Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach
In this paper, we prove the existence of three positive and concave solutions, by means of an elementary simple approach, to the 2th order two-point boundary-value problem x''(t) = α(t)f(t, x(t), x'(t)), 0 < t < 1, x(0) = x(1) = 0,. We rely on a combination of the analysis of...
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irk-123456789-1755892021-02-02T01:28:13Z Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach Palamides, P.K. Palamides, A.P. In this paper, we prove the existence of three positive and concave solutions, by means of an elementary simple approach, to the 2th order two-point boundary-value problem x''(t) = α(t)f(t, x(t), x'(t)), 0 < t < 1, x(0) = x(1) = 0,. We rely on a combination of the analysis of the corresponding vector field on the phase-space along with Kneser’s type properties of the solutions funnel and the Schauder’s fixed point theorem. The obtained results justify the simplicity and efficiency (one could study the problem with more general boundary conditions) of our new approach compared to the commonly used ones, like the Leggett – Williams Fixed Point Theorem and its generalizations. З допомогою елементарного пiдходу до двоточкової граничної задачi другого порядку x''(t) = α(t)f(t, x(t), x'(t)), 0 < t < 1, x(0) = x(1) = 0,. доведено iснування трьох додатних та вгнутих розв’язкiв. При цьому використано аналiз вiдповiдного векторного поля на фазовому просторi, кнессеровськi властивостi множини розв’язкiв та теорему Шаудера про нерухому точку. Отриманi результати пояснюють простоту та ефективнiсть розробленого нового пiдходу (можливiсть вивчати задачу з бiльш загальними граничними значеннями) в порiвняннi з методами, що використовувалися ранiше, наприклад теоремою Логгетт та Вiльямса про нерухому точку та її узагальнення 2012 Article Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach / P.K. Palamides, A.P. Palamides // Нелінійні коливання. — 2012. — Т. 15, № 2. — С. 233-243. — Бібліогр.: 21 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175589 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
In this paper, we prove the existence of three positive and concave solutions, by means of an elementary
simple approach, to the 2th order two-point boundary-value problem
x''(t) = α(t)f(t, x(t), x'(t)), 0 < t < 1, x(0) = x(1) = 0,.
We rely on a combination of the analysis of the corresponding vector field on the phase-space along with
Kneser’s type properties of the solutions funnel and the Schauder’s fixed point theorem. The obtained
results justify the simplicity and efficiency (one could study the problem with more general boundary
conditions) of our new approach compared to the commonly used ones, like the Leggett – Williams Fixed
Point Theorem and its generalizations. |
| format |
Article |
| author |
Palamides, P.K. Palamides, A.P. |
| spellingShingle |
Palamides, P.K. Palamides, A.P. Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach Нелінійні коливання |
| author_facet |
Palamides, P.K. Palamides, A.P. |
| author_sort |
Palamides, P.K. |
| title |
Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach |
| title_short |
Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach |
| title_full |
Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach |
| title_fullStr |
Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach |
| title_full_unstemmed |
Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach |
| title_sort |
triple positive solutions for a class of two-point boundary-value problems. a fundamental approach |
| publisher |
Інститут математики НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/175589 |
| citation_txt |
Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach / P.K. Palamides, A.P. Palamides // Нелінійні коливання. — 2012. — Т. 15, № 2. — С. 233-243. — Бібліогр.: 21 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT palamidespk triplepositivesolutionsforaclassoftwopointboundaryvalueproblemsafundamentalapproach AT palamidesap triplepositivesolutionsforaclassoftwopointboundaryvalueproblemsafundamentalapproach |
| first_indexed |
2025-07-15T12:54:07Z |
| last_indexed |
2025-07-15T12:54:07Z |
| _version_ |
1837717571075309568 |
| fulltext |
UDC 517.9
TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT
BOUNDARY-VALUE PROBLEMS. A FUNDAMENTAL APPROACH
ТРIЙКА ДОДАТНИХ РОЗВ’ЯЗКIВ ДЛЯ КЛАСУ ДВОТОЧКОВИХ
ГРАНИЧНИХ ЗАДАЧ. ЗАГАЛЬНИЙ ПIДХIД
P. K. Palamides
Naval Academy Greece
Piraeus, 451 10, Greece
e-mail: ppalam@otenet.gr
A. P. Palamides
Technol. Educat. Inst. Piraeus
Athens, Greece
e-mail: palamid@uop.gr
palamid@gmail.com
In this paper, we prove the existence of three positive and concave solutions, by means of an elementary
simple approach, to the 2th order two-point boundary-value problem
x
′′
(t) = α(t)f(t, x(t), x′(t)), 0 < t < 1,
x(0) = x(1) = 0.
