Abstract
The existence results of positive solutions are obtained for the fourth‐order periodic boundary value problem u(4) − βu′′ + αu = f(t, u, u′′), 0 ≤ t ≤ 1, u(i)(0) = u(i)(1), i = 0, 1, 2, 3, where f : [0, 1] × ℝ+ × ℝ → ℝ+ is continuous, α, β ∈ ℝ, and satisfy , β > −2π2, (α/π4) + (β/π2) + 1 > 0. The discussion is based on the fixed point index theory in cones.
Highlights
This paper concerns the existence of positive solutions for the fourth-order periodic boundary value problem PBVP u 4 t − βu t αu t f t, u t, u t, 0 ≤ t ≤ 1, 1.1 u i 0 u i 1, i 0, 1, 2, 3, where α, β ∈ R and f : 0, 1 ×R ×R → R is continuous, R 0, ∞
As an application of this strongly maximum principle, the author considered the existence of positive solutions for the special fourth-order periodic boundary problem u 4 t − βu t αu t g t, u t, 0 ≤ t ≤ 1, 1.5 u i 0 u i 1, i 0, 1, 2, 3, and obtained the following result
Since 3.35 and 3.36 have nonlinear terms of u , which are not in the range considered by 1–6 , the existence results in Example 3.1, and Example 3.2 cannot be obtained from 1–6
Summary
This paper concerns the existence of positive solutions for the fourth-order periodic boundary value problem PBVP u 4 t − βu t αu t f t, u t , u t , 0 ≤ t ≤ 1, 1.1 u i 0 u i 1 , i 0, 1, 2, 3, where α, β ∈ R and f : 0, 1 ×R ×R → R is continuous, R 0, ∞. As an application of this strongly maximum principle, the author considered the existence of positive solutions for the special fourth-order periodic boundary problem u 4 t − βu t αu t g t, u t , 0 ≤ t ≤ 1, 1.5 u i 0 u i 1 , i 0, 1, 2, 3, and obtained the following result.
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