Continuing work of E. Hernández and D. O’Regan [Proc. Am. Math. Soc. 141, No. 5, 1641–1649 (2013; Zbl 1266.34101)], this paper studies the following abstract differential equations with non-instantaneous impulses:
\[ \begin{aligned} & u'(t)=Au(t)+f(t,u(t)), \quad t\in(s_i,t_{i+1}],\quad i=0,\dots, N,\\ & u(t)=g_i(t,u(t)),\quad t\in(t_i,s_i],\quad i=1,\dots, N,\\ &u(0)=x_0, \end{aligned} \]
where \(A:D(A)\subset X \to X\) is the generator of an analytic \(C_0\)-semigroup of bounded linear operators \((T(t))_{t\geq 0}\) defined on an Banach space \((X, \|\cdot\|)\), \(x_0\in X\), \(0=t_0=s_0<t_1\leq s_1\leq t_2<\dots<t_{N-1}\leq s_{N}\leq t_N\leq t_{N+1}=a\) are fixed numbers, \(g_i\in C([t_i,s_i]\times X; X)\) for all \(i=1,\dots, N\), \(f\in C([0,a]\times X_\alpha;X)\), \(\alpha\in(0,1)\), and \(X_\alpha\) denotes the domain of the fractional power \((-A)^\alpha\) of \(-A\) endowed with the norm \(\|x\|_\alpha=\|(-A)^\alpha x\|\).
The paper uses the theory of analytic semigroups and fractional powers and obtains the existence of mild solutions in the space \(X_\alpha\) of the above impulsive problem via the contraction mapping principle and a fixed point theorem for condensing operators (Theorem 4.3.2 in the book [R. H. Martin, Nonlinear operations and differential equations in Banach spaces. (Reprint of the orig. ed. publ. 1976 by John Wiley & Sons, Inc., New York). Melbourne, Florida: Krieger Publishing Co., Inc. (1987; Zbl 0649.47039)]).
The authors claim that, in the case of instantaneous impulses (\(t_i=s_i\) for all \(i\)), their results can be applied to partial differential equations with impulses involving nonlinear expressions or partial derivatives of the solution. In the last section, they provide such an application.