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Zbl 1195.34033
Hao, Xinan; Liu, Lishan; Wu, Yonghong
Existence and multiplicity results for nonlinear periodic boundary value problems. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 9-10, 3635-3642 (2010). ISSN 0362-546X

Consider the following periodic boundary value problems $$\align -u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \ \ }0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \ \ }u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\endalign $$ and $$\align u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \ \ }0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \ \ }u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\endalign $$ where $a\in L_{1}\left( 0,2\pi \right) $, $f:\left[ 0,2\pi \right] \times \lbrack 0,+\infty )\rightarrow \lbrack 0,+\infty )$\ is continuous, $\lambda $ is a positive parameter. The criteria for the existence, nonexistence and multiplicity of positive solutions are established by using the Global continuation theorem, fixed point index theory and approximate method. The results obtained herein generalize and complement some previous findings of [{\it J. R. Graef, L. Kong} and {\it H. Wang}, J. Differ. Equations 245, No.~5, 1185--1197 (2008; Zbl 1203.34028)] and some other known results.
[Khanlar R. Mamedov (Mersin)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: periodic boundary value problem; positive solution; global continuation theorem; fixed point index; existence; multiplicity

Citations: Zbl 1203.34028

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