERENTIAL EQUATIONS WITH MAXIMA 105

for all t ∈ J, where U(t) = max
0≤ξ≤t

x(ξ), t ∈ J. Similarly, a function v ∈ C(J,R) is said to be an

upper solution of the QDE (1.1) if the function t 7→
v(t)

f(t,v(t))
is differentiable and satisfies 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 increasing for each t ∈ J.

(A1) f defines a function f : J ×R → R+.
(A2) There exists a constant Mf > 0 such that 0 < f(t,x) ≤ Mf for all t ∈ J and x ∈ R.
(A3) 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, x1 ≥ y1.
(A4) The function f(t,x) is periodic in t with period T for all x ∈ R, i.e., f(t,x) = f(t + T,x) for

each x ∈ R.
(B1) g defines a function g : J ×R×R → R+.
(B2) There exists a constant Mg > 0 such that g(t,x,y) ≤ Mg for all t ∈ J and x,y ∈ R.
(B3) g(t,x,y) is nondecreasing in x and y for all t ∈ J.
(B4) The QDE (1.1) has a lower solution u ∈ C1(J,R).

Remark 3.1. Notice that if hypothesis (A0) holds, then the function x 7→
x

f(t,x)
is injective for each

t ∈ J.

The following useful lemma is obvious and may be found in Nieto and Lopez [20].

Lemma 3.2. For any h ∈ L1(J,R+) and σ ∈ L1(J,R), x is a solution to the differential equation

(3.3)
x′(t) + h(t)x(t) = σ(t), t ∈ J,

x(0) = x(T),

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

(3.4) x(t) =

∫ T
0

Gh(t,s) σ(s) ds

where,

(3.5) Gh(t,s) =



eH(s)−H(t)+H(T)

eH(T) − 1
, if 0 ≤ s ≤ t ≤ T,

eH(s)−H(t)

eH(T) − 1
, if 0 ≤ t < s ≤ T.

and H(t) =

∫ t
0

h(s) ds.

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

Kh := max{|Gh(t,s)| : t,s ∈ [0,T]}
exists for all h ∈ L1(J,R+). In particular, if h(t) = λ for all t ∈ J, then for the sake of convenience we
write Gλ(t,s) = G(t,s) and Kλ = K.

An application of above Lemma 3.2 we obtain

Lemma 3.3. Suppose that hypothesis (A1) holds. Then a function u ∈ C(J,R) is a solution of the
PBVP (1.1) if and only if it is a solution of the nonlinear integral equation,

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

](∫ T
0

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

)



106 S.B. DHAGE AND B.C. DHAGE

for all t ∈ J, where

(3.7) G(t,s) =



eλs−λt+λT

eλT − 1
, if 0 ≤ s ≤ t ≤ T,

eλs−λt

eλT − 1
, if 0 ≤ t < s ≤ T.

Theorem 3.1. Assume that hypotheses (A0)-(A4) and (B1)-(B3) hold. Furthermore, assume that

(3.8) K Mg T ϕ(r) < r, r > 0,

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

(3.9) xn+1(t) =
[
f(t,xn(t))

](∫ T
0

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

)
, t ∈ J,

where x1 = u, converges monotonically to x
∗.

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

Define two operators A and B on E by
(3.10) Ax(t) = f(t,x(t)), t ∈ J,
and

(3.11) Bx(t) =
∫ T
0

G(t,s)g(s,x(s),X(t)) ds, t ∈ J.

From the continuity of the integral, it follows that A and B define the maps A,B : E → E. Now
by Lemma 3.3, the QDE (1.1) is equivalent to the operator equation

(3.12) Ax(t)Bx(t) = x(t), t ∈ J.
We shall show that the operators A and B satisfy all the conditions of Theorem 2.1. This is achieved

in the series of following steps.

Step I: A and B are nondecreasing on E.
Let x,y ∈ E be such that x ≥ y. Then x(t) ≥ y(t) for all t ∈ J. Since y is continuous on [a,t],

there exists a ξ∗ ∈ [a,t] such that y(ξ∗) = max
a≤ξ≤t

y(ξ). By definition of ≤, one has x(ξ∗) ≥ y(ξ∗).
Consequently, we obtain

X(t) = max
a≤ξ≤t

x(ξ) ≥ x(ξ∗) ≥ y(ξ∗) = max
a≤ξ≤t

y(ξ) = Y (t)

for each t ∈ J. Then by hypothesis (A3), we obtain
Ax(t) = f(t,x(t)) ≥ f(t,x(t)) = Ay(t),

for all t ∈ J. This shows that A is nondecreasing operator on E into E. Similarly using hypothesis
(B3), it is shown that the operator B is also nondecreasing on E into itself. Thus, A and B are
nondecreasing positive operators on E into itself.