We rely on a combination of the analysis of the corresponding vector field on the phase-space along with
Kneser’s type properties of the solutions funnel and the Schauder’s fixed point theorem. The obtained
results justify the simplicity and efficiency (one could study the problem with more general boundary
conditions) of our new approach compared to the commonly used ones, like the Leggett – Williams Fixed
Point Theorem and its generalizations.
З допомогою елементарного пiдходу до двоточкової граничної задачi другого порядку
x
′′
(t) = α(t)f(t, x(t), x′(t)), 0 < t < 1,
x(0) = x(1) = 0,
доведено iснування трьох додатних та вгнутих розв’язкiв. При цьому використано аналiз вiд-
повiдного векторного поля на фазовому просторi, кнессеровськi властивостi множини розв’яз-
кiв та теорему Шаудера про нерухому точку. Отриманi результати пояснюють простоту
та ефективнiсть розробленого нового пiдходу (можливiсть вивчати задачу з бiльш загальни-
ми граничними значеннями) в порiвняннi з методами, що використовувалися ранiше, наприклад
теоремою Логгетт та Вiльямса про нерухому точку та її узагальнення.
1. Introduction. In the past 20 years, there has been much attention focused on questions of
positive solutions for diverse nonlinear ordinary differential equation, difference equation,
and functional differential equation boundary-value problems without dependence on the first-
order derivative. To identify a few, we refer the reader to [15, 16] and references therein.
c© P. K. Palamides, A. P. Palamides, 2012
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2 233
234 P. K. PALAMIDES, A. P. PALAMIDES
The problem of existence of positive solutions for boundary-value problems generated by
applications in applied mathematics, physics, mechanics, chemistry, biology, etc. was extensively
studied in the literature, for details see the bibliography of this article. The main tools used
are fixed-point theorems, such as the well known Guo – Krasnosel’skii fixed-point theorem in
a cone. For example in [10, 14], this theorem plays an extremely important role. Fixed-point
theorems and their applications to nonlinear problems have a long history some of which is
documented in the recent book by Agarwal, O’Regan and Wong [1] which contains an excellent
summary of the current results and applications.
Concerning the question of multiplicity of solutions, most of the work done is based on the
well known Leggett – Williams Fixed Point Theorem [15]. Especially, an interest in triple soluti-
ons evolved from that theorem. Lately, several triple fixed-point theorems have been establi-
shed. See for example, Avery [2], Ren et al. [19] and Avery and Peterson [5] where such tri-
ple fixed-point theorems were applied in order to obtain triple solutions of certain boundary-
value problems for ordinary differential equations as well as for their discrete analogues. Also
recently, some new fixed-point theorems of cone expansion and compression of functional type,
due to Avery and Anderson (see [3]) have been proved. A different extension was given in [12].
Karakostas in [12], using the Leggett – Williams fixed point theorem in a cone proved the exi-
stence of triple positive solutions for a boundary-value problem governed by the φ-Laplacian
when the boundary conditions include nonlinear expressions at the end points. Also we refer to
the work in [13], where existence results for a countable set of solutions of a nonlocal boundary-
value problem were obtained.
