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Solvability problems for a linear homogeneous functional-differential equation of the pointwise type. (English. Russian original) Zbl 1378.34086

Differ. Equ. 53, No. 2, 145-156 (2017); translation from Differ. Uravn. 53, No. 2, 148-159 (2017).
Consider Cauchy problem for the linear functional-differential equation of pointwise type on the real axis of the form \[ \dot{x}(t)=\sum_{j=1}^s A_jx(t+n_j),\quad x(\bar{t})=\bar{x}, \quad \bar{t}\in\mathbb{Z}, \] where \(n_j\in\mathbb{Z}\). For \(\mu>0\) it is introduced the Banach space \[ \mathcal{L}_\mu^1C^{(k)}(\mathbb{R})=\{x(t)\in C^{(k)}(\mathbb{R}): \max_{0\leq r\leq k}\sup_\mathbb{R}|x^{(r)}(t)\mu^{|t|}|<+\infty\}. \] Let for given \(0<\mu<1\) there exist numbers \(r_1,r_2\in(\mu,\mu^{-1})\) such that \[ \left(\sum_{j=1}^sA_jr_1^{n_j}-\ln r_1\right) \left(\sum_{j=1}^sA_jr_2^{n_j}-\ln r_2\right)\leq0. \] It is proved that for any \(\bar{x}\in\mathbb{R}\) the Cauchy problem has a solution \(x(t)\in\mathcal{L}_\mu^1C^{(0)}(\mathbb{R})\). Moreover, \(x(t)\in\bigcap_{k=0}^\infty \mathcal{L}_\mu^1C^{(k)}(\mathbb{R})\).
Let \(g(\lambda):=\lambda^{-1}\exp(\sum_{j=1}^sA_j\lambda^{n_j})\) and \(\sigma=\{\lambda\in\mathbb{C}:\mu<|\lambda|<\mu^{-1}\}\). It is proved that the solution of the Cauchy problem is unique if and only if the equation \(g(\lambda)=1\) has a unique real root of multiplicity one in the ring \(\sigma\).

MSC:

34K06 Linear functional-differential equations
34K05 General theory of functional-differential equations
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