Abstract
In this note we apply Avery-Peterson multiple fixed point theorem to investigate the existence of multiple positive periodic solutions to the following non- linear non-autonomous functional differential system with feedback control ( dx dt = r(t)x(t) +F(t,xt,u(t �(t))), du dt = h(t)u(t) +g(t)x(t �(t)). We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on F.
Highlights
In this note, we obtain a new result for the existence of at least three positive ω-periodic solutions of the following system with feedback control: dx dt − δ(t))), (1)du dt where δ(t), σ(t) ∈ C(R, R), r(t), h(t), g(t) ∈ C(R, (0, +∞)), all of the above functions are ω-periodic functions and ω > 0 is a constant
We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on F
The map α is said to be a nonnegative continuous concave functional on the cone P provided that α : P → [0, ∞) is continuous and α(tx + (1 − t)y) ≥ tα(x) + (1 − t)α(y) for all x, y ∈ P and 0 ≤ t ≤ 1
Summary
We obtain a new result for the existence of at least three positive ω-periodic solutions of the following system with feedback control: dx dt. Some authors have studied the existence of at least one and two positive periodic solutions of (1) The following fixed point theorem due to Avery and Peterson [2] is crucial in the proof of our main result. In order to apply Theorem 2.1 to establish the existence of multiple positive solutions of system (1), we shall define an operator on a cone in a suitable Banach space. To this end, we first transform system (1) into a single equation. Q for all s ∈ [t, t + ω], where p, q are positive constants
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Electronic Journal of Qualitative Theory of Differential Equations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
