On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems
In this paper we investigate the existence of a positive solution of a second order singular Sturm – Liouville boundary-value problem, by constructing upper and lower solutions and combined them with properties of the consequent mapping. Applications to well known Emden – Fowler and Thomas – Ferm...
Збережено в:
| Дата: | 2001 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2001
|
| Назва видання: | Нелінійні коливання |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/174692 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems / S.K. Ntouyas, P.K. Palamides // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 326-344. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-174692 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1746922021-01-28T01:27:01Z On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems Ntouyas, S.K. Palamides, P.K. In this paper we investigate the existence of a positive solution of a second order singular Sturm – Liouville boundary-value problem, by constructing upper and lower solutions and combined them with properties of the consequent mapping. Applications to well known Emden – Fowler and Thomas – Fermi boundary-value problems are also presented. Further we generalize some of O’Regan’s results, allowing constants in the boundary conditions to be negative. 2001 Article On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems / S.K. Ntouyas, P.K. Palamides // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 326-344. — Бібліогр.: 10 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174692 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we investigate the existence of a positive solution of a second order singular Sturm –
Liouville boundary-value problem, by constructing upper and lower solutions and combined
them with properties of the consequent mapping. Applications to well known Emden – Fowler
and Thomas – Fermi boundary-value problems are also presented. Further we generalize some
of O’Regan’s results, allowing constants in the boundary conditions to be negative. |
| format |
Article |
| author |
Ntouyas, S.K. Palamides, P.K. |
| spellingShingle |
Ntouyas, S.K. Palamides, P.K. On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems Нелінійні коливання |
| author_facet |
Ntouyas, S.K. Palamides, P.K. |
| author_sort |
Ntouyas, S.K. |
| title |
On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems |
| title_short |
On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems |
| title_full |
On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems |
| title_fullStr |
On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems |
| title_full_unstemmed |
On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems |
| title_sort |
on sturm - liouville and thomas - fermi singular boundary-value problems |
| publisher |
Інститут математики НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/174692 |
| citation_txt |
On Sturm - Liouville and Thomas - Fermi Singular Boundary-Value Problems / S.K. Ntouyas, P.K. Palamides // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 326-344. — Бібліогр.: 10 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT ntouyassk onsturmliouvilleandthomasfermisingularboundaryvalueproblems AT palamidespk onsturmliouvilleandthomasfermisingularboundaryvalueproblems |
| first_indexed |
2025-07-15T11:44:35Z |
| last_indexed |
2025-07-15T11:44:35Z |
| _version_ |
1837713195982127104 |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
ON STURM – LIOUVILLE AND THOMAS – FERMI SINGULAR
BOUNDARY-VALUE PROBLEMS
S. K. Ntouyas
University of Ioannina
451 10 Ioannina, Greece
e-mail: sntouyas@cc.uoi.gr
P. K. Palamides
Naval Academy of Greece
Piraeus 185 03, Greece
In this paper we investigate the existence of a positive solution of a second order singular Sturm –
Liouville boundary-value problem, by constructing upper and lower solutions and combined
them with properties of the consequent mapping. Applications to well known Emden – Fowler
and Thomas – Fermi boundary-value problems are also presented. Further we generalize some
of O’Regan’s results, allowing constants in the boundary conditions to be negative.
AMS Subject Classification: 34B16, 34B24
1. Introduction
Singular boundary-value problem of the form
y′′ + q(t)y−σ = h(t), 0 < t < 1, (1.1)
y(0) = a, y(1) = b, (1.2)
y(0) = a, y′(1) = c, (1.3)
y(0) = a, αy′(1) + βy(1) = γ, (1.4)
where σ > 0, q, h ∈ C(0, 1) and q > 0 on (0, 1), a ≥ 0, b ≥ 0, c ≥ 0 and α, β, γ ∈ R,
appears in the study of many physical models. Problems (1.1), (1.2) (respectively (1.1) – (1.3) or
(1.1) – (1.4)) may be singular because q and h are allowed to have a suitable singularity at t = 0
or t = 1 and moreover a, b and c can be equal to zero.
Equation (1.1) with σ > 0 is known as the generalized Emden – Fowler equation with
negative exponent and arises frequently in applied mathematics (see [1] and the references
cited therein). Thomas [2] and Fermi [3] in order to determine the electrical potential in an
atom, derived the also singular equation
y′′ = t−
1
2 y
3
2 (1.5)
326 c© S. K. Ntouyas, P. K. Palamides, 2001
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 327
associated to next boundary conditions:
y(0) = 1, by′(1) = y(1) (neutral atom of Bohr radius 1),
y(0) = 1, y(1) = 0 (ionized atom),
y(0) = 1, lim
t→∞
y(t) = 0 (isolated neutral atom).
(1.6)
In the book of O’Regan [4] the study has been extended to the more general Sturm – Liouville
boundary-value problem
1
p(t)
(p(t)y′(t))′ + q(t)f(t, y(t), p(t)y′(t)) = 0, 0 < t < 1, (1.7)
-αy(0) + β lim
t→0+
p(t)y′(t) = 0, α > 0, β ≥ 0,
γy(1) + δ lim
t→1−
p(t)y′(t) = 0, γ ≥ 0, δ ≥ 0 and γ2 + δ2 > 0,
(1.8)
and the nonlinear function f may be singular at the point y = 0 but not at t = 0 or t = 1. (He
pointed out there, that there is no restriction in assuming homogeneous boundary data.)
