International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 1 (2015), 59-69
http://www.etamaths.com

DHAGE ITERATION METHOD FOR GENERALIZED

QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS

BAPURAO C. DHAGE

Abstract. In this paper we prove the existence as well as approximations of

the solutions for a certain nonlinear generalized quadratic functional integral

equation. An algorithm for the solutions is developed and it is shown that the
sequence of successive approximations starting at a lower or upper solution

converges monotonically to the solutions of related quadratic functional inte-

gral equation under some suitable mixed hybrid conditions. We rely our main
result on Dhage iteration method embodied in a recent hybrid fixed point the-

orem of Dhage (2014) in partially ordered normed linear spaces. An example
is also provided to illustrate the abstract theory developed in the paper.

1. Introduction

The quadratic integral equations have been a topic of interest since long time
because of their occurrence in the problems of some natural and physical processes
of the universe. See Argyros [1], Deimling [3], Chandrasekher [2] and the references
therein. The study gained momentum after the formulation of the hybrid fixed
point principles in Banach algebras due to Dhage [4, 5, 6, 7, 8]. The existence
results for such quadratic operators equations are generally proved under the mixed
Lipschitz and compactness type conditions together with a certain growth condition
on the nonlinearities involved in the quadratic operator or functional equations.
The hybrid fixed point theorems in Banach algebras find numerous applications in
the theory of nonlinear quadratic differential and integral equations. See Dhage
[5, 6, 7] and the references therein. The Lipschitz and compactness hypotheses
are considered to be very strong conditions in the theory of nonlinear differential
and integral equations but which still do not yield any algorithm to determine the
numerical solutions. Therefore, it is of interest to relax or weaken these condition in
the existence and approximation theory of quadratic integral equations. This is the
main motivation of the present paper. In this paper we prove the existence as well as
approximations of the solutions of a certain generalized quadratic integral equation
via an algorithm based on successive approximations under partially Lipschitz and
compactness type conditions.

2010 Mathematics Subject Classification. 45G10, 47H09, 47H10.
Key words and phrases. Quadratic functional integral equation; approximate solution; Dhage

iteration method; hybrid fixed point theorem.

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.

59



60 DHAGE

Given a closed and bounded interval J = [0,T] of the real line R for some T > 0,
we consider the quadratic functional integral equation (in short QFIE)

(1.1) x(t) = k(t,x(t)) +
[
f(t,x(t))

](
q(t) +

∫ t
0

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

)
, t ∈ J,

where q : J → R, v : J ×J → R and f,g,k : J ×R → R are continuous functions.

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

The QFIE (1.1) is well-known in the literature and studied earlier in the work
of Dhage [4]. If f(t,x) = 0 for all t ∈ J and x ∈ R the QFIE (1.1) reduces to the
nonlinear functional equation

(1.2) x(t) = k(t,x(t)), t ∈ J,

and if k(t,x) = 0 and f(t,x) = 1 for all t ∈ J and x ∈ R, it is reduced to nonlinear
usual Volterra integral equation

(1.3) x(t) = q(t) +

∫ t
0

v(t,s)g(s,x(s)) ds, t ∈ J.

Therefore, the QFIE (1.1) is general and the results of this paper include the
existence and approximations results for above nonlinear functional and Volterra
integral equations as special cases.

The paper is organized as follows: In the following section we give the prelimi-
naries and auxiliary results needed in the subsequent part of the paper. The main
result is included in Section 3. In Section 4 some concluding remarks are presented.

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 ‖ · ‖. 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 [13] and the
references therein.

We need the following definitions in the sequel.

Definition 2.1. 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.

Definition 2.2 (Dhage [9]). 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. A mapping T : E → E is called partially bounded if T (C) is
bounded for every chain C in E. T is called uniformly partially bounded if all



GENERALIZED QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS 61

chains T (C) in E are bounded by a unique constant. T is called bounded if T (E)
is a bounded subset of E.

