Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 93, pp. 1–16.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

MONOTONE SOLUTIONS OF FIRST ORDER NONLINEAR

DIFFERENTIAL SYSTEMS

LIANWEN WANG, ABDULRAHMAN MUBARAK

Abstract. This article concerns the classification, continuablity, bounded-
ness, and existence of solutions for a system of first order nonlinear differential

equations. First, we prove that all solutions of the system are eventually mono-
tonic and can be separated into two classes. Then we discuss the continuability

of solutions. After that we establish necessary and sufficient conditions for the

boundedness of all solutions. Also, we study the existence of monotone solu-
tions in certain classes.

1. Introduction

We study the system of first order nonlinear differential equations

x′(t) = p(t)f(y(t)),

y′(t) = q(t)g(x(t)),
(1.1)

where p(t), q(t), f(r), and g(r) are functions satisfying the conditions:

(H0) p(t), q(t) : [a,∞)→ (0,∞) are continuous; f(r) : R→ R is continuous and
rf(r) > 0 or r 6= 0; g(r) : R→ R is continuous and rg(r) > 0 for r 6= 0.

A pair of functions (x, y) is called a solution of (1.1) with maximal existence
interval [a, α), a < α ≤ ∞, if both x(t) and y(t) are differentiable and satisfy
system (1.1) on [a, α). A solution (x, y) is said to be eventually monotone if there
exists a t1 ≥ a such that both x(t) and y(t) are monotone on [t1, α). A solution (x, y)
is said to be eventually trivial if there exists a t1 ≥ a such that (x(t), y(t)) ≡ (0, 0)
on [t1, α). We only consider eventually nontrivial solutions.

The following lemma shows that all solutions of (1.1) are eventually monotone
and can be separated into two classes.

Lemma 1.1. If (x, y) is a solution of (1.1) with maximal existence interval [a, α),
a < α ≤ ∞, then (x, y) is eventually monotone and belongs to one of the two
classes:

A =
{

(x, y) : ∃ b ≥ a such that x(t)y(t) > 0,∀t ∈ [b, α)
}
,

B =
{

(x, y) : x(t)y(t) < 0,∀t ∈ [a, α)
}
.

2010 Mathematics Subject Classification. 34C11, 34C12.
Key words and phrases. Monotone solutions; system of differential equations; boundedness;

nonlinear; existence; continuability.
©2021. This work is licensed under a CC BY 4.0 license.

Submitted August 7, 2020. Published November 23, 2021.

1



2 L. WANG, A. MUBARAK EJDE-2021/93

Proof. The proof is similar to that of [3, Lemma 1] with minor modifications. Let
F (t) = x(t)y(t). In view of condition (H0) we have

F ′(t) = x′(t)y(t) + x(t)y′(t) = y(t)p(t)f(y(t)) + x(t)q(t)g(x(t)) ≥ 0.

Then F is increasing on [a, α) and there are three possible cases:

(1) F (t) < 0 for all t ∈ [a, α);
(2) There exists b ≥ a such that F (t) > 0 for all t ∈ [b, α);
(3) There exists b ≥ a such that F (t) ≡ 0 for all t ∈ [b, α).

Clearly, (x, y) ∈ B in the first case and (x, y) ∈ A in the second case. We claim
that (x, y) is eventually trivial in the third case. Otherwise, if there exists t1 ∈ [b, α)
such that x(t1) 6= 0, then y(t1) = 0 and there exists an open interval U1 containing
t1 such that x(t) 6= 0 for any t ∈ U1. This implies that y(t) ≡ 0 on U1. This
contradicts the fact that y′(t) = q(t)g(x(t)) 6= 0 for any t ∈ U1. Hence, x(t) ≡ 0 on
[b, α). Similarly, we can prove that y(t) ≡ 0 on [b, α). �

Remark 1.2. From (H0) and Lemma 1.1, each class A solution (x, y) of (1.1)
belongs to one of the two subclasses, one subclass is comprise of positive and in-
creasing functions x and y, and the other subclass is comprise of negative and
decreasing functions x and y. The former is called the subclass of positive class A
solutions and the latter is called the subclass of negative class A solutions.

The classification, boundedness, existence, asymptotic behavior, and other prop-
erties of solutions of special cases of system (1.1) (second order nonlinear differential
equations) have been extensively studied; see [1, 3, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17]
and other publications, but there is less discussion for differential system. A special
nonlinear differential system

x′(t) = p(t)yλ1(t),

y′(t) = q(t)xλ2(t)
(1.2)

is considered in [2, 11]. For system (1.1), classification and existence of positive
class A solutions are investigated in [10]. Nonoscillatory and existence of solutions
are explored in [4, 9] with the assumption that q(t) < 0.

The following assumptions are imposed for later discussions.

(H1) There exists a real number M > 0 such that

|f(uv)| ≤M |f(u)||f(v)|, |g(uv)| ≤M |g(u)||g(v)|, ∀u, v ∈ R.

