In the paper some existence results for periodic boundary value problems for the ordinary differential equation of the second order in a Hilbert space are given. Under some auxiliary assumptions the set of solutions is compact and connected or it is convex.
The paper deals with the properties of a monotone operator defined on a subset of an ordered Banach space. The structure of the set of fixed points between the minimal and maximal ones is described.
A periodic boundary value problem for nonlinear differential equation of the second order is studied. Nagumo condition is not assumed on a part of nonlinearity. Existence and multiplicity results are proved using the method of lower and upper solutions. Results are applied to the generalized Liénard oscillator.
In the paper it is proved that each generalized boundary value problem for the n-th order linear differential equation generates a Fredholm mapping of index zero.
In the paper it is proved that the generalized linear boundary value problem generates a Fredholm operator. Its index depends on the number of boundary conditions. The existence results of Landesman-Lazer type are given as an application to nonlinear problems by using dual generalized boundary value problems.
For a generalized pendulum equation we estimate the number of periodic solutions from below using lower and upper solutions and from above using a complex equation and Jensen’s inequality.
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