Definition 2.4. A 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 (C) is a uniformly partially bounded and
partially compact 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 [9]). The order relation � and the metric d on a non-empty
set E are said to be compatible if {xn}n∈N is a monotone, that is, monotone non-
decreasing or monotone nonincreasing sequence in E and if a subsequence {xnk}n∈N
of {xn}n∈N converges to x∗ implies that the whole sequence {xn}n∈N converges to
x∗. Similarly, given a partially ordered normed linear space (E,�,‖ · ‖), the order
relation � and the norm ‖·‖ are said to be compatible if � and the metric d defined
through the norm ‖ ·‖ are compatible.

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

Definition 2.6 (Dhage [6]). A 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 K are called the positive vectors of the normed linear algebra

E. The following lemma follows immediately from the definition of the set K and
which is often times used in the applications of hybrid fixed point theory in Banach
algebras.

Lemma 2.7 (Dhage [7]). If u1,u2,v1,v2 ∈ K are such that u1 � v1 and u2 � v2,
then u1u2 � v1v2.

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

The Dhage iteration principle or method (in short DIP or DIM) developed in
Dhage [9, 10, 11] may be formulated as “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”



62 DHAGE

and which is a powerful tool in the existence theory of nonlinear analysis. It is
clear that Dhage iteration method is different from the usual Picard’s successive
iteration method and embodied in the following applicable hybrid fixed point theo-
rems proved in Dhage [10] which forms a useful key tool for our work contained in
this paper. A few other hybrid fixed point theorems involving the Dhage iteration
method may be found in Dhage [9, 10, 11, 12].

Theorem 2.9 (Dhage [10]). Let
(
E,�,‖·‖

)
be a regular partially ordered complete

normed linear algebra such that the order relation � and the norm ‖ · ‖ in E are
compatible in every compact chain of E. Let A,B : E →K and B : E → E be three
nondecreasing operators such that

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

(b) B is partially continuous and uniformly partially compact, and
(c) MψA(r)+ψC(r) < r, r > 0, where M = sup{‖B(C)‖ : C is a chain in E},

and
(d) there exists an element x0 ∈ X such that x0 � Ax0 Bx0 + Cx0 or x0 �
Ax0 Bx0 + Cx0.

Then the operator equation

(2.3) AxBx + Cx = x

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

Remark 2.10. The compatibility of the order relation � and the norm ‖ · ‖ in
every compact chain of E holds if every partially compact subset of E possesses
the compatibility property with respect to � and ‖ · ‖.

3. Main Result

The QFIE (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
‖ ·‖ and ≤ are compatible in every partially compact subset of C(J,R).

Proof. The lemma mentioned in Dhage [10], but the proof appears in Dhage [11].
Since the proof is not well-known, we give the details of the proof. Let S be a



GENERALIZED QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS 63

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

(3.3) x1(t) ≤ x2(t) ≤ ···≤ xn(t) ≤ ··· ,
for each t ∈ R+.

Suppose that a subsequence {xnk}n∈N of {xn}n∈N is convergent and converges
to a point x in S. Then the subsequence {xnk (t)}n∈N of the monotone real se-
quence {xn(t)}n∈N is convergent. By monotone characterization, the whole se-
quence {xn(t)}n∈N is convergent and converges to a point x(t) in R for each t ∈ R+.
This shows that the sequence {xn(t)}n∈N converges point-wise in S. To show the
convergence is uniform, it is enough to show that the sequence {xn(t)}n∈N is e-
quicontinuous. Since S is partially compact, every chain or totally ordered set
and consequently {xn}n∈N is an equicontinuous sequence by Arzelá-Ascoli theo-
rem. 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 ∈ C(J,R) is said to be a lower solution of the QFIE
(1.1) if it satisfies

u(t) ≤ k(t,u(t)) +
[
f(t,u(t))

](
q(t) +

∫ t
0

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

)
(∗)

for all t ∈ J. Similarly, a function v ∈ C(J,R) is said to be an upper solution of
the QFIE (1.1) if it satisfies the above inequalities with reverse sign.

We consider the following set of assumptions in what follows:

(A1) f defines a function f : J ×R → R+.
(A2) There exists a constant Mf > 0 such that f(t,x) ≤ Mf for all t ∈ J and

x ∈ R.
(A3) There exists a D-function ψf such that

0 ≤ f(t,x) −f(t,y) ≤ ψf (x−y),
for all t ∈ J and x,y ∈ R, x ≥ y.