All of them can be regarded as extensions of the Leggett – Williams fixed-point theorem and
Guo – Krasnosel’skii fixed-point theorem. We notice however that, the most of the works on
positive solutions was done under the assumption that the first order derivative is not involved
explicitly in the nonlinear term. On the other hand the multiplicity of positive solutions with
dependence on derivatives is considered in very few cases, see [11]. For other literature regar-
ding the existence of triple solutions, that are not necessary positive, we refer reader to [5].
In [6] Bai, Wang and Ge using a fixed-point theorem of Avery and Peterson [5], obtained
sufficient conditions for the existence of at least three positive solutions for the equation
x′′(t) + q(t)f(t, x(t), x′(t)) = 0
subject to some boundary conditions. In [7] Bai and Ge generalized the Leggett – Williams
fixed-point theorem and proved the existence of triple positive solutions for the second-order
two point boundary-value problem
x′′(t) + f(t, x(t), x′(t)) = 0, 0 < t < 1, x(0) = x(1) = 0. (E)
Bai and Ge assumed except the positivity of the nonlinearity, that there exist constants r2 ≥
≥ 4b > b > r1 > 0 and L2 ≥ L1 > 0 such that 8b ≤ min{r2, L2}, and
f(t, u, v) < min{8r1, 2L1}, (t, u, v) ∈ [0, 1]× [0, r1]× [−L1, L1],
f(t, u, v) > 16b, (t, u, v) ∈
[
1
4
,
3
4
]
× [b, 4b]× [−L2, L2], (A0)
f(t, u, v) ≤ min{8r2, 2L2}, (t, u, v) ∈ [0, 1]× [0, r2]× [−L2, L2].
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT . . . 235
Motivated, by the above work and especially by the last two papers, we obtain sufficient
conditions for the existence of triple positive solutions of the simple boundary-value problem
(E). Although the conditions are very similar (actually a little bit more strict) than those in [7],
the proposed approach is quite different and very simple as it is of geometrical nature. Actually,
we use the flow generated by (E) (see Remark 1) and reduces boundary-value problem to the
algebraic problem of determining the initial values of solutions from the boundary conditions.
It turns out that such an approach, applied to a broad class of boundary-value problem, consi-
derably simplifies the proofs. We use a combination of the Schauder’s fixed point theorem, the
associating Green function and the Kneser’s property (the cross-section of the solutions funnel
is a continuum) under the light of the associated vector field.
We assume that the nonlinear function f is continuous and
f(t, u, v) ≥ 0, for almost all t ∈ [0, 1], and any u ≥ 0 and v ∈ R. (A1)
As we mentioned above, the presence of v in f(t, u, v) causes some difficulties. We overcome
this predicament, by modifying below suitably the assumption (A0).
Remark 1. We notice here that the differential equation (E) defines a vector field, the
properties of which will be crucial for our study. More specifically, let’s look at the (x, x′) face
semi-plane (x > 0). By the sign condition on f, we immediately see that x′′ ≤ 0. Thus any
trajectory (x(t), x′(t)), t ≥ 0, emanating from the semi-line
E0 := {(x, x′) : x′ > 0, x = 0}
"evolves"in a natural way (when x′(t) > 0), toward the positive x-semi-axis and then trends
toward the negative x′-semi-axis. Lastly, by setting a certain growth rate on f (say superlineari-
ty) we can control the vector field, so that some trajectory satisfies the given boundary condition
x(1) = 0
at the time t = 1. These properties will be referred as "The nature of the vector field” throu-
ghout the rest of paper.
So, the technique presented here is different to that given in the above mentioned papers,
but very close to the one given in [18]. Actually, we rely on the above "nature of the vector
field"and on the simple shooting method. Finally, we refer for completeness the well-known
Kneser’s theorem (see for example the Copel’s text-book [8]) as well as the Schauder’s fixed
point theorem.