His approach relied on the Leray – Schauder alternative for compact maps. More precise
existence results for (1.7), (1.8) were given (Theorem 5.1 and Corollary 5.1) for the case where
f ∈ C([0, 1]× (0,∞)× R,R), under the assumptions:
p ∈ C[0, 1] ∩ C1(0, 1), with p > 0 on (0, 1) and
1∫
0
1
p(s)
ds < ∞, (1.9)
q ∈ C(0, 1) with q > 0 on (0, 1) and
1∫
0
p(x)q(x)dx < ∞. (1.10)
It holds f(t, u, v) = g(t, u, v) + h(t, u, v) where g and h satisfy
|h(t, u, v)| ≤ K{|u|γ + |v|τ + 1}, 0 < γ, τ < 1,
ug(t, u, v) ≥ c |u|2 + d |uv| , d ≤ 0 and
|g(t, u, v)| ≤ A(t, u) |v|2 +B(t, u),
where A and B are bounded on bounded sets
(1.11)
and finally
p(t)
√
q(t) is bounded on [0, 1]. (1.12)
We must notice that (1.11) replaces in some sense the (necessary) growth rate condition of
Nagumo.
328 S.K. NTOUYAS, P.K. PALAMIDES
In this paper, we examine the same existence problem (1.7), (1.8) but we use the method of
upper and lower solutions combined with properties of the consequent mapping. More preci-
sely, we assume here only that
A) there exist m,K > 0 and n = 2k + 1, k = 1, 2, ... , such that
∣∣p(t)q(t)f(t, y(t), p(t)y′(t))
∣∣ ≤ K
p(t)
∣∣p(t)y′(t))∣∣n+1
, if
∣∣p(t)y′(t))∣∣ ≥ m. (1.13)
We observed that this condition leads to the construction of an “upper” and “lower” solution
of (1.7). Moreover we introduce the new concept of strong pair of upper and lower soluti-
ons and so (1.13) returns some bounds for the derivative p(t)y′(t) of any solution which stays
between lower and upper solutions, ensuring in this way that the consequent map is well-
defined. The properties of the last map give occasion of existence results.
In this paper, further of an existence result we give an estimation for the derivative y′,
assuming a rather mild growth rate on f (see (1.13)) and so we don’t use the Nagumo type
condition (1.11) at all. Further our main purpose is to extend the validity of existence results
and/or for some cases when constants α, β, γ and δ are negative (see Remark 4.1).
Furthermore motivated by the Thomas Fermi equation (1.5), we establish an existence result
to generalized Bohr boundary-value problem (1.7), (1.8), where
y(0) = a and
p(1)
1∫
0
ds
p(s)
y′(1) + y(1) = 0, (1.14)
under rather general assumptions (see Theorem 5.1). A similar problem has been studied by
O’Regan (see [4], Theorem 5.3) for the case when f is independent of its last argument.
The paper is organized as follows. In Section 2 we introduce notations, definitions and preli-
minary facts, which are used throughout this paper. The basic existence result in given in Secti-
on 3. Finally, two applications are given, the first for a singular second order Sturm – Liouville
boundary-value problem in Section 4 and the second for a generalized Bohr boundary-value
problem in Section 5.
2. Preliminaries
Consider the initial value problem (IVP for short)
1
p(t)
(p(t)x′(t))′ + q(t)f(t, x(t), p(t)x′(t)) = 0, 0 < t < 1, (2.1)
(τ, x(τ), p(τ)x′(τ)) = (τ, ξ, η) = P ∈ [0, 1]× R2 = Ω, (2.2)
and let X (P ) be the set of all (noncontinuable) solutions of IVP (2.1), (2.2), that is x(τ) = ξ
and lim
t→τ
p(t)x′(t) = η.
Definition 2.1. Consider two functions α and β ∈ C[0, 1] ∩ C2(0, 1) with pα′ and pβ′ ∈
C1[0, 1] and such that α(t) ≤ β(t), t ∈ [0, 1].
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 329
Then α is called lower solution to (2.1) if
1
p(t)
(p(t)α′(t))′ + q(t)f(t, α(t), p(t)α′(t)) > 0, 0 < t < 1,
and β is called upper solution to (2.1) if
1
p(t)
(p(t)β′(t))′ + q(t)f(t, β(t), p(t)β′(t)) < 0, 0 < t < 1.
Further (α, β) is a strong pair of lower and upper solutions, iff
β′(t) ≤ α′(t), 0 ≤ t ≤ 1,
where if lim
t→0+
p(t) = 0 and/or lim
t→1−
p(t) = 0, it may be
lim
t→0+
β′(t) = β′(0) = −∞ and/or lim
t→1−
β′(t) = β′(1) = −∞
and similarly
lim
t→0+
α′(t) = α′(0) = +∞ and/or lim
t→1−
α′(t) = α′(1) = +∞,
respectively.
Lemma 2.1. Let (α, β) be a strong pair of lower and upper solutions p and q satisfying (1.9),
(1.10) and further the function f(t, ., px′) be nonincreasing for any (t, px′) ∈ [0, 1]× R.
Then for any solution x = x(t) of (2.1) with
β′(0) ≤ x′(0) ≤ α′(0) and α(t) ≤ x(t) ≤ β(t), t ∈ [0, 1] ∩Dom (x),
the following inequality folds true:
β′(t) ≤ x′(t) ≤ α′(t), t ∈ [0, 1] ∩Dom (x).
