Systems of φ-Laplacian three-point boundary-value problems on the positive half-line
We study the existence of positive solutions to boundary-value problems for two systems of two second-order nonlinear three-point φ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions an...
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| author | Djebali, S. Mebarki, K. Дебалі, С. Мебаркі, К. |
| author_facet | Djebali, S. Mebarki, K. Дебалі, С. Мебаркі, К. |
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| description | We study the existence of positive solutions to boundary-value problems for two systems of two second-order nonlinear three-point φ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces. |
| first_indexed | 2026-03-24T02:18:40Z |
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UDC 517.9
S. Djebali (École Normale Supérieure, Algiers, Algeria),
K. Mebarki (Béjaia Univ., Algeria)
SYSTEMS OF φ-LAPLACIAN THREE-POINT
BOUNDARY-VALUE PROBLEMS ON THE POSITIVE HALF-LINE
СИСТЕМИ ТРИТОЧКОВИХ ГРАНИЧНИХ ЗАДАЧ
ДЛЯ φ-ЛАПЛАСIАНА НА ДОДАТНIЙ ПIВОСI
We study the existence of positive solutions to boundary-value problems for two systems of two second-order nonlinear
three-point φ-Laplacian equations defined on the positive half-line. The nonlinearities may change sign, exhibit time
singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we
prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces.
Вивчається iснування додатних розв’язкiв граничних задач для двох систем двох нелiнiйних триточкових φ-
лапласових рiвнянь другого порядку, що визначенi на додатнiй пiвосi. Нелiнiйностi можуть змiнювати знак, мати
часовi сингулярностi на початку координат та залежати як вiд розв’язкiв, так i вiд їх перших похiдних. Теорiю
нерухомих точок застосовано для доведення деяких результатiв щодо iснування нетривiальних додатних розв’язкiв
на вiдповiдних конусах в деяких звaжених банахових просторах.
1. Introduction. In this paper, we first consider the following general system of nonlinear second-
order φ-Laplacian three-point boundary-value problem posed on the positive half-line:
−(φ(y′1))
′(t) = m1(t)f1(t, y1(t), y2(t), y
′
1(t), y
′
2(t)), t ∈ I,
−(φ(y′2))
′(t) = m2(t)f2(t, y1(t), y2(t), y
′
1(t), y
′
2(t)), t ∈ I,
y1(0) = αy′1(η), lim
t→+∞
y′1(t) = 0,
y2(0) = αy′2(η), lim
t→+∞
y′2(t) = 0,
(1.1)
where α ≥ 0 and η > 0 are real parameters. For i = 1, 2, the nonlinear functions fi : R+ ×
× (R+)2 × R2 → R+ are nonnegative while mi : I → R+ are nonnegative, continuous functions
that are allowed to have a singularity at the origin t = 0. The interval I := (0,+∞) denotes the set
of positive real numbers and R+ : = [0,+∞).
The nonlinear operator of derivation φ : R → R is an increasing homeomorphism such that
φ(0) = 0 (φ is not necessarily odd), satisfies
|φ−1(x)| ≤ φ−1(|x|) ∀x ∈ R (1.2)
and is submultiplicative on R+, i.e.,
φ(αβ) ≤ φ(α)φ(β) ∀α, β ∈ R+. (1.3)
This implies that the inverse φ−1 is supermultiplicative:
φ−1(αβ) ≥ φ−1(α)φ−1(β) ∀α, β ∈ R+. (1.4)
φ extends the usual multiplicative p-Laplacian nonlinear operator φ(s) = |s|p−2s, p > 1.
In the second part of the paper, we focus on a system with a particular structure, namely
c© S. DJEBALI, K. MEBARKI, 2015
1626 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1627
−(φ(y′))′(t) = m(t)f(t, x(t)), t ∈ I,
−(φ(x′))′(t) = m(t)g(t, y(t)), t ∈ I,
y(0) = αy′(η), lim
t→+∞
y′(t) = 0,
x(0) = αx′(η), lim
t→+∞
x′(t) = 0,
(1.5)
where the nonnegative functions f, g : I×R+ → R are continuous but may change sign; the function
m : I → R+ is continuous with possible singularity at t = 0.
Singular differential systems arise in many branches of applied mathematics and physics such as
gas dynamics, Newtonian fluid mechanics, nuclear physics and have attracted many authors during
the last couple of years (see, e.g., [1, 8, 9, 12, 13]). The unknowns may represent a density, a
temperature, a velocity, . . . hence positivity of solutions is required.
Recently there has been so much work devoted to the investigation of positive solutions to systems
of boundary-value problems on finite intervals of the real line (see [11, 14, 18, 19] and the references
therein). Also, several methods have been employed to deal with boundary-value problems on the
positive half-line; we quote fixed point theorems in special Banach spaces, the fixed point index
theory on positive cones of functional Banach spaces, the upper and lower solutions techniques, and
the monotone iterative techniques [4, 18].
In the particular case of second-order differential equations corresponding to φ = Id, system (1.1)
has been widely studied in the literature. In 2009, by using the Krasnosel’skii fixed point theorem, Xi,
Jia, and Ji [15] studied the existence of positive solutions to the following boundary-value problem
with integral boundary conditions on the half-line:
y′′1(t) + f1(t, y1(t), y2(t)) = 0, t ∈ I,
y′′2(t) + f2(t, y1(t), y2(t)) = 0, t ∈ I,
y1(0) = 0, y′1(+∞) =
+∞∫
0
g1(s)y1(s)ds,
y2(0) = 0, y′2(+∞) =
+∞∫
0
g2(s)y2(s)ds.
In the same year, Zhang [17] investigated the existence of positive solutions for a singular multipoint
system of second-order differential equations posed on an infinite interval; he used the Mönch fixed
point theorem and a monotone iterative technique; the system considered reads:
x′′(t) + f(t, x(t), x′(t), y(t), y′(t)) = 0, t ∈ I,
y′′(t) + g(t, x(t), x′(t), y(t), y′(t)) = 0, t ∈ I,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1628 S. DJEBALI, K. MEBARKI
x(0) =
m−2∑
i=1
αix(ξi), x′(+∞) = x∞,
y(0) =
m−2∑
i=1
βix(ξi), y′(+∞) = y∞.
Motivated by the papers [4, 15, 17] and some related results for singular three-point φ-Laplacian
boundary-value problems posed on the positive half-line [6], we are first concerned in this paper with
the existence of positive solutions for the singular system (1.1). This system is different from those
considered in [4, 15, 17]. Firstly, the singular system (1.1) is posed on an infinite interval. Secondly,
the nonlinear operator of derivation extends the model case of the p-Laplacian nonlinear operator.
Finally, the function m presents a time-singularity at t = 0.
