International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 1 (2015), 53-62
http://www.etamaths.com

THE DHAGE ITERATION PRINCIPLE FOR COUPLED PBVPS

OF NONLINEAR SECOND ORDER DIFFERENTIAL

EQUATIONS

BAPURAO C. DHAGE

Abstract. The present paper proposes a new monotone iteration principle for

the existence as well as approximations of the coupled solutions for a coupled
periodic boundary value problem of second order ordinary nonlinear differen-

tial equations. An algorithm for the coupled solutions is developed and it is

shown that the sequences of successive approximations defined in a certain
way converge monotonically to the coupled solutions of the related differential

equations under some suitable hybrid conditions. A numerical example is also

indicated to illustrate the abstract theory developed in the paper. We claim
that the method as well as the results of this paper are new to literature on

nonlinear analysis and applications.

1. Introduction

Given a closed and bounded interval J = [0,T] of the real line R , consider the
coupled periodic boundary value problems (in short CPBVPs) of nonlinear second
order ordinary nonlinear differential equations (in short DEs) of the form

(1.1)
−x′′(t) + λ2x(t) = f(t,x(t),y(t)),

x(0) = x(T) , x′(0) = x′(T),




and

(1.2)
−y′′(t) + λ2x(t) = f(t,y(t),x(t)),

y(0) = y(T) , y′(0) = y′(T),




for all t ∈ J, where λ ∈ R, λ > 0 and f : J ×R×R → R is a continuous function.

By a coupled solution of the CPBVPs (1.1) and (1.2) we mean an ordered pair
of differentiable functions (u,v) ∈ C(J,R)×C(J,R) that satisfy the DEs (1.1) and
(1.2), where C(J,R) is the space of continuous real-valued functions defined on J.

The coupled PBVPs (1.1) and (1.2) are well-known and the existence of the cou-
pled solutions for them have been proved using the coupled fixed point theorems
based on the properties of cones in the solution space C(J,R). See Guo and Lak-
shmikantham [12], Heikkilä and Lakshmikantham [13] and the references therein.

2010 Mathematics Subject Classification. 34A12, 34A38.
Key words and phrases. coupled periodic boundary value problems; coupled fixed point theo-

rem; Dhage iteration principle; approximate coupled solutions.

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.

53



54 DHAGE

Recnetly, Bhaskar and lakshmikantham [11] proved the existence and uniqueness
results for the coupled solutions of the CBVPs (1.1) and (1.2) without using the
properties of the cones, however in this case the nonlinearity f involved in (1.1)
and (1.2) is required to satisfy a weak Lipschitz condition which is considered to be
strong in the theory of nonlinear differential and integral equations. Very recently,
Dhage and Dhage [7] proved the existence as well as approximations of the coupled
solutions for the coupled initial value problems (in short CIVPs) of the nonlinear
first order ordinary differential equations using the Dhage iteration principle which
does not require any type of Lipschitz condition as well as any property of the
cones in a appropriate Banach space. The aim of the present paper is to extend the
method involving the Dhage iteration principle to the CPBVPs (1.1) and (1.2) for
approximating the coupled solutions. Therefore, our approach to the considered
CPBVPs (1.1) and (1.2) is different from the earlier ones discussed in the litera-
ture. Moreover, when λ = 0, f(t,x,x) = −f1(t,x) and x = y in (1.1) or (1.2) for all
t ∈ J and x ∈ R, the results of this paper include the existence and approximations
results of Dhage et al. [10] as special cases.

2. Auxiliary Results

Let (E,�) be a partial ordered set and let d be a metric on E such that (E,�,d)
becomes a partially ordered metric space. By E×E we denote a metric space with
the metric d∗ defined by

(2.1) d∗
(
(x,y), (w,z)

)
= d(x,w) + d(y,z)

for (x,y), (w,z) ∈ E × E. We define a partial order � in E × E as follows. Let
(x1,x2), (y1,y2) ∈ E ×E. Then,

(2.2) (x1,x2) � (y1,y2) ⇐⇒ x1 � y1 and x2 � y2.

Then, the triplet (E×E,�,d∗) again becomes a partially ordered metric space.
Let F : E ×E → E and consider the coupled mapping equations,

(2.3) F(x,y) = x and F(y,x) = y.

A point (x∗,y∗) ∈ E ×E is said to be a coupled solution or coupled fixed point
for the coupled mapping equation (2.3) if

(2.4) F(x∗,y∗) = x∗ and F(y∗,x∗) = y∗.

We need the following definitions in what follows.

Definition 2.1. A partially ordered normed metric space (E,�,d) is called regular
if every nondecreasing (resp. nonincreasing) sequence {xn} converges to x∗, then
xn � x∗ (resp. xn � x∗) for all n ∈ N.

