Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems
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Інститут математики НАН України
2008
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| Цитувати: | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems / S. Stanek // Український математичний журнал. — 2008. — Т. 60, № 2. — С. 240–259. — Бібліогр.: 8 назв. — англ. |
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| citation_txt | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems / S. Stanek // Український математичний журнал. — 2008. — Т. 60, № 2. — С. 240–259. — Бібліогр.: 8 назв. — англ. |
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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.
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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
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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
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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
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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
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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
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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)
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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 ].
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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,
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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 =
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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.
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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
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Received 14.09.07
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 2
|
| id | nasplib_isofts_kiev_ua-123456789-164476 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-12-07T18:44:06Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Stanek, S. 2020-02-09T16:12:36Z 2020-02-09T16:12:36Z 2008 Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems / S. Stanek // Український математичний журнал. — 2008. — Т. 60, № 2. — С. 240–259. — Бібліогр.: 8 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164476 517.9 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. en Інститут математики НАН України Український математичний журнал Статті 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 Article published earlier |
| spellingShingle | Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems Stanek, S. Статті |
| 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 |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/164476 |
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