Step II: A is partially bounded and partially D-Lipschitz on E.
Let x ∈ E be arbitrary. Then by (A2),

|Ax(t)| ≤
∣∣f(t,x(t))∣∣ ≤ Mf,

for all t ∈ J. Taking the 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))



POSITIVE SOLUTIONS OF QUADRATIC DIFFERENTIAL EQUATIONS WITH MAXIMA 107

≤ ϕ(‖x−y‖),

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

‖Ax−Ay‖≤ ϕ(‖x−y‖),

for all x,y ∈ E with x ≥ y. Hence, A is a partial nonlinear D-Lipschitz on E which further also implies
that A is a partially continuous on operator on E.

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

convergence theorem, we have

lim
n→∞

Bxn(t) =
∫ T
0

G(t,s)
[

lim
n→∞

g(s,xn(s),Xn(s))
]
ds

=

∫ T
0

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

= Bx(t),

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

be arbitrary. Then, by hypothesis (B2),

|Bxn(t2) −Bxn(t1)| ≤
∣∣∣∣
∫ T
0

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

−
∫ T
0

G(t1,s)g(s,xn(s),Xn(s)) ds

∣∣∣∣
≤
∫ T
0

∣∣G(t2,s) −G(t1,s)∣∣ ∣∣g(s,xn(s),Xn(s))∣∣ds
≤ Mg

∫ T
0

∣∣G(t2,s) −G(t1,s)∣∣ds
→ 0 as t2 − t1 → 0

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

Step IV: B is a uniformly 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. Lety ∈B(C) be any element. Then there is
an element x ∈ C be such that y = Bx. Now, by hypothesis (B2),

|y(t)| ≤
∣∣∣∫ T

0

G(t,s)g(s,x(s),X(t)) ds
∣∣∣

≤ K
∫ T
0

|g(s,x(s),X(t))|ds

≤ K Mg T = M,

for all t ∈ J. Taking the supremum over t, we obtain ‖y‖ = ‖Bx‖≤ M for all y ∈B(C). Hence, B(C)
is a uniformly bounded subset of E. Moreover, ‖B(C)‖ ≤ M for all chains C in E. Hence, B is a
uniformly partially bounded operator on E.

Next, we will show that B(C) is an equicontinuous set in E. Let t1, t2 ∈ J be arbitrary. Then, for
any y ∈B(C), there is a x ∈ C such that y(t) = Bx(t). Hence, proceeding with arguments as in Step



108 S.B. DHAGE AND B.C. DHAGE

III,

|y(t2) −y(t1)| ≤
∣∣∣∫ T

0

G(t2,s)g(s,x(s),X(s)) ds−
∫ T
0

G(t1,s)g(s,x(s),X(s)) ds
∣∣∣

≤ Mg
∫ T
0

|G(t2,s) −G(t1,s)|ds

→ 0 as t2 − t1 → 0

uniformly for all y ∈ B(C). Hence B(C) is an equicontinuous subset of E. Now, B(C) is a uniformly
bounded and equicontinuous set of functions in E, so it is compact. Consequently, B is a uniformly
partially compact operator on E into itself.

Step V: u satisfies the operator inequality u ≤AuBu.
By hypothesis (B3), the QDE (1.1) has a lower solution u defined on J. Then, we have

(3.13)

d

dt

[
u(t)

f(t,u(t))

]
+ λ

[
u(t)

f(t,u(t))

]
≤ g(t,u(t),U(t)),

u(0)) ≤ x(T),




for all t ∈ J. Multiplying the first inequality in (3.13) by the integrating factor eλt, we obtain

(3.14)

(
eλt

u(t)

f(t,u(t))

)′
≤ eλtg(t,u(t),U(t)),

for all t ∈ J. A direct integration of (3.14) from 0 to t yields

(3.15) eλt
u(t)

f(t,u(t))
≤

u(0)

f(0,u(0))
+

∫ t
0

eλsg(s,u(s)) ds,

for all t ∈ J. Therefore, in particular,

(3.16) eλT
u(T)

f(T,u(T))
≤

u(0)

f(0,u(0))
+

∫ T
0

eλsg(s,u(s),U(s)) ds.

Now u(0) ≤ u(T), so by hypothesis (A0) one has

(3.17)
u(0)

f(0,u(0))
eλT ≤

u(T)

f(T,u(T))
eλT .

From (3.15) and (3.17) it follows that

(3.18) eλT
u(0)

f(0,u(0))
≤

u(0)

f(0,u(0))
+

∫ T
0

eλsg(s,u(s),U(s)) ds

which further yields

(3.19)
u(0)

f(0,u(0))
≤
∫ T
0

eλs

(eλT − 1)
g(s,u(s),U(s)) ds

Substituting (3.19) in (3.15) we obtain

u(t) ≤
[
f(t,u(t))

](∫ T
0

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

)
, t ∈ J.