(B0) q defines a continuous function q : J → R+.
(B1) v defines a continuous and nonnegative function on J ×J.
(B2) g defines a function g : J ×R → R+.
(B3) There exists a constant Mg > 0 such that g(t,x) ≤ Mg for all t ∈ J and

x ∈ R.
(B4) g(t,x) is nondecreasing in x for all t ∈ J.
(C1) There exists a constant Mk > 0 such that |k(t,x)| ≤ Mk for all t ∈ J and

x ∈ R.
(C2) There exists a D-function ψk, such that

0 ≤ k(t,x) −k(t,y) ≤ ψk(x−y),
for all t ∈ J and x,y ∈ R, x ≥ y.

(C3) The QFIE (1.1) has a lower solution u ∈ C(J,R).

Theorem 3.3. Assume that hypotheses (A1)-(A3), (B0)-(B4) and (C1)-(C3) hold.
Furthermore, assume that

(3.4)
(
‖q‖ + Mg T

)
ψf (r) + ψk(r) < r, r > 0,



64 DHAGE

then the QFIE (1.1) has a solution x∗ defined on J and the sequence {xn}n∈N∪{0}
of successive approximations defined by

(3.5) xn+1(t) = k(t,xn(t)) +
[
f(t,xn(t))

](
q(t) +

∫ t
t0

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

)
,

for all t ∈ J, where x0 = u, converges monotonically to x∗.

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

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

(3.7) Bx(t) = q(t) +
∫ t
t0

v(t,s)g(s,x(s)) ds, t ∈ J,

and

(3.8) Cx(t) = k(t,x(t)), t ∈ J.
From the continuity of the integral and the hypotheses (A1) and (B0)-(B2), it

follows that A, B and C define the maps A,B : E → K and C : E → E. Now
by definitions of the operators A, B and C, the QFIE (1.1) is equivalent to the
quadratic operator equation

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

2.9. This is achieved in the series of following steps.

Step I: A, B and B are nondecreasing on E.

Let x,y ∈ E be such that x ≥ y. Then by hypothesis (A3) and (C2), we obtain
Ax(t) = f(t,x(t)) ≥ f(t,y(t)) = Ay(t),

and
Cx(t) = k(t,x(t)) ≥ k(t,y(t)) = Cy(t),

for all t ∈ J. This shows that A and C are nondecreasing operators on E into E.
Similarly, using hypothesis (B4),

Bx(t) = q(t) +
∫ t
0

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

≤ q(t) +
∫ t
0

v(t,s)g(s,y(s)) ds

= By(t)
for all t ∈ J. Hence, it is follows that the operator B is also nondecreasing on E
into itself. Thus, A, B and C are nondecreasing operators on E into itself.

Step II: A and C are partially bounded and partially D-Lipschitz on E.

Let x ∈ E be arbitrary. Then by (A2),
|Ax(t)| ≤

∣∣f(t,x(t))∣∣ ≤ Mf,



GENERALIZED QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS 65

for all t ∈ J. Taking supremum over t, we obtain ‖Ax‖≤ Mf and so, A is bounded.
This further implies that A is partially bounded on E. Similarly, using hypothesis
(C1), it is shown that ‖Cx‖≤ Mk and consequently C is partially bounded on E.

Next, let x,y ∈ E be such that x ≥ y. Then, by hypothesis (A3),

|Ax(t) −Ay(t)| =
∣∣f(t,x(t)) −f(t,y(t))∣∣ ≤ ψf (|x(t) −y(t)|) ≤ ψf (‖x−y‖),

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

‖Ax−Ay‖≤ ψf (‖x−y‖)

for all x,y ∈ E with x ≥ y. Similarly, by hypothesis (C2),

‖Cx−Cy‖≤ ψk(‖x−y‖)

for all x,y ∈ E with x ≥ y. Hence A and C are partially nonlinear D-Lipschitz
operators on E which further implies that they are also a partially continuous on
E into itself.

