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Second-order nonlinear three point boundary-value problems on time scales

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We consider a second order three point boundary value problem for dynamic equations on time scales and establish criteria for the existence of at least two positive solutions of an eigenvalue problem by an application of a fixed point theorem in cones. Existence result for non-eigenvalue problem is also given by the monotone method. | Turk J Math 30 (2006) , 9 – 24. ¨ ITAK ˙ c TUB Second-Order Nonlinear Three Point Boundary-Value Problems on Time Scales S. G¨ ul¸san Topal Abstract We consider a second order three point boundary value problem for dynamic equations on time scales and establish criteria for the existence of at least two positive solutions of an eigenvalue problem by an application of a fixed point theorem in cones. Existence result for non-eigenvalue problem is also given by the monotone method. Key Words: Dynamic equations, cone, positive solutions, upper and lower solutions. 1. Introduction We are concerned with the three point boundary value problem −y4∇ (t) = λf(t, y(t)), αy(ρ(a)) − βy4 (ρ(a)) = 0, t ∈ [a, b], y(σ(b)) − δy(η) = 0, (1.1) (1.2) where α, β ≥ 0 and α + β > 0, λ > 0, 0 t (β + α(t − ρ(a)))(σ(b) − s), and D = β(1 − δ) + α(σ(b) − ρ(a) − δ(η − ρ(a))). Lemma 1.1 For h(t) ∈ C[ρ(a), σ(b)], the BVP Ly ≡ −y∆∇ (t) = h(t), αy(ρ(a)) − βy4 (ρ(a)) = 0, t ∈ [a, b], (1.5) y(σ(b)) − δy(η) = 0 (1.6) has a unique solution y(t) = 10 β+α(t−ρ(a)) R σ(b) (σ(b) ρ(a) RtD − ρ(a) (t − s)h(s)∇s. − s)h(s)∇s − δ(β+α(t−ρ(a))) R η (η D ρ(a) − s)h(s)∇s TOPAL Lemma 1.2 Let 0 0. So this contradicts the assertion y(t) is a monotone decreasing function. 2. If y(σ(b)) 0 and h(σ(b)) = 0. σ(b) − ρ(a) Since h4∇ (t) ≤ 0 then h(t) ≥ 0 on [η, σ(b)]. So y(t) y(σ(b)) ≤ . For the function h(t), σ(b) t since h(η) > 0, h(σ(b)) = 0 and h4∇ (t) ≤ 0 then the function h(t) is decreasing on [η, σ(b)]. So y(η) y(t) ≤ for all t ∈ [η, σ(b)]. t η 2 Lemma 1.4 Let 0 0 and that Φ : Pr → P is compact operator such that Φx 6= x for x ∈ ∂Pr := {x ∈ P : kxk = r}. Then, the following assertions hold: (i) If kxk ≤ kΦxk, for all x ∈ ∂Pr , then i(Φ, Pr , P ) = 0. (ii)If kxk ≥ kΦxk, for all x ∈ ∂Pr , then i(Φ, Pr , P ) = 1. Thus, if there exist r1 > r2 > 0 such that condition (i) holds for x ∈ ∂Pr1 and (ii) holds for x ∈ ∂Pr2 (or (ii) and (i)), then, from the additivity properties of the index, we .