Moreover any such solution x = x(t) can be defined on [0, 1] and∣∣p(t)x′(t)∣∣ ≤ M = max
t∈[0,1]
{
∣∣p(t)α′(t)∣∣ , ∣∣p(t)β′(t)∣∣}.
Proof. Let’s suppose that there are s, τ ∈ (0, 1) such that
α(t) ≤ x(t) ≤ β(t), t ∈ [0, τ ], β′(t) ≤ x′(t) ≤ α′(t), t ∈ [0, s],
and β′(t) > x′(t), t ∈ (s, τ ]. Then since β is an upper solution to (2.1), by the monotonicity
of f ,
1
p(s)
(p(s)β′(s))′(s) < −q(s)f(s, β(s), p(s)β′(s))
= −q(s)f(s, β(s), p(s)x′(s))
≤ −q(s)f(s, x(s), p(s)x′(s)) =
1
p(s)
(p(s)x′(s))′(s).
330 S.K. NTOUYAS, P.K. PALAMIDES
Consequently we get
β′′(s) < x′′(s) and β′(s) = x′(s)
and thus inequality β′(t) > x′(t), t ∈ (s, τ ] cannot be true.
By a similar argument we can easily prove that x′(t) ≤ α′(t), t ∈ [0, 1] ∩ Dom (x). The last
conclusion follows by the well-known theorem of extensibility of solutions of (2.1), since these
and their derivatives px′ are bounded.
The lemma is proved.
Consider now a subset ω̂ of Ω such that
ω̂◦ 6= ∅ and Ω− ω̂ 6= ∅,
and let
ω̂(τ) := {(t, x, y) ∈ ω̂ : t = τ}
be its cross-section at t = τ.
A point P = (τ, ξ, η) ∈ ∂ω̂ is a point of egress of ω̂ (with respect to the system (2.1)), iff for
any solution x ∈ X (P ) there exists ε > 0 such that the graph of the restriction x|[τ − ε, τ ] is in
ω̂◦, i.e.,
G(x|[τ − ε, τ);P ) := {(t, x(t), x′(t)) : τ − ε ≤ t < τ} ⊆ ω̂◦.
If moreover for all solutions x ∈ X (P ) there is ε > 0 such that
G(x|(τ, τ + ε];P ) ⊆ Ω− ω̂,
then P is called a strict egress point (see [5] or [6]). As usual the set of egress (strict egress)
points of ω̂ will be denoted by ω̂e (respectively ω̂se).
Remark 2.1. Let now consider the set
ω := {(t, x, x′) : 0 ≤ t ≤ 1, α(t) ≤ x ≤ β(t), x′ ∈ R},
and let
Q′α = Q′α [0, 1] := {(t, x, x′) ∈ ∂ω : 0 ≤ t ≤ 1, x = α(t) and x′ ≤ α′(t)} and
Q′β = Q′β [0, 1] := {(t, x, x′) ∈ ∂ω : 0 ≤ t ≤ 1, x = β(t) and x′ ≥ β′(t)}.
Then we can easily see [7] that every egress point of ω is a strict egress one and to be more
specific we have
ωe = ωse = Q′α ∪Q′β ∪ ω(1). (2.3)
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 331
A point P = (τ, ξ, η) ∈ ωe is a consequent of another one P0 = (t0, x0, x
′
0) ∈ ω◦, t0 ≤ τ ,
iff there exists a solution x ∈ X (P0, P ) := X (P0) ∩X (P ) such that
G(x|(t0, τ);P0, P ) ⊆ ω◦.
The set of consequent points of P0 ∈ ω◦ will denoted by K(P0) and the so defined (set-valued)
mapping
K : S(ω) ⊆ ω◦ → ωe, S(ω) = {Q ∈ ω◦ : K(Q) 6= 0}
will be referred as the consequent mapping.
Consider a set-valued mapping F, which maps the points of a topological space X into
compact subsets of another one Y. F is upper semi-continuous (u.s.c.) at x0 ∈ X iff for any
open subset V in Y with F(x) ⊆ Y, there exists a neighbourhood U of x0 such that F(x) ⊆ V ,
for every x ∈ U.
The next lemmas (see [8]) give sufficient conditions for the upper semicontinuity of the
consequent mapping and some useful properties for a class of u.s.c. mappings. We notice that
the consequent mapping is included in this class [9].
Lemma 2.2. If P ∈ S(ω) and every solution x ∈ X (P ) egresses strictly from ω, then the
consequent mapping K is u.s.c. at the point P and furthermore the image K(P ) is a continuum in
∂ω.
Lemma 2.3. Let X and Y be metric spaces and let F : X → 2Y be an u.s.c. mapping. If A
is a continuum subset of X , such that for every x ∈ A the image F (x) is a continuum, then the
image F (A) = ∪{F (x) : x ∈ A} is also a continuum subset of Y.
Notice that, if an IVP has a unique solution, then the consequent map is simply continuous.
We shall need another Lemma (see [9] or [10], Chapter V, Theorem 2).
Lemma 2.4. If A is an arbitrary proper subset of a continuum C and if S is a connected
component of A, then
clS ∩ cl(C\A) 6= ∅, i.e. S̄ ∩ ∂A 6= ∅.
3. Existence Results
Theorem 3.1. Suppose that assumptions (1.9), (1.10) are fulfilled and further that there is a
constant M > 0 such that
|f(t, u, v)| ≤ M, 0 ≤ t ≤ 1, (u, v) ∈ R2. (3.1)
Then for any y0 ∈ [α(0), β(0)], the equation (2.1) has a global solution y = y(t), 0 ≤ t ≤ 1 (i.e.