In this paper, we first prove an existence result of nontrivial nonnegative solutions for system
(1.1) under suitable conditions on the positive functions fi and mi, i = 1, 2, and thus we extend
some of the above works. The boundary-value problem is formulated as a fixed point problem of
a nonlinear operator. The index fixed point theory on cones of Banach spaces is then employed.
This is the object of Section 2. Then Section 3 is devoted to investigating the singular system (1.5)
where the nonlinearities are allowed to change sign. We prove the existence of positive solutions on
a translate of a cone in a Banach space. Each existence result is illustrated by means of an example
of application.
2. The general case of system (1.1). Some preliminaries including the main assumptions and
a compactness criterion are presented in Subsection 2.1. We construct a special cone in the Banach
space of continuously differentiable functions with vanishing derivatives at positive infinity; then we
study the properties of a corresponding fixed point operator. Subsection 2.2 is devoted to proving our
main existence theorem. We end this section with an example of application in Subsection 2.3. By a
nonnegative solution, we mean a couple of functions (y1, y2) ∈ C1[0,+∞) × C1[0,+∞) such that
φ(y′i) ∈ C1(0,+∞) with yi(t) ≥ 0 on [0,+∞) for i = 1, 2 and such that the equations in (1.1) are
satisfied.
2.1. The general framework. Consider the space
X =
y = (y1, y2) | yi ∈ C1(R+,R), yi(0) = αy′i(η),
and lim
t→+∞
y′i(t) = 0, i = 1, 2
.
This is a Banach space with the norm
‖y‖ = ‖y1‖+ ‖y2‖, where ‖yi‖ = sup
t∈R+
|y′i(t)|, i = 1, 2.
In order to transform problem (1.1) into a fixed point problem, the following auxiliary lemma is
needed; the proof which is immediate is skipped.
Lemma 2.1. Let v ∈ L1(I). Then u ∈ C1(I) is a solution of
−(φ(u′))′(t) = v(t), t ∈ I,
u(0) = αu′(η), lim
t→+∞
u′(t) = 0
(2.1)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1629
if and only if
u(t) = C +
t∫
0
φ−1
+∞∫
s
v(τ)dτ
ds, t ≥ 0, (2.2)
where C = αφ−1
(∫ +∞
η
v(τ)dτ
)
.
In order to study the compactness of the fixed point operator, we first recall a classical result.
Given the Banach space
Cl([0,+∞),R) = {x ∈ C([0,+∞),R) | lim
t→+∞
x(t) exists}
with the norm ‖x‖l = supt∈[0,+∞) |x(t)|, we have the following lemma.
Lemma 2.2 [3, p. 62]. A set M ⊆ Cl(R+,R) is relatively compact if the following conditions
hold:
(a) M is uniformly bounded in Cl(R+,R).
(b) The functions belonging to M are almost equicontinuous on R+, i.e., equicontinuous on
every compact interval of R+.
(c) The functions from M are equiconvergent, that is, given ε > 0, there corresponds T (ε) > 0
such that |x(t)− x(∞)| < ε, for all t ≥ T (ε) and x ∈M.
From this lemma, we derive the following lemma.
Lemma 2.3. Let M ⊆ X. Then M is relatively compact in X if the following conditions hold:
(a) M is uniformly bounded in X.
(b) The functions belonging to the set A = {z | z(t) = y′(t), y ∈M} are almost equicontinuous
on R+.
(c) The functions from A are equiconvergent at +∞.
Now, consider the following hypotheses where i = 1, 2:
(G1) The functions fi : R+ × (R+)2 × R2 → R+ are continuous and when y1, y2, z1, z2 are
bounded fi(t, (t+ α)y1, (t+ α)y2, z1, z2) are bounded on [0,+∞).
(G2) The functionsmi : I −→ R+ are continuous and do not vanish identically on any subinterval
of I. They may be singular at t = 0 but are integrably bounded, that is
Ai :=
+∞∫
0
mi(s)ds <∞.
2.2. Fixed point formulation. Let P be the nonnegative cone defined by
P = {y ∈ X | y(t) ≥ 0 ∀ t ≥ 0} .
By y ≥ 0, it is meant yi ≥ 0 for each i = 1, 2. We start with a simple observation:
Remark 2.1. If, for i = 1, 2, yi ∈ P, then the mean value theorem yields
sup
t≥0
yi(t)
t+ α
≤ ‖yi‖.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1630 S. DJEBALI, K. MEBARKI
Now let Ω ⊂ X be a bounded subset. Then, there exists M > 0 such that ‖y‖ ≤ M, for all
y = (y1, y2) ∈ Ω. According to Assumption (G1), for i = 1, 2, let
S
(i)
M = sup {fi(t, (t+ α)y1, (t+ α)y2, z1, z2) ,
for t ≥ 0, (y1, y2) ∈ [0,M ]2, and (|z1|, |z2|) ∈ [0,M ]2
}
.
Then for every t ≥ 0, |y′i(t)| ≤M and thus 0 ≤ yi(t)
t+ α
≤M, i = 1, 2. Hence
+∞∫
0
mi(s)fi(s, y1(s), y2(s), y
′
1(s), y
′
2(s))ds =
=
+∞∫
0
mi(s)fi
(
s,
(s+ α)y1(s)
s+ α
,
(s+ α)y2(s)
s+ α
, y′1(s), y
′
2(s)
)
ds ≤
≤ S(i)
M
+∞∫
0
mi(s)ds <∞, i = 1, 2.
As a consequence, for i = 1, 2, the integrals
+∞∫
0
mi(s))fi(s, y1(s), y2(s), y
′
1(s), y
′
2(s))ds
are convergent. From Lemma 2.1, we know that the boundary-value problem (1.1) is equivalent to
yi(t) = Ci +
t∫
0
φ−1
+∞∫
s
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
)
dτ
ds,
where
Ci = αφ−1
+∞∫
η
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
)
dτ
, i = 1, 2.
Let Ω ⊂ X be a bounded subset and define the integral operators
Fi : Ω ∩ P −→ C1(R+,R+),
y = (y1, y2) 7−→ Fiy(t),
(2.3)
where, for i = 1, 2,
Fiy(t) = Ci +
t∫
0
φ−1
+∞∫
s
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
)
dτ
ds.
Next, we study the properties of the fixed point operator F defined by
Fy(t) = (F1y(t), F2y(t)) .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1631
Lemma 2.4. Under Assumptions (G1) and (G2) the operator F maps the set Ω ∩ P into P.
Proof. First we show that F : Ω∩ P→ X is well defined. Let y = (y1, y2) ∈ Ω∩ P. Then there
exists M > 0 such that ‖y‖ ≤ M, for all y = (y1, y2) ∈ Ω ∩ P. By Assumptions (G1) and (G2), we
have, for i = 1, 2,
+∞∫
0
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
)
dτ ≤ AiS(i)
M .
