EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 1, 2020, 33-47
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Higher order nonlocal boundary value problems at
resonance on the half-line

S.A. Iyase1, A.A. Opanuga2,∗

1 Department of Mathematics, College of Science and Technology, Covenant University,
Ota, Ogun State, Nigeria

Abstract. This paper investigates the solvability of a class of higher order nonlocal boundary
value problems of the form

u(n)(t) = g(t, u(t), u′(t) · · ·u(n−1)(t)), a.e. t ∈ (0,∞)

subject to the boundary conditions

u(n−1)(0) =
(n− 1)!

ξn−1
u(ξ), u(i)(0) = 0, i = 1, 2, . . . , n− 2,

u(n−1)(∞) =

∫ ξ

0

u(n−1)(s)dA(s)

where ξ > 0, g : [0,∞)×<n −→ < is a Caratheodory’s function,
A : [0, ξ] −→ [0, 1) is a non-decreasing function with A(0) = 0, A(ξ) = 1. The differential operator
is a Fredholm map of index zero and non-invertible. We shall employ coicidence degree arguments
and construct suitable operators to establish existence of solutions for the above higher order
nonlocal boundary value problems at resonance.

2020 Mathematics Subject Classifications: 34B40, 34B15

Key Words and Phrases: Higher order, Resonance, Coincidence degree, Nonlocal boundary
value problem, Half-Line

1. Introduction

In this paper, we study the existence of solutions for the higher order boundary value
problems.

u(n)(t) = g(t, u, u′(t)...un−1(t)) a.e. t ∈ (0,∞) (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3539

Email addresses: samuel.iyase@covenantuniversity.edu.ng (S.A. Iyase),
abiodun.opanuga@covenantuniversity.edu.ng (A.A. Opanuga)

http://www.ejpam.com 33 c© 2020 EJPAM All rights reserved.



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 34

subject to the boundary conditions

u(n−1)(0) =
(n− 1)!

ξn−1
u(ξ), u(i)(0) = 0, i = 1, 2, . . . , n− 2,

u(n−1)(∞) =

∫ ξ

0
u(n−1)(s)dA(s) (2)

where ξ > 0, g : [0,∞)×<n −→ < is a Caratheodory’s function and
A : [0, ξ] −→ [0, 1) is a non-decreasing function with A(0) = 0, A(ξ) = 1. The integral is
the Riemann-Stieltjes integral. Boundary value problems such as (1)-(2) with nontrivial
Kernels are called resonance problems. To the best of our knowledge higher order boundary
value problems on infinite intervals at resonance have not received much attention in the
Literature especially those with integral boundary conditions. Most papers focused on
boundary value problems at resonance on finite intervals especially for second and third
order boundary value problems. For some results in this direction see [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 17, 18,19] and references therein. Nonlocal boundary value problems
were first studied in [3] by Bicadze and Samarskii. In a recent paper [10] Karakostas and
Tsamatos considered the following nonlocal boundary value problem.

x
′′
(t) + q(t)f(x(t), x

′
(t)) = 0, t ∈ (0, 1)

x(0) = 0, x
′
(1) =

∫ 1

0
x
′
(s)dg(s)

under the nonresonance condition 0 = g(0) ≤ g(1) < 1. They used Krasnoselskii’s fixed
point theorem in establishing existence of solutions. In [13] Lin derived existence results
for the nonlocal boundary value problem.

x
′′
(t) = f(t, x(t), x

′
(t)), t ∈ (0, 1)

x(0) = α(ξ), x
′
(1) =

∫ 1

0
x
′
(s)dg(s)

under the resonant condition g(1) = 1
The main purpose of this paper is to provide new sufficient conditions that guarantees

existence of solutions to (1)-(2). Our investigation will be based on the coincidence degree
theory of Mawhin [17].

