Solubility of the Cauchy problem for differential equations in scales of Banach spaces with completely continuous embeddings
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 \(W(A, p, q, \delta)\).
nonlinear differential equations in scales of Banach spaces
scientific article; zbMATH DE number 771970