Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 139, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
POSITIVE SOLUTIONS FOR SYSTEMS OF THIRD-ORDER
GENERALIZED STURM-LIOUVILLE BOUNDARY-VALUE
PROBLEMS WITH (p, q)-LAPLACIAN
NEMAT NYAMORADI
Abstract. In this work, we use the Leggett-Williams fixed point theorem to
prove the existence of at least three positive solutions to a system of third-order
ordinary differential equations with (p, q)-Laplacian
1. Introduction
In this article, we prove the existence of at least three positive solutions to a
boundary-value problem with the (p, q)-Laplacian:
(φp (u00 (t)))0 + a1 (t)f1 (t, u(t), v(t)) = 0
00
0
(φq (v (t))) + a2 (t)f2 (t, u(t), v(t)) = 0
0
0 ≤ t ≤ 1,
0 ≤ t ≤ 1,
0
α1 u(0) − β1 u (0) = µ11 u(ξ1 ),
γ1 u(1) + δ1 u (1) = µ12 u(η1 ),
u00 (0) = 0,
α2 v(0) − β2 v 0 (0) = µ21 v(ξ2 ),
γ2 v(1) + δ2 v 0 (1) = µ22 u(η2 ),
v 00 (0) = 0,
(1.1)
where φp (s) = |s|p−2 s and φq (s) = |s|q−2 s are p, q-Laplacian operators; p > 1,
q > 1, 0 < ξi < 1, 0 < ηi < 1, for i = 1, 2.
Il’in and Moiseev [4] studied the existence of solutions for a linear multi-point
boundary-value problem. Gupta [3] studied certain three-point boundary-value
problems for nonlinear ordinary differential equations. Since then, more general
nonlinear multi-point boundary-value problems have been studied by several authors because multi-point boundary-value problems describe many phenomena of
applied mathematics and physics (see [2, 5, 12, 11]). There is much current interest in questions of positive solutions of boundary-value problems for ordinary
differential equations, on may see [1, 8, 9, 10, 13, 14, 15] and references therein.
Motivated by the works [7, 16], in this paper we will show the existence of three
positive solutions for the problem (1.1).
The basic space used in this paper is a real Banach space E = (C[0, 1], R) ×
(C[0, 1], R) with the norm k(u, v)k := kuk + kvk, where kuk = maxt∈[0,1] |u(t)|. For
convenience, we make the following assumptions:
2000 Mathematics Subject Classification. 34L30, 34B18, 34B27.
Key words and phrases. Positive solution; fixed point theorem; (p, q)-Laplacian;
Sturm-Liouville; boundary value problem.
c
2011
Texas State University - San Marcos.
Submitted August 15, 2011. Published October 25, 2011.
Research supported the Razi University.
1
2
N. NYAMORADI
EJDE-2011/139
(H1) αi ≥ 0, βi ≥ 0, γi ≥ 0, δi ≥ 0, ρi = αi γi + βi γi + αi δi > 0, ρi − µi2 ψ(ηi ) > 0,
ρi − µi1 ϕ(ξi ) > 0, µi1 , µi2 > 0, ∆i < 0, for i = 1, 2, and σ ∈ (0, 1/2),
−µi1 ψ(ξi )
ρ − µi1 ϕ(ξi )
, i = 1, 2,
∆i = ρ − µi2 ψ(ηi ) −µi2 ϕ(ηi ) where
ψi (t) = βi + αi t,
ϕi (t) = γi + δi − γi t,
t ∈ [0, 1],
i = 1, 2,
(1.2)
are linearly independent solutions of the equation x00 (t) = 0, t ∈ [0, 1].
Obviously, ψi is non-decreasing on [0, 1] and ϕi is non-increasing on [0, 1].
(H2) fi ∈ C([0, 1] × [0, +∞) × [0, +∞), [0, +∞)), and ai : (0, 1) → [0, +∞) is
continuous and ai (t) 6= 0, for i = 1, 2 on any subinterval of (0, 1), and
Z 1
ai (s)ds < +∞.
0<
0
For the convenience of the reader, we present here the Leggett-Williams fixed
point theorem [6].
