Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part

Abstract

We study nonlinear ODE problems in the complex Euclidean space, with the right-hand side being polynomial with nonconstant periodic coefficients. As the coefficients functions, we admit only functions with vanishing Fourier coefficients for negative indices. This leads to an existence theorem which relates the number of solutions with the number of zeros of the averaged right-hand side polynomial. A priori estimates of the norms of solutions are based on the Wirtinger-Poincaré-type inequality. The proof of existence theorem is based on the continuation method of Krasnosielski et al., Mawhin et al., and the Leray-Schauder degree. We give a few applications on the complex Riccati equation and some others.

Periodic solutions of quaternionic-valued ordinary differential equations

“…The more general case of Floquet boundary conditions has been considered in [28,29,39]. The case of polynomial equations with coefficients belonging to the algebra of T-periodic functions with Fourier coefficients of negative index equal to zero has been recently initiated by Borisovich and Marzantowicz [3,4] and developed by Taddei [41].…”

Planar nonautonomous polynomial equations: The Riccati equation

“…In contrast to the rest of the paper and similarly to [7,8], the existence of periodic solutions can be proved for short periods. It also can be done for small |b|.…”

“…In those papers the coefficients are real. The complex ones were considered in [7,8]. The problem of nonexistence of periodic solutions was investigated in [9][10][11][12][13].…”

“…But there are no criteria to determine how many periodic solutions a particular equation has. The ones from [7,8] can be applied only when the coefficients of a, b and c in Fourier expansion of negative indices are equal to zero. They do not provide any information about the equations containing other terms, e.g.…”

About K-Positivity Properties of Time-Invariant Linear Systems Subject to Point Delays

“…In particular, the results presented in the paper could be extended to singular dynamic systems as well as to hybrid systems composed of coupled continuous-time and digital states. They could be also potentially extended to more general descriptions involving ODE problems in the complex Euclidean space with 26 Journal of Inequalities and Applications the right-hand side being polynomials with, in general, nonconstant periodic coefficients [43][44][45][46][47].…”