The details of the regularity property of the ordered sets may be found in Heikkilä
and Lakshmikantham [13] and the references therein.

Definition 2.2 (Dhage [2]). A mapping F : E ×E → E is called partially contin-
uous at a point (a,b) ∈ E ×E if for � > 0 there exists a δ > 0 such that

d∗
(
F(x,y),F(a,b)

)
< �

whenever (x,y) is comparable to (a,b) and

d∗
(
(x,y), (a,b)

)
< δ.



NONLINEAR SECOND ORDER DIFFERENTIAL EQUATIONS 55

If F is partially continuous at every point of E × E, we say that F is partially
continuous on E ×E.

Remark 2.1. If F is partially continuous on E×E, then it is continuous on every
totally ordered set or chain in E ×E.

Definition 2.3. A mapping F : E×E → E is called partially compact if F(C1×C2)
is a relatively compact subset of E for all chains C1 and C2 in E.

The details of compact and continuous operators may be found in the monograph
by Heikkilä and Lakshmikantham [13] and the references therein.

Definition 2.4. A mapping F is called mixed monotone if F(x,y) is nondecreasing
in x for each y ∈ E and nonincreasing in y for each x ∈ E with respect to the order
relation � in E.

Remark 2.2. If F is mixed monotone, then it is a nondecreasing mapping on
E ×E with respect to the order relation � defined in E ×E.

Definition 2.5 (Dhage [2, 3]). 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, mono-
tone nondecreasing or monotone nonincreasing sequence in E and if a subsequence
{xnk}n∈N of {xn}n∈N converges to x

∗ implies that the whole sequence {xn}n∈N con-
verges 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 metric
defined by the absolute value function has this property. Similarly, every finite
dimensional Euclidean space Rn is compatible with respect to usual componentwise
order relation and the standard norm in it.

The Dhage iteration principle which states that the sequence of successive ap-
proximations of a nonlinear equation beginning with a lower or an upper solution
converges monotonically to its solution forms a powerful tool in th existence theo-
ry of such equations. The details of Dhage iteration principle are given in Dhage
[2, 3, 4] and Dhage and Dhage [9]. The following applicable coupled hybrid fixed
point theorem is a slight improvement of the coupled hybrid fixed point theorem
proved in Dhage and Dhage [8] containing the Dhage iteration principle.

Theorem 2.1. Let (E,�,d) be a regular partially ordered complete metric space
such that the metric d and the order relation � are compatible in every compact
chain C of E. Let F : E × E → E is a mixed monotone, partially continuous
and partially compact mapping. If there exist elements x0 ∈ E and y0 ∈ E such
that x0 � F(x0,y0) and y0 � F(y0,x0), then F has a coupled fixed point (x∗,y∗)
and the sequences {xn} and {yn} defined by xn = F(xn−1,yn−1) = Fn(x0,y0) and
yn = F(yn−1,xn−1) = Fn(y0,x0) converge monotonically to x∗ and y∗respectively.

Proof. Define the sequences {xn} and {yn} of points in E as follows. Choose

x1 = F(x0,y0) and y1 = F(y0,x0).

Then, x0 � x1 and y1 � y0. Again, choose

x2 = F2(x0,y0) = F(x1,y1) = F(F(x0,y0),F(y0,x0)) �F(x0,y0) = x1.



56 DHAGE

Similarly, choose y2 = F2(y0,x0) = F(y1,x1) so that
y2 �F(F(y0,x0),F(x0,y0)) �F(y0,x0) = y1.

Proceeding in this way, by induction, define

(2.5) xn+1 = F(xn,yn) = Fn(x0,y0) and yn+1 = F(yn,xn) = Fn(y0,x0),
for n = 0, 1, 2, ..., so that

(2.6) x0 � x1 � ···� xn � ··· ,
and

(2.7) y0 � y1 � ···� yn � ··· .
Thus, {xn} and {yn} are respectively monotone nondecreasing and monotone

nonincreasing sequences and so are chains in E. From the construction of {xn} and
{yn}, it follows that

{xn}⊆F ({xn},{yn}) ⊆F ({xn}×{yn}) .
Since F is partially compact on E ×E, one has F({xn}×{yn}) is a relatively

compact subset of E. As a result, F({xn}×{yn}) is compact and that {xn} has
a convergent subsequence converging to a point, say x∗ ∈ E. Since d and � are
compatible in every compact chain C of E, the whole sequence {xn} converges to
x∗. Similarly, the sequence {yn} converges to a point say y∗ ∈ E. Equivalently,
(xn,yn) → (x∗,y∗) in the topology of the norm in E×E. As E is a regular, we have
that xn � x∗ and yn � y∗ for all n ∈ N. Therefore, we obtain (xn,yn) � (x∗,y∗)
for all n ∈ N. Finally, by the partial continuity of F, we obtain

x∗ = lim
n→∞

xn+1 = lim
n→∞

F(xn,yn) = F(x∗,y∗)

and
y∗ = lim

n→∞
yn+1 = lim

n→∞
F(yn,xn) = F(y∗,x∗).

