Solubility of the Cauchy problem for differential equations in scales of Banach spaces with completely continuous embeddings
type: Item
instance of: scholarly article
title: Solubility of the Cauchy problem for differential equations in scales of Banach spaces with completely continuous embeddings
zbMATH Open document ID: 0827.34051
DOI: 10.1007/BF02112476
author: V. I. Nazarov
published in: Mathematical Notes
publication date: 1995-07-10T00:00:00Z
review text: If \(A\) is a linear continuous operator in a scale \(\{E_s, s\in (a, b)\}\), \((- \infty\leq a< b< \infty)\), \(\delta\in (0, b- a)\); \(p,q\in N\) and \(\lambda> 0\), one introduces \[ c_{A, p, q, \delta}(\lambda)= \sup_{s\in (a, b- \delta)} \sum^\infty_{k= 0} {\lambda^k|A^{kq}|\over (kp)!} \] and \(W(A, p, q, \delta)= \{\lambda; c_{A, p, q, \delta}(\lambda)< \infty\}\) the norm being taken in \({\mathcal L}(E_{s+ \delta}, E_s)\), \(E_s\) Banach spaces. One supposes also that the family of nonlinear continuous operators \(F(t)\) in the considered scale and depending on the real parameter \(t\) satisfies, for all \(x\in E_{s+ \delta}\), \(s\in (a, b- \delta)\), \(|A^k F(t)x|_{E_s}\leq \nu(s, \delta)|A^{k+ q} x|_{E_s}+ \eta(s, \delta, k)\); \(k= 0,1,\dots\), where \(\nu(s, \delta)\geq 0\), \(\eta(s, \delta, k)\geq 0\). Sufficient solvability conditions are considered for the Cauchy problem for nonlinear differential equations in scales of Banach spaces with completely continuous embeddings \(x^{(p)}(t)= F(t) x(t)\), \(t\in [t_0, T]\); \(x^{(l)}(t_0)= x_l\in E_{s_0}\); \(s_0\in (a+ \delta, b)\), \(l= 0, 1,\dots, p- 1\). The investigation of the solvability of this Cauchy problem is reduced to the investigation of the nonemptiness of \(W(A, p, q, \delta)\).
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 771970
zbMATH Keywords: Cauchy problem
zbMATH Keywords: nonlinear differential equations in scales of Banach spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 771970