The main motivation for this article is the recent paper of Frioui, Guezane-Lakoud and
Khaldi [7]. The authors obtained existence results for the problem

x(n)(t) = f(t, x(t)), t ∈ (0,∞) (3)

x(i)(0) = 0, i := 0, 1 . . . n− 2, x(n−1)(∞) =
n!

ξn

∫ ξ

0
x(t)dt (4)

where f : [0,∞) × < −→ < is a given function satisfying certain conditions. There is so
far little research with regards to (1)-(2), therefore it is important to investigate it. We



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 35

also note that (1)-(2) is more general than (3)-(4).
In section 2, we provide some background definitions, Lemmas and the coincidence degree
theorem of Mawhin [17], section 3 will be devoted to proving the main existence results.

2. Preliminaries

In this section, we recall some background definitions and the coincidence degree the-
orem [17]. We shall also provide compactness criterion for continuous vector-valued func-
tions on unbounded domains. First, we give some background results from coincidence
degree theory.

Definition 2.1:

Definition 1. Let X and Z be real Banach Spaces. A linear mapping L : domL ⊂ X −→
Z is said to be a Fredholm mapping if

(i) KerL has finite dimension.

(ii) ImL is closed and has a finite codimension.

In this case, the Fredholm index is the integer

IndL = dim kerL− codimImL.

In this work, we shall utilise Fredholm mappings of index zero. If L is a Fredholm
mapping of index zero, then there exist continuous projections
P : X −→ X and Q : Z −→−→ Z
such that
ImP = kerL, kerQ = ImL and X = kerL⊕ kerP
Z = ImL⊕ ImQ and the mapping
L|domL∩kerP : domL ∩ kerP −→ ImL is invertible.‘
We denote the inverse of L|domL∩kerP by Kp : ImL −→ domL ∩ kerP . We designate the
generalised inverse of L given by KP,Q : Z −→ domL ∩ kerP as KP,Q = KP (I −Q).

Definition 2.2: The map g : [0,∞) × <n −→ < is L1[0,∞)-Caratheodory, if the
following conditions are satisfied.

(i) For each u ∈ <n, f(t, u) is Lebesgue measurable

(ii) For a.e. t ∈ [0,∞), there exists ϕr ∈ L1[0,∞) such that for a.e.
t ∈ [0,∞) and every u such that |u| ≤ r we have |f(t, u)| ≤ ϕr(t).

Let X = {u ∈ Cn−1[0,∞), limt→∞ e
−t|u(i)(t)| exists, 0 ≤ i ≤ n− 1, u(n)(t) ∈ L1[0,∞)}

endowed with the norm

‖x‖ = max
0≤i≤n−1

(
sup

t∈[0,∞)
e−t|u(i)(t)|

)



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 36

Then X is a Banach space.

Theorem 2.1 [2]a Let F be a subset of C∞ = {y ∈ C ([0,∞)) , limt→∞) y(t) exists
that is equipped with the norm ‖y‖∞ = supt∈[0,∞) |y(t)|
Then F is relatively compact if the following conditions hold.

(i) F is bounded in X

(ii) The functions belonging to F are equicontinuous on any compact subinterval of
[0,∞)

(iii) The functions from F are equiconvergent at infinity.

The following adaptation of the above theorem will be used to establish the compact-
ness of Kp (I −Q)

Lemma 2.1 [7]:
Let D ⊂ X, then D is relatively compact in X if the following conditions hold.

(i) D is bounded in X

(ii) The family W i = {ψi : ψi(t) = e−tu(i)(t), t ≥ 0, u ∈ D} is equicontinuous on any
compact subinterval of [0,∞) for i = 0, . . . , n− 1.

(iii) The family W i = {ψi : ψi(t) = e−tu(i)(t), t ≥ 0, u ∈ D} is equiconvergent at infinity
for i = 0, 1, . . . , n− 1.

