ON EXPONENTIALLY BOUNDED ＄￥alpha＄ -TIMES INTEGRATED C-COSINE FUNCTIONS

"In this paper we apply some basic properties concerning ＄￥alpha＄-times integrated C-cosine functions to deduce a characterization of an exponentially bounded ＄￥alpha＄-times integrated C-cosine function in terms of its Laplace transform, and then use it to show that for each ＄x￥in(￥lambda^{2}-A)^{-1}CX＄ the second order abstract Cauchy problem: ＄t^{￥alpha-1}＄ ＄u^{￥prime￥prime}(t)=Au(t)+_{￥overline{￥Gamma(￥alpha)}}x＄ for ＄t>0,u(O)=u^{￥prime}(0)=0＄ has a unique solution ＄u(￥cdot)＄ which satisfies ＄||u(t)￥Vert,||u^{￥prime￥prime}(t)||￥in O(e^{￥omega t})＄ as ＄ t￥rightarrow￥infty＄ when the closed linear operator ＄A＄ : ＄D(A)￥subset X￥rightarrow X＄ which generates an exponentially bounded ＄￥alpha＄-times integrated C-cosine function ＄C(￥cdot)＄ on a Banach space ＄X＄ with ＄||C(t)||￥leq Me^{￥{vt}＄ for all ＄t￥geq 0＄ and for some fixed ＄M￥omega￥geq 0.Moreover＄ , we show that a closed linear operator in ＄X＄ generates an exponentially bounded ＄￥alpha＄-times integrated C-cosine function on ＄X＄ also generates an exponentially bounded ＄￥underline{￥alpha}＄times integrated C-semigroup on X.