(H2) There exists a real number m > 0 such that f(r) and g(r) are increasing
for |r| ≥ m.

(H3A) There exists a real number r0 > 0 such that∫ ±∞
±r0

dr

f(g(r))
=∞.

(H3B) There exists a real number r0 > 0 such that∫ ±∞
±r0

dr

g(f(r))
=∞.

(H4A) There exists a real number r1 > 0 such that∫ ±r1
0

dr

f(g(r))
=∞.



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 3

(H4B) There exists a real number r1 > 0 such that∫ ±r1
0

dr

g(f(r))
=∞.

Remark 1.3. (H1) holds for all homogeneous functions such as the so-called p-
Laplacian operator

f(r) = Φp(r) = |r|p−2r, p > 1.

(H1) is also satisfied for some non-homogeneous functions as well, for example,

f(r) =

{
1+r2

r , |r| > 1,

2r, |r| ≤ 1.

It is easy to check that f is non-homogeneous, continuous, and increasing on
(−∞,∞), that rf(r) > 0 for all r 6= 0, and that

|f(uv)| ≤ |f(u)||f(v)|, ∀u, v ∈ R.

Our goal is to investigate the classification, continuability, boundedness, and ex-
istence of solutions of (1.1). This article is organized in the follows: Section 1 is
for the introduction. The background, motivation, classification, and assumptions
are addressed in this section. Continuability of solutions is discussed in Section 2.
After that, necessary and sufficient conditions for boundedness of all solutions are
established in Section 3. Moreover, examples are provided to illustrate that the
boundedness conditions are optimal in some sense. Finally, the existence of mono-
tone solutions in certain classes is obtained in Section 4.

2. Continuability of solutions

The maximal existence interval of a solution of (1.1) might be finite or infinite.
Since many applications require the infinite existence interval of solutions, the next
theorem provides conditions for the continuability of solutions.

Theorem 2.1. Suppose that conditions (H1), (H2), (H3A), (H3B) hold. Then all
solutions of (1.1) can be extended to [a,∞).

Proof. We focus on class A solutions since all class B solutions can be extended to
infinity. Without loss of generality we consider a positive class A solution (x, y), in
other words, x(t) > 0 and y(t) > 0 for b ≤ t < α and both are increasing. If α <∞,
we claim limt→α− x(t) = ∞ and limt→α− y(t) = ∞. Indeed, if limt→α− x(t) < ∞,
it follows from

y(t) = y(a) +

∫ t

a

q(s)g(x(s))ds

that

lim
t→α−

y(t) = y(a) +

∫ α

a

q(t)g(x(t))dt <∞.

So (x, y) can be extended to [a, α] and further to a small neighborhood at the
right of α. This contradicts the assumption that [a, α) is the maximal existence
interval of (x, y). Therefore, limt→α− x(t) = ∞. Similarly, we can show that
limt→α− y(t) =∞. Thus, there exists a real number c > a such that x(t) ≥ m and



4 L. WANG, A. MUBARAK EJDE-2021/93

y(t) ≥ m for all c ≤ t < α, where the number m is defined in (H2). By (H2) both
f(x(t)) and g(y(t)) are increasing on [c, α), then

y(t) = y(c) +

∫ t

c

q(s)g(x(s))ds

≤ y(c) + g(x(t))

∫ t

c

q(s)ds

= g(x(t))
( y(c)

g(x(t))
+

∫ t

c

q(s)ds
)

≤ g(x(t))
( y(c)

g(x(c))
+

∫ t

c

q(s)ds
)
.

Choosing k > 1 and t1 ≥ c such that for t ≥ t1,

y(c)

g(x(c))
+

∫ t

c

q(s)ds ≤ k
∫ t

c

q(s)ds,

we have

y(t) ≤ kg(x(t))

∫ t

c

q(s)ds,

p(t)f(y(t)) ≤ p(t)f
(
kg(x(t))

∫ t

c

q(s)ds
)
.

By (H1)

x′(t) = p(t)f(y(t)) ≤M2f(k)p(t)f
(
g(x(t)

)
f
(∫ t

c

q(s)ds
)
.

Then

x′(t)

f
(
g(x(t)

) ≤M2f(k)p(t)f
(∫ t

c

q(s)ds
)
.

Integrating from t1 to t we have∫ x(t)

x(t1)

dr

f(g(r))
=

∫ t

t1

x′(s)ds

f(g(x(s))
≤M2f(k)

∫ t

t1

p(s)f
(∫ s

c

q(σ)dσ
)
ds.

Letting t→ α− yields∫ ∞
x(t1)

dr

f(g(r))
≤M2f(k)

∫ α

t1

p(s)f
(∫ s

c

q(σ)dσ
)
ds,

which contradicts (H3A), and hence, x(t) can be extended to infinity.
Following the similar procedure, we have∫ ∞

y(t1)

dr

g(f(r))
≤M2g(k)

∫ α

t1

q(s)g
(∫ s

c

p(σ)dσ
)
ds.

