ON EXISTENCE AND APPROXIMATION OF SOLUTIONS OF ABSTRACT CAUCHY PROBLEM

Abstract

Let $A$ be the generator of a nondegenerate $\alpha$-times integrated $C$-semigroup $T(\cdot)$ on a complex Banach space $X$ for some $\alpha\geq 0$, $x\in X$ and $f\in L_{loc}^1([0,\infty),X)\cap C((0,\infty),X)$. We first show that the abstract Cauchy problem $ACP (A,C f,C x)$: $u'(t)=A u(t)+C f(t)$ for $t\gt 0$ and $u(0)=C x$ has a strong solution is equivalent to the function $v(\cdot)=T(\cdot)x+T *f(\cdot)\in C^\alpha ([0,\infty),X)$ and $D^\alpha v(\cdot)\in C^1((0,\infty),X)$, and then use it to prove some new existence and approximation theorems concerning strong solutions of $ACP(A,C y+j_{\alpha-1} *C g,C x)$ in $C^1([0,\infty),X)$ and mild solutions of $ACP(A,C y+j_{\alpha-2} *C g,C x)$ (for $\alpha\geq 1$) in $C([0,\infty),X)$ when vectors $x$ and $y$ both satisfy some suitable regularity assumptions and $T(\cdot)$ is locally Lipschitz continuous.