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.

How to Cite this Article:
Dan Li, Minghe Pei, Zan Yang, Existence and multiplicity of positive solutions of a third-order periodic boundary value problem with one-parameter, Journal of Nonlinear Functional Analysis, Vol. 2020 (2020), Article ID 14, pp. 1-15.