Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems
The paper presents existence principles for the nonlocal boundary-value problem $$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$ where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$ is an increasing and odd homeomorphism, $g$ is a Caratheodory functi...
Saved in:
| Date: | 2008 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3153 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509193705881600 |
|---|---|
| author | Staněk, S. Станєк, С. |
| author_facet | Staněk, S. Станєк, С. |
| author_sort | Staněk, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:46:54Z |
| description | The paper presents existence principles for the nonlocal boundary-value problem
$$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$
where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$
is an increasing and odd homeomorphism, $g$ is a Caratheodory function which is either regular or has singularities in its space variables
and $\alpha_k: C^{p-1}[0,T]\rightarrow{\mathbb R}$ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems
$(-1)^n(\phi(u^{(2n-1)}))' = f (t,u,...,u^{(2n-1)}),\quad u^{(2k)}(0) = 0,\quad$
$a_ku^{(2k)}(T) + b_k u^{(2k+1)}(T)=0,\quad 0\leq k\leq n-1$ is given. |
| first_indexed | 2026-03-24T02:37:13Z |
| format | Article |
| fulltext |
UDС 517.9
S. Staněk (Palacký Univ., Czech Republic)
EXISTENCE PRINCIPLES FOR HIGHER ORDER
NONLOCAL BOUNDARY-VALUE PROBLEMS
AND THEIR APPLICATIONS
TO SINGULAR STURM – LIOUVILLE PROBLEMS*
ПРИНЦИПИ IСНУВАННЯ ДЛЯ НЕЛОКАЛЬНИХ
ГРАНИЧНИХ ЗАДАЧ ВИЩОГО ПОРЯДКУ
ТА ЇХ ЗАСТОСУВАННЯ
ДО СИНГУЛЯРНИХ ЗАДАЧ ШТУРМА – ЛIУВIЛЛЯ
The paper presents existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′ =
= g(t, u, . . . , u(p−1)), αk(u) = 0, 1 ≤ k ≤ p − 1, where p ≥ 2, φ : R → R is an increasing and
odd homeomorphism, g is a Carathéodory function which is either regular or has singularities in its space
variables and αk : Cp−1[0, T ] → R is a continuous functional. An application of the existence princi-
ples to singular Sturm – Liouville problems (−1)n(φ(u(2n−1)))′ = f(t, u, . . . , u(2n−1)), u(2k)(0) = 0,
aku(2k)(T ) + bku(2k+1)(T ) = 0, 0 ≤ k ≤ n− 1, is given.
Наведено принципи iснування для нелокальної граничної задачi (φ(u(p−1)))′ = g(t, u, . . . , u(p−1)),
αk(u) = 0, 1 ≤ k ≤ p − 1, де p ≥ 2, φ : R → R — гомеоморфiзм, що зростає i є непарним, g —
функцiя Каратеодорi, що або є регулярною, або має особливостi за своїми просторовими змiнними,
а αk : Cp−1[0, T ] → R — неперервний функцiонал. Показано застосування принципiв iснування
до сингулярних задач Штурма – Лiувiлля (−1)n(φ(u(2n−1)))′ = f(t, u, . . . , u(2n−1)), u(2k)(0) = 0,
aku(2k)(T ) + bku(2k+1)(T ) = 0, 0 ≤ k ≤ n− 1.
1. Introduction. Let T > 0 and let R− = (−∞, 0), R+ = (0,∞) and R0 = R \ {0}.
As usual, Cj [0, T ] denotes the set of functions having the jth derivative continuous on
[0, T ]. AC[0, T ] and L1[0, T ] is the set of absolutely continuous functions on [0, T ] and
Lebesgue integrable functions on [0, T ], respectively. C0[0, T ] and L1[0, T ] is equipped
with the norm
‖x‖ = max
{
|x(t)| : t ∈ [0, T ]
}
and ‖x‖L =
T∫
0
|x(t)| dt,
respectively.
Assume that G ⊂ Rp, p ≥ 2. Car
(
[0, T ] × G
)
stands for the set of functions
f : [0, T ]×G→ R satisfying the local Caratéodory conditions on [0, T ]×G, that is: (i) for
each (x0, . . . , xp−1) ∈ G, the function f(·, x0, . . . , xp−1) : [0, T ] → R is measurable;
(ii) for a.e. t ∈ [0, T ], the function f(t, ·, . . . , ·) : G → R is continuous; (iii) for each
compact set K ⊂ G, sup{|f(t, x0, . . . , xp−1)| : (x0, . . . , xp−1) ∈ K} ∈ L1[0, T ].
Let p ∈ N, p ≥ 2. Denote by A the set of functionals α : Cp−1[0, T ] → R which are
(a) continuous and
(b) bounded, that is, α(Ω) is bounded for any bounded Ω ⊂ Cp−1[0, T ].
*Supported by grant No. A100190703 of the Grant Agency of the Academy of Science of the Czech
Republic and by the Council of Czech Government MSM 6198959214.
c© S. STANĚK, 2008
240 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 241
Let φ : R → R is an increasing and odd homeomorphism and let either g ∈ Car([0, T ]×
×Rp) or g ∈ Car([0, T ]×D∗), D∗ ⊂ Rp and has singularities only at the value 0 of its
space variables. Consider the nonlocal boundary-value problem(
φ(u(p−1))
)′ = g(t, u, . . . , u(p−1)), (1.1)
αk(u) = 0, αk ∈ A, 0 ≤ k ≤ p− 1, (1.2)
where αk satisfy a compatibility condition that for each µ ∈ [0, 1] there exists a solution
of the problem
(φ(u(p−1)))′ = 0, αk(u)− µαk(−u) = 0, 0 ≤ k ≤ p− 1.
This problem is equivalent to the fact that the system
αk
(
p−1∑
i=0
Ait
i
)
− µαk
(
−
p−1∑
i=0
Ait
i
)
= 0, 0 ≤ k ≤ p− 1, (1.3)
has a solution (A0, . . . , Ap−1) ∈ Rp for each µ ∈ [0, 1].
We say that u ∈ Cp−1[0, T ] is a solution of problem (1.1), (1.2) if φ(u(p−1)) ∈
∈ AC[0, T ], u satisfies (1.2) and fulfils
(
φ(u(p−1)(t))
)′ = g
(
t, u(t), . . . , u(p−1)(t)
)
for
a.e. t ∈ [0, T ].
The aim of this paper is
1) to present existence principles for problem (1.1), (1.2) in a regular and a singular
case and
2) to give an application of these existence principles to singular Sturm – Liouville
boundary-value problems.
Notice that our existence principles stand a generalization of those obtained for
second-order differential equations with φ-Laplacian in [1, 2].
Our Sturm – Liouville problem consisting of the differential equation
(−1)n
(
φ(u(2n−1))
)′ = f(t, u, . . . , u(2n−1)) (1.4)
and the boundary conditions
u(2k)(0) = 0, aku
(2k)(T ) + bku
(2k+1)(T ) = 0, 0 ≤ k ≤ n− 1. (1.5)
Here n ≥ 2, φ : R → R is an increasing homeomorphism, f ∈ Car([0, T ] × D) is
positive where
D =
R+ × R0 × R− × R0 × . . .× R+ × R0︸ ︷︷ ︸
4`−2
if n = 2`− 1,
R+ × R0 × R− × R0 × . . .× R− × R0︸ ︷︷ ︸
4`
if n = 2`,
f may be singular at the value 0 of all its space variables and
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
242 S. STANĚK
ak > 0, bk > 0, akT + bk = 1 for 0 ≤ k ≤ n− 1. (1.6)
We say that a function u ∈ C2n−1[0, T ] is a solution of problem (1.4), (1.5) if
φ(u(2n−1)) ∈ AC[0, T ], u satisfies the boundary conditions (1.5) and fulfils the equality
(−1)n
(
φ(u(2n−1)(t))
)′ = f
(
t, u(t), . . . , u(2n−1)(t)
)
for a.e. t ∈ [0, T ].