Thus (x∗,y∗) is a coupled fixed point of the mapping F on E × E into itself.
This completes the proof. �

Remark 2.3. The regularity of the partially ordered metric space E may be replaced
with a stronger condition of continuity than the partial continuity of the mappings
F on E ×E. Again, the condition of 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 ‖ ·‖.

The simple fact concerning the compactibility of the order relation and the norm
mentioned in Remark 2.3 has been used in formulating the main results of this
paper. In the following sectin we prove the main existence and approximation
results for the CBVP (1.1) and (1.2) defined on J.

3. Existence and Approximations Results

We place our considerations of the CBVPs (1.1) and (1.2) in the function space
C(J,R). 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)



NONLINEAR SECOND ORDER DIFFERENTIAL EQUATIONS 57

for all t ∈ J. Clearly, (C(J,R),‖ · ‖,≤) is a partially ordered complete normed
linear space and has compatibility property with respect to the norm ‖ ·‖ and the
order relation ≤ in certain subsets of of it. The following lemma in this connection
is useful in what follows.

Lemma 3.1 (Dhage [4]). Let
(
C(J,R),≤,‖·‖

)
be a partially ordered Banach space

with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2). Then ‖ · ‖
and ≤ are compatible in every partially compact subset S of C(J,R).

Proof. Let S be a partially compact subset of C(J,R) and let {xn}n∈N be a mono-
tone 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 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 S for each t ∈ J.
This shows that the sequence {xn(t)}n∈N converges to x(t) 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 the sequel.

Definition 3.1. An ordered pair of differentiable functions (u,v) ∈ C(J,R) ×
C(J,R) is said to be a coupled lower solution of the CPBVPs of coupled differential
equations (1.1) and (1.2) if

−u′′(t) + λ2u(t) ≤ f(t,u(t),v(t)),

u(0) ≤ u(T) , u′(0) ≤ u′(T),


 ,

and
−v′′(t) + λ2v(t) ≥ f(t,v(t),u(t)),

v(0) ≥ v(T) , v′(0) ≥ v′(T),


 ,

for all t ∈ J. Similarly, an ordered pair of differentiable functions (p,q) ∈ C(J,R)×
C(J,R) is said to be a coupled upper solution of the CPBVPs (1.1) and (1.2) if the
above inequalities are satisfied with reverse sign.

We consider the following set of hypotheses in what follows.

(H1) f is bounded on J ×R×R with bound M.
(H2) The function f(t,x,y) is nondecreasing in x and nonincreasing in y for each

t ∈ J.
(H3) The CPBVPs (1.1) and (1.2) have a lower coupled solution (u,v) ∈ C(J,R)×

C(J,R.
(H4) The CPBVPs (1.1) and (1.2) have a lower coupled solution (p,q) ∈ C(J,R)×

C(J,R).



58 DHAGE

The following useful lemma is obvious and may be found in Dhage [1] and the
references therein.

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

(3.3)
−x′′(t) + λ2x(t) = σ(t), t ∈ J,

x(0) = x(T) , x′(0) = x′(T),




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

(3.4) x(t) =

∫ T
0

G(t,s) σ(s) ds

where, G(t,s) is the Green’s function associated to the PBVP

(3.5)
−x′′(t) + λ2x(t) = 0, t ∈ J,

x(0) = x(T) , x′(0) = x′(T).




Notice that the Green’s function G is continuous and nonnegative on J ×J and
therefore, the number K := max{|G(t,s)| : t,s ∈ [0,T]} exists.

An application of above Lemma 3.2 we obtain

Lemma 3.3. A pair of function (u,v) ∈ C(J,R)×C(J,R) is a coupled solution of
the CPBVPs (1.1) and (1.2) if and only if u and v are the solutions of the nonlinear
integral equations,

(3.6) x(t) =

∫ T
0

G(t,s)f(s,x(s),y(s)) ds

and

(3.7) y(t) =

∫ T
0

G(t,s)f(s,y(s),x(s)) ds

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

Theorem 3.1. Assume that the hypotheses (H1) through (H3). Then the CPBVPs
(1.1) and (1.2) have a coupled solution (x∗,y∗) defined on J and the sequences {xn}
and {yn} defined by

(3.8) xn+1(t) =

∫ T
0

G(t,s)f(s,x(s),yn(s)) ds

and

(3.9) yn+1(t) =

∫ T
0

G(t,s)f(s,yn(s),xn(s)) ds

for each t ∈ J converge monotonically to x∗ and y∗ respectively.