Given a cone K in a real Banach space E, a map α is said to be a nonnegative
continuous concave (resp. convex) functional on K provided that α : K → [0. + ∞)
is continuous and
α(tx + (1 − t)y) ≥ tα(x) + (1 − t)α(y),
(resp.α(tx + (1 − t)y) ≤ tα(x) + (1 − t)α(y)),
for all x, y ∈ K and t ∈ [0, 1].
Let 0 < a < b be given and let α be a nonnegative continuous concave functional
on K. Define the convex sets Pr and P (α, a, b) by
Pr = {x ∈ K|kxk < r},
P (α, a, b) = {x ∈ K|a ≤ α(x), kxk ≤ b}.
Theorem 1.1 (Leggett-Williams fixed point theorem). . Let A : Pc → Pc be
a completely continuous operator and let α be a nonnegative continuous concave
functional on K such that α(x) ≤ kxk for all x ∈ Pc . Suppose there exist 0 < a <
b < d ≤ c such that
(A1) x ∈ P (α, b, d)|α(x) > b 6= ∅, and α(Ax) > b for x ∈ P (α, b, d);
(A2) kAxk < a for kxk ≤ a; and
(A3) α(Ax) > b for x ∈ P (α, b, c) with kAxk > d.
Then A has at least three fixed points x1 , x2 , and x3 and such that kx1 k < a, b <
α(x2 ) and kx3 k > a, with α(x3 ) < b.
Inspired and motivated by the works mentioned above, in this work we will
consider the existence of positive solutions to (1.1). We shall first give a new
form of the solution, and then determine the properties of the Green’s function
for associated linear boundary-value problems; finally, by employing the LeggettWilliams fixed point theorem, some sufficient conditions guaranteeing the existence
of three positive solutions. The rest of the article is organized as follows: in Section
2, we present some preliminaries that will be used in Section 3. The main results
and proofs will be given in Section 3. Finally, in Section 4, an example are given
to demonstrate the application of our main result.
EJDE-2011/139
POSITIVE SOLUTIONS
3
2. Preliminaries
In this section, we present some notations and preliminary lemmas that will be
used in the proof of the main result.
Definition 2.1. Let X be a real Banach space. A non-empty closed convex set
P ⊂ X is called a cone of X if it satisfies the following conditions:
(1) x ∈ P, µ ≥ 0 implies µx ∈ P ;
(2) x ∈ P, −x ∈ P implies x = 0.
Let y1 (t) = −φp (u00 (t)), y2 (t) = −φq (v 00 (t)), then for i = 1, 2, the following two
boundary-value problems
(φp (u00 (t)))0 + a1 (t)f1 (t, u(t), v(t)) = 0,
0 ≤ t ≤ 1,
00
u (0) = 0,
and
(φq (v 00 (t)))0 + a2 (t)f2 (t, u(t), v(t)) = 0,
0 ≤ t ≤ 1,
00
v (0) = 0,
are turned into the following two boundary-value problems
yi0 (t) − ai (t)fi (t, u(t), v(t)) = 0,
0 ≤ t ≤ 1,
yi (0) = 0,
Lemma 2.2. Problem (2.1) has a unique solution
Z t
yi (t) =
ai (s)fi (s, u(s), v(s))ds,
i = 1, 2.
i = 1, 2.
