Abstract
On a superlinear periodic boundary value problem with vanishing Green's function
Highlights
In this paper, we consider the existence of nonnegative solutions for the periodic boundary value problem y + a(t)y = λg(t) f (y), 0 ≤ t ≤ 2π, (1.1)y(0) = y(2π), y (0) = y (2π), where the associated Green’s function is nonnegative and f is allowed to change sign
We prove the existence of positive solutions for the boundary value problem y + a(t)y = λg(t) f (y), 0 ≤ t ≤ 2π, y(0) = y(2π), y (0) = y (2π), where λ is a positive parameter, f is superlinear at ∞ and could change sign, and the associated Green’s function may have zeros
We consider the existence of nonnegative solutions for the periodic boundary value problem y + a(t)y = λg(t) f (y), 0 ≤ t ≤ 2π, (1.1)
Summary
When a ∈ L∞(0, 2π), the Green’s function is positive if a ∞ < 1/4 and nonnegative if a ∞ ≤ 1/4, which have been obtained in [12] when a is a constant These conditions were extended to sign-changing a(t) with nonnegative average in [5]. Existence results for positive solutions of (1.1) when the associated Green’s function is positive have been obtained in [2, 4, 7, 8, 11, 13, 14, 18] using Krasnosel’skii’s fixed point theorem on the cone. In this paper, motivated by the results in [6, 16], we shall establish the existence of positive solutions to (1.1) when the Green’s function is nonnegative, and f is superlinear at ∞ without assuming convexity of f. The results in [6, 16] cannot be applied here
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