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Existence and multiplicity results for nonlinear periodic boundary value problems. (English) Zbl 1195.34033

Consider the following periodic boundary value problems \[ \begin{aligned} -u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \;\;}0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \;\;}u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\end{aligned} \] and \[ \begin{aligned} u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \;\;}0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \;\;}u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\end{aligned} \] where \(a\in L_{1}\left( 0,2\pi \right) \), \(f:\left[ 0,2\pi \right] \times [ 0,+\infty )\rightarrow [ 0,+\infty )\) 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 [J. R. Graef, L. Kong and H. Wang, J. Differ. Equations 245, No. 5, 1185–1197 (2008; Zbl 1203.34028)] and some other known results.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations

Citations:

Zbl 1203.34028
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