Existence of three periodic solutions for a quasilinear periodic boundary value problem

Zhongqian Wang; Dan Liu; Mingliang Song; Zhongqian Wang; Dan Liu; Mingliang Song

doi:10.3934/math.2020389

AIMS Mathematics

2020, Volume 5, Issue 6: 6061-6072. doi: 10.3934/math.2020389

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Research article

Existence of three periodic solutions for a quasilinear periodic boundary value problem

Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing, 210013, P. R. China

Received: 02 June 2020 Accepted: 15 July 2020 Published: 24 July 2020
MSC : 34B15, 34C25, 70H12

In this paper, we prove the existence of at least three periodic solutions for the quasilinear periodic boundary value problem $ \begin{eqnarray} \left\{ \begin{array}{ll} -p(x')x''+\alpha(t)x = \lambda f(t,x) ~{\rm a.e.} ~t\in[0,1], \\ x(1) -x(0) = x'(1)-x'(0) = 0 \end{array} \right. \end{eqnarray} $ under appropriate hypotheses via a three critical points theorem of B. Ricceri. In addition, we give an example to illustrate the validity of our result.
- periodic solutions,
- three solutions,
- critical point,
- periodic boundary problem
Citation: Zhongqian Wang, Dan Liu, Mingliang Song. Existence of three periodic solutions for a quasilinear periodic boundary value problem[J]. AIMS Mathematics, 2020, 5(6): 6061-6072. doi: 10.3934/math.2020389

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Abstract

In this paper, we prove the existence of at least three periodic solutions for the quasilinear periodic boundary value problem $ \begin{eqnarray} \left\{ \begin{array}{ll} -p(x')x''+\alpha(t)x = \lambda f(t,x) ~{\rm a.e.} ~t\in[0,1], \\ x(1) -x(0) = x'(1)-x'(0) = 0 \end{array} \right. \end{eqnarray} $ under appropriate hypotheses via a three critical points theorem of B. Ricceri. In addition, we give an example to illustrate the validity of our result.

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