Singular problems of the Sturm – Liouville type for higher order differential equations
were considered in [3 – 5]. In [3] the authors discuss the differential equation u(n) +
+ h1(t, u, . . . , u(n−2)) = 0 together with the boundary conditions
u(j)(0) = 0, 0 ≤ j ≤ n− 3,
αu(n−2)(0)− βu(n−1)(0) = 0, γu(n−2)(1) + δu(n−1)(1) = 0,
(1.7)
where αγ+αδ+βγ > 0, β, δ ≥ 0, β+α > 0, δ+γ > 0 and h1 ∈ C0
(
(0, 1)×Rn−1
+
)
is positive. The existence of a positive solution u ∈ Cn−1[0, 1] ∩ Cn(0, 1) is proved
by a fixed point theorem for mappings that are decreasing with respect to a cone in a
Banach space. Paper [4] deals with the problem u(n) + h2(t, u, . . . , u(n−1)) = 0, (1.7),
where h2 ∈ Car
(
[0, T ]×D∗
)
, D∗ = Rn−1
+ ×R0, is positive. The existence of a positive
solution u ∈ ACn−1[0, T ] is proved by a combination of regularization and sequential
techniques with a Fredholm type existence theorem. In [5], by constructing some special
cones and using a Krasnoselskii fixed point on a cone, the existence of a positive solution
u ∈ C4n−2[0, 1] ∩ C4n(0, 1) is proved for problem u(4n) = h3(t, u, u(4n−2)), u(0) =
= u(1) = 0, au(2k)(0)−bu(2k+1)(0) = 0, cu(2k)(1)+du(2k+1)(1) = 0, 1 ≤ k ≤ 2n−1.
Here h3 ∈ C
(
[0, 1]×R+×R−
)
is nonnegative, a, b, c, d are nonnegative constants and
ac+ ad+ bc > 0.
To the best our knowledge, there is no paper considering singular problems of the
Sturm – Liouville type in our generalization (1.4), (1.5). In addition, any solution u of
problem (1.4), (1.5) has the maximal smoothness, u and its even derivatives (≤ 2n− 2)
‘start’ at the singular points of f and its odd derivatives (≤ 2n − 1) ‘go throughout’
singularities of f somewhere inside of [0, T ].
Throughout the paper we work with the following conditions on the functions φ and
f in equation (1.4):
(H1) φ : R → R is an increasing and odd homomorphism such that φ(R) = R,
(H2) f ∈ Car([0, T ]×D) and there exists a > 0 such that
a ≤ f(t, x0, . . . , x2n−1)
for a.e. t ∈ [0, T ] and all (x0, . . . , x2n−1) ∈ D,
(H3) f(t, x0, . . . , x2n−1) ≤ h
(
t,
∑2n−1
j=0
|xj |
)
+
∑2n−1
j=0
ωj(|xj |) for a.e. t ∈
∈ [0, T ] and all (x0, . . . , x2n−1) ∈ D, where h ∈ Car([0, T ] × [0,∞)) is positive and
nondecreasing in the second variable, ωj : R+ → R+ is nonincreasing,
lim sup
v→∞
1
φ(v)
T∫
0
h(t, 2n+Kv) dt < 1 (1.8)
with
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 243
K =
2n if T = 1,
T 2n − 1
T − 1
if T 6= 1,
(1.9)
and
1∫
0
ω2n−1(φ−1(s)) ds <∞,
1∫
0
ω2j(s) ds <∞ for 0 ≤ j ≤ n− 1,
1∫
0
ω2j+1(s2) ds <∞ for 0 ≤ j ≤ n− 2.
Remark 1.1. If φ satisfies (H1) then φ(0) = 0. Under assumption (H3) the functi-
ons ω2n−1(φ−1(s)), ω2j(s), 0 ≤ j ≤ n− 1, and ω2i+1(s2), 0 ≤ i ≤ n− 2, are locally
Lebesgue integrable on [0,∞) since ωk, 0 ≤ k ≤ 2n− 1, is nonincreasing and positive
on R+.
The rest of the paper is organized as follows. In Section 2, we present existence
principles for a regular and a singular problem (1.1), (1.2). The regular existence
principle is proved by the Leray – Schauder degree (see, e.g., [6]). An application of
both principles is given in Section 3 to the Sturm – Liouville problem (1.4), (1.5).
2. Existence principles. The following result states conditions for solvability of
problem (1.1), (1.2) where g in equation (1.1) is regular.
Theorem 2.1. Let (H1) hold. Let g ∈ Car([0, T ]×Rp) and ϕ ∈ L1[0, T ]. Suppose
that there exists a positive constant L independent of λ such that
‖u(j)‖ < L, 0 ≤ j ≤ p− 1,
for all solutions u of the differential equations
(φ(u(p−1)))′ = (1− λ)ϕ(t), λ ∈ [0, 1], (2.1)
(φ(u(p−1)))′ = λg(t, u, . . . , u(p−1)) + (1− λ)ϕ(t), λ ∈ [0, 1], (2.2)
satisfying the boundary conditions (1.2). Also assume that there exists a positive constant
Λ such that
|Aj | < Λ, 0 ≤ j ≤ p− 1, (2.3)
for all solutions (A0, . . . , Ap−1) ∈ Rp of system (1.3) with µ ∈ [0, 1].
Then problem (1.1), (1.2) has a solution u ∈ Cp−1[0, T ], φ(u(p−1)) ∈ AC[0, T ].
Proof. Let
Ω =
{
x ∈ Cp−1[0, T ] : ‖x(j)‖ < max{L,ΛK1} for 0 ≤ j ≤ p− 1
}
,
where
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
244 S. STANĚK
K1 =
p if T = 1,
T p − 1
T − 1
if T 6= 1.
Then Ω is an open and symmetric with respect to 0 ∈ Cp−1[0, T ] subset of the Banach
space Cp−1[0, T ]. Define an operator P : [0, 1]× Ω → Cp−1[0, T ] by the formula
P(ρ, x)(t) =
t∫
0
(t− s)p−2
(p− 2)!
φ−1
φ(x(p−1)(0) + αp−1(x)) +
s∫
0
V (ρ, x)(v) dv
ds+
+
p−2∑
j=0
x(j)(0) + αj(x)
j!
tj (2.4)
where V (ρ, x)(t) = ρg(t, x(t), . . . , x(p−1)(t)) + (1− ρ)ϕ(t). It follows from the conti-
nuity of φ and αj , 0 ≤ j ≤ p − 1, g ∈ Car([0, T ] × Rp) and from the Lebesgue
dominated convergence theorem that P is a continuous operator. We now prove that
P
(
[0, T ]×Ω
)
is relatively compact in Cp−1[0, T ]. Notice that the boundedness of Ω in
Cp−1[0, T ] guarantees the existence of a positive constant r and a ψ ∈ L1[0, T ] such
that |αk(x)| ≤ r and
∣∣g(t, x(t), . . . , x(p−1)(t))
∣∣ ≤ ψ(t) for a.e. t ∈ [0, T ] and all x ∈ Ω,
0 ≤ k ≤ p− 1. Then
∣∣(P(ρ, x))(j)(t)
∣∣ ≤ (r + max{L,ΛK1}
) p−j−2∑
i=0
T i
i!
+
+
T p−j−1
(p− j − 2)!
