Global solutions for abstract differential equations with non-instantaneous impulses. (English) Zbl 1353.34071

Pierri, Michelle; Henríquez, Hernán R.; Prokopczyk, Andréa

Mediterr. J. Math. 13, No. 4, 1685-1708 (2016).

The paper deals with the following problem involving a semilinear differential equation subject to the action of non-istantaneous impulses:
\[ \begin{aligned} & u'(t)=Au(t)+f(t,u(t))\;,\;t\in [s_i,t_{i+1}],\, i\in \mathbb{N},\\ & u(t)=g_i(t,N_i(t)(u))\;,\;t\in (t_i,s_i],\, i\in \mathbb{N},\\ & u(0)=x_0, \end{aligned} \] where: \(A:D(A)\subseteq X\to X\) is the infinitesimal generator of a \(C_0\)-semigroup \(\{T(t)\}_{t\geq 0}\); \(X\) is a Banach space; \(f:[0,+\infty)\times X\to X\) is a suitable function; \(0=t_0=s_0<t_1<s_1<\dots<t_i<s_i<\dots\) are fixed real numbers; for every \(i\in \mathbb{N}\), \(g_i:[t_i,s_i]\times X\to X\) is continuous, \(N_i(t):C([t_i,s_i];X)\to X\) are continuous maps

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for \(t\in [t_i,s_i]\), and the function \(t\mapsto N_i(t)(u)\) is continuous for each \(u\in C([t_i,s_i];X)\); \(x_0\in X\).
The existence of mild solutions on the half-line

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\([0,\infty)\) is established, as well as the existence of \(\mathcal{S}\)-asymptotically \(\omega\)-periodic mild solutions for the problem.
Finally, some applications involving the heat equation are given.

Reviewer: Paola Rubbioni (Perugia)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34C25	Periodic solutions to ordinary differential equations
34A37	Ordinary differential equations with impulses
35R12	Impulsive partial differential equations
47D06	One-parameter semigroups and linear evolution equations

Keywords:

first-order abstract Cauchy problem; impulsive differential equations; mild solutions; asymptotically periodic solutions

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