Hence for every t ∈ [0,+∞)
t∫
0
φ−1
+∞∫
s
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
)
dτ
ds ≤
t∫
0
φ−1
(
AiS
(i)
M
)
ds <∞.
In addition, we can easily prove that for all y ∈ Ω ∩ P,
Fiy ∈ C1([0,+∞),R), Fiy(t) ≥ 0, t ∈ [0,+∞),
Fiy(0) = Ci = αφ−1
+∞∫
η
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
) = α(Fiy)′(η),
and
lim
t→+∞
(Fiy)′(t) = lim
t→+∞
φ−1
+∞∫
t
mi(τ)fi
(
τ, y1(τ), y2(τ), y′1(τ), y′2(τ)
) = φ−1(0) = 0,
ending the proof of the lemma.
The proof of the following lemma is a consequence of Lemma 2.3. The proof is omitted.
Lemma 2.5. Assume that (G1) and (G2) hold. Then, the mapping F : Ω ∩ P→ P is completely
continuous.
2.3. Existence result. Two auxiliary lemmas are needed in this section. More details concerning
the theory of the fixed point index on cones of Banach spaces can be found for instance in [5, 10, 16].
Lemma 2.6. Let Ω be a bounded open subset of a real Banach space E, P a cone of E, θ ∈ Ω,
and A : Ω ∩ P → P a completely continuous operator. Suppose that
Ax 6= λx ∀x ∈ ∂Ω ∩ P, λ ≥ 1.
Then the fixed point index i (A,Ω ∩ P,P) = 1.
Lemma 2.7. Let Ω be a bounded open set in a real Banach space E, P be a cone of E, and A :
Ω ∩ P → P be a completely continuous mapping. Assume that
Ax 6≤ x ∀x ∈ ∂Ω ∩ P.
Then the fixed point index i (A,Ω ∩ P,P) = 0.
We are now in position to prove the main existence result of this section.
Theorem 2.1. Assume that the following assumptions hold for i = 1, 2 :
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1632 S. DJEBALI, K. MEBARKI
(G3)
0 ≤ fi(t, y1, y2, z1, z2) ≤
≤ ai(t)
(
φ
(
y1
t+ α
)
+ φ
(
y2
t+ α
))
+ bi(t) (φ(z1) + φ(z2)) + ci(t),
for all (t, y1, y2, z1, z2) ∈ I × (R+)2 × R2, with
miai, mibi, mici ∈ L1(R+).
(G4) There exists R > 0 such that
φ−1 (2 (|miai|L1 + |mibi|L1)φ(R) + |mici|L1) < R/2. (2.4)
(G5) There exist 0 < γ < δ such that for all (t, y1, y2, z1, z2) ∈ [γ, δ]× (R+)2 × R2, we have
fi(t, y1, y2, z1, z2) ≥ gi
(
t,
y1
t+ α
,
y2
t+ α
)
,
where gi ∈ C([γ, δ]× (R+)2) satisfies
lim inf
y1+y2→0
min
t∈[γ,δ]
gi
(
t,
y1
t+ α
,
y2
t+ α
)
φ(y1 + y2)
≥ `i (2.5)
with constants `i satisfying φ−1
(
`i
∫ δ
γ
mi(t)dt
)
> 1/γ.
Then, problem (1.1) has at least one nonnegative solution y = (y1, y2) such that
0 < ‖y‖ < R .
Remark 2.2. It is easily seen that Assumption (G3) is a substitution of Assumptions (G1), (G2)
and Lemmas 2.4 and 2.5 still remain valid.
Proof. Define the open ball
ΩR = {y ∈ X : ‖y‖ < R}.
From Lemma 2.5, F : ΩR ∩ P→ P is completely continuous.
Claim 1. Fy 6= λy, for y ∈ ∂ΩR ∩ P and λ ≥ 1. To see this, let y = (y1, y2) ∈ ∂ΩR ∩ P. By
Assumption (G3) and Remark 2.1, the following estimates hold for positive t and i = 1, 2:
|(Fiy)′(t)| = φ−1
+∞∫
t
m(τ)fi(τ, y1(τ), y2(τ), y′1(τ), y′2(τ))dτ
≤
≤ φ−1
+∞∫
0
mi(τ)
(
ai(τ)
(
φ
(
y1(τ)
τ + α
)
+ φ
(
y2(τ)
τ + α
))
+
+bi(τ)
(
φ(y′1(τ)) + φ(y′2(τ))
)
+ ci(τ)
)
dτ
)
≤
≤ φ−1 (|miai|L1 [φ(‖y1‖) + φ(‖y2‖)] + |mibi|L1 [φ(‖y1‖) + φ(‖y2‖)] + |mici|L1) ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1633
≤ φ−1 (2|miai|L1φ(‖y‖) + 2|mibi|L1φ(‖y‖) + |mici|L1) ≤
≤ φ−1 (2 (|miai|L1 + |mibi|L1)φ(R) + |mici|L1) <
R
2
.
Taking the least upper bound over t yields ‖Fiy‖ < ‖y‖/2, for all y ∈ ∂ΩR ∩ P and all i = 1, 2.
Moreover
‖Fy‖ = ‖F1y‖+ ‖F2y‖ < ‖y‖ ∀ y ∈ ∂ΩR ∩ P. (2.6)
As a consequence
Fy 6= λy ∀ y ∈ ∂ΩR ∩ P ∀λ ≥ 1. (2.7)
If not there would exist some y0 ∈ ∂ΩR ∩ P and λ0 ≥ 1 such that Fy0 = λ0y0. Hence ‖Fy0‖ =
= λ0‖y0‖ ≥ ‖y0‖, contradicting (2.6). This implies that (2.7) holds. Therefore, Lemma 2.6
guarantees
i (F,ΩR ∩ P,P) = 1. (2.8)
Finally (2.8) and the existence property of the fixed point index imply that the operator F has a fixed
point y = (y1, y2) which belongs to ΩR ∩ P with 0 ≤ ‖y‖ < R.
Claim 2. By (2.5), there exists an r0 > 0 such that for i = 1, 2, we have
gi
(
t,
y1
t+ α
,
y2
t+ α
)
≥ `iφ(y1 + y2), for 0 ≤ y1 + y2 ≤ r0 and t ∈ [γ, δ]. (2.9)
Let 0 < r < min
(
R,
r0
δ + α
)
and consider the open set
Ωr =
{
y ∈ X : ‖y‖ < r
}
.