(2.1)
(2.2)
0
Lemma 2.3. If (H1) holds, then for yi (t) ∈ C([0, 1]), the following two boundaryvalue problems
u00 (t) + φp−1 (y1 (t)) = 0, 0 ≤ t ≤ 1,
α1 u(0) − β1 u0 (0) = µ11 u(ξ1 ),
(2.3)
0
γ1 u(1) + δ1 u (1) = µ12 u(η1 ),
and
v 00 (t) + φq−1 (y2 (t)) = 0,
0 ≤ t ≤ 1,
0
α2 v(0) − β2 v (0) = µ21 v(ξ2 ),
(2.4)
0
γ2 v(1) + δ2 v (1) = µ22 v(η2 ),
have a unique solutions
Z 1
−1
−1
u(t) =
G1 (t, s)φ−1
p (y1 (s))ds + A1 (φp (y1 ))ψ1 (t) + B(φp (y1 ))ϕ1 (t),
Z
(2.5)
0
1
v(t) =
−1
−1
G2 (t, s)φ−1
q (y2 (s))ds + A2 (φq (y2 ))ψ2 (t) + B2 (φq (y2 ))ϕ2 (t),
(2.6)
0
where
(
1 ϕi (t)ψi (s), s ≤ t,
Gi (t, s) =
i = 1, 2,
ρi ϕi (s)ψi (t), t ≤ s,
R1
−1
1
G
(ξ
,
s)φ
(y
)ds
ρ
−
µ
ϕ
(ξ
)
µ
11 0
1
1
11 1 1 p
R1 1 1
A1 (φ−1
,
p (y1 )) =
∆1 µ12 0 G1 (η1 , s)φ−1
−µ12 ϕ1 (η1 ) p (y1 )ds
(2.7)
(2.8)
4
N. NYAMORADI
1
B1 (φp−1 (y1 )) =
∆1
EJDE-2011/139
R1
µ11 0 G1 (ξ1 , s)φ−1
(y
)ds
−µ11 ψ1 (ξ1 )
1
p
R1
,
ρ1 − µ12 ψ1 (η1 ) µ12 0 G1 (η1 , s)φ−1
p (y1 )ds
(2.9)
A2 (φ−1
q (y2 ))
1
=
∆2
R1
µ21 0 G2 (ξ2 , s)φ−1
q (y2 )ds
R1
µ22 0 G2 (η2 , s)φ−1
q (y2 )ds
ρ2 − µ21 ϕ2 (ξ2 )
,
−µ22 ϕ2 (η2 ) (2.10)
B2 (φ−1
q (y2 ))
1
=
∆2
R1
µ21 0 G2 (ξ2 , s)φ−1
(y2 )ds
−µ21 ψ2 (ξ2 )
q
R1
.
ρ2 − µ22 ψ2 (η2 ) µ22 0 G2 (η2 , s)φ−1
q (y2 )ds
(2.11)
and
The proof of the above theorem follows by routine calculations. We omit it.
Remark 2.4. For a fixed integrable function y, it is obvious that A(φ−1
p (y)) and
B(φ−1
q (y)) are constant.
For convenience, let
n ϕ (1 − σ) ψ (σ) ϕ (1 − σ) ψ (σ) o
1
2
2
1
,
,
,
,
ϕ1 (σ)
ψ1 (1)
ϕ2 (σ)
ψ2 (1)
n
o
Λ1 = max 1, kψ1 k, kϕ1 k, kψ2 k, kϕ2 k ,
Λ0 = min
Λ2 = min
n
min
t∈[σ,1−σ]
ϕ1 (t),
min
t∈[σ,1−σ]
ψ1 (t),
min
t∈[σ,1−σ]
ϕ2 (t),
min
t∈[σ,1−σ]
o
ψ2 (t), 1 ,
n
Λ2 o
.
λ = min Λ0 ,
Λ1
If (H1) and (H2) hold, then from Lemmas 2.2 and 2.3, we know that (u(t), v(t)) is
a solution of (1.1) if and only if
Z 1
−1
−1
u(t) =
G1 (t, s)φ−1
p (W1 (s))ds + A1 (φp (W1 (s)))ψ1 (t) + B1 (φp (W1 (s)))ϕ1 (t),
0
Z
v(t) =
1
−1
−1
G2 (t, s)φ−1
q (W2 (s))ds + A2 (φq (W2 (s)))ψ2 (t) + B2 (φq (W2 (s)))ϕ2 (t),
0
Rs
where 0 ≤ t ≤ 1, and Wi (s) = 0 ai (τ )fi (τ, u(τ ), v(τ ))dτ , for i = 1, 2.
We need some properties of the functions Gi , i = 1, 2, in order to discuss the
existence of positive solutions. For the Green’s functions Gi (t, s), we have the
following two Lemmas [7].