φ−1
(
φ(r + max{L,ΛK1}
)
+ ‖ψ‖L + ‖ϕ‖L),
∣∣(P(ρ, x))(p−1)(t)
∣∣ ≤ φ−1
(
φ
(
r + max{L,ΛK1}
)
+ ‖ψ‖L + ‖ϕ‖L
)
,
∣∣∣φ((P(ρ, x))(p−1)(t2))− φ((P(ρ, x))(p−1)(t1))
∣∣∣ ≤
∣∣∣∣∣∣
t2∫
t1
(ψ(s) + |ϕ(s)|) ds
∣∣∣∣∣∣
for t, t1, t2 ∈ [0, T ], (ρ, x) ∈ [0, T ] × Ω and 0 ≤ j ≤ n − 2. Hence P
(
[0, T ] ×
× Ω
)
is bounded in Cp−1[0, T ] and the set {φ((P(ρ, x))(p−1)) : (ρ, x) ∈ [0, 1] × Ω}
is equicontinuous on [0, T ]. Since φ : R → R is increasing and continuous, the set{
(P(ρ, x))(p−1) : (ρ, x) ∈ [0, 1] × Ω
}
is equicontinuous on [0, T ] too. Now, by the
Arzelà – Ascoli theorem, P([0, 1] × Ω ) is relatively compact in Cp−1[0, T ]. We have
proved that P is a compact operator.
Suppose that x∗ is a fixed point of the operator P(1, ·). Then
x∗(t) =
p−2∑
j=0
x
(j)
∗ (0) + αj(x∗)
j!
tj +
t∫
0
(t− s)p−2
(p− 2)!
φ−1×
×
φ(x(p−1)
∗ (0) + αp−1(x∗)) +
s∫
0
g(v, x∗(v), . . . , x
(p−1)
∗ (v))dv
ds
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 245
for t ∈ [0, T ]. Hence αk(x∗) = 0 for 0 ≤ k ≤ p−1 and x∗ is a solution of equation (1.1).
Consequently, x∗ is a solution of problem (1.1), (1.2). In order to prove the assertion of
our theorem it suffices to show that
deg (I − P(1, ·),Ω, 0) 6= 0 (2.5)
where “deg” stands for the Leray – Schauder degree and I is the identical operator on
Cp−1[0, T ]. To show this let the compact operator K : [0, 2]×Ω → Cp−1[0, T ] be defied
by
K(µ, x)(t) =
p−1∑
j=0
[
x(j)(0) + αj+1(x)− (1− µ)αj(−x)
] tj
j!
if µ ∈ [0, 1],
t∫
0
(t− s)p−2
(p− 2)!
φ−1
(
φ(x(p−1)(0) + αp−1(x))+
+(µ− 1)
s∫
0
ϕ(v) dv
)
ds+
p−2∑
j=0
x(j)(0) + αj(x)
j!
tj if µ ∈ (1, 2].
Then K(0, ·) is odd (that is K(0,−x) = −K(0, x) for x ∈ Ω) and
K(2, x) = P(0, x) for x ∈ Ω. (2.6)
Assume that K(µ0, u0) = u0 for some (µ0, u0) ∈ [0, 1]× Ω. Then
u0(t) =
p−1∑
j=0
[
u
(j)
0 (0) + αj(u0)− (1− µ0)αj(−u0)
] tj
j!
, t ∈ [0, T ],
and therefore u0(t) =
∑p−1
j=0
Ãj
tj
j!
where Ãj = u
(j)
0 (0) + αj(u0)− (1− µ0)αj(−u0).
Consequently, u(j)
0 (0) = Ãj and so αj(u0)− (1− µ0)αj(−u0) = 0 for 0 ≤ j ≤ p− 1,
which means
αk
p−1∑
j=0
Ãj
tj
j!
− (1− µ0)αk
− p−1∑
j=0
Ãj
tj
j!
= 0, 0 ≤ k ≤ p− 1.
Then, by our assumption,
∣∣∣∣ Ãj
j!
∣∣∣∣ < Λ for 0 ≤ j ≤ p− 1 and we have
∥∥u(j)
0
∥∥ < Λ
p−1∑
j=0
T j = ΛK1, 0 ≤ j ≤ p− 1.
Hence u0 6∈ ∂Ω and therefore, by the Borsuk antipodal theorem and the homotopy
property,
deg (I − K(0, ·),Ω, 0) 6= 0 (2.7)
and
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
246 S. STANĚK
deg (I − K(0, ·),Ω, 0) = deg (I − K(1, ·),Ω, 0). (2.8)
We come to show that
deg (I − K(1, ·),Ω, 0) = deg (I − K(2, ·),Ω, 0). (2.9)
If K(µ1, u1) = u1 for some (µ1, u1) ∈ (1, 2]× Ω then
u1(t) =
p−2∑
j=0
u
(j)
1 (0) + αj(u1)
j!
tj+
+
t∫
0
(t− s)p−2
(p− 2)!
φ−1
φ(u(p−1)
1 (0) + αp−1(u1)) + (µ1 − 1)
s∫
0
ϕ(v) dv
ds
for t ∈ [0, T ]. Hence u1 satisfies the boundary conditions (1.2) and u1 is a solution of the
differential equation (2.1) with λ = 2−µ1 ∈ [0, 1). By our assumptions, ‖u(j)
1 ‖ < L for
0 ≤ j ≤ p−1. Therefore u1 6∈ ∂Ω and equality (2.9) follows from the homotopy property.
Finally, suppose that P(ρ̃, ũ) = ũ for some (ρ̃, ũ) ∈ [0, 1] × Ω. Then ũ is a solution of
problem (2.2), (1.2) with λ = ρ̃ and therefore ‖ũ(j)‖ < L for 0 ≤ j ≤ p − 1. Hence
ũ 6∈ ∂Ω and, by the homotopy property, deg (I−P(0, ·),Ω, 0) = deg (I−P(1, ·),Ω, 0).
From this and from (2.6) – (2.9) it follows that (2.5) holds, which completes the proof.
Remark 2.1. If functional αk ∈ A is linear for 0 ≤ k ≤ p − 1 then system (1.3)
has the form
p−1∑
j=0
Ajαk(tj) = 0, 0 ≤ k ≤ p− 1.
All of its solutions (A0, . . . , Ap−1) ∈ Rp are bounded exactly if det (αk(tj))p−1
k,j=0 6= 0
(and then Aj = 0 for 0 ≤ j ≤ p − 1), which is equivalent to the fact that problem(
φ(u(p−1))
)′ = 0, (1.2) has only the trivial solution.
If the function g ∈ Car([0, T ] × D∗), D∗ ⊂ Rp in equation (1.1) has singularities
only at the value 0 of its space variables, then the following result for the solvability of
problem (1.1), (1.2) holds.
Theorem 2.2. Let condition (H1) hold. Let g ∈ Car([0, T ] × D∗), D∗ ⊂ Rp,
have singularities only at the value 0 of its space variables. Let the function gm ∈
∈ Car
(
[0, T ]× Rp
)
in the differential equation(
φ(u(p−1))
)′ = gm(t, u, . . . , u(p−1)) (2.10)
satisfy
0 ≤ νgm(t, x0, . . . , xp−1) ≤ q
(
t, |x0|, . . . , |xp−1|
)
for a.e. t ∈ [0, T ] and all (x0, . . . , xp−1) ∈ Rp
0, m ∈ N,
where q ∈ Car([0, T ]× Rp
+) and ν ∈ {−1, 1}.
(2.11)
Suppose that for each m ∈ N, the regular problem (2.10), (1.2) has a solution um and
there exists a subsequence {ukm} of {um} converging in Cp−1[0, T ] to some u.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 247
Then φ(u(p−1)) ∈ AC[0, T ] and u is a solution of the singular problem (1.1), (1.2)
if u(j) has a finite number of zeros for 0 ≤ j ≤ p− 1 and
lim
m→∞
gkm
(
t, ukm
(t), . . . , u(p−1)
km
(t)
)
= g
(
t, u(t), . . . , u(p−1)(t)
)
(2.12)
for a.e. t ∈ [0, T ].
