ON EXISTENCE AND APPROXIMATION OF SOLUTIONS OF SECOND ORDER ABSTRACT CAUCHY PROBLEM

Abstract

Let $A$ be the generator of a nondegenerate local $\alpha$-times integrated $C$-cosine function $C(\cdot)$ on a Banach space $X$ for some $\alpha \geq 0$, $f \in L_{loc}^1([0,T_0),X) \cap C((0,T_0),X)$, and $x,y \in X$. We first show that the abstract Cauchy problem : $ACP (A,C f, C x,C y) \quad u''(t) = A u(t) + C f(t)$, $u(0) = Cx$ and $u'(0) = C y$, has a strong solution is equivalent to the function $v(\cdot) = C(\cdot)x + j_0*C(\cdot) y + j_0*C*f(\cdot) \in C^{\alpha+1}([0,T_0),X)$ and $D^{\alpha+1} v(\cdot) \in C^1((0,T_0),X)$, and then use it to prove some new existence and approximation theorems concerning strong solutions of $ACP(A, C z + j_{\alpha-1}*C g,C x, C y)$ and mild solutions of $ACP(A, C x + j_1 C y + j_2 C z + j_{\alpha-1}*C g,0,0)$ (for $\alpha\geq 2$) in $C^2([0,T_0),X)$ when $C(\cdot)$ is locally Lipschitz continuous, and vectors $x$, $y$ and $z$ satisfy some suitable regularity assumptions. Here $0 \lt T_0 \leq \infty$ is fixed.