Multiplicity of Positive Solutions for an Even-order Nonhomogeneous Boundary Value Problem
by
Britney Hopkins
Baylor University

In this talk, we focus on the existence of multiple positive solutions for the 2nth order ordinary differential equation,

u(2n)=lh(t, u, u", ..., u(2(n-1))),  t ∈ (0, 1),

satisfying the boundary conditions u^(2k)(0)=0,  k=0, ..., n-1
u^(2k+1)(1)=(-1)^k a_k,  k=0, ..., n-1 where l, a₀, ..., a_n-1 ≥ 0, with ∑_k=0^n-1a_k > 0, and h:[0, 1]×∏_i=0^n-1(-1)ⁱ[0, ∞)→ (-1)ⁿ [0, ∞) is continuous. We transform the even order boundary value problem into a system of second order differential equations satisfying homogeneous right focal boundary conditions. Then, by applying the Guo-Krasnosel'skii Fixed Point Theorem several times, we show the existence of multiple positive solutions.

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Date received: September 6, 2008