#### Dan Li, Minghe Pei, Zan Yang, Existence and multiplicity of positive solutions of a third-order periodic boundary value problem with one-parameter, Vol. 2020 (2020), Article ID 14, pp. 1-15

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DOI: 10.23952/jnfa.2020.14

Received December 4, 2019; Accepted April 2, 2020; Published April 16, 2020

Abstract. In this paper, we study the solvability of a third-order periodic boundary value problem with one-parameter of the form
$\begin{cases} x'''(t)+\rho^3x(t)= \lambda a(t)f(t,x(t)),\quad 0\leq t \leq2\pi,\\ x^{(i)}(0)=x^{(i)}(2\pi),\quad i=0,1,2, \end{cases}$
where $\rho\in(0,1/\sqrt{3})$ is a constant, and $\lambda>0$ is a parameter. By applying the fixed point theorem of cone compression and expansion of norm type, we establish a series of criteria for the above one-parameter problems to have one, two, an arbitrary number, and even an infinite number of positive solutions. Criteria for the nonexistence of positive solutions are also derived.