Theorem 1. Consider a system
x′ = f(t, x, x′), (t, x, x′) ∈ Ω := [α, β]× R2n, (1.1)
where the function f is continuous. Let Ê0 be a continuum (compact and connected) set in Ω0 :=
:= {(t, x, x′) ∈ Ω : t = α} and let X (Ê0) be the family of all solutions of (1.1) emanating from
Ê0. If any solution x ∈ X (Ê0) is defined on the interval [α, τ ], then the set (cross-section at the
point τ)
X (τ ; Ê0) :=
{
(x(τ), x′(τ)) : x ∈ X (Ê0)
}
is a continuum in R2n.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
236 P. K. PALAMIDES, A. P. PALAMIDES
Theorem 2 (Schauder’s FPT). If X is a Banach space, C ⊆ X is nonempty, bounded, closed
and convex and T : C → C is completely continuous, then T has a fixed point.
2. Main results. We are concerned with the existence of triple positive solutions for the
second-order two point boundary-value problem (E), where f : [0, 1]× [0,∞)×R → [0,∞) is
a continuous function.
Let X = C1{0, 1] be endowed with the ordering x ≤ y if x(t) ≤ y(t), for all t ∈ [0, 1],
and the maximum norm, ‖x‖ = max{max0≤t≤1 |x(t)|,max0≤t≤1 |x′(t)|}. From the fact x′′(t) =
= −f(t, x, x′) ≤ 0, we know that x is concave on [0, 1]. So, we define the cone P ⊆ X as
P = {x ∈ X| x(t) ≥ 0, x is concave on [0, 1]} ⊆ X.
Denote by G(t, s) the Green’s function for the boundary-value problem
x′′(t) = 0, 0 < t < 1,
x(0) = x(1) = 0,
then G(t, s) ≥ 0 for 0 ≤ t, s ≤ 1 and
G(t, s) =
{
t(1− s), 0 ≤ t ≤ s ≤ 1,
s(1− t), 0 ≤ s ≤ t ≤ 1.
Let
δ = min
3/4∫
1/4
G
(
1
4
, s
)
ds,
3/4∫
1/4
G
(
3
4
, s
)
ds
=
1
16
.
We set now
u∗ =
0, u ≤ 0,
u, 0 < u ≤ r2,
r2, r2 < u,
and v∗ =
−L2, v ≤ −L2,
v, −L2 < v ≤ L2,
L2, L2 < v,
and consider the modified boundary-value problem
x′′(t) + g(t, x(t), x′(t)) = 0, 0 < t < 1, x(0) = x(1) = 0, (2.1)
where
g(t, u, v) = f(t, u∗, v∗), (t, u, v) ∈ [0, 1]× R2.
Remark 2. It is obvious that the map g is nonnegative and satisfies the conditions (A1) – (A3)
below, on the entire region [0, 1]× R2.
Theorem 3. Suppose that there exist constants L1, L2 and r2 > b ≥ r1 > L1 > 0, such that
352
5
b ≥ L2 ≥
32
15
b, 8b ≥ 1, and the following assumptions hold:
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT . . . 237
(A1) f(t, u, v) < min{8r1, 2L1}, (t, u, v) ∈ [0, 1]× [0, r1]× [−L1, L1];
(A2) f(t, u, v) > 24b+
3L2
4
, (t, u, v) ∈
[
1
4
,
3
4
]
× [b, 4b]× [−L2, L2];
(A3) f(t, u, v) ≤ min{8r2, L2}, (t, u, v) ∈ [0, 1]× [0, r2]× [−L2, L2].
Then, the boundary-value problem (E) has at least three positive and concave solutions x1,
x2 and x3 satisfying
max
0≤t≤1
x1(t) ≤ r1, max
0≤t≤1
x′1(t) ≤ L1;
b < min
1/4≤t≤3/4
x2(t) < max
0≤t≤1
x2(t) < r2, max
0≤t≤1
|x′2(t)| < L2;
max
0≤t≤1
x3(t) ≤ 4b, max
0≤t≤1
|x′3(t)| ≤ L2.