Proof. Assume that (2.12) holds for a.e. t ∈ [0, T ] and let 0 ≤ ξ1 < . . . < ξ` ≤ T
are all zeros of u(j) for 0 ≤ j ≤ p − 1. Since ‖u(j)
km
‖ ≤ L for each m ∈ N and
0 ≤ j ≤ p− 1, where L is a positive constant, it follows that
T∫
0
νgkm
(
t, ukm
(t), . . . , u(p−1)
km
(t)
)
dt = ν
[
φ
(
u
(p−1)
km
(T )
)
− φ
(
u
(p−1)
km
(0)
)]
≤ 2φ(L)
for m ∈ N. Now (2.11), (2.12) and the Fatou lemma [7, 8] give
T∫
0
νg(t, u(t), . . . , u(p−1)(t)) dt ≤ 2φ(L).
Hence νg
(
t, u(t), . . . , u(p−1)(t)
)
∈ L1[0, T ] and so g
(
t, u(t), . . . , u(p−1)(t)
)
∈ L1[0, T ].
Put ξ0 = 0 and ξ`+1 = T. We show that the equality
φ(u(p−1)(t)) = φ
(
u(p−1)
(
ξi+1 + ξi
2
))
+
t∫
(ξi+1+ξi)/2
g(s, u(s), . . . , u(p−1)(s)) ds
(2.13)
is satisfied on [ξi, ξi+1] for each i ∈ {0, . . . , `} such that ξi < ξi+1. Indeed, let i ∈
∈ {0, . . . , `}, ξi < ξi+1. Choose an arbitrary ρ ∈
(
0,
ξi+1 + ξi
2
)
and let us look at the
interval [ξi + ρ, ξi+1 − ρ]. We know that |u(j)| > 0 on (ξi, ξi+1) for 0 ≤ j ≤ p− 1 and
therefore |u(j)(t)| ≥ ε for t ∈ [ξi + ρ, ξi+1− ρ] and 0 ≤ j ≤ p− 1 where ε is a positive
constant. Hence there exists m0 ∈ N such that
∣∣u(j)
km
(t)
∣∣ ≥ ε
2
for t ∈ [ξi + ρ, ξi+1 − ρ],
0 ≤ j ≤ p− 1 and m ≥ m0. This gives (see (2.11))∣∣gkm
(t, ukm
(t), . . . , u(p−1)
km
(t))
∣∣ ≤
≤ sup
{
q(t, x0, . . . , xp−1) : t ∈ [0, T ], xj ∈
[
ε
2
, L
]
for 0 ≤ j ≤ p− 1
}
∈ L1[0, T ]
for a.e. t ∈ [ξi + ρ, ξi+1 − ρ] and all m ≥ m0. Letting m→∞ in
φ
(
u
(p−1)
km
(t)
)
= φ
(
u
(p−1)
km
(
ξi+1 + ξi
2
))
+
+
t∫
(ξi+1+ξi)/2
gkm
(
s, ukm(s), . . . , u(p−1)
km
(s)
)
ds
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
248 S. STANĚK
yields (2.13) for t ∈ [ξi +ρ, ξi+1 +ρ] by the Lebesgue dominated convergence theorem.
Since ρ ∈
(
0,
ξi+1 + ξi
2
)
is arbitrary, equality (2.13) holds on the interval (ξi, ξi+1)
and using the fact that g
(
t, u(t), . . . , u(p−1)(t)
)
∈ L1[0, T ], (2.13) is satisfied also at
t = ξi and ξi+1. From equality (2.13) on [ξi, ξi+1] (for 0 ≤ i ≤ `), we deduce that
φ(u(p−1)) ∈ AC[0, T ] and u is a solution of equation (1.1). Finally, it follows from
αj(ukm) = 0 for 0 ≤ j ≤ p − 1 and m ∈ N, and from the continuity of αj that
αj(u) = 0 for 0 ≤ j ≤ p− 1. Consequently, u is a solution of problem (1.1), (1.2).
The theorem is proved.
3. Sturm – Liouville problem. 3.1. Auxiliary results. Throughout the next part
of this paper we assume that numbers ak, bk in the boundary conditions (1.5) fulfil
condition (1.6). For each j ∈ {0, . . . , n − 2}, denote by Gj the Green function of the
Sturm – Liouville problem
−u′′ = 0, u(0) = 0, aju(T ) + bju
′(T ) = 0.
Then
Gj(t, s) =
s(1− ajt) for 0 ≤ s ≤ t ≤ T,
t(1− ajs) for 0 ≤ t < s ≤ T.
Hence Gj(t, s) > 0 for (t, s) ∈ (0, T ] × (0, T ] and Gj(t, s) = Gj(s, t) for (t, s) ∈
∈ [0, T ]× [0, T ]. Put G[1](t, s) = Gn−2(t, s) for (t, s) ∈ [0, T ]× [0, T ] and define G[j]
recurrently by the formula
G[j](t, s) =
T∫
0
Gn−j−1(t, v)G[j−1](v, s) dv, (t, s) ∈ [0, T ]× [0, T ], (3.1)
for 2 ≤ j ≤ n− 1. It follows from the definition of the function G[j] that the equalities
u(2n−2j)(t) = (−1)j−1
T∫
0
G[j−1](t, s)u(2n−2)(s) ds, 2 ≤ j ≤ n, (3.2)
are true on [0, T ] for each u ∈ C2n−2[0, T ] satisfying the boundary conditions (1.5).
Lemma 3.1. For 1 ≤ j ≤ n− 1, the inequality
G[j](t, s) ≥ T 2j−3(1− αT )j
3j−1
ts for (t, s) ∈ [0, T ]× [0, T ] (3.3)
holds where
α = max{ak : 0 ≤ k ≤ n− 2}
(
<
1
T
)
. (3.4)
Proof. Since
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 249
Gj(t, s) =
s(1− ajt) ≥ s(1− ajT ) for 0 ≤ s ≤ t ≤ T,
t(1− ajs) ≥ t(1− ajT ) for 0 ≤ t < s ≤ T
for 0 ≤ j ≤ n−2,we haveGj(t, s) ≥
1− ajT
T
st ≥ 1− αT
T
st for (t, s) ∈ [0, T ]×[0, T ]
and 0 ≤ j ≤ n − 2. Consequently, G[1](t, s) = Gn−2(t, s) ≥
1− αT
T
st for (t, s) ∈
∈ [0, T ] × [0, T ] and therefore inequality (3.3) is true for j = 1. We now proceed by
induction. Assume that (3.3) is true for j = i (< n− 1). Then
G[i+1](t, s) =
T∫
0
Gn−i−2(t, v)G[i](v, s) dv ≥
≥
T∫
0
1− αT
T
tv
T 2i−3(1− αT )i
3i−1
vs dv =
=
T 2i−4(1− αT )i+1
3i−1
ts
T∫
0
v2ds =
T 2i−1(1− αT )i+1
3i
ts
for (t, s) ∈ [0, T ]× [0, T ]. Therefore (3.3) is true with j = i+ 1.
The lemma is proved.
Let φ satisfy (H1). Choose an arbitrary a > 0 and put
Ba =
{
u ∈ C2n−1[0, T ] : φ(u(2n−1)) ∈ AC[0, T ], (−1)n
(
φ(u(2n−1)(t))
)′ ≥ a
for a.e. t ∈ [0, T ] and u satisfies (1.5)
}
. (3.5)
The properties of functions belonging to the set Ba are given in the following lemma.
Lemma 3.2. Let u ∈ Ba. Then there exists {ξ2j+1}n−1
j=0 ⊂ (0, T ) such that
u(2j+1)(ξ2j+1) = 0, 0 ≤ j ≤ n− 1, (3.6)
and
∣∣u(2n−1)(t)
∣∣ ≥ φ−1
(
a|t− ξ2n−1|
)
, (3.7)
∣∣u(2n−2j+1)(t)
∣∣ ≥ T 2j−4S
2 · 3j−2
(1− αT )j−2(t− ξ2n−2j+1)2, 2 ≤ j ≤ n, (3.8)
(−1)n+ju(2n−2j)(t) ≥ T 2j−2S
3j−1
(1− αT )j−1t, 1 ≤ j ≤ n, (3.9)
for t ∈ [0, T ], where
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
250 S. STANĚK
S =
1
T
min
bn−1
T/2∫
0
φ−1(at) dt,
bn−1
an−1
φ−1
(
aT
2
) (3.10)
and α is given in (3.4).