Step III: B is a partially continuous 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) = lim
n→∞

q(t) + lim
n→∞

∫ t
0

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

= q(t) +

∫ t
0

v(t,s)
[

lim
n→∞

g(s,xn(s))
]
ds

= q(t) +

∫ t
0

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

= Bx(t),

for all t ∈ J. This shows that Bxn converges monotonically 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 with t1 < t2. Then

|Bxn(t2) −Bxn(t1)|

≤
∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣

∫ t2
0

v(t1,s)g(s,xn(s)) ds−
∫ t1
0

v(t1,s)g(s,xn(s)) ds

∣∣∣∣
≤
∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣

∫ t2
0

v(t2,s)g(s,xn(s)) ds−
∫ t2
0

v(t1,s)g(s,xn(s)) ds

∣∣∣∣
+

∣∣∣∣
∫ t2
0

v(t1,s)g(s,xn(s)) ds−
∫ t1
0

v(t1,s)g(s,xn(s)) ds

∣∣∣∣
≤
∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣

∫ t2
0

|v(t2,s) −v(t1,s)| |g(s,xn(s))|ds
∣∣∣∣

+

∣∣∣∣
∫ t2
t1

|v(t1,s)| |g(s,xn(s))|ds
∣∣∣∣



66 DHAGE

≤
∣∣q(t1) −q(t2)∣∣ +

∣∣∣∣∣
∫ T
0

|v(t2,s) −v(t1,s)|Mg ds

∣∣∣∣∣
+ V Mg|t2 − t1|.(3.10)

Since the function q is continuous on compact interval J and v is continuous on
compact set J × J, they are uniformly continuous there. Therefore, from above
inequality (3.10) it follows that

|Bxn(t2) −Bxn(t1)|→ 0 as n →∞
uniformly for all n ∈ N. This shows that the convergence Bxn → Bx is uniform
and hence B is 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. Let
y ∈ B(C) be any element. Then there is an element x ∈ C be such that y = Bx.
Now, by hypothesis (B2),

|y(t)| ≤ |q(t)| +
∫ t
0

v(t,s)|g(s,x(s))|ds ≤‖q‖ + V Mg T ≤ r,

for all t ∈ J. Taking supremum over t, we obtain ‖y‖ = ‖Bx‖≤ r for all y ∈B(C).
Hence, B(C) is a uniformly bounded subset of E. Moreover, ‖B(C)‖ ≤ r 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 with t1 < t2. Then, for any y ∈B(C), one has
|y(t2) −y(t1)| = |Bx(t2) −Bx(t1)|

≤
∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣

∫ t2
0

v(t1,s)g(s,x(s)) ds−
∫ t1
0

v(t1,s)g(s,x(s)) ds

∣∣∣∣
≤
∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣

∫ t2
0

v(t2,s)g(s,x(s)) ds−
∫ t2
0

v(t1,s)g(s,x(s)) ds

∣∣∣∣
+

∣∣∣∣
∫ t2
0

v(t1,s)g(s,x(s)) ds−
∫ t1
0

v(t1,s)g(s,x(s)) ds

∣∣∣∣
≤
∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣

∫ t2
0

|v(t2,s) −v(t1,s)| |g(s,x(s))|ds
∣∣∣∣

+

∣∣∣∣
∫ t2
t1

|v(t1,s)| |g(s,x(s))|ds
∣∣∣∣

≤
∣∣q(t1) −q(t2)∣∣ +

∣∣∣∣∣
∫ T
0

|v(t2,s) −v(t1,s)|Mg ds

∣∣∣∣∣
+ V Mg|t2 − t1|

→ 0 as n →∞,
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



GENERALIZED QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS 67

compact. Consequently, B is a uniformly partially compact operator on E into
itself.

Step V: u satisfies the operator inequality u ≤AuBu + Cu.

By hypothesis (C3), the QFIE (1.1) has a lower solution u defined on J. Then,
we have

(3.11) u(t) ≤ k(t,u(t)) +
[
f(t,u(t))

](
q(t) +

∫ t
0

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

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

Step VI: The D-functions ψA and ψC satisfy the growth condition
MψA(r) + ψC(r) < r, r > 0.