Proof. It is well known that the Problem (E) has a solution x = x(t) if and only if x solves
the operator equation
x(t) = Tx(t) =
1∫
0
G(t, s)f(s, x(s), x′(s)) ds
and that T : P → X is completely continuous.
Consider the set
Ω =
{
x ∈ P : max
0≤t≤1
x(t) ≤ r1 and 0 ≤ max
0≤t≤1
|x′(t)| ≤ L1
}
.
By the positivity of the functions G(t, s) and f(t, u, v), we immediately get that T : P → P.
Furthermore we will show that T (Ω) ⊆ Ω. Indeed, for any x ∈ Ω, by the assumption (A1), we
get
max
0≤t≤1
Tx(t) = max
0≤t≤1
1∫
0
G(t, s)f(s, x(s), x′(s)) ds ≤ 8r1 max
0≤t≤1
1∫
0
G(t, s) ds = r1.
In view of Remark 1 and the inequality
(Tx)′′(t) = −f(t, x(t), x′(t)) ≤ 0, 0 ≤ t ≤ 1,
the map (Tx)′(t) is decreasing and thus max0≤t≤1(Tx)′(t) = max{|(Tx)′(0)|, |(Tx)′(1)|}. Con-
sequently,
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
238 P. K. PALAMIDES, A. P. PALAMIDES
max
0≤t≤1
|(Tx)′(t)| = max
0≤t≤1
∣∣∣∣∣∣−
t∫
0
sf(s, x(s), x′(s)) ds+
1∫
t
(1− s)f(s, x(s), x′(s)) ds
∣∣∣∣∣∣ =
= max
1∫
0
sf(s, x(s), x′(s)) ds,
1∫
0
(1− s)f(s, x(s), x′(s)) ds
≤
≤ 2L1 max
1∫
0
s ds,
1∫
0
(1− s) ds
≤ 2L1
1
2
= L1.
Thus T (Ω) ⊆ Ω. In addition, the set Ω is nonempty, bounded, closed and convex. Thus by
the Schauder’s fixed point Theorem 2, we conclude the existence of a solution x1 ∈ Ω of the
boundary-value problem (E).
Consider now the boundary-value problem (2.1), as well as the initial value problem
x′′(t) + g(t, x(t), x′(t)) = 0, 0 < t < 1, x(0) = 0, x′(0) = ξ ≥ 0. (2.2)
For ξ = L2, via the Taylor’s theorem, the assumption (A3) and the definition of the modification
g, we get a τ ∈ [0, 1] such that
x(1) = x′(0)− 1
2
g
(
τ, x(τ), x′(τ)
)
≥ L2 −
L2
2
≥ 0. (2.3)
Similarly for ξ = 4b+
L2
8
and any x ∈ X ([ξ, L2]) we have
x
(
1
4
)
=
(
1
4
)
x′(0)−
(
1
2
) (
1
4
)2
g
(
τ, x(τ), x′(τ)
)
≥
≥
(
1
4
)(
4b+
L2
8
)
− 1
2.16
L2 = b. (2.4)
Let x ∈ X(ξ). Then we obtain an α ∈
[
0,
1
4
)
such that x(α) = b. We assert that
x
(
3
4
)
< b. (2.5)
Indeed, supposing on the contrary that x
(
3
4
)
≥ b, we get via the assumption (A2), the
contradiction
x
(
3
4
)
= x(α) +
(
3
4
− α
)
x′(α)−
(
1
2
)(
3
4
− α
)2
g
(
τ, x(τ), x′(τ)
)
<
< b+
3
4
(
4b+
L2
8
)
− 1
2
(
1
2
)2(
24b+
3L2
4
)
= b.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT . . . 239
On the other hand, we have for any x ∈ X (L2),
x
(
3
4
)
=
3
4
x′(0)− 1
2
(
3
4
)2
g
(
τ, x(τ), x′(τ)
)
≥ 3
4
L2 −
1
2
(
3
4
)2
2L2 ≥ b,
given that L2 ≥
32
15
b.