This contracts (H3B) and implies that y(t) can be extended to infinity. �

Theorem 2.1 generalizes part of [14, Theorem 1] for system (1.1).



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 5

3. Boundedness of solutions

In this section we consider the boundedness of all solutions of (1.1) and assume
that all solutions can be extended to infinity. Notice that it is possible only one
component of a solution is bounded. For example, consider a system defined on
[1,∞)

x′(t) =
1

(t3 + t)
y(t),

y′(t) =
1

arctan t
x(t).

(3.1)

All the conditions of Theorem 2.1 are satisfied, so all solutions can be extended to
infinity. It is easy to verify that (x(t), y(t)) = (arctan t, t) is a class A solution of
(3.1) with bounded x component and unbounded y component.

Now we provide conditions for the boundedness of the first component of all
solutions of (1.1).

Theorem 3.1. Suppose that conditions (H1), (H2), (H3A) hold. Then the x com-
ponents of all solutions of (1.1) are bounded if and only if∫ ∞

a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞.

Proof. We consider class A solutions since all class B solutions are bounded. With-
out loss of generality we assume that (x, y) is a positive class A solution.

If x is bounded, then limt→∞ x(t) = L1 <∞. Define

K1 = min
x(b)≤r≤L1

g(r) > 0.

Then

y(t) = y(b) +

∫ t

b

q(s)g(x(s))ds ≥ K1

∫ t

b

q(s)ds.

In view of (H1), we have

f
(∫ t

b

q(s)ds
)
≤ f

(y(t)

K1

)
≤Mf

( 1

K1

)
f(y(t)).

Thus

p(t)f
(∫ t

b

q(s)ds
)
≤Mf

( 1

K1

)
x′(t).

Integrating from b to infinity implies∫ ∞
b

p(t)f
(∫ t

b

q(s)ds
)
ds ≤Mf

( 1

K1

)
(L1 − x(b)) <∞.

Therefore, ∫ ∞
a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞.

For sufficiency, using arguments similar to the proof of Theorem 2.1, we have∫ x(t)

x(t1)

dr

f(g(r))
≤M2f(k)

∫ t

t1

p(s)f
(∫ s

c

q(σ)dσ
)
ds. (3.2)

If x is unbounded, then limt→∞ x(t) =∞. Letting t→∞ to (3.2) we have∫ ∞
x(t1)

dr

f(g(r))
≤M2f(k)

∫ ∞
t1

p(t)f
(∫ t

c

q(s)ds
)
ds <∞.



6 L. WANG, A. MUBARAK EJDE-2021/93

This contradicts to (H3A) and hence, x is bounded. �

The following corollary is directly derived from the proof of Theorem 3.1.

Corollary 3.2. Let (H1), (H2), (H3A) hold. If the x component of one class A
solution of (1.1) is bounded, then the x components of all solutions are bounded.
On the other hand, if the x component of one class A solution is unbounded, then
the x components of all class A solutions are unbounded.

Remark 3.3. Condition (H3A) in Theorem 3.1 is sharp. For example consider the
differential system, defined for t ≥ 1,

x′(t) =
1

t4/3
y1/3(t),

y′(t) =
4

t2
x5(t).

(3.3)

Here, p(t) = 1/t4/3, q(t) = 4/t2, f(r) = r1/3, and g(r) = r5. Clearly, (H1), (H1),
and (H2) are satisfied, but (H3A) does not hold because∫ ±∞

±1

dr

f(g(r))
=

∫ ±∞
±1

dr

r
5
3

<∞.

Note that ∫ ∞
1

p(t)f
(∫ t

1

q(s)ds
)
dt =

3
√

4

∫ ∞
1

1

t4/3

(
1− 1

t

)1/3
<∞.

However, (x, y) = (t, t4) is an unbounded solution of (3.3).

Remark 3.4. Theorem 3.1 can be applied to some differential equations but [3,
Theorem 8] is not applicable. For example, let p(t) = Φp∗(1/a(t)), f(r) = Φp∗(r),
q(t) = b(t), and g(r) = f(r) (the function f is defined in [3]), where Φp(r) is the
p-Laplacian operator and p∗ = p

p−1 . Then system (1.1) becomes equation (1) in

[3],

(a(t)Φp(x
′))′ = b(t)f(x). (3.4)

Define

f(r) =

{
Φp(r ln |r|), |r| > e,

ep−2r, |r| ≤ e.
It is easy to check that f is continuous and increasing on (−∞,∞) and that rf(r) >
0 for all r 6= 0. The major condition (22) in [3] does not hold because

lim
|r|→∞

f(r)

Φp(r)
=∞.

Thus, [3, Theorem 8] cannot be applied to equation (3.4). However, (H3A) is
satisfied because ∫ ±∞

±e

1

f(g(r))
dr =

∫ ∞
e

dr

r ln r
=∞.

