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Periodic solution of certain nonlinear differential equations: via topological degree theory and matrix spectral theory. (English) Zbl 1258.34114
Summary: The main purpose of this article is to establish the existence and stability of a periodic solution of nonlinear differential equation connected with a problem from mathematical biology. The existence and stability conditions are given in terms of spectral radius of explicit matrices, which are better than conditions obtained by using classic norms. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory and Lyapunov functional. It should be noted that the new problem appears due to the introduction of Gilpin-Ayala effect. The standard methods used in the previous literature cannot be used to analyze the asymptotic stability of such systems. To handle this problem, two novel techniques should be employed. One is to rescale the system by $w_i=\frac{y_i}{d_i}$, not $w_i=\frac{y_i}{d}$. The other is to analyze the maximal eigenvalue of a matrix. Finally, some examples and their simulations show the feasibility of our results.
34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
47N20Applications of operator theory to differential and integral equations
34D20Stability of ODE
92D25Population dynamics (general)
34D23Global stability of ODE
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