Proof. Since φ is increasing and
(
φ((−1)nu(2n−1)(t))
)′ = (−1)n
(
φ(u(2n−1)(t))
)′ ≥
≥ a for a.e. t ∈ [0, T ], it follows that (−1)nu(2n−1) is increasing on [0, T ] and
(−1)n−1u(2n−2) is concave on this interval. If u(2n−1)(t) 6= 0 for t ∈ (0, T ), then∣∣an−1u
(2n−2)(T ) + bn−1u
(2n−1)(T )
∣∣ =
=
∣∣∣∣∣∣an−1
T∫
0
u(2n−1)(t)dt+ bn−1u
(2n−1)(T )
∣∣∣∣∣∣ > 0,
contrary to an−1u
(2n−2)(T ) + bn−1u
(2n−1)(T ) = 0 by (1.5) with k = n − 1. Hence
u(2n−1)(ξ2n−1) = 0 for a unique ξ2n−1 ∈ (0, T ). Now integrating the equality(
φ((−1)nu(2n−1)(t))
)′ ≥ a over [t, ξ2n−1] and [ξ2n−1, t] gives
(−1)n−1u(2n−1)(t) ≥ φ−1
(
a(ξ2n−1 − t)
)
, t ∈ [0, ξ2n−1], (3.11)
(−1)nu(2n−1)(t) ≥ φ−1
(
a(t− ξ2n−1)
)
, t ∈ [ξ2n−1, T ], (3.12)
which shows that (3.7) holds. In order to prove inequality (3.9) for j = 1 we consider
two cases, namely ξ2n−1 <
T
2
and ξ2n−1 ≥
T
2
.
Case 1. Let ξ2n−1 <
T
2
. Then (see (3.12))
(−1)nu(2n−1)(T ) ≥ φ−1(a(T − ξ2n−1)) > φ−1
(
aT
2
)
,
and therefore (see (1.5) with k = n− 1)
(−1)n−1u(2n−2)(T ) = (−1)n bn−1
an−1
u(2n−1)(T ) >
bn−1
an−1
φ−1
(
aT
2
)
. (3.13)
Case 2. Let ξ2n−1 ≥
T
2
. Then (3.11) yields
(−1)n−1u(2n−2)
(
T
2
)
= (−1)n−1
T/2∫
0
u(2n−1)(t) dt ≥
T/2∫
0
φ−1
(
a(ξ2n−1 − t)
)
dt ≥
≥
T/2∫
0
φ−1
(
a
(
T
2
− t
))
dt =
T/2∫
0
φ−1(at) dt =: L.
Let ε := (−1)nu(2n−1)(T ). We know that (−1)nu(2n−1) is increasing on [0, T ] and
u(2n−1)(ξ2n−1) = 0. Hence ε > 0 and
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 251
(−1)n−1u(2n−2)(t) = (−1)n−1u(2n−2)(ξ2n−1) + (−1)n−1
t∫
ξ2n−1
u(2n−1)(s) ds >
> (−1)n−1u(2n−2)(ξ2n−1)− ε(t− ξ2n−1) ≥
≥ (−1)n−1u(2n−2)
(
T
2
)
− ε(t− ξ2n−1)
for t ∈ (ξ2n−1, T ]. Consequently, (−1)n−1u(2n−2)(T ) > L− ε(T − ξ2n−1) > L− εT.
Then
bn−1
an−1
ε = (−1)n bn−1
an−1
u(2n−1)(T ) = (−1)n−1u(2n−2)(T ) > L − εT, and so
(see (1.6)) ε > L
(
bn−1
an−1
+ T
)−1
= an−1L. It follows that
(−1)n−1u(2n−2)(T ) = (−1)n bn−1
an−1
u(2n−1)(T ) =
bn−1
an−1
ε > bn−1L. (3.14)
Now (3.13) and (3.14) imply that (−1)n−1u(2n−2)(T ) > ST where S is given in
(3.10). This and u(2n−2)(0) = 0 and the fact that (−1)n−1u(2n−2) is concave on [0, T ]
guarantee that (−1)n−1u(2n−2)(t) ≥ St for t ∈ [0, T ], which proves (3.9) for j = 1.
Combining (3.2), (3.3) and (3.9) (with j = 1), we get
(−1)n+ju(2n−2j)(t) = (−1)n−1
T∫
0
G[j−1](t, s)u(2n−2)(s) ds ≥
≥ T 2j−5S
3j−2
(1− αT )j−1t
T∫
0
s2 ds =
T 2j−2S
3j−1
(1− αT )j−1t
for t ∈ [0, T ] and 2 ≤ j ≤ n. We have proved that (3.9) is true.
Since, by (3.9), |u(2n−2j)| > 0 on (0, T ] for 1 ≤ j ≤ n and u satisfies (1.5), essenti-
ally the same reasoning as in the beginning of this prove shows that u(2j+1)(ξ2j+1) = 0
for a unique ξ2j+1 ∈ (0, T ), 0 ≤ j ≤ n− 2. Using (3.9) we obtain
∣∣u(2n−2j+1)(t)
∣∣ =
∣∣∣∣∣∣∣
t∫
ξ2n−2j+1
u(2n−2j+2)(s) ds
∣∣∣∣∣∣∣ ≥
≥ T 2j−4S
3j−2
(1− αT )j−2
∣∣∣∣∣∣∣
t∫
ξ2n−2j+1
s ds
∣∣∣∣∣∣∣ =
=
T 2j−4S
2 · 3j−2
(1− αT )j−2|t2 − ξ22n−2j+1| ≥
T 2j−4S
2 · 3j−2
(1− αT )j−2(t− ξ2n−2j+1)2
for t ∈ [0, T ] and 2 ≤ j ≤ n. Hence (3.8) is true, which finishes the proof.
3.2. Auxiliary regular problems. Let (H2) and (H3) hold. For each m ∈ N,
define χm, ϕm, τm ∈ C0(R) and Rm ⊂ R by the formulas
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
252 S. STANĚK
χm(v) =
v for v ≥ 1
m
,
1
m
for v <
1
m
,
ϕm(v) =
− 1
m
for v > − 1
m
,
v for v ≤ − 1
m
,
τm =
χm if n = 2k − 1,
ϕm if n = 2k,
Rm = R \
(
− 1
m
,
1
m
)
.
Choose m ∈ N and use the function f to define fm ∈ Car
(
[0, T ] × R2n
)
by the
formula
fm(t, x0, x1, x2, x3, . . . , x2n−2, x2n−1) =
=
f(t, χm(x0), x1, ϕm(x2), x3, . . . , τm(x2n−2), x2n−1)
for (t, x0, x1, x2, x3, . . . , x2n−2, x2n−1) ∈
∈ [0, T ]× R× Rm × R× Rm × . . .× R× Rm,
m
2
[
fm
(
t, x0,
1
m
,x2, x3, . . . , x2n−2, x2n−1
)(
x1 +
1
m
)
−
−fm
(
t, x0,−
1
m
,x2, x3, . . . , x2n−2, x2n−1
)(
x1 −
1
m
)]
for (t, x0, x1, x2, x3, . . . , x2n−2, x2n−1) ∈
∈ [0, T ]× R×
[
− 1
m
,
1
m
]
× R× Rm × . . .× R× Rm,
m
2
[
fm
(
t, x0, x1, x2,
1
m
, . . . , x2n−2, x2n−1
)(
x3 +
1
m
)
−
−fm
(
t, x0, x1, x2,−
1
m
, . . . , x2n−2, x2n−1
)(
x3 −
1
m
)]
for (t, x0, x1, x2, x3, . . . , x2n−2, x2n−1) ∈
∈ [0, T ]× R3 ×
[
− 1
m
,
1
m
]
× . . .× R× Rm,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m
2
[
fm
(
t, x0, x1, x2, . . . , x2n−2,
1
m
)(
x2n−1 +
1
m
)
−
−fm
(
t, x0, x1, x2, . . . , x2n−2,−
1
m
)(
x2n−1 −
1
m
)]
for (t, x0, x1, x2, . . . , x2n−2, x2n−1) ∈ [0, T ]× R2n−1 ×
[
− 1
m
,
1
m
]
.
Then conditions (H2) and (H3) give
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 253
a ≤ (1− λ)a+ λfm(t, x0, . . . , x2n−1) (3.15)
for a.e. t ∈ [0, T ] and all (x0, . . . , x2n−1) ∈ R2n, λ ∈ [0, 1], and
(1− λ)a+ λfm(t, x0, . . . , x2n−1) ≤ h
t, 2n+
2n−1∑
j=0
|xj |
+
2n−1∑
j=0
ωj
(
|xj |
)
(3.16)
for a.e. t ∈ [0, T ] and all (x0, . . . , x2n−1) ∈ R2n
0 , λ ∈ [0, 1].
Consider the family of approximate regular differential equations
(−1)n
(
φ(u(2n−1))
)
= λfm(t, u, . . . , u(2n−1)) + (1− λ)a, λ ∈ [0, 1]. (3.17)
Lemma 3.3. Let (H1) – (H3) hold. Then there exists a positive constant W
independent of m ∈ N and λ ∈ [0, 1] such that
‖u(j)‖ < W, 0 ≤ j ≤ 2n− 1, (3.18)
for all solutions u of problem (3.17), (1.5).
Proof. Let u be a solution of problem (3.17), (1.5). Then (−1)n
(
φ(u(2n−1)(t))
)′ ≥
≥ a for a.e. t ∈ [0, T ] by (3.15) and consequently, u ∈ Ba where the set Ba is given
in (3.5). Hence, by Lemma 3.2, u satisfies (3.6) and (3.7) where ξ2j+1 ∈ (0, T ) is the
unique zero of u(2j+1), 0 ≤ j ≤ n− 1, and∣∣u(2n−2j+1)(t)
∣∣ ≥ Qj(t− ξ2n−2j+1)2, 2 ≤ j ≤ n,
(−1)n+iu(2n−2i)(t) ≥ Pit, 1 ≤ i ≤ n,
for t ∈ [0, T ], where
Qj =
T 2j−4S
2 · 3j−2
(1− αT )j−2, Pi =
T 2i−2S
3i−1
(1− αT )i−1 (3.19)
with α and S given in (3.4) and (3.10), respectively. Accordingly,
2n−1∑
j=0
T∫
0
ωj
(
|u(j)(t)|
)
dt ≤
n∑
j=1
T∫
0
ω2n−2j(Pjt) dt+
+
n∑
j=2
T∫
0
ω2n−2j+1
(
Qj(t− ξ2n−2j+1)2
)
dt+
T∫
0
ω2n−1(φ−1(a|t− ξ2n−1|)
)
dt <
<
n∑
j=1
1
Pj
PjT∫
0
ω2n−2j(s) ds+ 2
n∑
j=2
1√
Qj
√
QjT∫
0
ω2n−2j+1(s2) ds+
+
2
aT
aT∫
0
ω2n−1(φ−1(s)) ds =: Λ. (3.20)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
254 S. STANĚK
By (H3), Λ < ∞. Since u(2j)(0) = 0 and u(2j+1)(ξ2j+1) = 0 for 0 ≤ j ≤ n − 1, we
have
‖u(j)‖ ≤ T 2n−j−1‖u(2n−1)‖, 0 ≤ j ≤ 2n− 2. (3.21)
Combining (3.16), (3.20), (3.21) and u(2n−1)(ξ2n−1) = 0, we obtain
φ
(
|u(2n−1)(t)|
)
=
∣∣∣∣∣∣∣
t∫
ξ2n−1
[(1− λ)a+ λfm(s, u(s), . . . , u(2n−1)(s))] ds
∣∣∣∣∣∣∣ <
<
T∫
0
h
t, 2n+
2n−1∑
j=0
|u(j)(t)|
dt+
2n−1∑
j=0
T∫
0
ωj
(
|u(j)(t)|
)
dt <
<
T∫
0
h
t, 2n+ ‖u(2n−1)‖
2n−1∑
j=0
T j
dt+ Λ =
=
T∫
0
h(t, 2n+K‖u(2n−1)‖) dt+ Λ
for t ∈ [0, T ], where K is given in (1.9). Hence
φ
(
‖u(2n−1)‖
)
<
T∫
0
h
(
t, 2n+K‖u(2n−1)‖
)
dt+ Λ. (3.22)
It follows from condition (1.8) that there exists a positive constant W∗ such that∫ T
0
h(t, 2n + Kv) dt < φ(v) whenever v ≥ W∗. This and (3.22) yields ‖u(2n−1)‖ <
< W∗.Consequently, (3.21) shows that (3.18) is fulfilled withW = W∗max
{
1, T 2n−1
}
.
The lemma is proved.
Remark 3.1. Let c > 0. If follows from the proof of Lemma 3.3 that any soluti-
on u of problem (−1)n
(
φ(u(2n−1))
)′ = c, (1.5) satisfies the inequality ‖u(j)‖ <
< φ−1(cT ) max{1, T 2n−1} for 0 ≤ j ≤ 2n− 1.
We are now in a position to show that for each m ∈ N there exists a solution um of
the regular differential equation
(−1)n
(
φ(u(2n−1))
)′ = fm(t, u, . . . , u(2n−1)) (3.23)
satisfying the boundary conditions (1.5).
Lemma 3.4. Let (H1) – (H3) hold. Then for each m ∈ N there exists a solution
um ∈ C2n−1[0, T ], φ(u(2n−1)) ∈ AC[0, T ], of problem (3.23), (1.5) and
‖u(j)
m ‖ < W for m ∈ N and 0 ≤ j ≤ 2n− 1, (3.24)
where W is a positive constant. In addition, the sequence {u(2n−1)
m } is equicontinuous
on [0, T ].
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 255
Proof. Choose an arbitrary m ∈ N. Let W be a positive constant in Lemma 3.3. In
order to prove the existence of a solution of problem (3.23), (1.5) we use Theorem 2.1
with p = 2n, g = (−1)nfm and ϕ = (−1)na in equations (2.1), (2.2) and with
α2k(u) = u(2k)(0), α2k+1(u) = aku
(2k)(T ) + bku
(2k+1)(T ), 0 ≤ k ≤ n− 1,
(3.25)
in the boundary conditions (1.2).
Due to Lemma 3.3 and Remark 3.1, all solutions u of problems (3.17), (1.5) and
(−1)n
(
φ(u(2n−1))
)′ = λa, (1.5) (0 ≤ λ ≤ 1) satisfy inequality (3.18). Moreover, αk
(defined in (3.25)) belongs to the set A (with p = 2n) for 0 ≤ k ≤ 2n− 1. The system
(see (1.3))
αk
(
2n−1∑
i=0
Ait
i
)
− µαk
(
−
2n−1∑
i=0
Ait
i
)
= 0, 0 ≤ k ≤ 2n− 1, (3.26)
has the form (see (3.25))
(1 + µ)
(
2n−1∑
i=0
Ait
i
)(2k) ∣∣∣∣
t=0
= 0, 0 ≤ k ≤ n− 1, (3.27)
(1 + µ)
[
ak
(
2n−1∑
i=0
Ait
i
)(2k) ∣∣∣∣
t=T
+
+bk
(
2n−1∑
i=0
Ait
i
)(2k+1) ∣∣∣∣
t=T
]
= 0, 0 ≤ k ≤ n− 1. (3.28)
It follows from (3.27) that A2k = 0 for 0 ≤ k ≤ n − 1 and then we deduce from
(3.28) and from akT + bk = 1 that A2j+1 = 0 for 0 ≤ j ≤ n − 1. Consequently,
(A0, . . . , A2n−1) = (0, . . . , 0) ∈ R2n is the unique solution of (3.26) for each µ ∈ [0, 1].