We claim that Fy 6≤ y, for every y ∈ ∂Ωr ∩ P. Otherwise, let y0 = (y0,1, y0,2) ∈ ∂Ωr ∩ P be such
that
Fy0 ≤ y0. (2.10)
Then, by virtue of (1.4), (2.9), and (2.10), for all t ∈ [γ, δ] and i = 1, 2, we have the estimates:
y0,i(t) = Ci +
t∫
0
φ−1
+∞∫
s
mi(τ)fi(τ, y0,1(τ), y0,2(τ), y′0,1(τ), y′0,2(τ))dτ
ds ≥
≥
γ∫
0
φ−1
+∞∫
s
mi(τ)fi(τ, y0,1(τ), y0,2(τ), y′0,1(τ), y′0,2(τ))dτ
ds ≥
≥
γ∫
0
φ−1
δ∫
γ
mi(τ)fi(τ, y0,1(τ), y0,2(τ), y′0,1(τ), y′0,2(τ))dτ
ds ≥
≥
γ∫
0
φ−1
δ∫
γ
mi(τ)gi(τ, y0,1(τ)/(τ + α), y0,2(τ)/(τ + α))dτ
ds ≥
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1634 S. DJEBALI, K. MEBARKI
≥ γ φ−1
δ∫
γ
mi(τ)`iφ(y0,1(τ) + y0,2(τ)))dτ
≥
≥ γ φ−1
(
φ( min
t∈[γ,δ]
(y0,1(t) + y0,2(t))
)
φ−1
`i δ∫
γ
mi(τ)dτ
≥
≥ γ φ−1
`i δ∫
γ
mi(τ)dτ
min
t∈[γ,δ]
(y0,1(t) + y0,2(t)) >
> min
t∈[γ,δ]
(y0,1(t) + y0,2(t)) ≥ min
t∈[γ,δ]
y0,i(t),
contradicting the continuity of the functions y0,i, i = 1, 2, on the compact interval [γ, δ], where Ci
is given by (2.2). Therefore, Lemma 2.7 yields
i (F,Ωr ∩ P,P) = 0. (2.11)
Combining (2.8), (2.11), and the fact that Ωr ⊂ ΩR, we find
i (F, (ΩR \ Ωr) ∩ P,P) = 1. (2.12)
Finally, the fixed point y = (y1, y2) ∈ P satisfies r < ‖y‖ < R .
Theorem 2.1 is proved.
2.4. Example. Let
a1(t) = e−90t, a2(t) = e−500t, b1(t) = e−200t, b2(t) = e−80t,
c1(t) =
1
100
, c2(t) = 50, m1(t) = e−3t, m2(t) = e−10t,
and let the increasing homeomorphism φ be defined by φ(x) = x3. In order to check the inequality
(2.4) in Assumption (G4), take α = 3/2 and η = 10. Then we can choose θ = 1 and R = 15 and so
we have
φ−1 (2 (|m1a1|L1 + |m1b1|L1)φ(R) + |m1c1|L1) =
3569
743
<
R
2
and
φ−1 (2 (|m2a2|L1 + |m2b2|L1)φ(R) + |m2c2|L1) =
4403
971
<
R
2
.
Consider the nonlinearities fi, i = 1, 2, defined in (R+)
3 × R2 by
fi(t, y1, y2, z1, z2) = ai(t)
(
φ
(
y1
t+ α
)
+ φ
(
y2
t+ α
))
+ bi(t) (φ(z1) + φ(z2)) + ci(t).
Then for all 0 < γ < δ and all (t, y1, y2, z1, z2) ∈ [γ, δ]× (R+)2 × R2, we have
fi(t, y1, y2, z1, z2) ≥ ai(t)
(
φ
(
y1
t+ α
)
+ φ
(
y2
t+ α
))
+ ci(t) =
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SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1635
= gi
(
t,
y1
t+ α
,
y2
t+ α
)
,
where
gi
(
t,
y1
t+ α
,
y2
t+ α
)
φ(y1 + y2)
≥ ci(t)
φ(y1 + y2)
−→ +∞, as y1 + y2 → 0.
Therefore Assumptions (G3) – (G5) in Theorem 2.1 are satisfied. All the computations have been
undertaken using Matlab 7.9. As a consequence, we have proved that the problem
−((y′1)
3)′(t) =
e−93t
(t+ 3/2)3
(y31 + y32) + e−203t((y′1)
3 + (y′2)
3) +
e−3t
100
, t > 0,
−((y′2)
3)′(t) =
e−510t
(t+ 3/2)3
(y31 + y32) + e−90t((y′1)
3 + (y′2)
3) + 50e−10t, t > 0,
y1(0) =
3
2
y′1(10), lim
t→+∞
y′1(t) = 0,
y2(0) =
3
2
y′2(10), lim
t→+∞
y′2(t) = 0,
has at least one nontrivial nonnegative solution y = (y1, y2) ∈ X satisfying r < ‖y‖ < 15, for some
0 < r < 15.
3. The particular case of system (1.5). In this section, we prove an existence result of
positive solutions for problem (1.5) under new conditions on the functions f, g, and m. In particular
the nonlinearities f, g may change sign but we still obtain existence of positive solutions with
precise information on the lower bounds. By a positive solution we mean a solution (y, x) ∈
∈ C1[0,+∞) × C1[0,+∞) such that φ(y′) ∈ C1(0,+∞) and φ(x′) ∈ C1(0,+∞) with x(t) > 0
and y(t) > 0 on [0,+∞) and the equations in (1.5) are satisfied. Some preliminaries including the
main assumptions, the problem transformation, and a compactness criterion of a fixed point operator
are collected in Subsection 3.1. Then we prove an existence theorem by constructing a special cone
in a weighted Banach space in Subsection 3.2. We end the section with an example of application in
Subsection 3.3.
3.1. Problem setting and main assumptions. For some real parameter θ > 0, let
Y =
{
y ∈ C([0,+∞),R) | y(0) = αy′(η), lim
t→+∞
e−θty(t) = 0
}
which is a weighted Banach space with the Bielecki-type norm [2]
‖y‖θ = sup
t∈[0,+∞)
e−θt|y(t)|·
Remark 3.1. Notice that if limt→+∞ y
′(t) = 0 then y has a sublinear growth at positive infinity
and thus for each positive θ, limt→+∞ e
−θty(t) = 0. Thus Y is larger than the space X used in
Section 2. However, we will still obtain existence of positive solutions with vanishing derivatives at
positive infinity.
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1636 S. DJEBALI, K. MEBARKI
The nonlinearities are assumed to satisfy the following hypotheses:
(H1) The functions f : R+×R+ → R and g : R+×R+ → R are continuous and when y, z are bo-
unded, the function g(t, eθty) is bounded on [0,+∞) and f satisfies
∫ +∞
0
m(s)f(s, k(α+s))ds <∞,
for any positive constant k. For instance, one may take g(t, y) = 1 +
e−θt
1 + t
y.
(H2) The functionm : I −→ R+ is continuous and does not vanish identically on any subinterval
of I . It may be singular at t = 0 but is integrably bounded, i.e.,
A :=
+∞∫
0
m(s)ds <∞.