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.

Consider the mapping F on E ×E defined as

(3.10) F(x,y)(t) =
∫ T
0

G(t,s)f(s,x(s),y(s)) ds, t ∈ J



NONLINEAR SECOND ORDER DIFFERENTIAL EQUATIONS 59

and

(3.11) F(y,x)(t) =
∫ T
0

G(t,s)f(s,y(s),x(s)) ds, t ∈ J.

Since Green’s function G is continuous on J×J, we have that F(x,y),F(y,x) ∈
E. As a result, F defines a mapping F : E × E → E. We shall show that F
satisfies the conditions of Theorem 2.1. This will be achieved in a series of following
steps.

Step I : F is a mixed monotone operator on E ×E.
Let x1,x2 ∈ S be such that x1 ≤ x2. Then, by hypothesis (H2),

F(x1,y)(t) =
∫ T
0

G(t,s)f(s,x1(s),y(s)) ds

≤
∫ T
0

G(t,s)f(s,x2(s),y(s)) ds

= F(x2,y)(t)

for all t ∈ J. This shows that F(x,y) is monotone nondecreasing in x for all t ∈ J
and y ∈ S. Next, let y1,y2 ∈ E be such that y1 ≤ y2. Then,

F(x,y1)(t) =
∫ T
0

G(t,s)f(s,x(s),y1(s)) ds

≥
∫ T
0

G(t,s)f(s,x(s),y2(s)) ds

= F(x,y2)(t)

for all t ∈ J and x ∈ S. Hence F(x,y) is monotone nonincreasing in y for all x ∈ E.
Thus F is a mixed monotone mapping on E ×E.

Step II: F is partially continuous mixed monotone operator on E ×E.
Let {Xn}n∈N = {(xn,yn)} be a monotone nondecreasing sequence in a chain

C = C1 ×C2 of E ×E such that Xn = (xn,yn) → (x,y) = X and Xn ≤ X for all
n ∈ N. Then, by dominated convergence theorem,

lim
n→∞

F(Xn)(t) =
∫ T
0

G(t,s)
[

lim
n→∞

f(s,xn(s),yn(s))
]
ds

=

∫ T
0

G(t,s)f(s,x(s),y(s)) ds

= F(X)(t),

for all t ∈ J. This shows that F(Xn) converges monotonically to F(X) pointwise
on J.

Next, we will show that {F(Xn)}n∈N is an equicontinuous sequence of functions
in E. Let t1, t2 ∈ J be arbitrary. Then, by hypothesis (B2),

|F(Xn)(t2) −F(Xn)(t1)|



60 DHAGE

≤

∣∣∣∣∣
∫ T
0

G(t2,s)f(s,xn(s),yn(s)) ds−
∫ T
0

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

∣∣∣∣∣
≤
∫ T
0

∣∣G(t2,s) −G(t1,s)∣∣ ∣∣f(s,xn(s),yn(s))∣∣ds
≤ Mf

∫ T
0

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

uniformly for all n ∈ N. This shows that the convergence F(Xn) → F(X) is
uniform and hence F is a partially continuous on E ×E.

Step III: F is a partially compact mixed monotone operator on E ×E.
Let C1 and C2 be two arbitrary chains in E. We show that F(C1 × C2) is a

relatively compact subset of E. To finish it is enough to prove that F(C1 × C2)
is uniformly bounded and equicontinuous set in E. Let x ∈ C1 and y ∈ C2 be
arbitrary. Then, by (H1),

|F(x,y)(t)| ≤
∫ T
0

G(t,s)|f(s,x(s),y(s))|ds ≤ MfK T = r

for all t ∈ J. Taking the supremum over t, we obtain ‖F(x,y)‖≤ r for all x ∈ C1
and y ∈ C2. Hence, F(C1×C2) is a uniformly bounded subset of E. Next, we show
that F(C1 ×C2) is an equicontinuous set in E. Let t1, t2 ∈ J be arbitrary. Then,
for any z ∈ F(C1 × C2), there exist x ∈ C1 and y ∈ C2 such that z = F(x,y).
Without loss of generality, we may assume that x(t1) ≥ x(t2) and y(t1) ≤ y(t2).
Therefore, by the definition of F,