Let Z = L1[0,∞) with the norm ‖y‖1 =
∫∞

0 |y(t)|dt for y ∈ Z. We denote ACloc[0,∞)
as the space of locally absolutely continuous functions on [0,∞). We define L to be the
linear operator from domL ⊂ X → Z with

domL =
{
u ∈ X : u(n−1)(t) ∈ ACloc[0,∞), u(i)(0) = 0, i = 1, 2, . . . , n− 2

u(n−1)(0) =
(n− 1)!

ξn−1
u(ξ), u(n−1)(∞) =

∫ ξ

0
u(n−1)(s)dA(s), u(n) ∈ Z

}
and Lu(t) = u(n)(t), u ∈ domL, t ∈ [0,∞). We define N : X −→ Z by setting

Nu(t) = g(t, u(t), u′(t) · · ·u(n−1)(t)), t ∈ [0,∞)

We can then write (1)-(2) as
Lu = Nu (5)

Definition 2.3:

Definition 2. Let L : domL ⊂ X −→ Z be a Fredholm mapping, E a metric space and
N : E −→ Z be a mapping: N is said to be L-compact on E if QN : E −→ Z and
KP,QN : E −→ Z are compact on E. N is called completely continuous if it is L-compact
on every bounded E ⊂ X.



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 37

The existence of solution to (5) will be guaranteed by the following theorem of Mawhin
[17].

Theorem 2.2 [16]: Let Ω ⊂ X be , and L be a Fredholm mapping of index zero
and N be L-compact on Ω̄. Assume that the following conditions are satisfied.

(1) Lu 6= Nu for every (u, λ) ∈ [(domL\ kerL) ∩ ∂Ω]× (0, 1)

(2) Nu /∈ ImL for every u ∈ kerL ∩ ∂Ω

(3) deg(QN |∂Ω∩kerL,Ω ∩ kerL, 0) 6= 0 where Q : Z −→ Z is a projection such that
ImL = kerQ

Then the equation Lu = Nu has at least one solution in domL ∩ Ω̄.
Lemma 2.2: If u(n−1)(0) = (n−1)!

ξn−1 u(ξ), A(ξ) = 1, A(0) = 0,

u(n−1)(∞) =
∫ ξ

0 u
(n−1)(s)dA(s), u(i)(0) = 0, i = 1, 2, . . . , n− 2 then

(i) kerL = {u ∈ domL : u = dtn−1, d ∈ <, t ∈ (0,∞)}

(ii) ImL =
{
y ∈ Z :

∫∞
0 y(τ)dτ −

∫ ξ
0

∫ s
0 y(v)dvdA(s) = 0

}
.

Proof:

(i) Let u ∈ kerL, then u(n)(t) = 0 for a.e. t ∈ [0,∞). Since u(i)(0) = 0 for i =
1.2. . . . , n− 2 and using (5) we derive that u(t) = dtn−1. Thus

kerL = {u ∈ X : u(t) = dtn−1}

(ii) We next show that

ImL =

{
y ∈ Z :

∫ ∞
0

y(s)ds−
∫ ξ

0

∫ s

0
y(τ)dτdA(s) = 0

}
.

We consider the problem
u(n)(t) = y(t), y ∈ Z (6)

We prove that (6) has a solution u(t) satisfying

u(n−1)(0) =
(n− 1)!

ξn−1
u(ξ), u(i)(0) = 0, i = 1, 2, . . . , n− 2, un−1(∞) =

∫ ξ

0
u(n−1)(s)dA(s)

If and only if ∫ ∞
0

y(τ)dτ −
∫ ξ

0

∫ s

0
y(v)dvdA(s) = 0 (7)



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 38

Suppose (6) has a solution u(t) satisfying (5). We show that this solution satisfies (7).
From (5) we obtain

u(t) = u(0) +
u(n−1)