Let
K =
α, if θα ≥ 1,
1
θ
eθα−1, if 0 < θα < 1,
(3.1)
and, for positive t, define the function
ω(t) = αφ−1
+∞∫
η
m(τ)dτ
+
t∫
0
φ−1
+∞∫
s
m(τ)dτ
ds. (3.2)
Remark 3.2. The properties of φ and m imply that ω is positive. Moreover ω is the unique
solution of problem (2.1) for v ≡ m.
This function ω is now used to define the translate of the positive natural cone as:
K = P + ω = {y ∈ Y | y(t) ≥ ω(t) ∀ t ≥ 0} ,
where
P = {y ∈ Y | y(t) ≥ 0 ∀ t ≥ 0} .
In [7], we have proved that the classical fixed point theory for compact mappings defined on
cones of Banach spaces is still valid on translates of cones. In particular, Lemmas 2.6 and 2.7 hold
when the cone P is replaced by its translate K and thus we have the following lemma.
Lemma 3.1 ([7], Proposition 4). Let Ω be a bounded open subset of a real Banach space E,
P a cone of E, K = P + ω a translate of P with ω ∈ Ω. Let A : Ω ∩ K → K be a completely
continuous mapping satisfying
Ax− ω 6= λ(x− ω) ∀x ∈ ∂Ω ∩ K, λ ≥ 1.
Then the index i (A,Ω ∩ K,K) = 1.
Let Ω = B(ω, r) be an open ball centered at ω with radius r in a real Banach space E.
Lemma 3.2 ([7], Lemma 3). Let P a cone of E, K = P+ω a translate of P. Let A : Ω∩K → K
be a completely continuous mapping satisfying
Fx 6≤ x ∀x ∈ ∂Ω ∩ K.
Then the fixed point index i (A,Ω ∩ K,K) = 0.
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SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1637
For Ω ⊂ Y a bounded subset, define the nonlinear operator
F : Ω ∩ K −→ C([0,+∞),R+)
by
Fy(t) = C +
t∫
0
φ−1
+∞∫
s
m(τ)f (τ, Ty(τ)) dτ
ds, t ∈ R+, (3.3)
where
C = αφ−1
+∞∫
η
m(τ)f(τ, Ty(τ))dτ
,
T y(τ) = D +
τ∫
0
φ−1
+∞∫
s
m(σ)g(σ, y(σ))dσ
ds,
D = αφ−1
+∞∫
η
m(σ)g(σ, y(σ))dσ
.
As in Lemma 2.1, it is easy to see that if F has a fixed point y, then problem (1.5) admits the coupley,D +
t∫
0
φ−1
+∞∫
s
m(τ)g(τ, y(τ))dτ
ds
as a solution. In the following three lemmas, we study some properties of the fixed point operator F .
Lemma 3.3. Under Assumptions (H1), (H2) and
(H3)
{
f(t, x) ≥ 1, for t ∈ R+, x ≥ D,
g(t, y) ≥ 0, for t ∈ R+, y ≥ ω(t),
(3.4)
F maps the set Ω ∩ K into K.
Proof. First we show that the mapping F : Ω ∩ K → Y is well defined. For y ∈ Ω ∩ K, there
exists M > 0 such that ‖y‖θ ≤M. Using Assumption (H1), let
SM = sup{g(t, eθty) | t ∈ R+, y ∈ [0,M ]}.
For any t ≥ 0, we have 0 ≤ y(t)e−θt ≤M , and so Assumption (H1) implies
+∞∫
0
m(τ)g(τ, y(τ))dτ =
+∞∫
0
m(τ)g
(
τ, eθτy(τ)e−θτ
)
dτ ≤ ASM .
Hence for each fixed t ∈ [0,+∞), we have
t∫
0
φ−1
+∞∫
s
m(τ)f(τ, Ty(τ))dτ
ds ≤
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1638 S. DJEBALI, K. MEBARKI
≤
t∫
0
φ−1
+∞∫
0
m(τ)f(τ, φ−1(ASM )(α+ τ))dτ
ds <∞.
In addition, it is easily proved that, for all y ∈ Ω ∩ K, we obtain
Fy ∈ C([0,+∞),R), Fy(t) ≥ 0, t ∈ R+, Fy(0) = α(Fy)′(η),
and then (see Remark 2.1 in [6])
0 ≤ lim
t→+∞
e−θtFy(t) ≤ lim
t→+∞
t+ α
eθt
sup
t∈R+
|(Fy)′(t)| ≤
≤ lim
t→+∞
t+ α
eθt
φ−1
+∞∫
0
m(τ)f (τ, Ty(τ)) dτ
= 0.
Now, we claim that Fy(t) ≥ ω(t) on R+. On the contrary we assume
sup
t∈R+
{ω(t)− Fy(t)} > 0
and consider two cases:
Case 1. supt∈R+{ω(t)− Fy(t)} = limt→+∞{ω(t)− Fy(t)} > 0. From (3.4), we have
lim
t→+∞
{ω(t)− Fy(t)} =
= αφ−1
+∞∫
η
m(τ)dτ
+
+∞∫
0
φ−1
+∞∫
s
m(τ)dτ
ds−
−αφ−1
+∞∫
η
m(τ)f(τ, Ty(τ))dτ
−
−
+∞∫
0
φ−1
+∞∫
s
m(τ)f(τ, Ty(τ))dτ
ds ≤ 0,
leading to a contradiction.
Case 2. There exists a real number t1 ≥ 0 such that
sup
t∈R+
{ω(t)− Fy(t)} = ω(t1)− Fy(t1) > 0.
Arguing as in Case 1, we can see that ω(t1)−Fy(t1) ≤ 0 which is again a contradiction, ending the
proof of the lemma.
To prove the compactness of the operator F , we need the following result which is a direct
consequence of Lemma 2.2.
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SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1639
Lemma 3.4. Let M ⊆ Y. Then M is relatively compact in Y if the following conditions hold:
(a) M is uniformly bounded in Y.
(b) The functions belonging to the set A =
{
x | x(t) = y(t)/eθt, y ∈ M
}
are almost equicon-
tinuous on R+.
(c) The functions from A are equiconvergent at +∞.
Since m may be singular at the origin, we first consider the regular case; then we argue by
approximation.
Lemma 3.5. Assume that m : (0,+∞) → [0,+∞) is continuous and bounded at the origin.
Then, the mapping F : Ω ∩ K → K is completely continuous.
Proof. Claim 1. F is continuous on K. Let (yn)n be some sequence converging to some limit
y in K; then there exists N > 0 independent of n such that
max
{
‖y‖θ, sup
n≥1
‖yn‖θ
}
≤ N.
Letting
SN = sup
{
g(t, eθty) | t ∈ [0,+∞), y ∈ [0, N ]
}
,
we get
+∞∫
0
m(s)[g(s, yn(s))− g(s, y(s))]ds ≤ 2ASN .
