Integrated semigroups and higher-order abstract equations (Q1269057)

Let \(E\) be a Banach space and let \(B_0,B_1,\dots, B_{n-1}\) be closed linear operators on \(E\). Moreover, let \(f:[0,\infty)\to E\) be a given function. The authors study the wellposedness of the Cauchy problem for a higher-order abstract differential equation \[ u^{(n)}(t)+ \sum^{n- 1}_{i= 0} B_i u^{(i)}(t)= f(t)\quad (t\geq 0),\tag{1} \] \[ u^{(k)}(0)= u_k,\quad 0\leq k\leq n-1.\tag{2} \] The main tool used is the theory of \(r\)-times integrated semigroups. A few examples of applications to the theory of partial differential equations are presented.