A new approach to the stability of an abstract system in the presence of infinite history

In this paper, we consider the following problem { u t t ( t ) + A u ( t ) − ∫ 0 + ∞ g ( s ) A u ( t − s ) d s = 0 , ∀ t > 0 u ( − t ) = u 0 ( t ) , ∀ t ⩾ 0 u t ( 0 ) = u 1 , where A is a self-adjoint positive definite operator and g is a positive nonincreasing function. We adopt the method introduced in [19], for finite history, with some modifications imposed by the nature of our problem, to establish a general decay result which depends only on the behavior of the relaxation function. Our result extends the decay result obtained for problems with finite history to those with infinite history. In addition, it improves, in some cases, some decay results obtained earlier in [15].