Then, the Lebesgue dominated convergence theorem together with the continuity of f, g, and φ−1
yield the estimates:
e−θt|(Fyn)(t)− (Fy)(t))| =
= e−θt
∣∣∣∣∣∣αφ−1
+∞∫
η
m(τ)f(τ, Tyn(τ))dτ
+
+
t∫
0
φ−1
+∞∫
s
m(τ)f (τ, Tyn(τ)) dτ
ds−
−αφ−1
+∞∫
η
m(τ)f(τ, Ty(τ))dτ
−
−
t∫
0
φ−1
+∞∫
s
m(τ)f (τ, Ty(τ)) dτ
ds
∣∣∣∣∣∣ −→ 0, as n→ +∞.
Consequently,
‖Fyn − Fy‖θ −→ 0, as n→ +∞,
which proves the claim.
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1640 S. DJEBALI, K. MEBARKI
Claim 2. F is completely continuous, i.e., it maps bounded sets into relatively compact sets. Let
B be a bounded subset of Y; then there exists M > 0 such that ‖y‖θ ≤ M , for all y ∈ B ∩ K. On
one hand, we obtain
‖Fy‖θ = sup
t∈[0,+∞)
e−θt
C +
t∫
0
φ−1
+∞∫
s
m(τ)f (τ, Tyn(τ)) dτ
ds
≤
≤ K φ−1
+∞∫
0
m(τ)f
(
τ, φ−1(ASM )(α+ τ)
)
dτ
<∞ ∀ y ∈ B ∩ K,
which implies that the set F (B ∩ K) is uniformly bounded. On the other hand, for all y ∈ B ∩ K,
β ∈ (0,+∞) and t1, t2 ∈ [0, β] (t1 < t2), we have the estimates∣∣∣∣Fy(t2)
eθt2
− Fy(t1)
eθt1
∣∣∣∣ =
=
∣∣∣∣∣∣C
(
e−θt2 − e−θt1
)
+ e−θt2
t2∫
0
φ−1
+∞∫
s
m(τ)f(τ, Ty(τ))dτ
ds −
−e−θt1
t1∫
0
φ−1
+∞∫
s
m(τ)f(τ, Ty(τ))dτ
ds
∣∣∣∣∣∣ ≤
≤
∣∣∣e−θt2 − e−θt1∣∣∣
C +
t1∫
0
φ−1
+∞∫
0
m(τ)f(τ, φ−1(ASM )(α+ τ)) dτ
ds
+
+e−θt1
∣∣∣∣∣∣
t2∫
t1
φ−1
+∞∫
0
m(τ)f(τ, φ−1(ASM )(α+ τ) dτ
ds
∣∣∣∣∣∣ .
The terms in the last two lines tend to 0, as |t1 − t2| → 0. Hence F (B ∩ K) is equicontinuous.
Finally, Assumption (H1) yields
+∞∫
0
m(s)f(s, φ−1(ASM )(α+ s))ds <∞,
then
lim
t→+∞
∣∣∣∣Fy(t)
eθt
− lim
s→+∞
Fy(s)
eθs
∣∣∣∣ ≤
≤ lim
t→+∞
t+ α
eθt
φ−1
+∞∫
0
m(τ)f(τ, φ−1(ASM )(α+ τ))dτ
= 0.
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SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1641
This means that F (B ∩K) is equiconvergent at +∞. Using Lemma 3.4, we conclude that F (B ∩K)
is relatively compact, ending the proof of the lemma.
Lemma 3.6. Let m be singular at t = 0. Then the mapping F given by (3.3) is completely
continuous.
Proof. Let B ⊂ Y be a bounded subset. For each n ≥ 1, define the approximating operator Fn
on B ∩ K by
Fny(t) = C +
t∫
1
n
φ−1
+∞∫
s
m(τ)f(τ, Ty(τ))dτ
ds, t ∈ I.
It suffices to prove that Fn converges uniformly to F on B ∩ K. For every t ∈ I and y ∈ B ∩ K
satisfying ‖y‖θ ≤M, the following estimates follow from (H1) and (H2):
e−θt|Fny(t)− Fy(t))| =
=
∣∣∣∣∣∣∣
1
n∫
0
e−θtφ−1
+∞∫
s
m(τ)f(τ, Ty(τ))dτ
ds
∣∣∣∣∣∣∣ ≤
≤ 1
n
φ−1
+∞∫
0
m(τ)f(τ, φ−1(ASM )(α+ τ))dτ
.
Consequently, Assumptions (H1) and (H2) together with the Cauchy criterion for convergence of
integrals imply that
‖Fny − Fy‖θ −→ 0, as n→ +∞.
Since, from Lemma 3.5, for each n ≥ 1, the operator Fn : B ∩K → K is completely continuous and
Fn converges to F uniformly on closed bounded subsets of B ∩ K, the uniform limit operator F is
completely continuous, ending the proof of the lemma.
3.2. Existence result. Let K be defined by (3.1) and for each M > 0
SM = sup{g(t, eθty) | t ∈ R+, y ∈ [0,M ]}.
Our main existence result in this section is the following theorem.
Theorem 3.1. Assume that Assumptions (H1), (H2) hold together with
(H4)
{
f(t, eθty) ≤ a1(t) (φ(y) + y) + b1(t),
g(t, eθty) ≤ a2(t) (φ(y − ω) + y − ω) + b2(t),
for all (t, y) ∈ R+ × R+, where mai, mbi belong to L1(I), i = 1, 2.
(H5) There exists R > 0 such that
Kφ−1 {|ma1|L1 (φ(KR) +KR) + |mb1|L1}+ ‖ω‖θ < R, (3.5)
φ−1 {|ma2|L1 (φ(R) +R) + |mb2|L1} < R. (3.6)
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1642 S. DJEBALI, K. MEBARKI
(H6) There exist positive constants ‖ω‖θ < ρ < R and γ, δ (η < γ < δ) such that, for all
(t, y) ∈ [γ, δ]× [ω(t), ρ],
g(t, eθty) > `φ2(eθty),
and for all t ∈ [γ, δ], y ≥ µ,
f(t, y) ≥ λy,
where
` :=
φ2(1/γ)
+∞∫
γ
m(t)dt
, λ =
γ +∞∫
γ
m(t)dt
−1 ,
and
µ = (α+ γ)φ−1
Λ
δ∫
γ
m(t)dt
,
where Λ = mint∈[γ,δ], y∈[0,ρ] g(t, eθty).
Then problem (1.5) has at least one positive solution (y, x) such that
0 < ‖y − ω‖θ < R, ‖x‖θ ≤ Kφ−1(ASR),
and for all t ∈ [0,+∞),
y(t) > αφ−1
δ∫
γ
m(τ)dτ
+
t∫
0
φ−1
+∞∫
s
m(τ)dτ
ds,
x(t) > αφ−1
δ∫
γ
m(s)g(s, y(s))ds
.