(n− 1)!
tn−1 +

∫ t

0

∫ τn

0
· · ·
∫ τ2

0
y(τ)dτ1, . . . , dτn

u(n−1)(t) = u(n−1)(0) +

∫ t

0
y(τ)dτ

u(n−1)(∞) = u(n−1)(0) +

∫ ∞
0

y(τ)dτ =

∫ ξ

0

[
u(n−1)(0) +

∫ s

0
y(τ)dτ

]
dA(s)

= u(n−1)(0)A(ξ) +

∫ ξ

0

∫ s

0
y(τ)dτdA(s)

= u(n−1)(0) +

∫ ξ

0

∫ s

0
y(τ)dτdA(s)

and hence ∫ ∞
0

y(τ)dτ −
∫ ξ

0

∫ s

0
y(v)dvdA(s) = 0.

If however, (7) holds then setting

u(t) =
u(n−1)(0)

(n− 1)!
tn−1 +

∫ t

0

∫ τn

0
· · ·
∫ τ2

0
y(τ1)dτ1, · · · , dτn

We conclude that u(t) is a solution of (6) satisfying (7).

Lemma 2.3 The mapping L : domL ⊂ X −→ Z is a Fredholm mapping of index zero
and furthermore the linear continuous projector Q : Z −→ Z can be defined as

Qy = h(t)

[∫ ∞
0

y(s)ds−
∫ ξ

0

∫ s

0
y(τ)dτdA(s)

]
where

h(t) =
e−t∫ ξ

0 e
−sdA(s)

The linear operator Kp : ImL → domL ∩ kerP , the inverse of L|domL ∩ kerp can be
defined as

Kpy =

∫ t

0

∫ τn

0
· · ·
∫ τ2

0
y(τ1), dτ1 . . . dτn −

∫ ξ

0

∫ τn

0
· · ·
∫ τ2

0
y(τ1)dτ1 · · · dτn

with

‖Kpy‖ ≤ Dn‖y‖1 (8)



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 39

where

Dn = max

[
2 sup
t∈[0,∞)

e−ttn−1, max
1∈i≤n−1

( sup
t∈[0,∞)

e−ttn−1−i)

]

Proof: For y ∈ Z, we define the projection Qy as:

Qy = h(t)

[∫ ∞
0

y(v)dv −
∫ ξ

0

∫ s

0
y(τ)dτdA(s)

]
where

h(t) =
e−t∫ ξ

0 e
−sdA(s)

6= 0

Then we have

Q2y = Q(Qy) = h(t)

[∫ ∞
0

y(v)dv −
∫ ξ

0

∫ s

0
y(τ)dτdA(s)

]
·
∫ ξ

0 e
−sdA(s)∫ ξ

0 e
−sdA(s)

= Qy

This implies that Q is a projection.
Let y1 = y −Qy i.e. y1 ∈ kerQ then∫ ∞

0
y1(v)dv −

∫ ξ

0

∫ s

0
y1(τ)dτdA(s)

=

[∫ ∞
0

y(v)dv −
∫ ξ

0

∫ s

0
y(τ)dτdA(s)

][
1−

∫ ξ
0 e
−sdA(s)∫ ξ

0 e
−sdA(s)

]
= 0

Therefore, y1 ∈ ImL and hence Z = ImL+ ImQ.
Since ImL ∩ ImQ = {0} we obtain Z = ImQ⊕ ImL.
This implies that dim kerL = dim ImQ = 1.
Hence, L is a Fredholm operator of index zero.
Let P : X −→ X be defined by

Pu =
u(n−1)(0)tn−1

(n− 1)!
(9)

We define
Kp : ImL −→ domL ∩ kerP as

Kpy =

∫ t

0

∫ τn

0
· · ·
∫ τn

0
y(τ1)dτ1 · · · dτn −

∫ ξ

0

∫ τn

0
· · ·
∫ τ2

0
y(τ1)dτ1 · · · dτn



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 40

For y ∈ ImL, (LKp)y(t) = [(Kpy)(t)]n = y(t) and for u ∈ domL ∩ kerP and noting that
u(i)(0) = 0, i = 1.2, . . . , n− 2

(KpL)u(t) = u(t)− u(n−1)(0)tn−1

(n− 1)!
− u(ξ) +

u(n−1)(0)ξn−1

(n− 1)!