[0, 1] = Dom (x)) with y(0) = y0. (Here α(0), β(0) are the values at t = 0 of an upper solution
α(t) and a lower solution β(t) of (2.1) respectively.)
332 S.K. NTOUYAS, P.K. PALAMIDES
Proof. Consider the Banach space
K1[0, 1] = {u ∈ C[0, 1] : pu′ ∈ C[0, 1] with norm ‖u‖1},
where
‖u‖1 = max{‖u‖ ,
∥∥pu′∥∥}
and ‖u‖ is the usual sup-norm of u on [0, 1]. Let also
T = K1[0, 1] → K1[0, 1]
be an operator defined by
(Ty)(t) := y(0)−
t∫
0
1
p(s)
s∫
0
p(x)q(x)f(x, y(x), p(x)y′(x))dxds.
We claim that T has a fixed point in
K1
0 [0, 1] = {u ∈ K1[0, 1] : u(0) ∈ [α(0), β(0)]}.
To prove the compactness of T , we notice that there exist K0 and K1 such that
|(Ty)(t)| ≤ |y(0)|+M
t∫
0
1
p(s)
s∫
0
p(x)q(x)dxds ≤ K0
and
∣∣p(t)(Ty)′(t)
∣∣ ≤ M
t∫
0
p(x)q(x)dx ≤ K1,
i.e.,
‖(Ty)‖1 ≤ K = max{K0,K1}, y ∈ K1
0 [0, 1].
Furthermore TK1
0 [0, 1] is an equicontinuous family, since
∣∣(Ty)(t)− (Ty)(t′)
∣∣ <
∣∣∣∣∣∣
t∫
t′
1
p(s)
s∫
0
p(x)q(x)f(x, y(x), p(x)y′(x))dxds
∣∣∣∣∣∣
≤ M
∣∣φ(t)− φ(t′)
∣∣
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 333
and
∣∣p(t)(Ty)′(t)− p(t)(Ty)(t′)
∣∣ <
∣∣∣∣∣∣
t∫
t′
p(x)q(x)f(x, y(x), p(x)y′(x))dx
∣∣∣∣∣∣
≤ M
∣∣φ∗(t)− φ∗(t′)∣∣ , y ∈ K1
0 [0, 1],
because the functions
φ(t) =
t∫
0
1
p(s)
s∫
0
p(x)q(x)dxds and φ∗(t) =
t∫
0
p(x)q(x)dx
are absolutely continuous. Finally by an application of the standard Schauder fixed point theorem,
we get a solution y ∈ X (P ), P = (0, y0, 0), i.e., X (P ) 6= ∅ and any y ∈ X (P ) is defined over
the entire interval [0, 1].
The theorem is proved.
Let now V be any subset of Ω and J a subset of the interval [0, 1]. Consider the cross-section
of V ,
V (J) = {(t, y, y′) ∈ V : t ∈ J},
over the interval J (J may be a single-point set). Next assume that
0 ≤ α(t) ≤ β(t), t ∈ [0, 1] (3.2)
and consider a line E0 = {(0, y, y′) ∈ ω(0) : y = y0 ∈ [α(0), β(0)]}, where we recall that
ω = {(t, y, y′) : 0 ≤ t ≤ 1, α(t) ≤ y ≤ β(t), y′ ∈ R}.
Theorem 3.2. Assume that hypotheses of Theorem 3.1 are still holding. Then there exist
points P1 = (0, y0, y
′
1) ∈ E0 and P2 = (0, y0, y
′
2) ∈ E0 such that any solution y ∈ X (P1)
egresses from ω through the surface
Sα := {(t, y, y′) ∈ ω : y = α(t)}
and similarly any solution y ∈ X (P2) egresses from ω through the surface
Sβ := {(t, y, y′) ∈ ω : y = β(t)}.
Proof. For any x ∈ X (P2) we have
y(t) = y′2
t∫
0
1
p(x)
dx+ y0 −
t∫
0
1
p(s)
s∫
0
p(x)q(x)f(x, y(x), p(x)y′(x))dxds.
334 S.K. NTOUYAS, P.K. PALAMIDES
Consequently by the conditions (3.1) and (3.2) we get
y(t) ≥ y′2
t∫
0
1
p(x)
dx− |y0| −M
∣∣∣∣∣∣
t∫
0
1
p(s)
s∫
0
p(x)q(x)dxds
∣∣∣∣∣∣ .
Thus, if we choose y′2 large enough we get y(t) > β(t) for some 0 < t ≤ 1, given that (1.9),
(1.10) hold true.
The theorem is proved.
We are ready to state and prove a general existence result. Let E0 ⊆ ω(0) and E1 ⊆ ω(1) be
two continua.
Theorem 3.3. Suppose that there is a strong pair (α, β) of upper and lower solutions and
assumptions (1.9), (1.10) and (3.1) are fulfilled. Assume further that the function f(t, ., px′) is
nonincreasing for any (t, px′) ∈ [0, 1]× R. Moreover
K(E0) ∩Q′α[0, 1]6= ∅ and K(E0) ∩Q′β[0, 1]6= ∅,
E1 ∩Q′α(1)= ∅ and E1 ∩Q′β(1)= ∅.
(3.3)
Then the boundary-value problem
1
p(t)
(p(t)y′(t))′ + q(t)f(t, y(t), p(t)y′(t)) = 0, 0 < t < 1,
y ∈ X (E0) ∩ X (E1)
has at least a solution y = y(t) such that
α(t) ≤ y(t) ≤ β(t) and β′(t) ≤ y′(t) ≤ α′(t), 0 ≤ t ≤ 1.