Thus, by the Kneser’s property and (2.4), we obtain a point ξ0 ∈
(
4b+
L2
8
, L2
)
and x ∈
∈ X (ξ0) such that
x(t) ≥ b, t ∈
(
1
4
,
3
4
)
and x
(
3
4
)
= b.
Moreover, by the monotonicity of x′(t), we conclude that
x′
(
3
4
)
= x′
(
1
4
)
− 1
2
g
(
τ, x(τ), x′(τ)
)
≤ L2 −
1
2
(
24b+
3L2
2
)
=
1
4
L2 − 12b.
Consequently, since L2 ≤
352
5
b, the above solution x ∈ X (ξ0) satisfies in addition
x(1) = x
(
3
4
)
+
1
4
x′
(
3
4
)
− 1
2
(
1
4
)2
g
(
τ, x(τ), x′(τ)
)
≤
≤ b+
1
4
(
1
4
L2 − 12b
)
− 1
2
(
1
4
)2(
24b+
3L2
4
)
≤ 0. (2.6)
Applying now the Kneser’s Theorem 1, in view of (2.3) and (2.6), we obtain another soluti-
on x2 ∈ X ((ξ0, L2)) ⊂ X
([
4b+
L2
8
, L2
])
of the boundary-value problem (2.1). Following
similar resonance, as for the previous solution x1(t), we may easily prove that
0 ≤ x2(t) ≤ r2 and − L2 ≤ x′2(t) ≤ L2, 0 ≤ t ≤ 1.
Thus x2(t), 0 ≤ t ≤ 1, is actually a solution of the initial boundary-value problem (E).
Furthermore, we assert that
min
1/4≤t≤3/4
x2(t) ≥ b. (2.7)
Indeed, since (2.4) is true for any x ∈ X
([
4b+
L2
8
, L2
])
, we obviously have x2
(
1
4
)
> b
and assuming on the contrary that there exists β ∈
(
1
4
,
3
4
)
such that x2(β) = b and of course
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
240 P. K. PALAMIDES, A. P. PALAMIDES
x2
(
3
4
)
< b, we get
x2(β) =
1∫
0
G(β, s)f(s, x2(s), x
′
2(s))ds ==
β∫
0
G(β, s)f(s, x2(s), x
′
2(s)) ds+
+
1∫
β
G(β, s)f(s, x2(s), x
′
2(s)) ds ≥
β∫
1/4
G(β, s)f(s, x2(s), x
′
2(s)) ds >
>
β∫
1/4
G
(
1
4
, s
)
f(s, x2(s), x
′
2(s)) ds ≥
1
16
(
24b+
3L2
4
)
> b,
a contradiction.
We seek now for the third solution x3(t). The existence of x3(t) follows directly by (2.6) and
since, by Remark 2, for any x ∈ X (L1),
x(1) = x′(0)− 1
2
g
(
τ, x(τ), x′(τ)
)
≥ 4r1 −
1
2
8r1 = 0.
The application of the condition (A1) is possible, since for any x ∈ X (L1),(
x(t), x′(t)
)
∈ [0, r1]× [−L1,L1].
In fact, let’s assume that there exists a τ ∈ [0, 1] such that
0 ≤ x(t) ≤ x(τ) = r1, 0 ≤ t ≤ τ,
and then, in view of the vector field, we know that
0 ≤ x′(t) ≤ L1, 0 ≤ t ≤ τ.
Consequently, we get the contradiction
r1 = x(τ) = τL1 −
1
2
τ2g
(
τ, x(τ), x′(τ)
)
< τL1 < L1.
Then the Kneser’s theorem may be applied once again to get a solution x3 ∈ X ((L1, ξ0)).
We notice that since L1 < 4b+
L2
8
, clearly x3 is different than the other two obtained solutions
x1 and x2.