Proof. Consider the open ball
ΩR = {y ∈ Y : ‖y − ω‖θ < R}.
From Lemma 3.5, the operator F : ΩR ∩ K → K is completely continuous.
Claim 1. Fy − ω 6= λ(y − ω), for all y ∈ ∂ΩR ∩K and λ ≥ 1. Let y ∈ ∂ΩR ∩K. Using (H4)
and the inequality (3.6), for all positive t, the following estimates hold true:
Ty(t) = αφ−1
+∞∫
η
m(σ)g(σ, y(σ))dσ
+
+
t∫
0
φ−1
+∞∫
s
m(σ)g(σ, y(σ))dσ
ds ≤
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SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1643
≤ αφ−1
+∞∫
0
m(σ)
(
a2(σ)
(
φ(e−θσ(y(σ)− ω(σ))) +
+e−θσ(y(σ)− ω(σ))
)
+ b2(σ)
)
dσ
+
+
t∫
0
φ−1
+∞∫
0
m(σ)
(
a2(σ)
(
φ(e−θσ(y(σ)− ω(σ))) +
+e−θσ(y(σ)− ω(σ))
)
+ b2(σ)
)
dσ
ds ≤
≤ (t+ α)φ−1 (|ma2|L1 (φ(‖y − ω‖θ) + ‖y − ω‖θ)) + |mb2|L1) =
= (t+ α)φ−1 (|ma2|L1 (φ(R) +R) + |mb2|L1) < R(t+ α).
Hence
e−θtTy(t) ≤ t+ α
eθt
R ≤ KR ∀ t ≥ 0.
Using (3.5), for all positive t, we deduce that
e−θt|(Fy)(t)− ω(t)| ≤
≤ αe−θtφ−1
+∞∫
η
m(τ)f(τ, Ty(τ))dτ
+
+e−θt
t∫
0
φ−1
+∞∫
s
φ−1
+∞∫
t
m(τ)f(τ, Ty(τ), )dτ
ds+ e−θt|ω(t)| ≤
≤ αe−θtφ−1
+∞∫
0
m(τ)
(
a1(τ)
(
φ(e−θτTy(τ)) + e−θτTy(τ)
)
+ b1(τ)
)
dτ
+
+e−θt
t∫
0
φ−1
+∞∫
0
m(τ)
(
a1(τ)
(
φ(e−θτTy(τ)) + e−θτTy(τ)
)
+ b1(τ)
)
dτ
ds+ ‖ω‖θ ≤
≤ t+ α
eθt
φ−1 ((|ma1|L1 (φ(KR) +KR) + |mb1|L1)) + ‖ω‖θ ≤
≤ Kφ−1 ((|ma1|L1 (φ(KR) +KR) + |mb1|L1)) + ‖ω‖θ < R.
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1644 S. DJEBALI, K. MEBARKI
Taking the least upper bound over t, we get
‖Fy − ω‖θ < ‖y − ω‖θ ∀ y ∈ ∂ΩR ∩ K. (3.7)
As a consequence
Fy − ω 6= λ(y − ω) ∀ y ∈ ∂ΩR ∩ K ∀λ ≥ 1. (3.8)
Indeed, on the contrary there would exist some y0 ∈ ∂ΩR ∩ K and λ0 ≥ 1 such that Fy0 − ω =
= λ0(y0 − ω). Hence
‖Fy0 − ω‖θ = λ0‖y0 − ω‖θ ≥ ‖y0 − ω‖θ = R,
contradicting (3.7). This implies that (3.8) holds. Therefore, Lemma 3.1 yields
i (F,ΩR ∩ K,K) = 1. (3.9)
Finally (3.9) and the existence property of the fixed point index imply that the operator F has at least
one nonnegative fixed point y which belongs to ΩR ∩ K.
Claim 2. Let 0 < R̂ < ρ− ‖ω‖θ and consider the open ball
Ω
R̂
:= {y ∈ Y : ‖y − ω‖θ < R̂}.
We claim that
Fy 6≤ y, for all y ∈ ∂Ω
R̂
∩ K. (3.10)
Otherwise, let y0 ∈ ∂Ω
R̂
∩ K be such that Fy0 ≤ y0. Then, in one hand,
0 ≤ e−θty0(t) ≤ R̂+ ‖ω‖θ < ρ ∀ t ∈ [γ, δ].
In the other one, by (H6) and the definition of Λ, we obtain the estimates
Ty0(t) = D +
t∫
0
φ−1
+∞∫
s
m(σ)g(σ, y0(σ))dσ
ds =
= αφ−1
+∞∫
η
m(σ)g(σ, y0(σ))dσ
+
+
t∫
0
φ−1
+∞∫
s
m(σ)g(σ, y0(σ))dσ
ds ≥
≥ αφ−1
δ∫
γ
m(σ)g(σ, y0(σ))dσ
+
+
γ∫
0
φ−1
δ∫
γ
m(σ)g(σ, y0(σ))dσ
ds ≥
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1645
≥ (α+ γ)φ−1
Λ
δ∫
γ
m(σ)dσ
= µ.
Making use of Assumption (H6), we have also the following estimates valid for every t ∈ [γ, δ]:
y0(t) ≥ C +
t∫
0
φ−1
+∞∫
s
m(τ)f(τ, Ty0(τ))dτ
ds ≥
≥
γ∫
0
φ−1
+∞∫
γ
m(τ)f(τ, Ty0(τ))dτ
ds ≥
≥
γ∫
0
φ−1
+∞∫
γ
m(τ)λ Ty0(τ)dτ
ds ≥
≥
γ∫
0
φ−1
+∞∫
γ
m(τ)λ γφ−1
+∞∫
γ
m(σ)g(σ, y0(σ))dσ
dτ
ds >
> γφ−1
λ γ +∞∫
γ
m(τ)dτφ−1
+∞∫
γ
m(σ)`φ2(y0(σ))dσ
.
Using the property (1.4) of φ−1 and the definition of λ, `, we find a lower bound for y0:
y0(t) > γ φ−1
φ−1(φ2( min
t∈[γ,δ]
y0(t))
)
φ−1
` +∞∫
γ
m(σ)dσ
≥
≥ γ min
t∈[γ,δ]
y0(t) (φ−1)2
` +∞∫
γ
m(σ)dσ
= min
t∈[γ,δ]
y0(t).
Hence for every t ∈ [γ, δ], y0(t) > mint∈[γ,δ] y0(t), contradicting the continuity of the function y0
on the compact interval [γ, δ]. This implies that (3.8) holds. As a consequence, Lemma 3.2 yields
i (F,Ω
R̂
∩ K,K) = 0. (3.11)
To sum up, from (3.10), (3.11), and the fact that Ω
R̂
⊂ ΩR, we conclude that
i (F, (ΩR \ Ω
R̂
) ∩ K,K) = 1.