Proof. By Lemma 2.1, Remark 2.1 and Theorem 3.1, any solution y ∈ X (E0) is defined on
entire the interval [0, 1]. Thus the consequent mapping K is well defined and in view of Lemma
2.2 its image K(E0) is a continuum subset of ∂ω. Thus
Sα(1) ∩ K(E0) 6= ∅ and Sβ(1) ∩ K(E0) 6= ∅.
Since now both the sets Sα(1) ∩K(E0) and Sβ(1) ∩K(E0) are clearly compact by Lemma 2.4,
we conclude that there is a connected component E∗1 of ω(1) ∩ K(E0) such that
E∗1 ∩ Sα(1) 6= ∅ and E∗1 ∩ Sβ(1) 6= ∅,
i.e.,
E∗1 ∩Q′α(1) 6= ∅ and E∗1 ∩Q′β(1) 6= ∅.
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 335
Consequently in view of (2.3), by assumptions (3.3) we get
E∗1 ∩ E1 6= ∅,
that is, there exists a solution y ∈ X (E0) ∩ X (E1) such that
α(t) ≤ y(t) ≤ β(t) and β′(t) ≤ y′(t) ≤ α′(t), 0 ≤ t ≤ 1,
where the last estimates follow by the definition of the consequent mapping K and Lemma 2.1.
4. Sturm – Liouville Boundary-Value Problems
In this section an existence result for the singular Sturm – Liouville boundary-value problem
1
p(t)
(p(t)y′(t))′ + q(t)f(t, y(t), p(t)y′(t)) = 0, 0 < t < 1, (4.1)
−αy(0) + β lim
t→0+
p(t)y′(t) = c, α 6= 0, β > 0,
γy(1) + δ lim
t→1−
p(t)y′(t) = d, γ 6= 0, δ > 0
(4.2)
will be given, where we do not assume that α and/or γ are necessarily positive. The singularity of
the nonlinear function f may occurs at y = 0 and the functions 1/p, q and 1/q may be singular
at t = 0 or/and t = 1.
Assume the next conditions:
(A1) q ∈ C(0, 1) with q > 0 on (0, 1);
(A2) p ∈ C[0, 1] ∩ C1(0, 1) with p > 0 on (0, 1),
1∫
0
dt
p(t)
< ∞ and
1∫
0
p(t)q(t)dt < ∞;
(A3) f : [0, 1] × (0,∞) × R → R is continuous and further the function f(t, ., px′) is
nonincreasing for any (t, px′) ∈ [0, 1]× R.
Theorem 4.1. Suppose that conditions (A1)− (A3) are satisfied. Moreover assume:
(A4) There exist constants K > 0, 0 < a0 < b0, a natural number n = 2k + 1, k = 1, 2, ...,
and further
0 < m <
1
n
√√√√√Kn
1∫
0
dt
p(t)
336 S.K. NTOUYAS, P.K. PALAMIDES
such that
|p(t)q(t)f(t, u, v)| < K
p(t)
|v|n+1 , 0 < t < 1, a0 ≤ u ≤ b0, and |v| ≥ m, (4.3)
and finally
(A5) The constants of the Sturm – Liouville boundary conditions satisfy
−αb0 − βm ≤ c ≤ βm− αa0, if p0 > 0,
γα̂0 + δm̂0 < d < γb̂0 + δm̂1, if p1 > 0, and (4.4)
a0 < −
c
α
< b0, if p0 = 0 and α̂0 <
d
γ
< b̂0, if p1 = 0.
Then the boundary-value problem (4.1), (4.2) has a solution y ∈ C[0, 1] ∩ C1(0, 1) (with
px′ ∈ AC([0, 1]) such that
α(t) ≤ y(t) ≤ β(t), 0 ≤ t ≤ 1.
Furthermore its derivative satisfies also the estimate
β′(t) ≤ y′(t) ≤ α′(t), 0 ≤ t ≤ 1,
where the constants α̂0, b̂0, m̂0, m̂1, p0, p1 and the functions α(t) and β(t), 0 ≤ t ≤ 1, will be
defined below.
Remark 4.1. For α = 0 = c and δ = 0 = d, the boundary-value problem (4.1), (4.2) has
also a solution y ∈ C[0, 1]∩C1(0, 1) as above and this has been proved in a recent paper of the
authors (see [7]).
By (4.4) we can see that the slope α/β of the line
E0 = {(t, y, y′) ∈ ω(0) : −αy(0) + βp0y
′(0) = c}
must satisfy α/β ∈ [−(βm+ c)/(βb0), (βm− c)/(βa0)] and so it may accept negative values
and this extends the usual case of α, β ≥ 0. At the same time, since
−γ
δ
∈
(
δm̂− d
δa0
,
δm̂− d
δb0
)
,
the slope −γ/δ of the line
E1 = {(t, y, y′) ∈ ω(1) : γy(1) + δp1y
′(1) = d}
may accept positive values and this also extends the usual case of γ, δ ≥ 0.