We replace now the set Ω by the next one:
Ω∗ =
{
x ∈ P : max
0≤t≤1
x(t) ≤ r2 and − L2 ≤ max
0≤t≤1
x′(t) ≤ L2
}
.
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TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT . . . 241
Then obviously T (Ω∗) ⊆ Ω∗, and by the formula
x(t) =
1∫
0
G(t, s)f(s, x(s), x′(s)) ds,
we conclude that x3 is actually a solution of the original boundary-value problem (E).
Theorem 3 is proved.
Remark 3. In view of (2.5), we may choose the initial value ξ0 as a minimal one, in the sense
that any solution x3 ∈ X ((L1, ξ0)) of the boundary-value problem (E) satisfies the inequalities
max
t∈[0,1]
x3(t) ≤ L2 and min
x∈[ 14 ,
3
4 ]
x3(t) ≤ b.
Because of the additional relation
min
x∈[ 14 ,
3
4 ]
x3(t) ≥
1
4
max
t∈[0,1]
x3(t)
we immediately conclude that maxt∈[0,1] x3(t) ≤ 4b.
Finally we present an example to illustrate our results.
Example 1. Consider the boundary-value problem (E), where
f(t, u, v) =
u3
4
+
( v
3200
)2
, for u ≤ 1
32
, |v| ≤ 70,
5114.6u− 159.81 +
( v
3200
)2
, for
1
32
≤ u ≤ 1
24
, |v| ≤ 70,
53.338 +
( v
3200
)2
, for
1
24
≤ u ≤ 4
24
, |v| ≤ 70,
53.29 + 0.06u+
( v
3200
)2
, for
4
24
≤ u ≤ 120, |v| ≤ 70.
Choose L1 =
1
64
, r1 =
1
32
, b =
1
24
, r2 = 120 and L2 = 70 ≤ 352
5
. Then the nonlinearity
satisfies the assumptions (A1) – (A3), namely
f(t, u, v) ≤ 1
4
(
1
32
)3 1
32
+
(
640
3200
)2
< min{8r1, 2L1} ≤
1
4
,
(t, u, v) ∈ [0, 1]×
[
0,
1
32
]
×
[
− 1
64
,
1
64
]
;
f(t, u, v) ≥ 53.338 > 24b+
3L2
4
= 53.25,
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
242 P. K. PALAMIDES, A. P. PALAMIDES
(t, u, v) ∈
[
1
4
,
3
4
]
×
[
1
24
,
1
6
]
× [−70, 70];
f(t, u, v) ≤ 53.338 ≤ min{8r2, L2} = 70,
(t, u, v) ∈ [0, 1]× [0, 120]× [−70, 70].
An application of Theorem 3, guarantees the existence of three positive solutions x1, x2, x3
such that
0 ≤ x1(t) ≤
1
32
, − 1
64
≤ x′1(t) ≤
1
64
, 0 ≤ t ≤ 1,
1
24
≤ min
1
4
≤t≤ 3
4
x2(t) ≤ max
0≤t≤1
x2(t) ≤ 120, −70 ≤ x′1(t) ≤ 70, 0 ≤ t ≤ 1,
0 ≤ x3(t) ≤
4
24
, −70 ≤ x′1(t) ≤ 70, 0 ≤ t ≤ 1.
Remark 4. The proposed method except its simplicity, has several advantages. For example
the well-known upper-lower solutions method requires a Nagumo type growth rate in the thi-
rd variable of the nonlinearity [21]. In order to avoid such a restriction, in several previous
papers [1 – 5, 17, 20] the nonlinearity does not depends on it. Mainly, we may study more general
boundary value problems, as we indicate in the next remark (see also [9]).
Remark 5. By the above analysis, it is obvious that we are able to replace the boundary
conditions in (E), for example, by the next ones
αx(0)− βx′(0) = 0, γx(1) + δx′(1) = 0,
under the assumption
αb− βL2 ≥ 0 ≥ γb− δL2.
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Received 14.02.11,
after revision — 28.11.11
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
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