Hence all the assumptions of Theorem 2.1 are satisfied and therefore for each m ∈ N,
there exists a solution um ∈ C2n−1[0, T ], φ(u(2n−1)) ∈ AC[0, T ], of problem (3.23),
(1.5) fulfilling inequality (3.24).
It remains to show that the sequence {u(2n−1)
m } is equicontinuous on [0, T ]. Notice
that um ∈ Ba for all m ∈ N where the set Ba is given in (3.5). Then, by Lemma 3.2,
there exists {ξ2j+1,m}n−1
j=0 ⊂ (0, T ), m ∈ N, such that
u(2j+1)
m (ξ2j+1,m) = 0, 0 ≤ j ≤ n− 1, m ∈ N, (3.29)
and ∣∣u(2n−1)
m (t)
∣∣ ≥ φ−1
(
a|t− ξ2n−1,m|
)
,
∣∣u(2n−2j+1)
m (t)
∣∣ ≥ Qj(t− ξ2n−2j+1,m)2, 2 ≤ j ≤ n, (3.30)
(−1)n+ju(2n−2j)
m (t) ≥ Pjt, 1 ≤ j ≤ n,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
256 S. STANĚK
for t ∈ [0, T ] and m ∈ N, where Qj , Pj are given in (3.19). Let 0 ≤ t1 < t2 ≤ T. Then
(see (3.16) with λ = 1, (3.24) and (3.30))∣∣∣φ(u(2n−1)
m (t2)
)
− φ
(
u(2n−1)
m (t1)
)∣∣∣ =
=
t2∫
t1
fm
(
t, um(t), . . . , u(2n−1)
m (t)
)
dt ≤
≤
t2∫
t1
h
t, 2n+
2n−1∑
j=0
‖u(j)
m ‖
dt+
2n−1∑
j=0
t2∫
t1
ωj
(
|u(j)
m (t)|
)
dt ≤
≤
t2∫
t1
h(t, 2n(1 +W )) dt+
t2∫
t1
ω2n−1
(
φ−1(a|t− ξ2n−1,m|
)
dt+
+
n∑
j=2
t2∫
t1
ω2n−2j+1
(
Qj(t− ξ2n−2j+1,m)2
)
dt+
+
n∑
j=1
t2∫
t1
ω2n−2j(Pjt) dt (3.31)
for m ∈ N. By (H3), h(t, 2n(1+W )) ∈ L1[0, T ] and ω2n−1(φ−1(s)), ω2j(s), 0 ≤ j ≤
≤ n − 1, ω2i+1(s2), 0 ≤ i ≤ n − 2, are locally integrable on [0,∞). From these facts
and from (3.31) and from the relations
t2∫
t1
ω2n−1
(
φ−1(a|t− ξ2n−1,m|)
)
dt =
=
1
a
a(ξ2n−1,m−t1)∫
a(ξ2n−1,m−t2)
ω2n−1
(
φ−1(t)
)
dt, if t2 ≤ ξ2n−1,m,
1
a
a(ξ2n−1,m−t1)∫
0
ω2n−1
(
φ−1(t)
)
dt+
+
a(t2−ξ2n−1,m)∫
0
ω2n−1
(
φ−1(t)
)
dt
if t1 < ξ2n−1,m < t2,
1
a
a(t2−ξ2n−1,m)∫
a(t1−ξ2n−1,m)
ω2n−1
(
φ−1(t)
)
dt if ξ2n−1,m ≤ t1,
t2∫
t1
ω2n−2j+1
(
Qj(t− ξ2n−2j+1,m)2
)
dt =
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 257
=
1√
Qj
√
Qj(ξ2n−2j+1,m−t1)∫
√
Qj(ξ2n−2j+1,m−t2)
ω2n−2j+1(t2) dt if t2 ≤ ξ2n−2j+1,m,
1√
Qj
√
Qj(ξ2n−2j+1,m−t1)∫
0
ω2n−2j+1(t2) dt +
+
√
Qj(t2−ξ2n−2j+1,m)∫
0
ω2n−2j+1(t2) dt
if t1 < ξ2n−2j+1,m < t2,
1√
Qj
√
Qj(t2−ξ2n−2j+1,m)∫
√
Qj(t1−ξ2n−2j+1,m)
ω2n−2j+1(t2) dt if ξ2n−2j+1,m ≤ t1,
it follows that
{
φ(u(2n−1)
m )
}
is equicontinuous on [0, T ]. We now deduce the equiconti-
nuity of {u(2n−1)
m } on [0, T ] from the equality∣∣u(2n−1)
m (t2)− u(2n−1)
m (t1)
∣∣ = ∣∣∣φ−1
(
φ(u(2n−1)
m (t2))
)
− φ−1
(
φ(u(2n−1)
m (t1))
)∣∣∣
for 0 ≤ t1 < t2 ≤T, m ∈ N, and the facts that
{
φ(u(2n−1)
m )
}
is bounded in C0[0, T ]
and φ−1 is continuous and increasing on R.
The lemma is proved.
3.3. Existence result and an example. The main result is presented in the following
theorem.
Theorem 3.1. Let (H1) – (H3) hold. Then problem (1.4), (1.5) has a solution u ∈
∈ C2n−1[0, T ], φ(u(2n−1)) ∈ AC[0, T ] and (−1)ku(2k) > 0 on (0, T ], u(2k+1)(ξ2k+1) =
= 0 for 0 ≤ k ≤ n− 1 where ξ2k+1 ∈ (0, T ).
Proof. By Lemma 3.4, for each m ∈ N there exists a solution um of problem (3.23),
(1.5). Consider the sequence {um}. Then inequality (3.24) is satisfied with a positive
constantW and since um ∈ Ba, Lemma 3.2 guarantees the existence of {ξ2j+1,m}n−1
j=0 ⊂
⊂ (0, T ) such that (3.29) and (30) hold for t ∈ [0, T ] and m ∈ N, where Qj and Pj
are given in (3.19). Moreover, the sequence {u2n−1
m } is equicontinuous on [0, T ] by
Lemma 3.4. Hence there exist a subsequence {ukm
} converging in C2n−1[0, T ] and a
subsequence {ξ2j+1,km
}, 1 ≤ j ≤ n − 1, converging in R. Let limm→∞ ukm
= u and
limm→∞ ξ2j+1,km = ξ2j+1, 1 ≤ j ≤ n−1. Letting m→∞ in (3.24), (3.29) and (3.30)
(with km instead of m) yields (for t ∈ [0, T ])∣∣u(2n−1)(t)
∣∣ ≥ φ−1
(
a|t− ξ2n−1|
)
,
u(2j+1)(ξ2j+1) = 0 for 0 ≤ j ≤ n− 1,∣∣u(2n−2j+1)(t)
∣∣ ≥ Qj(t− ξ2n−2j+1)2 for 2 ≤ j ≤ n− 1,
‖u(j)‖ ≤W for 0 ≤ j ≤ 2n− 1
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
258 S. STANĚK
and
(−1)n+ju(2n−2j)(t) ≥ Pjt for 1 ≤ j ≤ n. (3.32)
Hence u(j) has exactly one zero in [0, T ] for 0 ≤ j ≤ 2n− 1 and
lim
m→∞
fkm
(
t, ukm(t), . . . , u(2n−1)
km
(t)
)
=
= f
(
t, u(t), . . . , u(2n−1)(t)
)
for a.e. t ∈ [0, T ].
