Mild solutions to a class of nonlinear second order evolution equations

The purpose of this paper is to study the existence of mild solutions to a class of second order nonlinear evolution equations of the form {equation*} {cases} u''(t)+A(u'(t))+B(u(t)) f(t), &t∈(0,T),\\ u(0)=u_0, u'(0)=g(u') {cases} {equation*} where A D(A)⊆ X→ 2X is an m-accretive operator on a Banach space $X,$ B: X→ X is a lipschitz mapping, g C([0,T];X)→ X is a function and f∈ L¹(0,T,X). We obtain sufficient conditions for this problem to have at least a mild solution.