ON LOCAL INTEGRATED C-COSINE FUNCTION AND WEAK SOLUTION OF SECOND ORDER ABSTRACT CAUCHY PROBLEM

Let alpha be a nonnegative number, C : X -> X a bounded linear injection on a Banach space X and A : D(A) subset of X -> X a closed linear operator in X which satisfies C(-1)AC = A and may not be densely defined. We prove some equivalence relations between the generation of a local alpha-times integrated C-cosine function on X with generator A and the uniqueness existence of weak solutions of the abstract Cauchy problem: ACP(2)(A,f,x,y) { u ''(t) = Au(t) + f(t) for t is an element of (0, T(0)), u(0) = x, u'(0) = y, where x,y is an element of X are given and f is an X-valued function defined on a subset of R.