Positive solutions for a sixth-order boundary value problem with four parameters

Boundary Value Problems, Aug 2013

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.

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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 - 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 = –λλ – λλ – λλ > , c = λλλ <  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 = sinωi , 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)


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Ravi P Agarwal, B Kovacs, D O’Regan. Positive solutions for a sixth-order boundary value problem with four parameters, Boundary Value Problems, 2013, pp. 184, 2013,