From the definitions of the operators A and B it follows that u(t) ≤Au(t)Bu(t) for all t ∈ J. Hence
u ≤AuBu.

Step VI: D-function ϕ satisfies the growth condition MψA(r) < r, r > 0.
Finally, the D-function ϕ of the operator A satisfies the inequality given in hypothesis (d) of Theorem

2.1. Now from the estimate given in Step IV, it follows that MψA(r) ≤ K Mg T ϕ(r) < r for all r > 0.
Thus, A and B satisfy all the conditions of Theorem 2.1 and we apply it to conclude that the

operator equation AxBx = x has a positive solution x∗. Therefore, the integral equation (3.6) and
consequently the QDE (1.1) has a positive solution x∗ defined on J. Furthermore, the sequence



POSITIVE SOLUTIONS OF QUADRATIC DIFFERENTIAL EQUATIONS WITH MAXIMA 109

{xn}∞n=1 of successive approximations defined by (3.9) converges monotonically to x∗. This completes
the proof. �

Remark 3.2. The conclusion of Theorem 3.1 also remains true if we replace the hypothesis (B3) with
the following:

(B′3) The QDE (1.1) has an upper solution v ∈ C1(J,R).
The proof under this new hypothesis is similar to the proof of Theorem 3.1 with appropriate modifi-
cations.

Remark 3.3. We note that if the QDE (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 3.1 it has corresponding solutions x∗
and x∗ and these solutions satisfy x∗ ≤ x∗. Hence they are the minimal and maximal solutions of the
PBVP (1.1) in the vector segment [u,v] of the Banach space E = C1(J,R), where the vector segment
[u,v] is a set in C1(J,R) defined by

[u,v] = {x ∈ C1(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).

Finally, we give an example to illustrate our hypotheses (A0)-(A4) and (B1)-(B3) and the abstract
result formulated in Theorem 3.1.

Example 3.1. Given a closed and bounded interval J = [0, 1], consider the PBVP of QDEs,

(3.20)

d

dt

[
x(t)

f(t,x(t))

]
+

[
x(t)

f(t,x(t))

]
=

1

8

[
2 + tanh x(t) + tanh X(t)

]
,

x(0) = x(1),




for all t ∈ J, where the functions f : J ×R → R\{0} and g : J ×R×R → R are defined as

f(t,x,y) =




1, if x ≤ 0,

1 + x, if 0 < x < 3,

4, if x ≥ 3.

and

g(t,x,y) =
1

20

[
3 + tanh x

]
.

Clearly, the functions f and g are continuous nonnegative real-valued functions on J × R and

J × R × R respectively. As
∂

∂x

(
x

f(t,x)

)
> 0 for all t ∈ J and x ∈ R, the function x 7→

x

f(t,x)
is

increasing for each t ∈ J and so, the hypothesis (A0) is satisfied. Next, f satisfies the hypothesis (A3)
with ϕ(r) = r. To see this, we have

0 ≤ f(t,x) −f(t,y) ≤ x−y

for all x,y ∈ R, x ≥ y. Therefore, ϕ(r) = r. Moreover, the function f(t,x) is periodic in t for each
x ∈ R and is also bounded on J × R × R with bound Mf = 4 and so, the hypotheses (A2) and(A3)
are satisfied. Again, since g is positive and bounded on J × R × R by Mg = 14 , the hypothesis (B2)
holds. Furthermore, g(t,x,y) is nondecreasing in x and y for each t ∈ J, and thus hypothesis (B3) is
satisfied. Here, the Green’s function G(t,s) associated with the homogeneous PBVP

(3.21)
x′(t) + x(t) = 0, t ∈ [0, 1],

x(0) = x(T),






110 S.B. DHAGE AND B.C. DHAGE

is given by

(3.22) 0 ≤ G(t,s) =



es−t+1

e− 1
, if 0 ≤ s ≤ t ≤ 1,

es−t

e− 1
, if 0 ≤ t ≤ s ≤ 1

≤
e

e− 1
≤ 2 = K.

Also the condition (3.8) of Theorem 3.1 holds. Finally, after a simple computation, it is shown that

the function u(t) =
1

20

∫ 1
0

G(t,s) ds is a lower solution of the QDE (3.20) defined on J. Thus, all the

hypotheses of Theorem 3.1 are satisfied. Hence we apply Theorem 3.1 and conclude that the QDE
(3.20) has a positive solution x∗ defined on J and the sequence {xn}∞n=1 of successive approximations
defined by

x1(t) =
1

20

∫ 1
0

G(t,s) ds,

xn+1(t) =
1

20

[
f(t,xn(t))

](∫ 1
0

G(t,s)
[
2 + tanh xn(s) + tanh Xn(s)

]
ds

)
,

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

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∗Corresponding author: bcdhage@gmail.com