Therefore, there exists at least one positive fixed point y ∈ K satisfying
R̂ < ‖y − ω‖θ < R and y(t) ≥ ω(t), t ≥ 0.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1646 S. DJEBALI, K. MEBARKI
Moreover, for positive t,
x(t) = Ty(t) = αφ−1
+∞∫
η
m(σ)g(σ, y(σ))dσ
+
+
t∫
0
φ−1
+∞∫
s
m(σ)g(σ, y(σ))dσ
ds ≤
≤ (α+ t)φ−1
(
ASR+‖ω‖θ
)
.
Hence
x(t)
eθt
≤ α+ t
eθt
φ−1
(
ASR+‖ω‖θ
)
.
Passing to the least upper bound over t yields
‖x‖θ ≤ Kφ−1
(
ASR+‖ω‖θ
)
.
Finally, for all positive t,
x(t) ≥ αφ−1
+∞∫
η
m(σ)g(σ, y(σ))dσ
> αφ−1
δ∫
γ
m(σ)g(σ, y(σ))dσ
,
which completes the estimates of the solution (y, x).
Theorem 3.1 is proved.
3.3. Example. Let
a1(t) =
e−
4
5 , if 0 ≤ t ≤ 1,
e−
4
5 t, if t ≥ 1,
a2(t) =
1
1000
e−200000, if 0 ≤ t ≤ 1,
1
1000
e−200000t, if t ≥ 1,
b1(t) =
1
9
, b2(t) = 20, and m(t) =
t−
1
10 , if 0 < t ≤ 1,
t−
12
10 , if t ≥ 1,
and define the increasing homeomorphism φ by
φ(x) =
1
3
x2, if x ≥ 0,
0, if x ≤ 0.
In order to check the inequality (3.5) in Assumption (H5), choose α =
1
2
and η =
1
5
. Thus we can
take θ = 1 and R = 20. So K =
1213
2000
and ω(t) =
1
6
√
165− 6
10
√
5 +
10
√
15
9
10
√
t9, for t ≥ 0. Then
‖ω‖θ =
2267
849
,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
SYSTEMS OF φ-LAPLACIAN THREE-POINT BOUNDARY-VALUE PROBLEMS . . . 1647
Kφ−1 [|ma1|L1 (φ(KR) +KR) + |mb1|L1 ] + ‖ω‖θ =
3544
355
< R,
and
φ−1 [|ma2|L1 (φ(R) +R) + |mb2|L1 ] =
1934
101
< R.
Now consider the nonlinearities f, g defined for (t, y) ∈ (R+)2 by
f(t, y) = a1(t)
(
φ(e−θty) + e−θty
)
+ b1(t) ≥ 1,
g(t, y) = a2(t)
(
φ(e−θt(y − ω)) + e−θt(y − ω)
)
+ b2(t) ≥ 0.
Then, for every positive constant k, we have
+∞∫
0
m(s)f(s, k(α+ s))ds <∞.
If we let γ =
2
5
and δ =
1
2
, then γ > η and
+∞∫
γ
m(t)dt =
8287
1250
.
For ‖ω‖θ < ρ = 3 < R, we obtain
Λ = min
t∈[γ,δ], y∈R+
g(t, eθty) = b2(t) = 20.
Then
µ = (α+ γ)φ−1
Λ
δ∫
γ
m(t)dt
=
1402
611
.
Consequently
f(t, y)
y
≥ a1(t)
(
1
3
e−2θty + e−θt
)
≥ e−
4
5
(
1
3
e−2θδµ+ e−θδ
)
=
=
399
1000
≥ λ =
853
2262
, for all t ∈ [γ, δ], y ≥ µ.
Finally, for all (t, y) ∈ [γ, δ]× [ω(t), ρ], we have the estimates
g(t, eθty)
φ2(eθty)
≥ b2(t)
φ2(eθty)
≥ 20
φ2(eθδρ)
= 369/409 >
φ2(1/γ)∫ +∞
γ
m(τ)dτ
=
657
3011
.
Therefore Assumptions (H3) – (H6) in Theorem 3.1 are fulfilled. All the above computations have
been undertaken using Matlab 7.9. Therefore, the singular problem
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1648 S. DJEBALI, K. MEBARKI
−(φ(y′))′(t) = m(t)f(t, x(t)), t > 0,
−(φ(x′))′(t) = m(t)g(t, y(t)), t > 0,
y(0) = y′
(
1
5
)
, lim
t→+∞
y′(t) = 0,
x(0) = x′
(
1
5
)
, lim
t→+∞
x′(t) = 0,
has at least one positive solution (y, x) ∈ Y2 satisfying y(t) ≥ ω(t) ∀ t ∈ R+.
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Received 15.11.12
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
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| id | umjimathkievua-article-2096 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:40Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/22/aa93a5b93c5ea87b693833b1c5c08322.pdf |
| spelling | umjimathkievua-article-20962019-12-05T09:50:29Z Systems of φ-Laplacian three-point boundary-value problems on the positive half-line Системи триточкових граничних задач для φ-лапласiана на додатнiй пiвосi Djebali, S. Mebarki, K. Дебалі, С. Мебаркі, К. We study the existence of positive solutions to boundary-value problems for two systems of two second-order nonlinear three-point φ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces. Вивчається iснування додатних розв’язкiв граничних задач для двох систем двох нелiнiйних триточкових φ-лапласових рiвнянь другого порядку, що визначенi на додатнiй пiвосi. Нелiнiйностi можуть змiнювати знак, мати часовi сингулярностi на початку координат та залежати, як вiд розв’язкiв, так i вiд їх перших похiдних. Теорiю нерухомих точок застосовано для доведення деяких результатiв щодо iснування нетривiальних додатнiх розв’язкiв на вiдповiдних конусах в деяких звaжених банахових просторах. Institute of Mathematics, NAS of Ukraine 2015-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2096 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 12 (2015); 1626-1648 Український математичний журнал; Том 67 № 12 (2015); 1626-1648 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2096/1204 Copyright (c) 2015 Djebali S.; Mebarki K. |
| spellingShingle | Djebali, S. Mebarki, K. Дебалі, С. Мебаркі, К. Systems of φ-Laplacian three-point boundary-value problems on the positive half-line |
| title | Systems of φ-Laplacian three-point boundary-value problems on the positive half-line |
| title_alt | Системи триточкових граничних задач
для φ-лапласiана на додатнiй пiвосi |
| title_full | Systems of φ-Laplacian three-point boundary-value problems on the positive half-line |
| title_fullStr | Systems of φ-Laplacian three-point boundary-value problems on the positive half-line |
| title_full_unstemmed | Systems of φ-Laplacian three-point boundary-value problems on the positive half-line |
| title_short | Systems of φ-Laplacian three-point boundary-value problems on the positive half-line |
| title_sort | systems of φ-laplacian three-point boundary-value problems on the positive half-line |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2096 |
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