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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

-u '' +atu=λft,u,0t2π,u0=u2π,u ' 0=u ' 2π,


u '' +atu=λft,u,0t2π,u0=u2π,u ' 0=u ' 2π,

where aL 1 0,2π, f:0,2π×[0,+)[0,+) is continuous, λ 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.

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