Lemma 2.5. If (H1) and (H2) hold, then
0 ≤ Gi (t, s) ≤ Gi (s, s),
Gi (t, s) ≥ Λ0 Gi (s, s),
t, s ∈ [0, 1],
t ∈ [σ, 1 − σ], s ∈ [0, 1],
(2.12)
(2.13)
for i = 1, 2.
Denote
K = (u, v) ∈ E : u(t) ≥ 0, v(t) ≥ 0,
min (u(t) + v(t)) ≥ λk(u, v)k .
t∈[σ,1−σ]
It is obvious that K is cone. Define the operator T : E → E by
T (u, v)(t) = (T1 (u, v)(t), T2 (u, v)(t)),
∀t ∈ (0, 1),
(2.14)
EJDE-2011/139
POSITIVE SOLUTIONS
5
where
Z
T1 (u, v)(t) =
1
−1
G1 (t, s)φ−1
p (W1 (s))ds + A1 (φp (W1 (s)))ψ1 (t)
0
+ B1 (φ−1
0 ≤ t ≤ 1,
p (W1 (s)))ϕ1 (t),
Z 1
−1
T2 (u, v)(t) =
G2 (t, s)φ−1
q (W2 (s))ds + A2 (φq (W2 (s)))ψ2 (t)
(2.15)
0
+ B2 (φ−1
q (W2 (s)))ϕ2 (t),
0 ≤ t ≤ 1,
Rs
where Wi (s) = 0 ai (τ )fi (τ, u(τ ), v(τ ))dτ , for i = 1, 2. Evidently, (u(t), v(t)) is a
solution of (1.1) if and only if (u(t), v(t)) is a fixed point of operator T .
Lemma 2.6. If (H1) and (H2) hold, then the operator defined in (2.14) satisfies
T (K) ⊆ K.
Proof. For (u, v) ∈ K, then from properties of G1 (t, s) and G2 (t, s), T1 (u, v)(t) ≥ 0,
T2 (u, v)(t) ≥ 0, t ∈ [0, 1], and it follows form (2.15) that
T1 (u, v)(t)
Z 1
−1
−1
G1 (t, s)φ−1
=
p (W1 (s))ds + A1 (φp (W1 (s)))ψ1 (t) + B1 (φp (W1 (s)))ϕ1 (t)
0
1
Z
−1
−1
G1 (s, s)φ−1
p (W1 (s))ds + Λ1 [A1 (φp (W1 (s))) + B1 (φp (W1 (s)))].
≤
0
Thus,
1
Z
−1
−1
G1 (s, s)φ−1
p (W1 (s))ds + Λ1 [A1 (φp (W1 (s))) + B1 (φp (W1 (s)))].
kT1 (u, v)k ≤
0
On the other hand, for t ∈ [σ, 1 − σ], we have
min
t∈[σ,1−σ]
=
T1 (u, v)(t)
hZ
min
t∈[σ,1−σ]
Z
1
i
−1
−1
G1 (t, s)φ−1
p (W1 (s))ds + A1 (φp (W1 ))ψ1 (t) + B1 (φp (W1 ))ϕ1 (t)
0
1
−1
−1
G1 (s, s)φ−1
p (W1 (s))ds + A1 (φp (W1 ))ψ1 (t) + B1 (φp (W1 ))ϕ1 (t)
≥ Λ0
0
Z
1
Λ2
−1
.Λ1 .[A1 (φ−1
p (W1 )) + B1 (φp (W1 ))]
Λ1
0
Z
1
−1
−1
≥λ
G1 (s, s)φ−1
p (W1 (s))ds + Λ1 .[A1 (φp (W1 )) + B1 (φp (W1 ))]
≥ Λ0
G1 (s, s)φ−1
p (W1 (s))ds +
0
≥ λkT1 (u, v)k.
In this way, for any (u, v) ∈ K, we have
min
t∈[σ,1−σ]
T2 (u, v)(t) ≥ λkT2 (u, v)k.
Therefore,
min
t∈[σ,1−σ]
T1 (u, v)(t), T2 (u, v)(t) ≥ λkT1 (u, v)k + λkT2 (u, v)k
= λk(T1 (u, v), T2 (u, v))k.
From the above, we obtain T (K) ⊆ K. This completes the proof.