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 337
Proof. Returning to the proof of the theorem, we shall construct a strong pair (α, β) of
upper and lower solutions of (4.1). Consider the scalar differential equation
(p(t)y′(t))′ =
K
p(t)
(p(t)y′(t))n+1, t ∈ [0, 1]. (4.5)
Then a solution of (4.5) is
α(t) = a0 +
1
K(n− 1)mn−1
0
{
1− [1−Knmn
0P (t)]
n−1
n
}
, 0 ≤ t ≤ 1,
where P (t) =
t∫
0
ds
p(s)
> 0,
1
n
√
KnP (1)
≥ m0 ≥ m > 0, 0 ≤ t ≤ 1, and so
α′(t) =
m0
p(t) n
√
1−Knmn
0P (t)
≥ m0
p(t)
≥ m
p(t)
> 0. (4.6)
We choose now 0 < a0 < b0 so that
0 < α(t) < b0, 0 ≤ t ≤ 1,
and so by (4.3), for each 0 ≤ t ≤ 1 we get
(p(t)α′(t))′ =
K
p(t)
(p(t)α′(t))n+1 > −p(t)q(t)f(t, α(t), p(t)α′(t)),
i.e., the function x = α(t) is a lower solution of (4.1).
Similarly the map
β(t) = b0 −
1
K(n− 1)mn−1
1
{
1− [1 +Knmn
1P (t)]
n−1
n
}
, 0 ≤ t ≤ 1,
is a solution of
(p(t)y′(t))′ = − K
p(t)
(p(t)y′(t))n+1
such that for all 0 ≤ t ≤ 1,
β′(t) =
m1
p(t) n
√
1 +Knmn
1P (t)
≤ m1
p(t)
≤ − m
p(t)
< 0. (4.7)
Thus by (4.3), for each 0 ≤ t ≤ 1 we get (we recall that n = 2k + 1)
(p(t)β′(t))′ = − K
p(t)
(p(t)β′(t))n+1 < −p(t)q(t)f(t, β(t), p(t)β′(t)),
338 S.K. NTOUYAS, P.K. PALAMIDES
i.e., the function x = β(t) is an upper solution of (4.1).
Now by (4.7) and since we may choose 0 < α(t) < b0, 0 ≤ t ≤ 1, we get
0 < α(t) < β(t), 0 ≤ t ≤ 1,
and moreover by (4.6) and (4.7),
β′(t) < 0 < α′(t), 0 ≤ t ≤ 1,
that is the pair of upper and lower solution (α, β) is a strong one.
Consider the cross-section
ω(0) = {(t, x, x′) : t = 0, α(t) ≤ x ≤ β(t), x′ ∈ R},
and recall that the egress points of ω(τ), 0 ≤ τ ≤ 1, consist of the union Q′α(τ)∪Q′β(τ), where
Q′α(τ) := {(τ, α(τ), x′) : x′ ≤ α′(τ)} and
Q′β(τ) := {(τ, β(τ), x′) : x′ ≥ β′(τ)},
and notice that (set p0 := lim
t→0+
p(t) and p1 := lim
t→1−
p(t))
α′(0) =
m0
p0
> 0, β′(0) =
m1
p0
< 0 and
α′(1) :=
m̂0
p1
> 0, β′(1) :=
m̂1
p1
< 0.
We examine now two cases.
Both limits p0 and p1 exist and, of course, are positive.
Then the condition −αy(0) + β lim
t→0+
p(t)y′(0) = c reduces to
y′(0) =
c+ αy(0)
βp0
and thus, if y(0) = α(0) = a0, we get by assumption (4.4) and (4.6)
y′(0) =
c+ αa0
βp0
≤ m
p0
≤ m0
p0
= α′(0)
and similarly, if y(0) = β(0) = b0, we have
y′(0) =
c+ αb0
βp0
≥ −m
p0
≥ m1
p0
= β′(0).
Consequently we obtain
Q′α(0) ∩ E0 6= ∅ and Q′β(0) ∩ E0 6= ∅ (4.8)
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 339
where recall that
E0 = {(t, y, y′) ∈ ω(0) : −αy(0) + βp0y
′(0) = c}.
Now the condition γy(1) + δ lim
t→1−
p(t)y′(t) = d, which turns up to the cross-section ω(1) of ω
reduces to
y′(1) =
d− γy(1)
δp1
and thus, if y(1) = α(1) := â0 > 0, we get by assumption (4.4),
y′(1) =
d− γâ0
δp1
>
m̂0
p1
= α′(1)
and similarly, if y(1) = β(1) := b̂0, we have
y′(1) =
d− γb̂0
δp1
<
m̂1
p1
= β′(1).
Consequently by (2.3), we now obtain
Q′α(1) ∩ E1 = ∅ and Q′β(1) ∩ E1 = ∅ (4.9)
where
E1 = {(t, y, y′) ∈ ω(1) : γy(1) + δp1y
′(1) = d}.
To show that (4.1), (4.2) has a solution, we will now examine the modified problem
(E)
1
p(t)
(p(t)y′(t))′ + q(t)F (t, y(t), p(t)y′(t)) = 0, 0 < t < 1,
(C)
−αy(0) + β lim
t→0+
p(t)y′(t) = c,
γy(1) + δ lim
t→1−
p(t)y′(t) = d,
where
F (t, u, v) =
f(t, u, v), if α(t) ≤ u ≤ β(t), p(t)β′(t) ≤ v ≤ p(t)α′(t),
f(t, α(t), v), if u ≤ α(t), p(t)β′(t) ≤ v ≤ p(t)α′(t),
f(t, β(t), v), if u ≥ β(t), p(t)β′(t) ≤ v ≤ p(t)α′(t),
f(t, u, p(t)β′(t)), if α(t) ≤ u ≤ β(t), v ≤ p(t)β′(t),
f(t, u, p(t)α′(t)), if α(t) ≤ u ≤ β(t), p(t)α′(t) ≤ v,
f(t, α(t), p(t)β′(t)), if u ≤ α(t), v ≤ p(t)β′(t),
f(t, α(t), p(t)α′(t)), if u ≤ α(t), p(t)α′(t) ≤ v,
f(t, β(t), p(t)β′(t)), if u ≥ β(t), v ≤ p(t)β′(t),
f(t, β(t), p(t)α′(t)), if u ≥ β(t), p(t)α′(t) ≤ v.
