Beklaryan, L. A.; Beklaryan, A. L. 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 \(\sigma\). Reviewer: Nikita V. Artamonov (Moskva) Cited in 1 Document MSC: 34K06 Linear functional-differential equations 34K05 General theory of functional-differential equations Keywords:linear functional-differential equation; Cauchy problem PDFBibTeX XMLCite \textit{L. A. Beklaryan} and \textit{A. L. Beklaryan}, Differ. Equ. 53, No. 2, 145--156 (2017; Zbl 1378.34086); translation from Differ. Uravn. 53, No. 2, 148--159 (2017) Full Text: DOI References: [1] El’sgol’ts, L.E. and Norkin, S.B., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Deviating Argument), Moscow: Nauka, 1971. · Zbl 0224.34053 [2] Beklaryan, L.A., Introduction to the theory of functional-differential equations and their applications. A group approach, Sovrem. Mat. Fundam. Napravl., 2004, vol. 8, pp. 3-147. · Zbl 1095.34001 [3] Beklaryan, L.A. and Kruchenov, M.B., On the solvability of a linear homogeneous functional-differential equation of point type, Differ. Equations, 2008, vol. 44, no. 4, pp. 453-463. · Zbl 1172.34333 · doi:10.1134/S0012266108040010 [4] Beklaryan, L.A., The linear theory of functional differential equations: existence theorems and the problem of pointwise completeness of the solution, Sb. Math., 2011, vol. 202, no. 3, pp. 307-340. · Zbl 1228.34097 · doi:10.1070/SM2011v202n03ABEH004147 [5] Rudin, W., Functional Analysis, New York: McGraw-Hill, 1973. · Zbl 0253.46001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.