In addition, by (3.32), (−1)ku(2k) > 0 on (0, T ] and (−1)ku(2k+1)(0) ≥ Pn−k > 0
for 0 ≤ k ≤ n − 1. Hence (−1)ku(2k+1)(T ) < 0 for 0 ≤ k ≤ n − 1 by (1.5), which
combining with (−1)ku(2k+1)(0) > 0 implies ξ2k+1 ∈ (0, T ) for 0 ≤ k ≤ n − 1.
Finally, having in mind the definition of the function fm and inequality (3.16) we have
0 ≤ fm(t, x0, . . . , x2n−1) ≤ q
(
t, |x0|, . . . , |x2n−1|
)
for a.e. t ∈ [0, T ] and all (x0, . . . , x2n−1) ∈ R2n
0
where q(t, x0, . . . , x2n−1) = h
(
t, 2n+
∑2n−1
j=0
xj
)
+
∑2n−1
j=0 ωj(xj) for t ∈ [0, T ]
and (x0, . . . , x2n−1) ∈ R2n
+ . Clearly, q ∈ Car([0, T ] × R2n
+ ). Hence problem (1.4),
(1.5) satisfies the assumptions of Theorem 2.2 with p = 2n, g = (−1)nf, gm = fm
(that is ν = (−1)n in (2.11)) and with the boundary conditions (3.25) which are the
special case of the boundary conditions (1.2). Consequently, Theorem 2.2 guarantees
that φ(u(2n−1)) ∈ AC[0, T ] and u is a solution of problem (1.4), (1.5).
The theorem is proved.
Example 3.1. Let p > 1, α2n−1 ∈ (0, p − 1), α2j ∈ (0, 1) for 0 ≤ j ≤ n − 1,
α2j+1 ∈
(
0,
1
2
)
for 0 ≤ j ≤ n − 2, βk ∈ (0, p − 1), ck > 0, dk ∈ L1[0, T ] for
0 ≤ k ≤ 2n − 1, dk is nonnegative and r ∈ L1[0, T ], r(t) ≥ a > 0 for a.e. t ∈ [0, T ].
Consider the differential equation
(−1)n
(
|u(2n−1)|p−2u(2n−1)
)′ = r(t) +
2n−1∑
k=0
(
ck
|u(k)|αk
+ dk(t)|u(k)|βk
)
. (3.33)
Equation (3.33) satisfies conditions (H1) – (H3) with φ(v) = |v|p−2v, h(t, v) = r(t) +
+ (2n + vγ)
∑2n−1
j=0
dk(t) where γ = max{βk : 0 ≤ k ≤ 2n − 1} < p − 1 and
ωk(v) =
ck
vαk
, 0 ≤ k ≤ 2n − 1. Hence Theorem 3.1 guarantees that problem (3.33),
(1.5) has a solution u ∈ C2n−1[0, T ], φ(u(2n−1)) ∈ AC[0, T ] and (−1)ku(2k) > 0 on
(0, T ], u(2k+1)(ξ2k+1) = 0 for 0 ≤ k ≤ n− 1 where ξ2k+1 ∈ (0, T ).
1. Agarwal R. P., O’Regan D., Staněk S. General existence principles for nonlocal boundary value
problems with φ-Laplacian and their applications // Abstrs. Anal. Appl. – 2006. – 30 p.
2. Rach◦unková I., Staněk S., Tvrdý M. Singularities and Laplacians in boundary value problems for
nonlinear ordinary differential equations // Handb. Different. Equat. Ordinary Different. Equat. / Eds
A. Cañada, P. Drábek, A. Fonda. – 2006. – Vol. 3. – P. 607 – 723.
3. Agarwal R. P., Wong P. J. Y. Existence of solutions for singular boundary value problems for higher
order differential equations // Rend. Semin. mat. e fis. Milano. – 1995. – 65. – P. 249 – 264.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
EXISTENCE PRINCIPLES FOR HIGHER ORDER NONLOCAL BOUNDARY-VALUE ... 259
4. Rach◦unková I., Staněk S. Sturm – Liouville and focal higher order BVPs with singularities in phase
variables // Georg. Math. J. – 2003. – 10. – P. 165 – 191.
5. Zhao Ch., Yuan Y., Liu Y. A necessary and sufficient condition for the existence of positive solutions
to singular boundary-value problems of higher order differential equations // Electron. J. Different.
Equat. – 2006. – № 8. – P. 1 – 19.
6. Deimling K. Nonlinear functional analysis. – Berlin: Springer, 1985.
7. Bartle R. G. A modern theory of integration. – Providence, Rhode Island: AMS, 2001.
8. Lang S. Real and functional analysis. – New York: Springer Inc., 1993.
Received 14.09.07
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
|
| id | umjimathkievua-article-3153 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:13Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/00/2d1d950a48acebb37aae45a0f79aba00.pdf |
| spelling | umjimathkievua-article-31532020-03-18T19:46:54Z Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems Принципи існування для нелокальних граничних задач вищого порядку та їх застосування до сингулярних задач Штурма-Ліувілля Staněk, S. Станєк, С. The paper presents existence principles for the nonlocal boundary-value problem $$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$ where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$ is an increasing and odd homeomorphism, $g$ is a Caratheodory function which is either regular or has singularities in its space variables and $\alpha_k: C^{p-1}[0,T]\rightarrow{\mathbb R}$ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems $(-1)^n(\phi(u^{(2n-1)}))' = f (t,u,...,u^{(2n-1)}),\quad u^{(2k)}(0) = 0,\quad$ $a_ku^{(2k)}(T) + b_k u^{(2k+1)}(T)=0,\quad 0\leq k\leq n-1$ is given. Наведено принципи Снування для нелокальної граничної задачi $$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$, де $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$ — гомеоморфізм, що зростає i є непарним, $g$ — Функція Каратеодорі, що або є регулярною, або має особливості за своїми просторовими змінними, а $\alpha_k: C^{p-1}[0,T]\rightarrow{\mathbb R}$ — неперервний функціонал. Показано застосування принципів існування до сингулярних задач Штурма-Ліувілля $(-1)^n(\phi(u^{(2n-1)}))' = f (t,u,...,u^{(2n-1)}),\quad u^{(2k)}(0) = 0,\quad$ $a_ku^{(2k)}(T) + b_k u^{(2k+1)}(T)=0,\quad 0\leq k\leq n-1$. Institute of Mathematics, NAS of Ukraine 2008-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3153 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 2 (2008); 240–259 Український математичний журнал; Том 60 № 2 (2008); 240–259 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3153/3054 https://umj.imath.kiev.ua/index.php/umj/article/view/3153/3055 Copyright (c) 2008 Staněk S. |
| spellingShingle | Staněk, S. Станєк, С. Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems |
| title | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems |
| title_alt | Принципи існування для нелокальних граничних задач вищого порядку та їх застосування до сингулярних задач Штурма-Ліувілля |
| title_full | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems |
| title_fullStr | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems |
| title_full_unstemmed | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems |
| title_short | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems |
| title_sort | existence principles for higher-order nonlocal boundary-value problems and their applications to singular sturm-liouville problems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3153 |
| work_keys_str_mv | AT staneks existenceprinciplesforhigherordernonlocalboundaryvalueproblemsandtheirapplicationstosingularsturmliouvilleproblems AT stanêks existenceprinciplesforhigherordernonlocalboundaryvalueproblemsandtheirapplicationstosingularsturmliouvilleproblems AT staneks principiísnuvannâdlânelokalʹnihgraničnihzadačviŝogoporâdkutaíhzastosuvannâdosingulârnihzadačšturmalíuvíllâ AT stanêks principiísnuvannâdlânelokalʹnihgraničnihzadačviŝogoporâdkutaíhzastosuvannâdosingulârnihzadačšturmalíuvíllâ |