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

Let A be the generator of a nondegenerate local alpha-times integrated C-cosine function C(.) on a Banach space X for some alpha >= 0, f is an element of L(loc)(1)([0, T(0)), X) boolean AND C((0, T(0)), X), and x, y is an element of X. We first show that the abstract Cauchy problem : ACP (A, Cf, Cx, Cy) u ''(t) = Au(t) + Cf (t), u(0) = Cx and u'(0) = Cy, has a strong solution is equivalent to the function v(.) = C(.)x+j(0)*C(.)y+j(0)*C*f(.) is an element of C(alpha+1)([0, T(0)), X) and D(alpha+1)v(.) is an element of C(1)((0, T(0)), X), and then use it to prove some new existence and approximation theorems concerning strong solutions of ACP (A, Cz+j(alpha-1)*Cg, Cx, Cy) and mild solutions of ACP(A, Cx+j(1)Cy+j(2)Cz+j(alpha-1)*Cg, 0, 0) (for alpha >= 2) in C(2)([0, T(0)), X) when C(.) is locally Lipschitz continuous, and vectors x, y and z satisfy some suitable regularity assumptions. Here 0 < T(0) <= infinity is fixed.