On a class of semigroups
Let \(E\) be a Banach space and \(A\) be a linear operator on \(E\) with domain \(D(A)\subseteq E\) such that it generates a semigroup \(U(t)\) of class \(A(\alpha,\beta)\), i.e. \[ \|U(t)\|\leq Mt^{-\alpha} e^{-\omega t},\quad \|U'(t)\|\leq Mt^{-\beta} e^{-\omega t} \] for \(\omega> 0\), \(t> 0\), \(\alpha\geq 0\), \(\beta\geq 1\). The paper consists of four sections. Section 1 contains a description of the class \(A(\alpha,\beta)\). In section 2 there are given four examples of semigroups of class \(A(\alpha,\beta)\). A construction of fractional powers of operator \(A\) (from class \(A(\alpha,\beta)\)) is given in section 3. Finally, in section 4 the following theorem is stated: Suppose that the conditions hold: 1. there exists a bounded inverse \(A^{-1}\); 2. there is a semigroup \(U(t)\) of class \(A(\alpha,\beta)\) for some \(\omega> 0\), \(0\leq\alpha\leq 1\), \(\alpha +1\leq\beta\); 3. \(\|f(t+\Delta t)- f(t)\|\leq Ct^{-\mu}|\Delta t|^\varepsilon\) for some \(\varepsilon\in ((\beta- 1)/(\beta- \alpha), 1]\), \(\mu\in [0,1)\); 4. \(v_0\in D(A^\rho)\) for some \(\rho\in (\min\{2\alpha,(\beta- 1)/(\beta- \alpha)\},1]\); if \(\overline{D(A)}= E\) then \(v_0\in D(A^\rho)\) for \(\rho\in (\alpha, 1]\). Then the Cauchy problem \(v'(t)= Av(t)+ f(t)\), \(v(0)= v_0\), \(0< t\leq 1\), has the unique ``weakened'' solution \[ v(t)= U(t)v_0+ \int^t_0 U(t- s) f(s) ds,\quad 0\leq t\leq 1, \] i.e. the solution which is continuous for \(t\geq 0\) and \(v'(t)\), \(Av(t)\) are continuous for \(t>0\).
differential equations in abstract spaces
scientific article; zbMATH DE number 1452685