Note that the x component of a solution of (1.1) is a solution of equation (3.4).
Theorem 3.1 implies that all solutions of equation (3.4) are bounded.

Theorem 3.1 generalizes part of [14, Theorem 1] for the differential system (1.1).
Because of the symmetric manner of (1.1), we have the conditions for the bound-
edness of the y components of all solutions of (1.1).



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 7

Theorem 3.5. Suppose that conditions (H1), (H2), (H3B) hold. Then the y com-
ponents of all solutions of (1.1) are bounded if and only if∫ ∞

a

q(t)g
(∫ t

a

p(s)ds
)
dt <∞.

Corollary 3.6. Let (H1), (H2), (H3B) hold. If the y component of one class A
solution of (1.1) is bounded, then the y components of all solutions are bounded.
On the other hand, if the y component of one class A solution is unbounded, then
the y components of all class A solutions are unbounded.

Remark 3.7. Condition (H3B) in Theorem 3.5 is sharp. We already know that
(x, y) = (t, t4) is an unbounded solution of (3.3), but∫ ∞

1

q(t)g
(∫ t

a

p(s)ds
)
dt = 972

∫ ∞
1

1

t2

(
1− 1

t1/3

)5
<∞,

and (H3B) does not hold because∫ ±∞
±1

dr

g(f(r))
=

∫ ±∞
±1

dr

r
5
3

<∞.

Combining Theorem 3.1 and Theorem 3.5, we have the following theorem.

Theorem 3.8. Suppose that conditions (H1), (H2), (H3A), (H3B) hold. Then all
solutions of (1.1) are bounded if and only if∫ ∞

a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞,

and ∫ ∞
a

q(t)g
(∫ t

a

p(s)ds
)
dt <∞.

Corollary 3.9. Let (H1), (H2), (H3A), (H3B) hold. If system (1.1) has one
bounded class A solution, then all solutions are bounded. On the other hand, if
system (1.1) has one unbounded class A solution, then all class A solutions are
unbounded.

Clearly, (H1), (H2), (H3A), and (H3B) are satisfied for system (1.2) when λ1 > 0,
λ2 > 0, and λ1λ2 ≤ 1 because∫ ±∞

±r0

dr

f(g(r))
=

∫ ±∞
±r0

dr

g(f(r))
=

∫ ±∞
±r0

dr

rλ1λ2
=∞

for any r0 > 0. We have the following result for system (1.2).

Corollary 3.10. Suppose that λ1 > 0, λ2 > 0, and λ1λ2 ≤ 1. Then all solutions
of (1.2) can be extended to infinity. Moreover, the x components of all solutions
are bounded if and only if∫ ∞

a

p(t)
(∫ t

a

q(s)ds
)λ1

dt <∞,

the y components of all solutions are bounded if and only if∫ ∞
a

q(t)
(∫ t

a

p(s)ds
)λ2

dt <∞,



8 L. WANG, A. MUBARAK EJDE-2021/93

and all solutions are bounded if and only if∫ ∞
a

p(t)
(∫ t

a

q(s)ds
)λ1

dt <∞,

and ∫ ∞
a

q(t)
(∫ t

a

p(s)ds
)λ2

dt <∞.

If assumptions (H2), (H3A), (H3B) are dropped but the boundedness of functions
f and g is required, we have the following results.

Theorem 3.11. Let (H1) hold. Assume that there exists K > 0 such that |g(r)| ≤
K for all r ∈ R. Then the x components of all solutions of (1.1) are bounded if
and only if ∫ ∞

a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞.

Proof. The proof of necessity is similar to the proof of Theorem 3.1. For suffi-
ciency, without loss of generality we consider positive class A solution (x, y). By
the boundedness assumption of g we have

y(t) = y(b) +

∫ t

b

q(s)g(x(s))ds ≤ y(b) +K

∫ t

b

q(s)ds.

Choosing k > 1 and t1 ≥ b such that for t ≥ t1

y(b) +K

∫ t

b

q(s)ds ≤ k
∫ t

b

q(s)ds.

Then

p(t)f(y(t)) ≤ p(t)f
(
k

∫ t

b

q(s)ds
)
.

Applying (H1) we have

x′(t) = p(t)f(y(t)) ≤Mf(k)p(t)f
(∫ t

b

q(s)ds
)
.

Then

x(t)− x(t1) ≤Mf(k)

∫ t

t1

p(s)f
(∫ s

b

q(σ)dσ
)
ds,

and x is bounded. �

Theorem 3.11 generalizes part of [14, Theorem 2] for the differential system (1.1).

Corollary 3.12. Let (H1) hold. Assume that there exists K > 0 such that |g(r)| ≤
K for all r ∈ R. If the x component of one class A solution of (1.1) is bounded,
then the x components of all solutions are bounded. On the other hand, if the x
component of one class A solution is unbounded, then the x components of all class
A solutions are unbounded.