Since u ∈ domL ∩ kerP, Pu = u(n−1)(0)
(n−1)! t

n−1 = 0.

Also since u(n−1)(0) = (n−1)!
ξn−1 u(ξ) we derive

(KpL)u(t) = u(t)
Thus Kp = (L|domL∩kerP )−1

e−t|(Kpy)(t)| = e−t
∣∣∣∣[∫ t

0

∫ τn

0
· · ·
∫ τn

0
y(τ1)dτ1 · · · dτn −

∫ ξ

0

∫ τn

0
· · ·
∫ τ2

0
y(τ)dτ1 · · · dτn

]∣∣∣∣
≤ e−t

[∫ t

0

∫ τn

0
· · ·
∫ τ2

0
|y(τ1)|dτ1 · · · dτn +

∫ ξ

0

∫ τn

0
· · ·
∫ τ2

0
|y(τ1)|dτ1 · · · dτn

]
≤ sup

t∈[0,∞)
e−ttn−1

∫ ∞
0
|y(s)|ds+ sup

t∈[0,∞)
e−ttn−1

∫ ∞
0
|y(s)|ds

≤ 2 sup
t∈[0,∞)

e−ttn−1‖y‖1

For 1 ≤ i ≤ n− 1 we have

e−t|(Kpy)(i)(t)| =
e−t

(n− 1− i)!

∣∣∣∣∫ t

0
(t− s)n−1−iy(s)ds

∣∣∣∣
≤ e−ttn−1−i

∫ ∞
0
|y(s)|ds

We therefore conclude that

‖Kpy‖ ≤ max

[
2 sup
t∈[0,∞)

e−ttn−1, max
1≤i≤n−1

(
sup

t∈[0,∞)
e−ttn−1−i

)]
‖y‖1

= Dn‖y‖1 (10)

where Dn = max
[
2 supt∈[0,∞) e

−ttn−1,max1≤i≤n−1

(
supt∈[0,∞) e

−ttn−1−i
)]

�

Lemma 2.4 If g is a Caratheodory’s function and E ⊂ X is a bounded open subset
of X, such that domL ∩ Ē 6= φ then N is L-compact, where E denotes the closure of E

Proof: Let E ⊂ X with r = sup{‖u‖ : u ∈ Ē}. We consider Kp(I −Q)N(Ē). Since
g : [0,∞) × <n −→ < satisfies Caratheodory’s conditions with respect to L1[0,∞), there
exist a Lebesgue integrable function ϕr such that

|Nu(t)| = |g(t, u(t), u′(t) · · ·u(n−1)(t)| ≤ ϕr(t) a.e. t ∈ (0,∞)



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 41

‖Nu‖1 ≤
∫ ∞

0
ϕr(t)dt = ‖ϕr‖1 (11)

‖QNu‖1 ≤
∫ ∞

0
|QNu(s)|ds =

∫ ∞
0

∣∣∣∣h(t)

[∫ ∞
0

(g(τ, u(τ), u′(τ) . . . u(n−1)(τ))dτ

−
∫ ξ

0

∫ s

0
(g(τ, u(τ), u′(τ) · · ·u(n−1)(τ))dτdA(s)

]∣∣∣∣ dt
≤

∫ ∞
0
|h(t)|

[∫ ∞
0
|g(τ, u(τ), u′(τ) . . . u(n−1)(τ))|dτ

+

∫ ξ

0

∫ s

0
|g(τ, u(τ), u′(τ) . . . u(n−1)(τ))|dτdA(s)

]
dt

≤
∫ ∞

0
|h(t)

[∫ ∞
0

ϕr(τ)dτ +

∫ ξ

0

∫ s

0
ϕ(τ)dτdA(s)

