Generic results for the existence of nondegenerate periodic solutions of some differential systems with periodic nonlinearities (original) (raw)

On periodic solutions to nonlinear differential equations in Banach spaces

Filomat, 2016

Let A denote the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a complex Banach space H. In this work, an analog of the resolvent operator which is called quasi-resolvent operator and denoted by R? is defined for points of the spectrum, some equivalent conditions for compactness of the quasi-resolvent operators R? are given. Then using these, some theorems on existence of periodic solutions to the non-linear equations ?(A)x = f (x) are given, where ?(A) is a polynomial of A with complex coefficients and f is a continuous mapping of H into itself.

Periodic Solutions of Degenerate Differential Equations in Vector-valued Function Spaces

Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multipliers theorems, we obtain necessary and sufficient conditions to guarantee existence and uniqueness of periodic solutions to the equation d dt (M u(t)) = Au(t) + f (t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our results are obtained in the scales of periodic Besov and periodic Lebesgue vector-valued spaces. 2000 Mathematics Subject Classification. 35K65, 34G10, 34K13.

The existence of periodic solutions to nonautonomous differential inclusions

Proceedings of the American Mathematical Society, 1988

For an m-dimensional differential inclusion of the form ie A(t)x(t) + F[t,x(t)], with A and F T-periodic in t, we prove the existence of a nonconstant periodic solution. Our hypotheses require m to be odd, and require F to have different growth behavior for |i| small and |i| large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.

Existence and uniqueness of -almost periodic solutions to some ordinary differential equations

Nonlinear Analysis: Theory, Methods & Applications, 2007

In this paper we prove the existence and uniqueness of C (n)-almost periodic solutions to the ordinary differential equation x (t) = A(t)x(t) + f (t), t ∈ R, where the matrix A(t) : R → M k (C) is τ-periodic and f : R → C k is C (n)-almost periodic. We also prove the existence and uniqueness of an ultra-weak C (n)-almost periodic solution in the case when A(t) = A is independent of t. Finally we prove also the existence and uniqueness of a mild C (n)-almost periodic solution of the semilinear hyperbolic equation x (t) = Ax(t) + f (t, x) considered in a Banach space, assuming f (t, x) is C (n)-almost periodic in t for each x ∈ X, satisfies a global Lipschitz condition and takes values in an extrapolation space F A −1 associated to A.