Similar to Theorem 3.11 we have the following theorem.

Theorem 3.13. Let (H1) hold. Assume that there exists K > 0 such that |f(r)| ≤
K for all r ∈ R. Then the y components of all solutions of (1.1) are bounded if
and only if ∫ ∞

a

q(t)g
(∫ t

a

p(s)ds
)
dt <∞.



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 9

Corollary 3.14. Let (H1) hold. Assume that there exists K > 0 such that |f(r)| ≤
K for all r ∈ R. If the y component of one class A solution of (1.1) is bounded,
then the y components of all solutions are bounded. On the other hand, if the y
component of one class A solution is unbounded, then the y components of all class
A solutions are unbounded.

Combining Theorem 3.11 and Theorem 3.13 we have the following theorem.

Theorem 3.15. Let (H1) hold. Assume that there exists K > 0 such that |f(r)| ≤
K and |g(r)| ≤ K for all r ∈ R. Then all solutions of (1.1) are bounded if and only
if ∫ ∞

a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞,

and ∫ ∞
a

q(t)g
(∫ t

a

p(s)ds
)
dt <∞.

Corollary 3.16. Let (H1) hold. Assume that there exists K > 0 such that |f(r)| ≤
K and |g(r)| ≤ K for all r ∈ R. If system (1.1) has one bounded class A solu-
tion, then all solutions are bounded. On the other hand, if system (1.1) has one
unbounded class A solution, then all class A solutions are unbounded.

4. Existence of solutions

In this section we discuss the existence of solutions for each class and require
the existence and uniqueness of all initial value problems of system (1.1), in other
words, the IVP problem

x′(t) = p(t)f(y(t)), x(a) = x0,

y′(t) = q(t)g(x(t)), y(a) = y0
(4.1)

has a unique solution for any pair (x0, y0). It is well-known that the existence and
uniqueness of IVP problems of system (1.1) holds under general conditions.

Theorem 4.1. System (1.1) has both positive and negative class A solutions.

Proof. Let (x, y) be the solution of (4.1) with initial condition (x(a), y(a)) =
(x0, y0), where x0 > 0 and y0 > 0. We claim that (x, y) is a positive class A
solution of system (1.1). Indeed, from the proof of Lemma 1.1 we have F (t) > 0
for t ∈ [a, α), so (x, y) is a positive class A solution. Similarly, if x0 < 0 and y0 < 0,
(x, y) is a negative class A solution of system (1.1). �

Now, we focus on the existence of class B solutions.

Theorem 4.2. Let (H1), (H2), (H4A) hold. Assume that

lim
r→±∞

f(r) = ±∞. (4.2)

Then system (1.1) has class B solutions.

Proof. Let x0 > 0 and c be a real number. Denote the solution of (1.1) with the
initial condition (x(a), y(a)) = (x0, c) by (x(t, c), y(t, c)). We define two sets

U = {c ∈ R : there exists t1 ≥ a such that x(t1, c) < 0},
V = {c ∈ R : there exists t2 ≥ a such that y(t2, c) > 0}.



10 L. WANG, A. MUBARAK EJDE-2021/93

Clearly, both U and V are open sets and V is nonempty. We claim that U is also
nonempty. Indeed, by (4.2) we can select c ∈ R such that

c < −g(x0)

∫ a+1

a

q(s)ds

and

x0 + f
(
c+ g(x0)

∫ a+1

a

q(s)ds
)(∫ a+1

a

p(s)ds
)
< 0. (4.3)

Obviously, U is nonempty if there exists t̄ ∈ (a, a + 1] such that x(t̄, c) < 0, so we
assume x(t, c) ≥ 0 on [a, a+ 1] in the following discussion.

It is easy to check that y(t, c) < 0 for a ≤ t ≤ a + 1. If not, there exists
t1 ∈ (a, a+ 1] such that y(t1, c) = 0 and y(t, c) < 0 for t ∈ [a, t1). Consequently,

0 = y(t1, c) = c+

∫ t1

a

q(s)g(x(s, c))ds ≤ c+ g(x0)

∫ a+1

a

q(s)ds < 0.

