WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 14, 2015
Positive Periodic Solutions in Shifts δ± for a Class of Dynamic Equations with Feedback Control on Time Scales
Authors: ,
Abstract: Let $$\mathbb{T} ⊂ \mathbb{R}$$ be a periodic time scale in shifts $$δ±$$ with period $$P ∈ (t0, ∞)_{\mathbb{T}}$$ and $$t_{0} ∈ \mathbb{T}$$ is nonnegative
and fixed. By using a multiple fixed point theorem in cones, some criteria are established for the existence and
multiplicity of positive solutions in shifts $$δ±$$ for a class of higher-dimensional functional dynamic equations with
feedback control on time scales of the following form:
$$\begin{cases}
x^{∆}(t) = A(t)x(t) + b(t)f(t, x(τ (t)), u(t)),\\
u^{∆}(t) = −r(t)u(t) + g(t)x(t), t ∈ \mathbb{T},\end{cases}$$
where $$A(t) = (a_{ij} (t))_{n×n}$$ is a nonsingular matrix with continuous real-valued functions as its elements. Finally,
numerical examples are presented to illustrate the feasibility and effectiveness of the results.
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Keywords: Periodic solution, Functional dynamic equation, feedback control, Shift operator, Time scale
Pages: 10-19
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #2