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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)].

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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