This is a contradiction. Therefore, y(t, c) < 0 on a ≤ t ≤ a + 1 and x(t, c) is
decreasing on [a, a+ 1]. Moreover, for t ∈ [a, a+ 1] we have

y(t, c) = c+

∫ t

a

q(s)g(x(s, c))ds ≤ c+ g(x0)

∫ a+1

a

q(s)ds. (4.4)

In account of (4.3) and (4.4) we have

x(a+ 1, c) = x0 +

∫ a+1

a

p(t)f(y(t, c))dt

≤ x0 +

∫ a+1

a

p(t)f
(
c+ g(x0)

∫ a+1

a

q(s)ds
)
dt

≤ x0 + f
(
c+ g(x0)

∫ a+1

a

q(s)ds
)
dt

∫ a+1

a

p(s)ds < 0,

which contradicts the fact x(t, c) ≥ 0 on [a, a+ 1] and hence U is nonempty.
Clearly, U ∩V = ∅. Hence, R− (U ∪V ) 6= ∅. Let c ∈ R− (U ∪L). Then x(t, c) is

a nonincreasing nonnegative function and y(t, c) is a nondecreasing and nonpositive
function on [a,∞). We will show that x(t, c) > 0 and y(t, c) < 0 on [a,∞). If not,
there exists t∗ > a such that x(t, c) > 0, y(t, c) < 0 for t ∈ [a, t∗), and x(t, c) = 0,
y(t, c) = 0 for t ≥ t∗. Note that for t ∈ [a, t∗]

y(t, c) = y(t∗)−
∫ t∗

t

q(s)g(x(s, c))ds

≥ −
∫ t∗

t

q(s)g(x(s, c))ds

≥ −g(x(t))

∫ t∗

t

q(s)ds.

Then

x′(t) = p(t)f(y(t)) ≥ p(t)f
(
− g(x(t))

∫ t∗

t

q(s)ds
)

≥Mf(g(x(t)))p(t)f
(
−
∫ t∗

t

q(s)ds
)
.



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 11

Dividing both sides by f(g(x(t))) and integrating from a to t∗, we have∫ x0

0

dr

f(g(r))
≤ −M

∫ t∗

a

p(t)f
(
−
∫ t∗

t

q(s)ds
)
dt <∞,

which contradicts (H4A). Therefore, (x(t, c), y(t, c)) is a class B solution. �

Theorem 4.2 can be applied to some differential equations but [3, Theorem 6] is
not applicable. Consider equation (3.4) again and define

f(r) =


Φp(r), |r| > 1/e,

Φp(−r ln |r|), |r| ≤ 1/e, r 6= 0,

0, r = 0,

where p > 1. It is easy to check that f is continuous and nondecreasing on (−∞,∞),
that rf(r) > 0 for all r 6= 0, and that limr→±∞ f(r) = ±∞. Note that the major
condition (18) in [3] does not hold because

lim
r→0

f(r)

Φp(r)
=∞,

So, [3, Theorem 6] cannot be applied to equation (3.4). However, (H4A) is satisfied
because ∫ ±1/e

0

1

f(g(r))
dr = −

∫ ±1/e
0

dr

r ln |r|
=∞.

By Theorem 4.2 equation (3.4) has class B solutions.

Theorem 4.3. Let (H1), (H2), (H4B) hold. Assume that

lim
r→±∞

g(r) = ±∞. (4.5)

Then system (1.1) has class B solutions.

The proof of the above theorem is similar to Theorem 4.2 by switching the role
of x and y. We omit it here. We already know that class A solutions could be
bounded or unbounded. The next results provide the existence of certain bounded
class A solutions.

Theorem 4.4. Let (H1), (H2) hold. Then system (1.1) has a class A solution with
bounded x component if and only if∫ ∞

a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞.

Proof. The proof of necessity is similar to the proof of Theorem 3.1. For the
sufficiency, let

M2 = max
1≤r≤2

g(r) > 0.

From the assumption we can choose d ≥ b such that∫ ∞
d

p(s)f
( 1

M2
+

∫ s

d

q(σ)dσ
)
ds ≤ 1

Mf(M2)
.

Let CB[d,∞) be the Banach space of all bounded continuous functions defined
on [d,∞) with the supremum norm. Define a nonempty subset of CB[d,∞) as

X = {x ∈ CB[d,∞) : 1 ≤ x(t) ≤ 2, ∀t ≥ d}.



12 L. WANG, A. MUBARAK EJDE-2021/93

Clearly, X is a bounded, closed, and convex set. Define a mapping F : X →
CB[d,∞) by

(Fx)(t) = 1 +

∫ t

d

p(s)f
(

1 +

∫ s

d

q(σ)g(x(σ))dσ
)
ds.

It is a routine practice to show that F maps X into X, F is continuous in X, and
F (X) is pre-compact in CB[d,∞).
F maps X into X because for any x ∈ X we have

1 ≤ (Fx)(t) = 1 +

∫ t

d

p(s)f
(

1 +

∫ s

d

q(σ)g(x(σ))dσ
)
ds

≤ 1 +

∫ t

d

p(s)f
(

1 +M2

∫ s

d

q(σ)dσ
)
ds

≤ 1 +Mf(M2)

∫ ∞
d

p(s)f
( 1

M2
+

∫ s

d

q(σ)dσ
)
ds ≤ 2.