]
dt

≤ ‖h‖1[‖ϕr‖1 + ‖ϕr‖1A(ξ)] = 2‖ϕr‖1‖h‖1 (12)

This shows that QN(Ē) is bounded. We now apply Lemma 2.1 to prove the compactness
of Kp(I −Q)N(Ē). Let u ∈ Ē then from the definition of Kp(I −Q)N(u) together with
(10), (11) and (12) we derive

‖Kp(I −Q)Nu‖ ≤ Dn‖(I −Q)Nu‖1 ≤ Dn‖Nu‖1 + ‖QNu‖1
≤ Dn‖ϕr‖1 + 2‖ϕr‖1‖h‖1 (13)

Kp(I−Q)N(Ē) is therefore uniformly bounded in X. Let u ∈ E and t1, t2 ∈ [0, T ], t1 < t2
with T ∈ (0,∞). We prove that Kp(I − Q)N(Ē) is equicontinuous on every compact
subset [0, T ] of [0,∞). We have

|e−t2(Kp(I −Q)Nu)(i)(t2)− e−t1(Kp(I −Q)Nu)(i)(t1)|, 0 ≤ i ≤ n− 2

=

∣∣∣∣∫ t2

t1

[e−s(Kp(I −Q)Nu)(i)(s)]′ds

∣∣∣∣
=

∣∣∣∣∫ t2

t1

[−e−s(Kp(I −Q)Nu)(i)(s) + e−s(Kp(I −Q)Nu)i+1(s)]ds

∣∣∣∣
≤ 2| (t1 − t2) |‖Kp(I −Q)Nu‖ ≤ 2|t1 − t2|[Dn‖ϕr‖1 + 2‖ϕr‖1‖h‖1]

−→ 0 as t1 → t2

For i = n− 1, we obtain∣∣∣e−t2(Kp(I −Q)Nu)(n−1)(t2)− e−t1(Kp(I −Q)Nu)(n−1)(t1)
∣∣∣

=

∣∣∣∣e−t2 ∫ t2

0
(I −Q)Nu(s)ds− e−t1

∫ t1

0
(I −Q)Nu(s)ds

∣∣∣∣
≤
∫ t2

0

(
e−t1 − e−t2

) ∣∣∣∣(I −Q)Nu(s)|ds+

∫ t2

t1

e−t1
∣∣∣∣ (I −Q)Nu(s)|ds]



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 42

−→ 0 las t1 → t2

It follows that Kp(I −Q)N(Ē) is equicontinuous on every compact subset of [0,∞).
Next, we prove that Kp(I −Q)N(Ē) is equicontinuous at infinity. For u ∈ Ē we have

|e−t(Kp(I −Q)Nu)(t)| =

∣∣∣∣ e−t

(n− 1)!

∫ t

0
(t− s)n−1(I −Q)Nu(s)ds

− e−t

(n− 1)!

∫ ξ

0
(t− s)n−1(I −Q)Nu(s)ds

∣∣∣∣
≤ e−ttn−1

∫ ∞
0
|(I −Q)Nu(s)|ds+ e−ttn−1

∫ ∞
0
|(I −Q)Nu(s)|ds

≤ 2e−ttn−1(‖Nu‖1 + ‖QNu‖1) ≤ 2e−ttn−1[‖ϕr‖+ 2‖ϕr‖1‖h‖1]

−→ 0 as t→∞.

For i = 1, 2, . . . , n− 1 we have

|e−t(Kp(I −Q)Nu)(i)(t)| ≤ e−t

(n− 1− i)!

∫ t

0
(t− s)n−1−i|(I −Q)Nu(s)|ds

≤ e−ttn−1−i
∫ t

0
|(I −Q)Nu(s)|ds

≤ e−ttn−1−i‖(I −Q)Nu‖1
≤ e−ttn−1−i[‖ϕr‖1 + 2‖ϕr‖1‖h‖1]

−→ 0 as t→∞.