340 S.K. NTOUYAS, P.K. PALAMIDES
Since 0 < α(t) ≤ β(t), 0 ≤ t ≤ 1, we notice that the modification F is a bounded function
and let M be one of its bounds.
Finally by an application of our general principle (Theorem 3.3), we get a (positive) solution
y = y(t) of the Sturm – Liouville boundary-value problem (E), (C), i.e.,
0 ≤ α(t) ≤ y(t) ≤ β(t), 0 ≤ t ≤ 1,
which satisfies moreover the estimate
β′(t) ≤ y′(t) ≤ α′(t), 0 ≤ t ≤ 1.
Hence in view of definition of the modification F , y = y(t) is a solution of the original equa-
tion (4.1).
Assume now that
lim
t→0+
p(t) = p0 = 0 = p1 = lim
t→1−
p(t).
Then, obviously, by (4.6) and (4.7), we get
α′(0) = α′(1) = +∞, β′(0) = β′(1) = −∞.
Moreover by (4.4) and since for α 6= 0
E0 = {(t, y, y′) ∈ ω(0) : −αy(0) + β lim
t→0+
p(t)y′(0) = c}
⊇
{
(0, y, y′) : y = − c
α
∈ (a0, b0) and y′ ∈ R
}
,
in view of Theorem 3.2, we may set
E0 = {(0, y, y′) : y = − c
α
∈ (a0, b0) and y′1 ≤ y′ ≤ y′2},
since we may choose a suitable interval (a0, b0) for the case when αc < 0 (for αc ≥ 0, the
problem remains open (if c = 0, then lim
t→0+
y′(t) = ∞)).
Remark 4.2. Now if α = 0 = c, then y′(0) is undefined and therefore we may restrict our
consideration to the set
E0 = {(0, y, y′) ∈ ω(0) : y′(0) = 0}
and in this case, clearly, (4.8) still holds. However for α = 0 and c 6= 0, our existence problem
remains open.
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 341
A similar situation holds also at the other end point t = 1. Precisely
E1 = {(t, y, y′) ∈ ω(1) : γy(1) + δ lim
t→1−
p(t)y′(1) = d}
⊇
{
(0, y, y′) : y =
d
γ
∈ [â0, b̂0] and y′ ∈ R
}
,
and (4.9) clearly is true (for d/γ ∈ {â0, b̂0} we do not need to use (4.9) at all).
Then, if we use once again the boundary-value problem (E), (C), existence results follows
via Theorem 3.3 and this ends the proof.
Remark 4.3. Let us notice (see Remark 4.2) that for the case
α = c = 0, lim
t→0+
p(t) = p0 = 0,
the BVP (4.1), (4.2) has a 1-parametric family of solutions satisfying the properties given in the
above theorem.
Since for p0 = 0, we have
E0 = {(t, y, y′) ∈ ω(0) : αy(0) + β lim
t→0+
p(t)y′(0) = c} = {0} × [α(0), β(0)]× R
and further
Q′α(0) = {(0, α(0), y′) : y′ ≤ α′(0) = +∞}
and similarly
Q′β(0) = {(0, β(0), y′) : y′ ≥ β′(0) = −∞},
we can choose, in place of E0, any set
Eλ = {(0, y, y′) ∈ E0 : y′ = λ}.
Then obviously
Q′α(0) ∩ Eλ 6= ∅ and Q′β(0) ∩ Eλ 6= ∅.
Thus, by applying the previous Theorem we get the desired family {yλ(t)}, λ ∈ R, of solutions.
342 S.K. NTOUYAS, P.K. PALAMIDES
5. Thomas – Fermi Boundary-Value Problems
In this Section, motivated by the Thomas – Fermi equation (1.5), we consider the generali-
zed Bohr’s type boundary-value problem, namely,
1
p(t)
(p(t)y′(t))′ = q(t)f(t, y(t), p(t)y′(t)), 0 < t < 1,
(5.1)
y(0) = a and
p(1)
1∫
0
ds
p(s)
y′(1) + y(1) = 0.
A similar problem has been studied by O’Regan in [4] (Theorem 5.3) for the case when f is
independent of its last argument. We set P ∗(t) = p(t)
t∫
0
ds
p(s)
, t ∈ [0, 1].
Theorem 5.1. Let f : [0, 1]× (0,∞)× R →R be continuous. We assume:
p ∈ C[0, 1] ∩ C1(0, 1], p > 0 on (0, 1], p′(t) ≥ 0, t ∈ (0, 1),
(5.2)
and
1∫
0
1
p(s)
ds < ∞,
q ∈ C(0, 1) with q > 0 on (0, 1) and
1∫
0
p(x)q(x)dx < ∞, (5.3)
there exists a constant M ≥ 0 such that yf(t, y, 0) > 0
(5.4)
for |y| > M and all t ∈ [0, 1],
there is γ ∈
(
0,
M
P ∗(1)
)
such that −M − γ < a < M + γ
and both functions f(t, ., v) and f(t, u, .) are nonincreasing for each (5.5)
t ∈ [0, 1], u ∈ [−M − γ, M + γ]and v ∈ [−γ, γ].