To show the continuity of F in X, we need to prove that ‖Fxn − Fx∗‖ → 0 if
{xn}, x∗ ∈ X such that ‖xn−x∗‖ → 0 as n→∞. Indeed, for any s ∈ [d,∞), since
xn(s)→ x∗(s) as n→∞, we have∣∣∣p(s)f(1 +

∫ s

d

q(σ)g(xn(σ))dσ
)
− p(s)f

(
1 +

∫ s

d

q(σ)g(x∗(σ))dσ
)∣∣∣→ 0.

Moreover, for any s ∈ [d,∞)∣∣∣p(s)f(1 +

∫ s

d

q(σ)g(xn(σ))dσ
)
− p(s)f

(
1 +

∫ s

d

q(σ)g(x∗(σ))dσ
)∣∣∣

≤ 2Mf(M2)p(s)f
( 1

M2
+

∫ s

d

q(σ)dσ
)

:= J(s),

(4.6)

and ∫ ∞
d

J(s)ds = 2Mf(M2)

∫ ∞
d

p(s)f
( 1

M2
+

∫ s

d

q(σ)dσ
)
ds <∞. (4.7)

It follows from (4.6), (4.7), and the Dominated Convergence Theorem that

‖Fxn − Fx∗‖ = sup
b≤t<∞

|(Fxn)(t)− (Fx∗)(t)|

= sup
b≤t<∞

∣∣∣ ∫ t

d

p(s)f
(

1 +

∫ s

d

q(σ)g(xn(σ))dσ
)
ds

−
∫ t

d

p(s)f
(

1 +

∫ s

d

q(σ)g(x∗(σ))dσ
)
ds
∣∣∣

≤ sup
b≤t<∞

∫ t

d

∣∣∣p(s)f(1 +

∫ s

d

q(σ)g(xn(σ))dσ
)

− p(s)f
(

1 +

∫ s

d

q(σ)g(x∗(σ))dσ
)∣∣∣ds

≤
∫ ∞
d

∣∣∣p(s)f(1 +

∫ s

d

q(σ)g(xn(σ))dσ
)

− p(s)f
(

1 +

∫ s

d

q(σ)g(x∗(σ))dσ
)∣∣∣ds→ 0.



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 13

To prove that F (X) is pre-compact in CB[d,∞), we need to show that for any
sequence {xn} ∈ X, {Fxn} has a convergent subsequence in CB[d,∞). If we can
prove that {Fxn} has a convergent subsequence in C[b1, b2] for any closed and
bounded interval [b1, b2] of [d,∞), then {Fxn} has a convergent subsequence in
CB[d,∞) by the diagonal rule. By Arzelà–Ascoli Theorem, if we can show that
{Fxn} is uniformly bounded and equicontinuous in C[b1, b2], then {Fxn} has a
convergent subsequence in C[b1, b2].

Obviously, {Fxn} is uniformly bounded because 1 ≤ (Fxn)(t) ≤ 2 for all t ∈
[d,∞) and n ∈ N. Notice that

(Fxn)′(t) = p(t)f
(

1 +

∫ t

d

q(s)g(xn(s))ds
)

≤Mf(M2)p(t)f
( 1

M2
+

∫ t

d

q(s)ds
)
.

By the Mean Value Theorem, there exists a ξ ∈ [t1, t2] such that

|(Fxn)(t1)− (Fxn)(t2)|
= |(Fxn)′(ξ)(t1 − t2)|

≤Mf(M2) max
b1≤t≤b2

(
p(t)f

( 1

M2
+

∫ t

d

q(s)ds
))
|t1 − t2

∣∣.
Hence, {Fxn} is equicontinuous on [b1, b2] and therefore F (X) is pre-compact in
CB[d,∞). Applying Schauder Fixed-Point Theorem we claim that F has a fixed
point x̄ in X, that is

x̄(t) = 1 +

∫ t

d

p(s)f
(

1 +

∫ s

d

q(σ)g(x̄(σ))dσ
)
ds.

We define

ȳ(t) = 1 +

∫ t

d

q(s)g(x̄(s))ds.

It is easy to verify that (x̄(t), ȳ(t)) is a positive class A solution of system (1.1) with
bounded x component. �

Theorem 4.4 generalizes [3, Theorem 3] and [14, Theorem 3] for system (1.1).
Following the similar arguments we can prove the following Theorem.

Theorem 4.5. Suppose that (H1) and (H2) hold. Then system (1.1) has a class
A solution with bounded y component if and only if∫ ∞

a

q(t)g
(∫ t

a

p(s)ds
)
dt <∞.

Combining Theorems 4.4 and 4.5 we have the following result.

Theorem 4.6. Suppose that (H1), (H2) hold. Then system (1.1) has a bounded
class A solution if and only if∫ ∞

a

p(t)f
(∫ t

a

q(s)ds
)
dt <∞,

and ∫ ∞
a

q(t)g
(∫ t

a

p(s)ds
)
dt <∞.



14 L. WANG, A. MUBARAK EJDE-2021/93

If (x, y) is a class B solution, then both limt→∞ x(t) = ux and limt→∞ y(t) = uy
are finite. We will discuss the results of nonzero ux or uy.

