Positive solutions for a sixth-order boundary value problem with four parameters
Boundary Value Problems
Positive solutions for a sixth-order boundary value problem with four parameters
Ravi P Agarwal 0
B Kovacs 2
D O'Regan 1
0 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd. , Kingsville, 78363-8202 , USA
1 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland , Galway , Ireland
2 Department of Analysis, University of Miskolc , Egyetemvaros, 3515 , Hungary
This paper investigates the existence and multiplicity of positive solutions of a sixth-order differential system with four variable parameters using a monotone iterative technique and an operator spectral theorem. MSC: 34B15; 34B18
positive solutions; variable parameters; fixed point theorem; operator spectral theorem
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u() + u() + u = f (t, u),
u() = u() = u () = u () = u()() = u()() = .
u() = u() = u () = u () = .
They showed that there exists a λ > such that the above boundary value problem has at
least two, one, and no positive solutions for < λ < λ , λ = λ and λ > λ , respectively.
In this paper, we discuss the existence of positive solutions for the sixth-order boundary
value problem
u() = u() = u () = u () = u()() = u()() = ,
ϕ() = ϕ() = .
For this, we shall assume the following conditions throughout
(H) f (t, u) : [, ] × [, ∞) −→ [, ∞) is continuous;
(H) a, b, c ∈ R, a = λ + λ + λ > –π , b = –λλ – λλ – λλ > , c = λλλ <
where λ ≥ ≥ λ ≥ –π , ≤ λ < –λ and π + aπ – bπ + c > , and
A, B, C, D ∈ C[, ] with a = supt∈[,] A(t), b = inft∈[,] B(t) and c = supt∈[,] C(t).
Let K = max≤t≤[–A(t) + B(t) – C(t) – (–a + b – c)] and = π + aπ – bπ + c.
Assumption (H) involves a three-parameter nonresonance condition.
More recently Li [] studied the existence and multiplicity of positive solutions for a
sixth-order boundary value problem with three variable coefficients. The main difference
between our work and [] is that we consider boundary value problem not only with three
variable coefficients, but also with two positive parameters λ and μ, and the existence
of the positive solution depends on these parameters. In this paper, we shall apply the
monotone iterative technique [] to boundary value problem () and then obtain several
new existence and multiplicity results. In the special case, in [] by using the fixed point
theorem and the operator spectral theorem, we establish a theorem on the existence of
positive solutions for the sixth-order boundary value problem () with λ = .
2 Preliminaries
u ∈ X.
We also need the space X, equipped with the norm
u = max u , u
– u() + au() + bu + cu = h(t), < t < ,
u() = u() = u () = u () = u()() = u()() = ,
where a, b, c satisfy the assumption
and let = π + aπ – bπ + c. Inequality () follows immediately from the fact that =
π + aπ – bπ + c is the first eigenvalue of the problem –u() + au() + bu + cu = λu, u() =
u() = u () = u () = u()() = u()() = , and φ(t) = sin π t is the first eigenfunction, i.e.,
> . Since the line l = {(a, b, c) : π + aπ – bπ + c = } is the first eigenvalue line of
the three-parameter boundary value problem –u() + au() + bu + cu = , u() = u() =
u () = u () = u()() = u()() = , if (a, b, c) lies in l, then by the Fredholm alternative,
the existence of a solution of the boundary value problem () cannot be guaranteed.
Let P(λ) = λ + βλ – α, where β < π , α ≥ . It is easy to see that the equation P(λ) =
has two real roots λ, λ = –β±√β+α with λ ≥ ≥ λ > –π . Let λ be a number such
that ≤ λ < –λ. In this case, () satisfies the decomposition form
–u() + au() + bu + cu = – ddt + λ
u() = u() = .
We need the following lemmas.
Lemma [, ] Let ωi = √|λi|, then Gi(t, s) (i = , , ) can be expressed as
(i) when λi > ,
⎧ sinh ωit sinh ωi( – s)
Gi(t, s) = ⎪⎨⎪⎪ sinh ωωisissiinnhhωωii( – t)
⎪⎩ ωi sinh ωi
, ≤ s ≤ t ≤ ⎪⎭⎪
Gi(t, s) =
⎧ sin ωit sin ωi( – s)
Gi(t, s) = ⎨⎪⎪⎪ sin ωiωs issininωωi(i – t)
⎪⎩ ωi sin ωi
, ≤ s ≤ t ≤ ⎪⎭⎪
Lemma [] Gi(t, s) (i = , , ) has the following properties
(i) Gi(t, s) > , ∀t, s ∈ (, );
(ii) Gi(t, s) ≤ CiGi(s, s), ∀t, s ∈ [, ];
(iii) Gi(t, s) ≥ δiGi(t, t)Gi(s, s), ∀t, s ∈ [, ],
ω
where Ci = , δi = sinhiωi , if λi > ; Ci = , δi = , if λi = ; Ci = sinωi , δi = ωi sin ωi, if
–π < λi < .
In what follows, we let Di = maxt∈[,] Gi(t, s) ds.
Lemma [] Let X be a Banach space, K a cone and a bounded open subset of X. Let
θ ∈ and T : K ∩ ¯ → K be condensing. Suppose that Tx = υx for all x ∈ K ∩ ∂ and
υ ≥ . Then i(T , K ∩ , K ) = .
Lemma [] Let X be a Banach space, let K be a cone of X. Assume that T : K¯ r → K
(here Kr = {x ∈ K | x < r}, r > ) is a compact map such that Tx = x for all x ∈ ∂Kr. If
x ≤ Tx for x ∈ ∂Kr, then i(T , Kr, K ) = .
–u() + au() + bu + cu = – ddt + λ
the solution of boundary value problem () can be expressed as
u(t) =
Throughout th (...truncated)