Existence of nontrivial solutions for a nonlinear second order periodic boundary value problem with derivative term
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Journal of Fixed Point Theory and Applications
Existence of nontrivial solutions for a nonlinear second order periodic boundary value problem with derivative term Zhongyang Ming, Guowei Zhang
and Juan Zhang
Abstract. In this paper, we study the existence of nontrivial solutions to the following nonlinear differential equation with derivative term: u (t) + a(t)u(t) = f t, u(t), u (t) , t ∈ [0, ω], u(0) = u(ω), u (0) = u (ω), where a: [0, ω] → R+ R+ = [0, +∞) is a continuous function with a(t) ≡ 0, f : [0, ω] × R × R → R is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on nonlinearity, using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial solutions to the above periodic boundary value problem is established in C 1 [0, ω]. Finally, an example is given to demonstrate the validity of our main result. Mathematics Subject Classification. Primary 34C25; Secondary 34B15. Keywords. Nontrivial solution, Spectral radius, Topological degree, Fixed point.
1. Introduction Due to wide applications in physics and engineering, second-order periodic boundary value problems (PBVPs) have been extensively studied by many authors, see [1–12] and relevant references therein. In [1], the following problem was discussed by Atici and Guseinov − p(t)u (t) + q(t)u(t) = f t, u(t) , t ∈ [0, ω], u(0) = u(ω), p(0)u (0) = p(ω)u (ω), where p(x) and q(x) are real-valued measurable functions defined on [0, ω] satisfying p(x) > 0, q(x) ≥ 0, q(x) = 0 almost everywhere, and ω ω dx < +∞, q(x)dx < +∞, 0 p(x) 0 0123456789().: V,-vol
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f : [0, ω] × R → R is continuous and f (t, x) ≥ 0 for (t, x) ∈ [0, ω] × [0, ∞]. If there exist numbers 0 < r < R < +∞ such that for all t ∈ [0, ω], M 1 x for 0 ≤ x ≤ r and f (t, x) ≥ x for R ≤ x < +∞, ωM ωm2 where m = mint,s∈[0,ω] G(t, s), M = maxt,s∈[0,ω] G(t, s) and G(t, s) is the Green’s function according to its linear problem, the authors established the existence of positive solutions. Graef et al. [2] investigated the existence of positive solutions, under f (t, x) ≤
f (u) f (u) f (u) f (u) = +∞, lim = 0 or lim = 0, lim = +∞ u→+∞ u→+∞ u→0 u u→0 u u u lim
with f convex and nondecreasing, to u (t) + a(t)u(t) = g(t)f (u(t)), t ∈ [0, 2π], u(0) = u(2π), u (0) = u (2π), where f : [0, +∞) → [0, +∞), g : [0, 2π] → [0, +∞) are continuous such that mint∈[0,2π] g(t) > 0, and the Green’s function is nonnegative. Hai [3] proved the existence of positive solutions to u (t) + a(t)u(t) = λg(t)f (u(t)), t ∈ [0, 2π], u(0) = u(2π), u (0) = u (2π) for all λ > 0, where a : [0, 2π] → [0, +∞) is continuous with a(t) ≤ 1/4 for all t and a(t) ≡ 0, f : [0, +∞) → [0, +∞) is continuous, g ∈ L1 (0, 2π) with g ≥ 0 and g ≡ 0 on any subinterval of (0, 2π). Li and Liang in [4] obtained the existence of positive solutions for u (t) + a(t)u(t) = f (t, u(t)), t ∈ [0, ω], (1.1) u(0) = u(ω), u (0) = u (ω), where f : [0, ω] × [0, +
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