Theorem 4.7. Suppose that (H1), (H2) hold. Then system (1.1) has a class B
solution (x, y) with limt→∞ x(t) = ux 6= 0 if and only if∫ ∞

a

p(t)f
(∫ ∞

t

q(s)ds
)
dt <∞.

Proof. Let (x, y) be a class B solution. Without loss of generality we assume x(t) >
0, y(t) < 0, then x′(t) < 0, y′(t) > 0 for t ≥ a. If limt→∞ x(t) = ux > 0, then
limt→∞ y(t) = uy ≤ 0 and K1 := minx(b)≤r≤ux

g(r) > 0. Notice that

−y(t) ≥ uy − y(t) =

∫ ∞
t

q(s)g(x(s))ds ≥ K1

∫ ∞
t

q(s)ds.

In view of (H1), we have

f
(∫ ∞

t

q(s)ds
)
≤ f

(
− y(t)

K1

)
≤ −Mf

(
− 1

K1

)
f(y(t)).

Thus

p(t)f
(∫ ∞

t

q(s)ds
)
≤ −Mf

( 1

K1

)
x′(t).

Integrating from a to infinity,∫ ∞
a

p(t)f
(∫ ∞

t

q(s)ds
)
ds ≤ −Mf

( 1

K1

)
(ux − x(a)) <∞.

For the sufficiency, let

M2 = max
1≤r≤2

g(r) > 0.

From the assumption we can choose d ≥ a such that∫ ∞
d

p(s)f
(∫ ∞

s

q(σ)dσ
)
ds ≤ 1

Mf(M2)
.

Define

X = {x ∈ CB[d,∞) : 1 ≤ x(t) ≤ 2, ∀t ≥ d}
and F : X → CB[d,∞):

(Fx)(t) = 1−
∫ ∞
t

p(s)f
(
−
∫ ∞
s

q(σ)g(x(σ))dσ
)
ds.

Then F maps X into X as we have

1 ≤ (Fx)(t) ≤ 1−
∫ ∞
d

p(t)f
(
−
∫ ∞
t

q(s)g(x(s))ds
)
ds

≤ 1 +Mf(M2)

∫ ∞
d

p(t)f
(∫ ∞

t

q(σ)dσ
)
ds ≤ 2.

As in the proof of Theorem 4.4, we can show that F is continuous in X and F (X) is
pre-compact in CB[d,∞). By Schauder Fixed-Point Theorem F has a fixed point
x̄ in X, that is

x̄(t) = 1−
∫ ∞
t

p(s)f
(
−
∫ ∞
s

q(σ)g(x̄(σ))dσ
)
ds.



EJDE-2021/93 MONOTONE SOLUTIONS OF NONLINEAR DIFFERENTIAL SYSTEMS 15

Define

ȳ(t) = −
∫ ∞
t

q(s)g(x̄(s))ds.

It is easy to verify that (x̄(t), ȳ(t)) is a class B solution of (1.1) with limt→∞ x(t) =
1. �

Theorem 4.7 generalizes [3, Theorem 1] and [14, Theorem 5] for system (1.1).
Using the similar arguments we can prove the following Theorem.

Theorem 4.8. Suppose that (H1), (H2) hold. Then system (1.1) has a class B
solution (x, y) with limt→∞ y(t) = uy 6= 0 if and only if∫ ∞

a

q(t)g
(∫ ∞

t

p(s)ds
)
dt <∞.

Combining Theorem 4.7 and Theorem 4.8 we have the following theorem.

Theorem 4.9. Suppose that (H1), (H2) hold. Then system (1.1) has a class B
solution (x, y) with limt→∞ x(t) = ux 6= 0 and limt→∞ y(t) = uy 6= 0 if and only if∫ ∞

a

p(t)f
(∫ ∞

t

q(s)ds
)
dt <∞,

and ∫ ∞
a

q(t)g
(∫ ∞

t

p(s)ds
)
dt <∞.

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combined second-order differential equations on the half-line, Adv. Nonlinear Anal., 5 (2016),

205–222.
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value problems for Emden-Fowler-type differential systems, Differential Integral Equations,

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nary differential equations with sub-homogeneity, Hiroshima Math. J., 31 (2001) 35–49.

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[15] L. Wang, R. McKee; Nonoscillatory solutions of second-order differential equations without

monotonicity assumptions, J. Appl. Math., 2012, Article ID 313725.

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Lianwen Wang

School of Computer Science and Mathematics, University of Central Missouri, War-
rensburg, MO 64093, USA

Email address: lwang@ucmo.edu

Abdulrahman Mubarak
Department of Mathematics, Kuwait University, P.O. Box 5969, Safat- 3060, Kuwait

Email address: abdulrahman.mubarak@ku.edu.kw


	1. Introduction
	2. Continuability of solutions
	3. Boundedness of solutions
	4. Existence of solutions
	References