We conclude that Kp(I −Q)N(Ē) is equiconvergence at ∞. �

3. Existence Results

To establish the main existence results, we assume the following conditions

(R1) There exists functions ai(i = 0, 1, . . . , n − 1), b, r ∈ L1[0,∞) and constant θ ∈ [0, 1)
where ai, b, r : [0,∞) −→ [0,∞) are such that for all (u0, u1, . . . , un−1) ∈ <n, the
following inequality is satisfied

|g(t, u0(t) · · ·un−1(t)| ≤ e−t
(
n−1∑
i=0

ai(t)|ui(t)|+ b(t)|un−1(t)|θ
)

+ r(t) (14)

(R2) There exists a constant B1 > 0 such that for u ∈ domL if u(n−1)(t) > B1 for all
t ∈ [0,∞) we have QNu 6= 0

(R3) There exists a constant B2 > 0 such that for u(t) = dtn−1 ∈ kerL, d ∈ < with
|d| > B2

(n−1)! then either



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 43

d · h(t)

[∫ ∞
0

g
(
v, dvn−1, d(n− 1)vn−2 · · · (n− 1)!d

)
dv

−
∫ ξ

0

∫ s

0
g
(
τ, dτn−1, d(n− 1)τn−2 · · · (n− 1)!d

)
dτdA(s) > 0

(15)

or

d · h(t)
[∫∞

0 g
(
v, dvn−1, d(n− 1)vn−2 · · · (n− 1)!d

)
dv

−
∫ ξ

0

∫ s

0
g
(
τ, dτn−1, d(n− 1)τn−2 · · · (n− 1)!d

)
dτdA(s) < 0

(16)

Theorem 3.1: If (R1) − (R3) hold, then the boundary value problem (1)-(2) has at
least one solution in Cn−1[0,∞) provided

n−1∑
i=0

‖ai‖1 <
1

2Dn
(17)

Proof: Our goal is to construct an open bounded set Ω ⊂ X that satisfies assumption
(1)-(3) of theorem 2.2.
Let Ω1 = {u ∈ domL\ kerL, Lu = λNu for λ ∈ (0, 1]}.
For u ∈ Ω1, u /∈ kerL and therefore Nu ∈ ImL = kerQ. Thus QNu = 0 and by (R2)
there exist t0 ∈ [0,∞) such that |u(n−1)(t0)| ≤ B1. We have

|u(n−1)(0)| =

∣∣∣∣u(n−1)(t0)−
∫ t0

0
u(n)(s)ds

∣∣∣∣ ≤ B1 +

∫ ∞
0
|Nu(s)|ds

= B1 + ‖Nu‖1 (18)

For u ∈ Ω1, u ∈ domL\ kerL and hence (I − P )u ∈ domL ∩ kerP with LPu = 0. Thus
from (10) we obtain

‖(I − P )u‖ = ‖KpL(I − P )u‖ ≤ Dn‖L(I − P )u‖1 ≤ Dn‖Lu‖1 ≤ Dn‖Nu‖1 (19)

Using (18) and (19) we get

‖u‖ = ‖Pu+ (I − P )u‖ ≤ ‖Pu‖+ ‖‖(I − Pu)‖ ≤ Dn|u(n−1)(0)|+Dn‖Nu‖1
= Dn(B1 + ‖Nu‖1) +Dn‖Nu‖1
= DnB1 + 2Dn‖Nu‖1 (20)



Samuel A. Iyase, Abiodun A. Opanuga / Eur. J. Pure Appl. Math, 13 (1) (2020), 33-47 44

Using (14) we have

‖Nu‖1 =

∫ ∞
0
|g(s, u(s) · · ·un−1(s)|ds

≤
n−1∑
i=0

∫ ∞
0
|ai|e−s|u(i)(s)|ds+

∫ ∞
0
|b(s)|e−s|un−1(s)|θds+

∫ ∞
0
|r(s)|ds

≤
n−1∑
i=0

‖ai‖1, ‖u‖+ ‖b‖1, ‖u‖θ + ‖r‖1 (21)