6
N. NYAMORADI
EJDE-2011/139
3. Main results
In this section, we discuss the existence of positive solutions of (1.1). We define
the nonnegative continuous concave functional on K by
α(u, v) =
min (u(t) + v(t)).
σ≤t≤1−σ
It is obvious that, for each (u, v) ∈ K, α(u, v) ≤ k(u, v)k.
In this section, for convenience, we denote
µi1 fi = 1 µi1 ρi − µi1 ϕi (ξi ) , B
f0 = 1 −µi1 ψi (ξi )
A
,
−µi2 ϕi (ηi ) ∆i µi2
∆i ρi − µi2 ψi (ηi ) µi2 Z 1
Z 1−σ
Mi = max
Gi (t, s)ds, mi = min
Gi (t, s)ds, i = 1, 2.
0≤t≤1
σ≤t≤1−σ
0
σ
Also we use the following assumptions: There exist nonnegative numbers a, b, c
such that 0 < a < b ≤ min{λ, p1mM1 1 , p2mM2 2 }c, and fi (t, u, v) satisfy the following
conditions:
c
(H3) f1 (t, u, v) < R 1 a 1(t)dt φp
f
f , and
0
p1 M1 [1+Λ1 A1 +Λ1 B1 ]
1
f2 (t, u, v) < R 1
0
1
a2 (t)dt
φq
c
for any t ∈ [0, 1], u + v ∈ [0, c];
(H4) f1 (t, u, v) < R 1 a 1(t)dt φp
0
a
f1 +Λ1 B
f1 ]
p1 M1 [1+Λ1 A
1
f2 (t, u, v) < R 1
0
1
a2 (t)dt
φq
σ
σ
1
a2 (t)dt
,
,
, and
f2 + Λ1 B
f2 ]
p2 M2 [1 + Λ1 A
b
f1 +Λ2 B
f1 ]
m1 [1+Λ2 A
f2 (t, u, v) > R 1−σ
a
for any ∀t ∈ [0, 1], u + v ∈ [0, a];
(H5) f1 (t, u, v) > R 1−σ a1 (t)dt φp
1
f2 + Λ1 B
f2 ]
p2 M2 [1 + Λ1 A
φq
, or
b
f
f
m2 [1 + Λ2 A2 + Λ2 B2 ]
for any t ∈ [0, 1], u + v ∈ [b, λb ], where
1
p1
+
1
p2
≤ 1.
Theorem 3.1. Under assumptions (H1)–(H5), Problem (1.1) has at least three
positive solutions (u1 , v1 ), (u2 , v2 ), (u3 , v3 ) such that k(u1 , v1 )k < a, b < α((u2 , v2 )),
and k(u3 , v3 )k > a, with α((u3 , v3 )) < b.
Proof. First, we show that T : Pc → Pc is a completely continuous operator. If
(u, v) ∈ Pc , by condition (H3), we have
A1 (φp−1 (y))
R1
Rs
1 µ11 0 G1 (ξ1 , s)φp−1 ( 0 a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds
R
R
≤
s
∆1 µ12 01 G1 (ξ1 , s)φ−1
p ( 0 a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds
c
f,
≤
A
f
f1 ] 1
p1 [1 + Λ1 A1 + Λ1 B
and
B1 (φ−1
p (y))
ρ1 − µ11 ϕ1 (ξ1 )
−µ12 ϕ1 (η1 ) EJDE-2011/139
POSITIVE SOLUTIONS
7
R1
Rs
µ11 0 G1 (ξ1 , s)φ−1
( 0 a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds
−µ11 ψ(ξ1 )
p
R1
R
s
ρ1 − µ12 ψ1 (η1 ) µ12 0 G1 (ξ1 , s)φ−1
p ( 0 a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds
c
f.