Finally, the D-function φ of the operator A satisfies the inequality given in
hypothesis (d) of Theorem 2.9, viz.,

MψA(r) + ψC(r) ≤ (‖q‖ + V Mg T) ψf (r) + ψk(r) < r
for all r > 0.

Thus A, B and C satisfy all the conditions of Theorem 2.9 and we conclude that
the operator equation AxBx + Cx = x has a solution. Consequently the integral
equation and the QFIE (1.1) has a solution x∗ defined on J. Furthermore, the
sequence {xn}n∈N of successive approximations defined by (3.5) converges mono-
tonically to x∗. This completes the proof. �

The conclusion of Theorems 3.3 also remains true if we replace the hypothesis
(C3) with the following one:

(C′3) The QFIE (1.1) has an upper solution v ∈ C(J,R).
The proof of Theorem 3.3 under this new hypothesis is similar and can be obtained
by closely observing the same arguments with appropriate modifications.

Example 3.4. Given a closed and bounded interval J = [0, 1], consider the QFIE,

x(t) =
1

2

[
2 + tan−1 x(t)

]( t
t + 1

+

∫ t
0

1

t2 + 1
·

[1 + tanh x(s)]

4
ds

)
+

1

2
tan−1 x(t)(3.12)

for t ∈ J.

Here, q(t) =
t

t + 1
and v(t,s) =

1

t2 + 1
which are continuous and ‖q‖ =

1

2
and

V = 1. Similarly, the functions k, f and g are defined by k(t,x) =
1

2
tan−1 x,

f(t,x) =
1

2

[
2 + tan−1 x(t)

]
and g(t,x) =

1 + tanh x

4
.

The function f satisfies the hypothesis (A3) with ψf (r) =
1

2
·

r

1 + ξ2
for each

0 < ξ < r. To see this, we have

0 ≤ f(t,x) −f(t,y) ≤
1

2
·

1

1 + ξ2
· (x−y)



68 DHAGE

for all x,y ∈ R, x ≥ y and x > ξ > y. Moreover, the function f is nonnegative and
bounded on J ×R with bound Mf = 2 and so the hypothesis (A2) is satisfied.

Again, since g is nonnegative and bounded on J×R by Mg =
1

2
, the hypothesis

(B3) holds. Furthermore, g(t,x) is nondecreasing in x for all t ∈ J, and thus
hypothesis (B4) is satisfied.

Similarly, the function k satisfies the hypothesis (C2) with ψk(r) =
1

2
·

r

1 + ξ2

for every 0 < ξ < r. To see this, we have

0 ≤ k(t,x) −k(t,y) ≤
1

2
·

1

1 + ξ2
· (x−y)

for all x,y ∈ R, x ≥ y and x > ξ > y. Moreover, the function k is bounded on
J ×R with bound Mk =

π

4
and so the hypothesis (C1) is satisfied.

Also we have (
‖q‖ + Mg V T

)
ψf (r) + ψk(r) ≤

r

1 + ξ2
< r

for every r > 0. Thus, condition (3.4) of Theorem 3.3 is held. Finally, the QFIE
(3.12) has a lower solution u(t) = 0 on J. Thus all the hypotheses of Theorem 3.3
are satisfied. Hence we apply Theorem 3.3 and conclude that the QFIE (3.12) has
a solution x∗ defined on J and the sequence {xn}n∈N defined by

xn+1(t) =
1

2

[
2 + tan−1 xn(t)

]( t
t + 1

+

∫ t
0

1

t2 + 1
·

[1 + tanh xn(s)]

4
ds

)
+

1

2
tan−1 xn(t),(3.13)

for all t ∈ J, where x0 = 0, converges monotonically to x∗.

4. Conclusion

Finally, while concluding this paper we mention that the generalized quadratic
integral equation considered here is of very simple nature for which we have il-
lustrated the Dhage iteration method to obtain the algorithms for the solutions
under weaker partially Lipschitz and compactness conditions. However, an analo-
gous study could also be made for other complex quadratic integral equations as
well as other different types of quadratic integral equations using similar method
with appropriate modifications. Some of the results along this line will be reported
elsewhere.

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