|z(t1) −z(t2)| = |F(x,y)(t1) −F(x,y)(t2)|

=

∣∣∣∣
∫ t1
0

f(s,x(s),y(s)) ds−
∫ t2
0

f(s,x(s),y(s)) ds

∣∣∣∣
≤

∣∣∣∣
∫ t1
t2

|f(s,x(s),y(s))|ds
∣∣∣∣

≤ Mf |t1 − t2|

−→ 0 as t1 → t2,
uniformly for all x ∈ C1 and y ∈ C2. As a result, we have

|F(x,y)(t1) −F(x,y)(t2)| −→ 0 as t1 → t2,
uniformly for all (x,y) ∈ C1 ×C2. Consequently F(C1 ×C2) is an equi-continuous
set of E. We apply Arzeli-Ascoli theorem and deduce that F(C1×C2) is a relatively
compact subset of E. Hence F is partially relatively compact on E ×E.

Now F is a partially continuous and partially compact mixed monotone operator
on E ×E into E. Again, by hypothesis (H3), there exist elements x0 and y0 in S
such that x0 ≤ F(x0,y0) and y0 ≥ F(y0,x0). Thus all the conditions of Theorem
2.1 are satisfied and hence the coupled equations F(x,y) = x and y = F(y,x) have
a coupled solution (x∗,y∗) and the sequences {xn} and {yn} defined by (3.11) and



NONLINEAR SECOND ORDER DIFFERENTIAL EQUATIONS 61

(3.12) converge monotonically to x∗ and y∗ respectively. This completes the proof.
�

Remark 3.1. The conclusion of Theorem 3.1 also remains true if we replace the
hypothesis (H3) with (H4). The proof of Theorem 3.1 under this new hypothesis is
obtained using similar arguments with appropriate modifications.

Example 3.1. Given a closed and bounded interval J = [0, 1] in R, consider the
coupled PBVPs,

(3.12)
−x′′(t) + x(t) = tanh x(t) − tanh y(t),

x(0) = x(1) , x′(0) = x′(1),




and

(3.13)
−y′′(t) + y(t) = tanh y(t) − tanh x(t),

y(0) = y(1) , y′(0) = y′(1),




for all t ∈ [0, 1].

Here, the function f is given by

f(t,x,y) = tanh x− tanh y.
for all t ∈ [0, 1] and x,y ∈ R. Clearly, f is uniformly continuous and bounded on
J × R × R with bound Mf = 2. Furthermore, f(t,x,y) is nondecreasing in x for
each t ∈ J and y ∈ R and nonincreasing in y for each t ∈ J and x ∈ R. Finally,
there exist functions

x0(t) = −
[
e2(e−t −et)

(e− 1)
+
e(1 −e−t)

(e− 1)

]
and

y0(t) =

[
e2(e−t −et)

(e− 1)
+
e(1 −e−t)

(e− 1)

]
such that

(3.14)
−x′′0 (t) + x0(t) ≤ tanh x0(t) − tanh y0(t),

x0(0) ≤ x0(1) , x′0(0) ≤ x
′
0(1),




and

(3.15)
−y′′0 (t) + y0(t) ≥ tanh y0(t) − tanh x0(t),

y0(0) ≥ y0(1) , y′0(0) ≥ y
′
0(1),




for all t ∈ J. Thus, the nonlinearity f satisfies all the hypotheses (H1) through
(H3) of Theorem 3.1. Hence, the CPBVPs (3.12) and (3.13) have a coupled solution
(x∗,y∗) defined on [0, 1] and the sequences {xn}∞n=0 and {yn}∞n=0 of successive
approximations defined by

xn+1(t) =

∫ 1
0

G(t,s)
[

tanh xn(s) − tanh yn(s)
]
ds, t ∈ [0, 1],

and

yn+1(t) =

∫ 1
0

G(t,s)
[

tanh yn(s) − tanh xn(s)
]
ds, t ∈ [0, 1],



62 DHAGE

where G(t,s) is a Green’s function associated with the PBVP

(3.16)
−x′′(t) + x(t) = 0, t ∈ J,

x(0) = x(1) , x′(0) = x′(1),




given by

G(t,s) =
1

2(e− 1)

{
e1+s−t + et−s, 0 ≤ s ≤ t ≤ 1

e1+t−s + es−t, 0 ≤ t ≤ s ≤ 1,
converge monotonically to x∗ and y∗ respectively.

Remark 3.2. Finally, we mention that Theorem 2.1 may be applied to various
nonlinear initial and boundary value problems of ordinary coupled differential e-
quations for proving the existence as well as algorithms for the coupled solutions
under suitable mixed monotonic and partial compactness type conditions.

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