Unique strong and strict solutions for inhomogeneous hyperbolic differential equations in Banach space

Abstract

The purpose of the present paper is the improvement of results for solutions of the inhomogeneous differential equation: $$\begin{aligned} u'(t)+A(t)u(t)+f(t)= & {} 0, \quad t\in (t_1,t_2)\\ u(t_1)= & {} \varphi \end{aligned}$$ in reflexive Banach space X. For \(f(s)\in C^1\bigl ([t_1,t_2],X\bigr )\) , Kato obtained a unique strict solution under some conditions on the operator family \( \{A(t)\}_{t_1 \le t \le t_2} \) to ensure the hyperbolicity of the problem. In a previous paper, the author obtained in abstract Hilbert space H a unique strong solution \(u(t)\in C^{0,1}\bigl ([t_1,t_2],H\bigr )\cap D\) if \(f\in BV\bigl ([t_1,t_2],H \bigr )\) and a strict solution if additionally \(f\in C^{0}\bigl ([t_1,t_2],H\bigr )\). Here is \( D=D\bigl ((A(t)\bigr )\) independent of t. In the present paper we extend these results to reflexive Banach spaces X.