From (20) we derive

‖u‖ ≤ DnB1 + 2Dn

[
n−1∑
i=0

‖ai‖1‖u(i)‖∞|‖b‖1‖u(n−1)‖θ∞ + ‖r‖1

]

≤ DnB1 + 2Dn

[
n−1∑
i=0

‖ai‖1‖u‖+ ‖b‖1‖u‖θ + ‖r‖1

]

i.e.
(

1− 2Dn
∑n−1

i=0 ‖ai‖1
)
‖u‖ ≤ 2Dn‖b‖1‖u‖θ +DnB1 + 2Dn‖r‖1

Since θ ∈ [0, 1) and condition (17), we conclude that there exists constant M > 0 such
that ‖u‖ ≤M . Therefore, Ω1 bounded.
Let Ω2 = {u ∈ kerL : Nu ∈ ImL}.
For u ∈ Ω2, u ∈ kerL = {u ∈ domL : u = dtn−1, d ∈ <, t ∈ [0,∞)} and QNu = 0.
Therefore from (R2) there exist t0 ∈ [0,∞) such that |u(n−1)(t0)| < B1 i.e, (n− 1)!d ≤ B1

which implies that |d| ≤ B1
(n−1)! .

Now for u ∈ Ω2

‖u‖ = |d|max

(
sup

t∈[0,∞)
e−t|(tn−1)(i)|

)
≤ B1Dn <∞ (22)

Therefore Ω2 is bounded in X.
If (15) holds, we set

Ω3 = {u ∈ kerL : λJu+ (1− λ)QNu = 0} (23)

where J is the isomorphism, J : kerL→ ImQ defined by J(dtn−1) = de−t; d ∈ <.
For u ∈ Ω3, u = dtn−1 and from (23) we get
−λJu = (1− λ)QNu

−λde−t = (1− λ)h(t)
[∫∞

0 Nu(v)dv −
∫ ξ

0

∫ s
0 Nu(τ)dτdA(s)

]



REFERENCES 45

If λ = 1, then d = 0. However if |d| > B1
(n−1)! and 0 < λ < 1

then from (15) we obtain

−λd2e−t = (1− λ)h(t)d ·
[∫ ∞

0
Nu(v)dv −

∫ ξ

0

∫ s

0
Nu(τ)dτdA(s)

]
> 0

which is a contradiction.
Similarly if Ω3 = {u ∈ kerL : −λJu+ (1− λ)QNu = 0} we arrive at a similar contradic-
tion using (16). Therefore, Ω3 is bounded.
Let Ω be open and bounded such that ∪3

i=1Ωi ⊂ Ω. It is easily seen that assumptions (1)
and (2) of theorem 2.2 are satisfied. We now verify the third assumption. To do this, we
apply the invariance under a homotopy of the degree. We define

H(u, λ) = ±λJu+ (1− λ)QNu

Since ∪3
i=1Ωi ⊂ Ω, we have that H(u, λ) 6= 0 for u ∈ kerL ∩ ∂Ω. Hence

deg(QN |kerL∩∂Ω, Ω ∩ kerL, 0) = deg(H(0, 1),Ω ∩ kerL, 0)

= deg(±J,Ω ∩ kerL, 0) 6= 0

Therefore by theorem 2.1 Lu = Nu has at least one solution in domL ∩ Ω̄ i.e. (1) - (2)
has at least one solution in X �

4. Conclusion

This paper has established conditions for the existence of solutions for the resonant
boundary value problems (1) - (2); using coincidence degree theory. the results obtained
here are new and complements existing results for higher order boundary value problems
on infinite intervals.

Acknowledgements

Authors are grateful to Covenant University for financial assistance and the reviewers
for their useful comments.

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