≤
B
f
f1 ] 1
p1 [1 + Λ1 A1 + Λ1 B
1
≤
∆1
Thus,
kT1 (u, v)k
= max |T1 (u, v)(t)|
0≤t≤1
Z 1
−1
−1
= max
G1 (t, s)φ−1
(W
(s))ds
+
A
(φ
(W
))ψ
(t)
+
B
(φ
(W
))ϕ
(t)
1
1
1
1
1
1
1
p
p
p
0≤t≤1
0
c
c
f ϕ (t)
+
A
f1 ] p1 [1 + Λ1 A
f1 ] 1 1
f1 + Λ1 B
f1 + Λ1 B
p1 [1 + Λ1 A
c
f ϕ (t)
+
B
f
f1 ] 1 1
p1 [1 + Λ1 A1 + Λ1 B
c
f1 + Λ1 B
f1 ] = c .
≤
[1 + Λ1 A
f1 + Λ1 B
f1 ]
p1
p1 [1 + Λ1 A
≤
In the same way, for any (u, v) ∈ Pc , we have
kT2 (u, v)k ≤
c
.
p2
thus
c
c
+
≤ c.
p1
p2
Therefore, kT (u, v)k ≤ c, that is, T : Pc → Pc . The operator T is completely
continuous by an application of the Ascoli-Arzela theorem.
In the same way, the condition (H4) implies that the condition (A2) of Theorem
1.1 is satisfied. We now show that condition (A1) of Theorem 1.1 is satisfied.
Clearly, {(u, v) ∈ P (α, b, λb )|α(u, v) > b} =
6 ∅. If (u, v) ∈ P (α, b, λb ), then b ≤
b
u(s) + v(s) ≤ λ , s ∈ [σ, 1 − σ]. By condition (H5), we obtain
kT (u, v)k = kT1 (u, v)k + kT2 (u, v)k ≤
A1 (φp−1 (y))
R 1−σ
R 1−σ
1 µ11 σ G1 (ξ1 , s)φ−1
( σ a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds ρ1 − µ11 ϕ1 (ξ1 )
p
R 1−σ
R
≥
1−σ
∆1 µ12 σ G1 (ξ1 , s)φp−1 ( σ a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds
−µ12 ϕ1 (η1 ) ≥
b
f,
A
f
f1 ] 1
[1 + Λ2 A1 + Λ2 B
and
B(φ−1
p (y))
R 1−σ
R 1−σ
1 −µ11 ψ1 (ξ1 )
µ11 σ G1 (ξ1 , s)φp−1 ( σ a1 (τ )f1 (τ, u(τ ), v(τ )dτ )ds R 1−σ
R
≥
1−σ
∆1 ρ1 − µ12 ψ1 (η1 ) µ12 σ G1 (ξ1 , s)φ−1
a1 (τ )f1 (τ, u(τ ), v(τ ))dτ )ds
p ( σ
≥
b
f.
B
f
f1 ] 1
[1 + Λ2 A1 + Λ2 B
Thus,
α(T (u, v)(t))
8
=
≥
N. NYAMORADI
EJDE-2011/139
min (T1 (u, v)(t) + T2 (u, v)(t))
Z 1
−1
−1
min
G1 (t, s)φ−1
(W
(s))ds
+
A
(φ
(W
))ψ
(t)
+
B
(φ
(W
))ϕ
(t)
1
1
1
1
1
1
1
p
p
p
σ≤t≤1−σ
σ≤t≤1−σ
0
Z
1
−1
G2 (t, s)φ−1
q (W2 (s))ds + A2 (φq (W2 ))ψ2 (t)
0
(W
+ B2 (φ−1
2 ))ϕ2 (t)
q
+
min
σ≤t≤1−σ
b
b
b
f1 ψ1 (t) +
f ϕ (t)
+
A
B
f
f
f
f
f
f1 1 1
1 + Λ2 A1 + Λ2 B1
1 + Λ1 A1 + Λ2 B1
1 + Λ2 A1 + Λ2 B
b
f1 + Λ2 B
f1 ] = b
≥
[1 + Λ2 A
f
f1
1 + Λ 2 A1 + Λ 2 B
≥
Therefore, condition (A1) of Theorem 1.1 is satisfied.
Finally, we show that the condition (A3) of Theorem 1.1 is also satisfied. If
(u, v) ∈ P (α, b, c), and kT (u, v)k > b/λ, then
α(T (u, v)(t)) =
min
σ≤t≤1−σ
T (u, v)(t) ≥ λkT (u, v)k > b.
