The optimal control problem in coefficients for the pseudoparabolic variational inequality
The period function of a planar parameter-depending Hamiltonian system is examined. It is proved that, depending on the value of the parameter, it is either monotone or has exactly one critical point.
We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation , when is arbitrary and or when . The proof uses upper and lower solutions and the Leray–Schauder degree.
Sufficient conditions on the existence of periodic solutions for semilinear differential inclusions are given in general Banach space. In our approach we apply the technique of the translation operator along trajectories. Due to recent results it is possible to show that this operator is a so-called decomposable map and thus admissible for certain fixed point index theories for set-valued maps. Compactness conditions are formulated in terms of the Hausdorff measure of noncompactness.
We study the question of the unique solvability of the periodic type problem for the second order linear integro-differential equation with distributed argument deviation and on the basis of the obtained results by the a priori boundedness principle we prove the new results on the solvability of periodic type problem for the second order nonlinear functional differential equations, which are close to the linear integro-differential equations. The proved results are optimal in some sense.
We obtain, by means of Banach's Fixed Point Theorem, convergence for the Picard iterations associated to a general nonlinear system of measure differential equations. We study the existence of left-continuous solutions defined on maximal intervals and we establish some properties of these maximal solutions.