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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 $$\align -u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ }0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ }u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\endalign $$ 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.

34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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