Then (5.1) has at least a solution in C[0, 1] ∩ C2(0, 1].
ON STURM --LIOUVILLE AND THOMAS--FERMI SINGULAR BOUNDARY-VALUE PROBLEMS 343
Proof. Consider the functions
α(t) = −M − γ(1− t) and β(t) = M + γ(1− t), t ∈ [0, 1].
Then α(t) and β(t), t ∈ [0, 1], are lower and upper solutions of (5.1). Indeed, in view of (5.2) –
(5.4) and since f(t, u, .) is nonincreasing, we get
(p(t)α′(t))′ − p(t)q(t)f(t, α(t), p(t)α′(t))
= γp′(t)− p(t)q(t)f(t, α(t), γp(t))
≥ γp′(t)− p(t)q(t)f(t, α(t), 0) ≥ 0, 0 < t < 1,
because α(t) ≤ −M, 0 ≤ t ≤ 1. Similarly we can prove that β is an upper solution.
Furthermore (α, β) is a strong pair of lower and upper solutions. By using an adaptation F
of f similar to the one given in Theorem 4.1, clearly it is not a restriction to assume that the
function f is bounded on [0, 1]× R2.
If
E0 = {(0, y, y′) : y(0) = a, y′ ∈ R} and E1 = {(0, y, y′) : P ∗(1)y′ + y = 0}
by (5.5) the result of Theorem 3.2 holds and so we get
K(E0) ∩Q′α 6= ∅ and K(E0) ∩Q′β 6= ∅.
Further by (5.5) we can easily verify that
E1 ∩Q′α(1)= ∅ and E1 ∩Q′β(1)= ∅.
Existence now follows by an argument similar to the one given in the proof of the previous
theorem.
Corollary 5.1. Under the assumptions (5.2), (5.3) and (5.5) of the previous theorem and
f(t, 0, 0) = 0 and yf(t, y, 0) > 0, t ∈ [0, 1], y 6= 0,
the BVP (5.1) accepts a positive (negative) solution if α > 0 (α < 0 respectively).
Proof. All that we need to notice is that for α > 0 (α < 0 ) we can choose α(t) =
0 (β(t) = 0), 0 ≤ t ≤ 1, because f(t, 0, 0) = 0, 0 ≤ t ≤ 1.
We finally consider the Thomas – Fermi type singular equation
y′′ = −t−
1
2 y
3
2 := q(t)f(t, y) (5.6)
344 S.K. NTOUYAS, P.K. PALAMIDES
associated to one of the boundary conditions:
y(0) = 1, y(b) = 0 (b > 0), (5.7)
y(0) = 1, lim
t→∞
y(t) = 0. (5.8)
We notice at once that
α(t) = 0, β(t) = 2, t ≥ 0
is a strong pair of lower and upper solutions to (5.6). Considering a modification F of f, similar
to the one given in the proof of Theorem 4.1, we may assume that f is bounded and f(t, .) is
nonincreasing. By Theorem 3.1, any solution y = y(t) with y(0) = 1 is defined over the interval
[0,∞). Assume that there is no solution y ∈ X (E0) of (5.6) such that its graph
G(y|[0,∞);E0) = {(t, y(t)) : t ≥ 0} ⊆ ω.
Then any solution y ∈ X (E0) egresses strictly from ω and further the set K(E0) is a continuum.
Now the sets
K(E0) ∩ Sα and K(E0) ∩ Sβ
clearly consist of a closed partition of K(E0), and this is a contradiction. Let A := K(E0) ∩
Sα and let S be a connected component of A. By Lemma 2.4, S must be an unbounded set
(otherwise we easily get a similar contradiction) and this means that there exists a solution
y ∈ X (E0) such that lim
t→∞
y(t) = 0.
Similarly we may get a solution of (5.6), satisfying the boundary condition (5.7).
REFERENCES
1. Janus J. and Myjak J. “A generalized Emden – Fowler equation with a negative exponent,” Nonlinear Anal.,
23, No. 8, 953 – 970 (1994).
2. Thomas L. “The calculation of atomic fields,” Proc. Cambridge Phil. Soc., 23, 542 – 548 (1927).
3. Fermi E. “Un methodo statistico par la determinazione di alcune proprietà dell’ atoma,” Rend. Accad. naz.
del Lincei. sci., Mat. e nat., 6, 602 – 607 (1927).
4. O’Regan D. Theory of Singular Boundary-Value Problems, World Scientific, Singapore (1994).
5. Palamides P., Sficas Y., and Staikos V. “Wazewski’s topological method for Caratheodory systems,” Math.
Systems Theory, 7, 243 – 261 (1984).
6. Wazewski T. “Sur un principle topologique del’examen de l’allure asymptotique des integrales des equations
differentielles ordinaires,” Ann. Soc. Polon. Math., 20, 279 – 313 (1947).
7. Ntouyas S. K. and Palamides P. K. “On the existence of positive solutions of nonlinear singular boundary-
value problems,” Math. Comput. Modelling (to appear).
8. Berge C. Espaces Topologiques. Functions Multivoques, Dunod, Paris (1966).
9. Palamides P. “Singular points of the consequent mapping,” Ann. math. pura ed appl., 129, 383 – 395 (1981).
10. Kuratowski K. Topology II, Academic Press, New York (1968).
Received 15.05.2001
|