Therefore, the condition (A3) of Theorem 1.1 is also satisfied.
By Theorem 1.1, there exist three positive solutions (u1 , v1 ), (u2 , v2 ), (u3 , v3 ) such
that k(u1 , v1 )k < a, b < α((u2 , v2 )), and k(u3 , v3 )k > a, with α((u3 , v3 )) < b. we
have the conclusion.
4. Application
As an example, we consider the boundary-value problem
(φp (u00 (t)))0 + a1 (t)f1 (t, u(t), v(t)) = 0
00
0 ≤ t ≤ 1,
0
(φq (v (t))) + a2 (t)f2 (t, u(t), v(t)) = 0 0 ≤ t ≤ 1,
1 1
1
u(0) − u0 (0) = u( ), u(1) + u0 (1) = u( ), u00 (0) = 0,
4
2 2
1
1
1
v(0) − v 0 (0) = v( ), v(1) + v 0 (1) = v( ), v 00 (0) = 0,
4
2 2
where ai (t) = 1, αi = βi = γi = δi = 1, for i = 1, 2, and
f1 (t, u, v) = f2 (t, u, v)
 t
u+v
t ∈ [0, 1],

1000 + 1000 ,


 t + 2((u + v)2 − (u + v)) + 1 , t ∈ [0, 1],
1000
= 1000
t
u+v
1
 1000
+
2
log
(u
+
v)
+
+ 1000
, t ∈ [0, 1],

2

√ 2
 t
1
t ∈ [0, 1],
1000 + 4 u + v + 1000 ,
0≤u+v
1<u+v
2≤u+v
4<u+v
(4.1)
≤ 1,
< 2,
≤ 4,
< +∞,
Choose σ = 14 , p = q = 3, p1 = p2 = 2. Then by direct calculations, we obtain
ρi = 3ψi (t) = 1 + t, ϕi (t) = 2 − t, t ∈ [0, 1],
−µi1 ψi (ξi )
15
ρi − µi1 ϕ1 (ξ1 )
fi = 11 , B
fi = 23 ,
=− , A
∆i = ρi − µi2 ψi (ηi )
−µi2 ϕi (ηi ) 8
15
8
Z 1
Z 1−σ
4
25
Mi = max
Gi (t, s)ds = , mi = min
Gi (t, s)ds =
,
0≤t≤1 0
σ≤t≤1−σ σ
3
96
EJDE-2011/139
POSITIVE SOLUTIONS
Z
1
Z
9
1−σ
1
i = 1, 2,
2
0
σ
n ϕ (1 − σ) ψ (σ) ϕ (1 − σ) ψ (σ) o 5
1
2
2
2
= ,
Λ0 = min
,
,
,
ϕ1 (σ)
ψ2 (1)
ϕ2 (σ)
ψ2 (1)
7
n
o
Λ1 = max 1, kψ1 k, kϕ1 k, kψ2 k, kϕ2 k = 2,
n
o
Λ2 = min
min ϕ1 (t), min ψ1 (t), min ϕ2 (t), min ψ2 (t)1 = 1,
ai (t)dt = 1,
t∈[σ,1−σ]
ai (t)dt =
t∈[σ,1−σ]
t∈[σ,1−σ]
t∈[σ,1−σ]
n
Λ2 o 1
= ,
λ = min Λ0 ,
Λ1
2
1
45
1
2304
=
,
=
,
fi + Λ1 B
fi ]
fi + Λ2 B
fi ]
986
2765
pi Mi [1 + Λ1 A
mi [1 + Λ2 A
i = 1, 2.
So we choose a = 1, b = 2, c = 200, Then, by Theorem 3.1, system (4.1) has at
least three positive solutions (u1 , v1 ), (u2 , v2 ), (u3 , v3 ) such that k(u1 , v1 )k < a, b <
α((u2 , v2 )), and k(u3 , v3 )k > a, with α((u3 , v3 )) < b.
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10
N. NYAMORADI
EJDE-2011/139
Nemat Nyamoradi
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran
E-mail address: nyamoradi@razi.ac.ir