Three positive periodic solutions to nonlinear neutral functional differential equations with parameters on variable time scales.(English)Zbl 1235.34243

Summary: Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale $$(d/dt)(x(t) + c(t)x(t - \alpha)) = a(t)g(x(t))x(t) - \sum^n_{j=1} \lambda_j f_j (t, x(t - v_j(t))), (t, x) \in \mathbb T_0 (x), \Delta t|_{(t, x) \in \varsigma_{2i}} = \Pi^1_i (t, x) - t, \Delta x|_{(t, x) \in \varsigma_{2i}} = \Pi^2_i (t, x) - x$$, where $$\Pi^1_i (t, x) = t_{2i+1} + \tau_{2i+1} (\Pi^2_i(t, x))$$ and $$\Pi^2_i (t, x) = B_i x + J_i (x) + x, i = 1, 2, \dots$$. $$\lambda_j(j = 1, 2, \dots, n)$$ are parameters, $$\mathbb T_0 (x)$$ is a variable time scale with $$(\omega, p)$$-property, $$c(t), a(t), v_j(t)$$, and $$f_j(t, x) (j = 1, 2, \dots, n)$$ are $$\omega$$-periodic functions of $$t, B_{i+p} = B_i, J_{i+p}(x) = J_i(x)$$ uniformly with respect to $$i \in \mathbb Z$$.

MSC:

 34N05 Dynamic equations on time scales or measure chains 34A05 Explicit solutions, first integrals of ordinary differential equations
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