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On periodic solutions of first order nonlinear differential equations with deviating arguments. (English) Zbl 0927.34053
The author deals with existence and uniqueness of periodic solutions to scalar functional-differential equations of the type $x'(t)=f(t,x(\tau_1(t)),\dots,x(\tau_m (t))), \quad t\in\mathbb{R},\tag{1}$ where $$f:\mathbb{R}\times{\mathbb{R}}^m\to{\mathbb{R}}$$ is a Carathéodory function which is periodic in time with positive period $$\omega$$ $f(t+\omega,\cdot)=f(t,\cdot),\quad t\in\mathbb{R},$ and $$\tau=\tau(\cdot)=(\tau_1(\cdot),\dots,\tau_m (\cdot))$$ is an $$m$$-vectorial measurable function satisfying the periodicity condition $\tau(t+\omega)=\mu(t) \omega+\tau(t),\quad t\in\mathbb{R},$ where $$\mu$$ denotes an $$m$$-vectorial function with integer components.
Under the hypothesis SEQord(2)–SEQord(1) the problem $$(1)$$ is equivalent to the following one $y'(t)=f(t,y(\tau_{01}(t)),\dots,y( \tau_{0m}(t))), \qquad y(0)=y(\omega),\tag $$1'$$$ where $\tau(t)=[\tau](t) \omega+\tau_0(t), t\in\mathbb{R},$ ($$[\cdot]$$ denotes the $$m$$-vectorial function whose components are the respective greatest integer function of the components of $$\tau/\omega$$ – hence $$0\leq\tau_{0k}(t)<\omega$$ for $$t\in\mathbb{R}$$ and $$k=1,\dots,m$$).
The author proves the existence of a solution to problem $$(1)$$ whenever $$f$$ is of the form $f(t,y_1,\dots,y_m)=\sum_{k=1}^mp_k(t,y_1,\dots,y_m) y_k+q(t)$ where $$q(\cdot)$$ is a summable function and $$p(\cdot)=(p_1(\cdot),\dots,p_m(\cdot))$$ satisfies some technical conditions, namely, $$\sum_{k=1}^mp_k(t,y_1,\dots,y_m)$$ is of constant sign and its modulus is bounded from below by a summable positive function. Furthermore, if $$f(t,\cdot)$$ has a derivative in $${\mathbb{R}}^m$$ for $$t\in[0,\omega]$$ which is a Carathéodory function then the solution to problem $$(1)$$ is unique.
The proof is essentially done by reduction to the study of the case where $$q=0$$ and $$p$$ depends only on time, i.e., $$f$$ is of the form $f(t,y(\tau_{01}(t)),\dots,y(\tau_{0m}(t)))=\sum_{k=1}^mp_{0k}(t) y(\tau_{0k}(t)).$ In this particular case, the problem $$(1')$$ has only the trivial solution [see I. T. Kiguradze and B. Puzha, Differ. Equations 33, No. 2, 184-193 (1997; Zbl 0908.34046)].

##### MSC:
 34K13 Periodic solutions to functional-differential equations 34K10 Boundary value problems for functional-differential equations 34C25 Periodic solutions to ordinary differential equations 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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